Upfront Maths Correlation with UK National Numeracy Strategy 1998

Transcription

1 Upfront Maths Correlation with UK National Numeracy Strategy 1998 RECEPTION 7 Numbers and the Number System 7 Counting 7 Reading and writing numbers 7 Comparing and ordering numbers 7 Ordinal numbers 7 Adding and subtracting 7 Solving Problems 8 Reasoning about numbers or shapes 8 Measures 8 Length, mass and capacity 8 Time 8 2-D and 3-D shape 8 Patterns and symmetry 8 Position, direction and movement 9 YEAR 1 10 Numbers and the Number System 10 Counting 10 Place value and ordering 11 Calculations 12 Addition 12 Subtraction 13 Rapid recall of addition and subtraction facts 14 Solving Problems 15 Making decisions 15 Reasoning about numbers or shapes 15 Measures 17 Length, mass and capacity 17 Time 17 Shape and space 18 Properties of 3-D and 2-D shapes 18 Position and direction 18 Movement and angle 19 Handling data 19 Organising and using data 19 YEAR 2 20 Numbers and the Number System 20

2 Counting 20 Place value and ordering 21 Estimation and rounding 23 Fractions 23 Calculations 24 Addition 24 Subtraction 25 Rapid recall of addition and subtraction facts 26 Understanding multiplication 27 Understanding division 28 Solving Problems 29 Making decisions 29 Reasoning about numbers or shapes 29 Measures 31 Length, mass and capacity 31 Time 32 Shape and space 32 Properties of 3-D and 2-D shapes 32 Line symmetry 33 Position and direction 33 Movement and angle 34 Handling data 34 Organising and using data 34 YEAR 3 36 Numbers and the Number System 36 Counting 36 Place value and ordering 37 Estimation and rounding 39 Fractions 39 Calculations 40 Addition 40 Subtraction 41 Rapid recall of addition and subtraction facts 42 Understanding multiplication 44 Understanding division 45 Solving Problems 47 Making decisions 47 Reasoning about numbers or shapes 48 Measures 49 Length, mass and capacity 49 Time 50 Shape and space 51 Properties of 3-D and 2-D shapes 51 Line symmetry 52 Position and direction 52 Movement and angle 52

3 Handling data 53 Organising and using data 53 YEAR 4 54 Numbers and the Number System 54 Place value (whole numbers) 54 Ordering (whole numbers) 55 Rounding (whole numbers) 56 Negative Numbers 56 Properties of numbers and number sequences 56 Fractions and decimals 57 Calculations 59 Addition 59 Subtraction 59 Rapid recall of addition and subtraction facts 60 Mental calculation strategies (+ and -) 61 Pencil and paper procedures (addition) 62 Pencil and paper procedures (subtraction) 63 Understanding multiplication 63 Understanding division 64 Solving Problems 66 Making decisions 66 Reasoning about numbers or shapes 66 Measures 69 Length, mass and capacity 69 Time 70 Area and perimeter 72 Shape and space 72 Properties of 3-D and 2-D shapes 72 Reflective symmetry, reflection and translation 73 Position and direction 74 Angle and rotation 75 Handling data 75 Organising and interpreting data 75 YEAR 5 78 Numbers and the Number System 78 Place value (whole numbers) 78 Ordering (whole numbers) 79 Rounding (whole numbers) 79 Negative Numbers 80 Properties of numbers and number sequences 80 Fractions and decimals 82 Ratio and Proportion 84 Fractions, decimals and percentages 85 Calculations 86 Addition 86 Subtraction 86

4 Rapid recall of addition and subtraction facts 87 Mental calculation strategies (+ and -) 87 Pencil and paper procedures (addition) 89 Pencil and paper procedures (subtraction) 90 Understanding multiplication 90 Understanding division 91 Mental Calculation strategies (x and ) 93 Solving Problems 95 Making decisions 95 Reasoning about numbers or shapes 95 Measures 98 Length, mass and capacity 98 Time 100 Area and perimeter 101 Shape and space 102 Properties of 3-D and 2-D shapes 102 Reflective symmetry, reflection and translation 103 Position and direction 104 Angle and rotation 104 Handling data 105 Organising and interpreting data 105 YEAR Numbers and the Number System 106 Place value (whole numbers) 106 Rounding (whole numbers) 106 Negative Numbers 106 Properties of numbers and number sequences 107 Fractions and decimals 109 Ratio and proportion 111 Fractions, decimals and percentages 112 Calculations 113 Addition 113 Subtraction 113 Mental calculation strategies (+ and -) 114 Pencil and paper procedures (addition) 115 Pencil and paper procedures (subtraction) 116 Understanding multiplication 116 Understanding division 117 Mental Calculation strategies (x and ) 118 Solving Problems 121 Making decisions 121 Reasoning about numbers or shapes 121 Measures 125 Length, mass and capacity 125 Time 127 Area and perimeter 128

5 Shape and space 128 Properties of 3-D and 2-D shapes 128 Reflective symmetry, reflection and translation 130 Position and direction 131 Angle and rotation 131 Handling data 132 Organising and interpreting data 132 YEAR Using and applying mathematics to solve problems 133 Applying mathematics and solving problems 133 Numbers and the Number System 137 Place value, ordering and rounding 137 Integers, powers and roots 140 Fractions, decimals. percentages, ratio and proportion 143 Calculations 148 Number operations and the relationship between them 148 Mental methods and rapid recall of number facts 150 Written methods 154 Algebra 154 Equation, formulae and identities 154 Sequences and functions 158 Graphs and functions 160 Shape, Space and measures 162 Geometric reasoning: lines, angles and shapes 162 Transformations 165 Coordinates 169 Measures and mensuration 169 Handling data 172 Processing and representing data 172 Interpreting and discussing results 173 Probability 175 YEAR Using and applying mathematics to solve problems 178 Applying mathematics and solving problems 178 Numbers and the Number System 182 Place value, ordering and rounding 182 Integers, powers and roots 185 Fractions, decimals. percentages, ratio and proportion 187 Calculations 191 Number operations and the relationship between them 191 Mental methods and rapid recall of number facts 192 Written methods 196 Algebra 196 Equation, formulae and identities 196 Sequences and functions 201

6 Graphs and functions 204 Shape, Space and measures 205 Geometric reasoning: lines, angles and shapes 205 Transformations 209 Coordinates 212 Construction and loci 212 Measures and mensuration 212 Handling data 216 Processing and representing data 216 Interpreting and discussing results 217 Probability 217 YEAR Using and applying mathematics to solve problems 220 Applying mathematics and solving problems 220 Numbers and the Number System 224 Place value, ordering and rounding 224 Integers, powers and roots 226 Fractions, decimals. percentages, ratio and proportion 228 Calculations 231 Number operations and the relationship between them 231 Mental methods and rapid recall of number facts 232 Written methods 235 Algebra 235 Equation, formulae and identities 235 Sequences and functions 240 Graphs and functions 241 Shape, Space and measures 243 Geometric reasoning: lines, angles and shapes 243 Transformations 246 Coordinates 248 Construction and loci 248 Measures and mensuration 248 Handling data 251 Processing and representing data 251 Interpreting and discussing results 251 Probability 252

7 RECEPTION Numbers and the Number System Counting Children should: Say and use the number names in familiar contexts Recite the number names in order, continuing the count from a given number Recite the number names in order, counting back from a given number Count reliably a set of everyday objects Begin to recognise none and zero Count reliably in other contexts Count in tens Count in twos Reading and writing numbers Children should: Recognise and use numerals 1 to 9, extending 0 to 10, then beyond 10 Begin to record numbers Comparing and ordering numbers Children should: Understand and use language to compare two given numbers and say which is more or less Say the number that is one more or less than a given number Say a number lying between two given numbers Order a given set of numbers Order a given set of selected numbers Ordinal numbers Children should: Begin to understand and use ordinal numbers in different contexts Adding and subtracting Children should: Begin to use the vocabulary involved in adding and subtracting Find one more and one less than a given number Begin to relate addition to combining two groups of objects; extend to three groups Begin to relate addition to counting on Begin to relate the addition of doubles to counting on Find a total by counting on when one group of objects is hidden

8 Separate (partition) a given number of objects into two groups Select two groups of objects to make a given total Begin to relate subtraction to taking away, and counting how many are left Remove a smaller number from a larger and find how many are left by counting back from the larger number Begin to find out how many have been removed from a larger group of objects by counting up from a number Work out by counting how many more are needed to make a larger number Solving Problems Reasoning about numbers or shapes Children should: Recognise and recreate simple patterns Solve simple problems or puzzles in a practical context Make simple estimates and predictions Sort and match objects or pictures Begin to understand and use the vocabulary related to money Measures Length, mass and capacity Children should: Use language such as more or less, longer or shorter, heavier or lighter to compare directly two lengths, masses or capacities; extend to three or more quantities Time Children should: Begin to understand and use the vocabulary related to time; sequence familiar events; begin to know the days of the week in order and read o clock time 2-D and 3-D shape Children should: Use language to describe the shape and size of solids and flat shapes; begin to name shapes and use them to make models, pictures and patterns Patterns and symmetry Children should: Put sets of objects in order of size Talk about, recognise and recreate patterns

9 Position, direction and movement Children should: Use everyday words to describe position, direction and movement

10 YEAR 1 Numbers and the Number System Counting Pupils should be taught to: Know the number names and recite them in order, from and back to zero Count reliably a set of objects Describe and extend number sequences: count on or back in steps of 1, 10 or 100 from any number As outcomes, Year 1 pupils should, for example: Say the sequence: one, two, three.to 20 then beyond Say it backwards Respond to questions such as: What number comes after 6? After 17? Before 9? Before 14? Say the sequence: ten, twenty, thirty one hundred Recognise zero and none in stories and other contexts, including the counting sequence Using a 100 square, respond to questions such as: Count on in tens from zero from 30 from 3 Count back in tens from 100 from 80 from 63 Count on three tens from 50 from 20 from 70 Count round the circle in tens. Who will say 90? Describe this pattern: 80, 70, 60, 50 Say the next three numbers. Describe and extend number sequences: count on or back in twos, and recognise odd and even numbers As outcomes, Year 1 pupils should, for example: Understand and use in practical contexts: odd, even, every other Respond to questions such as: What numbers come next? 2, 4, 6, 8 15, 13, 11, 9 Describe the pattern. Describe and extend number sequences: count on or back in steps of any size As outcomes, Year 1 pupils should, for example: Respond to questions such as: Mark hops of 2 or 3 or 5 on a number track to at least 20 and say the numbers landed on What number comes next? 16, 14, 12. 5, 10, 15 3, 6, 9 Describe each pattern. Fill in the missing numbers: 2, 4, *, 8, 10, *

11 25, 20, 15, *, * Place value and ordering Read and write numbers in figures and words As outcomes, Year 1 pupils should, for example: Read and write numbers to at least 20 Respond to questions such as: Write a numeral to go with a dot pattern Write numbers to at least 20 in figures and words Know what each digit in a number represents, and partition a number into a multiple of 10 and ones (TU), or a multiple of 100, a multiple of 10 and ones (HTU) As outcomes, Year 1 pupils should, for example: Know what each digit represents in numbers from 10 to 20 Respond to questions such as: Say what the digit 1 in 14 stands for. And the digit 4? Say which number is the same as: one ten and seven ones; 2 tens and no ones In one step make 16 into 6; make 14 into 4 What number needs to go in each box? 14 = * = 2 + * Begin to partition larger numbers Understand and use the vocabulary of comparing and ordering numbers, including ordinal numbers; use the = sign to represent equality; compare two given numbers, say which is more or less, and give a number lying between them As outcomes, Year 1 pupils should, for example: Understand and use in practical contexts ordinal numbers: first, second, third How many As many as equal to more than less than fewer than Most, least, smallest, largest. Order, first, last, before, after, next, between Use the = sign to represent equality Respond to questions such as: Who is the first, last, third in this queue? Which is less: 15 or 19? Are there enough cups for these saucers? Pat has 6 pens. Alice has 8 pens. Who has fewer pens? How many more pens has Alice than Pat?

12 Say the number that is 1, 10, or 100 more or less than any given number As outcomes, Year 1 pupils should, for example: Use and apply knowledge of adding and subtracting 1 or 10 in a variety of contexts Respond to questions such as: What is 1 more than 6? Than 9? Than 19? Than 24? What is 1 less than 8? Than 20? Than 25? What number is one before 7? After 6? What number is 10 more than 6? 10 less than 17? What is 10 more than 17? 10 less than 30? Order a set of familiar numbers and position them on a number line and, where appropriate, a 100 square As outcomes, Year 1 pupils should, for example: Order numbers in real contexts in science, design and technology, geography, history, physical education Respond to questions such as: Which two numbers have been changed over? Put these in order, largest/smallest first: 7, 2, 9, 4 17, 6, 15, 7, 12, 22 Calculations Addition Understand the operation of addition and the related vocabulary, and recognise that addition can be done in any order As outcomes, Year 1 pupils should, for example: Understand and use in practical contexts: More, add, sum, total, altogether, equals, sign And read and write the plus (+) and equals (=) signs Understand addition as: Combining sets to make a total; Steps along a number track (counting on) Begin to understand that adding zero leaves a number unchanged Respond rapidly to oral questions phrased in a variety of ways such as: 3 add 1 Add 2 to 4 6 plus 3 What is the sum/total of 2 and 8? How many are 3 and 5 altogether? What must be added to 4 to make 10?

13 I think of a number. I add 3. The answer is 7. What is my number? Record simple mental additions in a number sentence using the + and = signs Recognise the use of symbols such as or to stand for unknown numbers, and complete, for example: = Understand, for example, that: = 2 + 5, but that 5 2 is not the same as = (5 + 2) + 6 or 5 + (2 + 6) and use these properties when appropriate Understand that more than two numbers can be added together As outcomes, Year 1 pupils should, for example: With the aid of apparatus Add three numbers Respond to questions such as: A plum costs 5p. What is the cost of three plums? using coins if necessary Mentally Add mentally three small numbers, within the range of 1 to about 12 Subtraction Understand the operation of subtraction and the related vocabulary As outcomes, Year 1 pupils should, for example: Understand and use in practical contexts: Take away, subtract, how many are left, how much less is than.., Difference between, how much more is than, how many more to make. and read and write the minus (-) sign Understand subtraction as: Taking away; Finding the difference between; How many more to make (complementary addition). Begin to understand that subtracting zero leaves a number unchanged Respond rapidly to oral questions phrased in a variety of ways such as: 4 take away 2 Take 2 from 7 7 subtract 3 Subtract 2 from 11 8 less than 9 What number must I take from 14 to leave 10? What is the difference between 14 and 12? How many more than 3 is 9?

14 How many less than 6 is 4? 6 taken from a number leaves 3. What is the number? I think of a number. I take away 3. The answer is 7. What is my number? Record simple mental subtractions in a number sentence using the - and = signs Recognise the use of symbols such as or to stand for unknown numbers, and complete, for example: 5-3 = Using rods, counters or cubes, coins or a number line, then mental strategies: 15 8 = 21 - = 10 Rapid recall of addition and subtraction facts Know by heart addition and subtraction facts As outcomes, Year 1 pupils should, for example: Know by heart all addition and subtraction facts for all numbers up to and including 5 For example, recall rapidly all number pairs for 4: = 4 4 = 0 = = = = = = = = = 2 Begin to know by heart number bonds for numbers up to 10, for both addition and subtraction Understand and use in practical contexts: Double, halve, half Know by heart addition doubles from to at least 5 +5 Begin to know doubles from to Know by heart all pairs of numbers that total 10 Use known number facts and place value to add or subtract a pair of numbers mentally As outcomes, Year 1 pupils should, for example: Add or subtract a single digit to or from a single digit, without crossing 10 (5 + 3; 9 4) Add or subtract a single digit to or from a teens number, without crossing 20 or 10 (13 + 6; 18 9)

15 Add or subtract a single digit to or from 10, then 20 (10 + 4; 10 3; ; 20 9) Begin to add a teens number to a teens number, without crossing the tens boundary ( ) Add 10 to a single-digit number and subtract 10 from a teens number (3 + 10; 18 10) Add or subtract a pair of numbers mentally (continued) by bridging through 10 or 100, or a multiple of 10 or 100, and adjusting As outcomes, Year 1 pupils should, for example: Begin to add a pair of single-digit numbers crossing ten Use two steps and cross 10 as a middle stage Begin to add a single digit to a teens number, crossing 20 Use two steps and cross 20 as a middle stage Solving Problems Making decisions Choose and use appropriate number operations and ways of calculating to solve problems As outcomes, Year 1 pupils should, for example: Understand and use in practical contexts: Operation, sign, number sentence Reasoning about numbers or shapes Solve mathematical problems or puzzles, recognise simple patterns or relationships, generalise and predict. As outcomes, Year 1 pupils should, for example: Solve puzzles and problems such as: Put 1, 2 or 3 in each circle so that each side adds up to 5. Solve simple word problems set in real life contexts and explain how the problem was solved. As outcomes, Year 1 pupils should, for example: Solve puzzles and problems such as: One-step operations I think of a number, then add 2. The answer is 7. What was my number?

16 Lisa has 5 pens and Tim has 2 pens. How many pens do they have altogether? How many more pens has Lisa than Tim? Tina rolled double six on her two dice. What was her score? Two-step operations Some hens lay 2 eggs, 4 eggs and 3 eggs. How many eggs did they lay altogether? Half of the cakes in this box of 10 are gone. How many are left? Solve simple word problems involving money and explain how the problem was solved. As outcomes, Year 1 pupils should, for example: Find totals and give change Solve puzzles and problems such as: How much altogether is 5p + 2p + 1p? Tim spent 4p. What was his change from 10p? Anil spent 6p and 3p on toffees. What change from 10p did he get? Rosie had 15p. She spent 6p. how much does she have left? Chews cost 5p. How much do 3 chews cost? Solve simple word problems involving measures and explain how the problem was solved. As outcomes, Year 1 pupils should, for example: Solve problems such as: Length, mass, capacity The classroom is 15 metres long. The library is 12 metres long. The classroom is longer than the library. How much longer is it? 8 bricks balance an apple. 10 bricks balance a pear. The apple and the pear are together on the scales. How many bricks will balance them? A full jug holds 6 cups of water. How many cups of water do 2 full jugs hold? Time How long is it from 2 o clock in the afternoon to 6 o clock in the evening? It is now half-past seven. What time was it 2 hours ago? It is 5 o clock. What time will it be 4 hours from now? What time was it 3 hours ago? If you go to bed at 8 o clock, how many hours until bed time?

17 Measures Length, mass and capacity Understand and use the vocabulary related to length, mass and capacity; begin to know the relationship between standard metric units. Measure and compare: by direct (side-by-side) comparison; using uniform non-standard units; using standard units. As outcomes, Year 1 pupils should, for example: Understand and use in practical contexts: Length and distance: long, short, tall, high, low, wide, narrow, deep, shallow, thick, thin, far, near, close. Mass: weight, weighs, heavy, light, balances Capacity: full, empty, holds And comparative words such as longer, longest.. Suggest suitable units to estimate or measure length, mass or capacity. As outcomes, Year 1 pupils should, for example: Understand and use in practical contexts: Guess, roughly, nearly, close to, about the same as. Too many, too few, enough, not enough Suggest a unit you could use to measure: The height of a table The width of a book Across the classroom The weight of a parcel How much a big saucepan holds Time Understand and use the vocabulary related to time; know and use units of time and the relationships between them; read the time from clocks; solve problems involving time. As outcomes, Year 1 pupils should, for example: Know that: 1 week = 7 days 1 day = 24 hours Know in order the days of the week. Order familiar events in a day or a week, or in a story. Read the time to the hour or half hour on an analogue clock

18 Shape and space Properties of 3-D and 2-D shapes Describe and classify common 3-D and 2-D shapes according to their properties As outcomes, Year 1 pupils should, for example: Use everyday language to name, sort and describe some features of familiar 3-D and 2-D shapes such as: Cube, cuboid, sphere, cone, cylinder Circle, triangle, rectangle, square Identify solid shapes in the environment. For example, find a cuboid (box) or a cylinder (baked bean tin) Sort 3-D shapes in different ways according to properties such as: Whether they have corners Whether all their edges are straight Whether they are solid or hollow Choose an example to match given properties and name the shape Make models, shapes and patterns with increasing accuracy, and describe their features. As outcomes, Year 1 pupils should, for example: Begin to relate 3-D shapes to pictures of them Make pictures and patterns using 2-D shapes Describe a picture or pattern and say which shapes have been used to make it Position and direction Describe positions and directions. As outcomes, Year 1 pupils should, for example: Use everyday language to describe positions. For example: Name an object which is above the door, behind the desk, between the window and the sink Describe the position of an object in a picture relative to another object. For example: the house is below the aeroplane, the window is above the door Use everyday language to describe directions

19 Movement and angle Describe movements (in a straight line and turning) and understand angle as a measure of turn. As outcomes, Year 1 pupils should, for example: Understand and use in practical contexts: Slide, roll, turn Whole, half Recognise and talk about movements. For example: Talk about things that turn about a point, such as a spinning top, taps, windmill arms, wheels, the hands of a clock, the blades of scissors Talk about things that turn about a line, such as a door, the pages of a book, a hinged lid Sort objects that will roll ( a ball, an orange, a wooden egg, a sphere); Slide (a book, a cuboid box, a cube, a pyramid); Both roll and slide (a cotton reel, a coin, a tin of soup, a cone, a cylinder ) Recognise whole turns and half turns Handling data Organising and using data Solve a given problem by collecting, sorting and organising information in simple ways. As outcomes, Year 1 pupils should, for example: Understand and use in practical contexts: Sort, list, set, count Make and organise a list, such as: All the counting numbers between 14 and 23 All the days of the week First names with 6 letters Organise a table and use it to respond to questions such as: Who can hold the most cubes? Who has one more cube than Mark? How many children did we ask?

20 YEAR 2 Numbers and the Number System Counting Pupils should be taught to: Know the number names and recite them in order, from and back to zero Count reliably a set of objects Describe and extend number sequences: count on or back in steps of 1, 10 or 100 from any number As outcomes, Year 2 pupils should, for example: Say forwards and backwards the sequences: zero, ten, twenty, thirty one hundred; zero, one hundred, two hundred, three hundred one thousand Respond to questions such as: Which tens number comes after 60? Before 30? Which hundreds number comes after 400? Before 900? Use zero when counting and understand the function of 0 as a place holder in two-digit numbers Count reliably to at least 100 Count larger collections by grouping in tens, then fives or twos Respond to such questions as: Here is part of a number track. Where would 42 be? Start at any two-digit number and count on in ones to 100, or back in ones to zero. Count on 6 from 63. Count back 6 from 78. Count on from 33 to 37. How many did you count? First with and then without a 100 square, respond to questions such as: Count on in tens from 30 from 26 Count back in tens from 80 from 72 Count back 40 in tens: from 80 from 72 Count on in tens from 30 to 70. How many tens did you count? Describe this sequence: 43, 53, 63, 73 Write the next three numbers. Describe and extend number sequences: count on or back in twos, and recognise odd and even numbers As outcomes, Year 2 pupils should, for example: Understand, use and begin to read: odd, even, sequence, predict, continue, rule Respond to questions such as: Select the even numbers:

21 Continue these sequences: 13, 15, 17, 19 26, 24, 22, 20 What odd number comes after 13? After 7? Make general statements about odd or even numbers such as: An even number divides exactly by 2. There is 1 left over when an odd number is divided by 2. Describe and extend number sequences: count on or back in steps of any size As outcomes, Year 2 pupils should, for example: Respond to questions such as: From zero and then from any small number, count on and back in 2s, 3s, 4s or 5s to 30 or more. 3, 6, 9, 12 16, 14, 12, 10 Describe each pattern. What is the rule? What are the next three numbers in each sequence? Fill in the missing number in this sequence: 3, 6, *, 12, 15 Recognise familiar multiples As outcomes, Year 2 pupils should, for example: Understand, use and begin to read: multiple Recognise that multiples of 10 end in 0 5 end in 0 or 5 Begin to recognise that multiples of 2 end in 0, 2, 4, 6,8 Begin to recognise two-digit multiples of 10, 5 or 2: for example, that 65 is a multiple of 5, or that 32 is a multiple of 2. Respond to questions such as: Select the numbers which are multiples of 10: Place value and ordering Read and write numbers in figures and words As outcomes, Year 2 pupils should, for example: Read and write numbers to at least 100 Respond to questions such as: Write numbers to 100 in figures and words Know what each digit in a number represents, and partition a number into a multiple of 10 and ones (TU), or a multiple of 100, a multiple of 10 and ones (HTU) As outcomes, Year 2 pupils should, for example: Know what each digit in a two-digit number represents Recognise 0 as a place holder in two-digit multiples of 10 such as 50, 90, 10 Respond to questions such as: Say what the digit 6 in 64 represents. And the 4?

22 Say which number is the same as: six tens and four ones; nine tens and no ones In one step make 5 into 75; change 49 to 9 What number needs to go in each box? 64 = * = 50 + * Understand and use the vocabulary of comparing and ordering numbers, including ordinal numbers; use the = sign to represent equality; compare two given numbers, say which is more or less, and give a number lying between them As outcomes, Year 2 pupils should, for example: Understand and use in practical contexts ordinal numbers: first, second, third How many As many as equal to more than les than fewer than Most, least, smallest, largest. Order, first, last, before, after, next, between Use the = sign to represent equality Respond to questions such as: Which is less: 36 or 63? Which is shorter: 18 metres or 15 metres? Ali has 16 pens. Ben has 28 pens. Who has fewer pens? How many more pens has Ben than Ali? Say the number that is 1, 10, or 100 more or less than any given number As outcomes, Year 2 pupils should, for example: Use and apply knowledge of adding and subtracting 1 or 10 in a variety of contexts Respond to questions such as: What is 1 more than 53? Than 89? Than 112? What is 1 less than 82? Than 60? Than 120? What number is 10 after 43? 10 before 78? What is 10 more than 96? 10 less than 102? Order a set of familiar numbers and position them on a number line and, where appropriate, a 100 square As outcomes, Year 2 pupils should, for example: Order numbers in real contexts in science, design and technology, geography, history, physical education Respond to questions such as: Put these in order, largest/smallest first: 27, 16, 85, 72, 52 Fill in the missing numbers on this number line: * 69 * 71 72

23 Write a number in each blank space so that the five numbers are in order: 85 * 91 * 102 Estimation and rounding Round a number to the nearest 10 or 100 As outcomes, Year 2 pupils should, for example: Begin to round numbers less than 100 to the nearest ten. For example: 33 is closer to 30 than 40. The nearest ten to 33 is is closer to 40 than 30. The nearest ten to 37 is is half way between 30 and 40. We say that the nearest ten to 35 is 40, because we round up when the number is half way between two tens. Fractions Recognise and find simple fractions; recognise the equivalence between them; compare two simple fractions in practical contexts As outcomes, Year 2 pupils should, for example: Understand, use and begin to read: part, fraction. One half, one whole, one quarter Recognise and write ½, ¼, as one half, one quarter. Respond to questions such as: What fraction of this shape is shaded? Click one half of this set of buttons. Say half of any even number up to 20. Find one quarter of 12 biscuits, of 8 pencils. Say what fraction of a cake each person will get when it is divided equally between two or four people. Recognise what is not one half or one quarter. Recognise and find simple fractions; recognise the equivalence between them; compare two simple fractions in practical contexts (continued) As outcomes, Year 2 pupils should, for example: Recognise that one whole can be broken into two identical halves or four identical quarters, and that two halves and four quarters will make one whole Recognise that: Two quarters are the same as one half; Three quarters and one quarter make one whole.

24 Calculations Addition Understand the operation of addition and the related vocabulary, and recognise that addition can be done in any order As outcomes, Year 2 pupils should, for example: Understand and use in practical contexts: More, add, sum, total, altogether, equals, sign And read and write the plus (+) and equals (=) signs Continue to develop understanding of addition as: Combining sets to make a total; Counting on steps along a number line Understand that adding zero leaves a number unchanged Begin to understand that addition reverses subtraction (addition is the inverse of subtraction) Respond rapidly to oral questions phrased in a variety of ways such as: 27 add 10 Add 60 to 30 4 plus 18 What is the sum/total of 18 and 4? How many are 5 and 14 altogether? What must be added to 14 to make 15? I think of a number. I add 10. The answer is 30. What is my number? Record simple mental additions in a number sentence using the + and = signs Recognise the use of symbols such as or to stand for unknown numbers, and complete, for example: = Understand, for example, that: = , but that 15 6 is not the same as = (15 + 2) + 7 or 15 + (2 + 7) and use these properties when appropriate Understand that more than two numbers can be added together As outcomes, Year 2 pupils should, for example: With the aid of apparatus Add three numbers Using coins if necessary, total a shopping bill such as: 29p 36p 18p Mentally

25 Add mentally three small numbers, within the range of 1 to about 20 Respond to oral/written questions such as: Add 5, 2 and plus 19 plus 1 What is the sum/total of 3, 6 and 7? How many altogether are 7, 4 and 2? Work mentally to complete written questions such as: = = 17 Subtraction Understand the operation of subtraction and the related vocabulary As outcomes, Year 2 pupils should, for example: Understand and use in practical contexts: Take away, subtract, how many are left, how much less is than.., Difference between, how much more is than, how many more to make. and read and write the minus (-) sign Continue to develop understanding of subtraction as: Taking away; Finding the difference between; Complementary addition Understand that: Subtracting zero leaves a number unchanged 4-2 (for example) is different from 2-4 Begin to understand the principle that subtraction reverses addition (subtraction is the inverse of addition). Respond rapidly to oral questions phrased in a variety of ways such as: 7 take away 3 Take 30 from subtract 2 Subtract 30 from 70 3 less than 7 What number must I take from 20 to leave 3? What is the difference between 10 and 18? How many more is 11 than 3? How many less is 7 than 18? 5 taken from a number leaves 11. What is the number? 8 added to a number is 18. What is the number? Find pairs of numbers with a difference of 10 with a difference of 9 Record simple mental subtractions in a number sentence using the - and = signs Recognise the use of symbols such as or to stand for unknown numbers, and complete, for example:

26 7-3 = - 6 = 2 Using rods, counters or cubes, coins or a number line, then mental strategies: 25 8 = 25 - = = - 40 = 28 Rapid recall of addition and subtraction facts Know by heart addition and subtraction facts As outcomes, Year 2 pupils should, for example: Know by heart all addition and subtraction facts for all numbers up to and including 10 Derive quickly these addition doubles: Doubles of numbers from to Doubles of multiples of 5 from to (e.g., ) Know by heart all pairs of numbers that total 20 Know by heart all pairs of multiples of 10 that total 100. For example: Put numbers in the boxes to make 100: + 20 = 100 Use knowledge that addition can be done in any order Find a small difference by counting up Identify near doubles Add or subtract 9, 19, 29 or 11, 21, 31 by adding or subtracting 10, 20, 30 and adjusting by 1 Use patterns of similar calculations Use the relationship between addition and subtraction As outcomes, Year 2 pupils should, for example: Mentally add or subtract 11 or 21, or 9 or 19, to/from any two-digit number Develop and recognise a pattern such as: = = = = = = 21 Recognise and use the pattern in, for example: = = = 700 Say and write the subtraction fact corresponding to a given addition fact, and vice versa. For example: = 19 implies that 19 4 = 15 You know that = 16 so: What is ?, or 16 4? Or 16 12? Given three numbers, say or write four different sentences relating these numbers. For example:

27 Given 2, 7 and 9, write that = 9; =9; 9 2 = 7; 9 7 =2 Use known number facts and place value to add or subtract a pair of numbers mentally As outcomes, Year 2 pupils should, for example: Add or subtract a single digit to or from any two-digit number, without crossing the tens boundary ( ; 37 4) Add a single digit to a multiple of 10 or 100 (40 + 8; ) Subtract a single digit from a multiple of 10 (70 8) Add/subtract a teens number to/from a two-digit number, without crossing the tens boundary or 100 ( ; 46 13) Add/subtract 10 to or from any two-digit number, without crossing 100 ( ; 97 10) Add or subtract a pair of multiples of 10, without crossing 100 ( ; 90 70) Find what must be added to a two-digit multiple of ten to make 100 (40 + = 100) Add or subtract a multiple of ten to or from a two-digit number, without crossing 100 ( ; Add or subtract a pair of multiples of 100, without crossing 1000 ( ; ; = 600, etc) Add or subtract a pair of numbers mentally (continued) by bridging through 10 or 100, or a multiple of 10 or 100, and adjusting As outcomes, Year 2 pupils should, for example: Add a pair of single-digit numbers, or subtract a single digit from a teens number, crossing 10 Use two steps and cross 10 as a middle stage Add a single digit to a teens number, or subtract a single digit from a twenties number, crossing 20 Use two steps and cross 20 as a middle stage Find a small difference between a pair of numbers lying either side of 20, or another multiple of ten Understanding multiplication Understand the operation of multiplication and the associated vocabulary, and that multiplication can be carried out in any order As outcomes, Year 2 pupils should, for example: Understand, use and begin to read: Double, times, multiply, multiplied by, multiple of Lots of, groups of.

28 Times as (big, long, wide ) Read and write the x sign Understand multiplication as: Repeated addition: for example, 5 added together 3 times is , or 3 lots of 5, or 3 times 5, or 5 x 3 (or 3 x 5) Describing an array: for example, (4 x 2 = 8) (2 x 4 = 8) Begin to recognise from arranging arrays that multiplication can be done in any order: for example, 4 lots of 2 and 2 lots are 4 are the same Understand and use the principle that doubling reversing halving (doubling is the inverse of halving) Respond rapidly to oral or written questions such as: Two fives; Double 5; 6 times 2; 5 multiplied by 2; Multiply 4 by 2 Recognise the use of symbols such as to stand for unknown numbers and complete with rapid mental recall, for example: 6 x 2 = x 2 = 14 Begin to interpret situations as multiplication calculations. For example: How many wheels are there on 3 cars? Understanding division Understand the operation of division and the associated vocabulary As outcomes, Year 2 pupils should, for example: Understand, use and begin to read: One each, two each share, halve, divide, left over, divided by, equal groups of, Read and write the sign Understand the operation of division as: Sharing equally: for example, 6 sweets are shared equally between 2 people. How many sweets does each person get? Grouping or repeated subtraction: for example, There are 18 apples in a box. How many bags of 3 apples can be filled? Interpret 8 2 as how many 2s make 8? Respond rapidly to oral or written questions phrased in a variety of ways such as: Share 18 between 2; Divide 6 by 3; How many tens make 80?; How many 2 coins do you get for 20?; How many 2cm lengths can you cut from 10cm of tape? Recognise the use of symbols such as to stand for unknown numbers and complete with rapid mental recall, for example: 6 2 = 20 = 2 10 = 3

29 Know simple multiplication and division facts by heart Derive doubles and halves quickly As outcomes, Year 2 pupils should, for example: Know by heart multiplication facts for 2 up to 2 x 10 and 10 up to 10 x 10 and derive quickly the corresponding division facts Begin to know multiplication facts for 5 up to 5 x 10 and derive quickly the corresponding division facts Use known facts to derive quickly: Doubles of numbers 1 to 15 Doubles of 5, 10, 15 to 50 Halves of even numbers to 20 Halves of multiples of 10 up to 100 Solving Problems Making decisions Choose and use appropriate number operations and ways of calculating to solve problems As outcomes, Year 2 pupils should, for example: Understand and use in practical contexts: Operation, sign, symbol, number sentence Make up number stories to reflect statements such as: 4 x 3 = 12 A tricycle has three wheels. Four tricycles have twelve wheels. What sign does each stand for? 24 8 = = 89 Reasoning about numbers or shapes Solve mathematical problems or puzzles, recognise simple patterns or relationships, generalise and predict. As outcomes, Year 2 pupils should, for example: Solve puzzles and problems such as: Find ways of rearranging digits in a number square so that the sum of each column, row and diagonal is the same

30 Solve simple word problems set in real life contexts and explain how the problem was solved. As outcomes, Year 2 pupils should, for example: Use mental addition or subtraction, or simple multiplication, and own strategies to solve story problems about numbers in real life. Solve puzzles and problems such as: One-step operations I think of a number, then halve it. The answer is 9. What was my number? There are 16 plums. 8 children share them equally. How many plums does each child have? Two people have 8 cakes each. How many cakes have they altogether? One person gives 2 cakes to the other. How many does each person have now? Two-step operations 7 people are on a bus. 8 more get on and 3 get off. How many people are on the bus now? There are 25 balls. Kim takes 11 and Anne takes 9. How many are left? There are 2 red buttons and 4 blue buttons on a card of buttons. How many buttons are there on 10 cards? Solve simple word problems involving money and explain how the problem was solved. As outcomes, Year 2 pupils should, for example: Use mental addition or subtraction, or simple multiplication, and own strategies to solve story problems about numbers in real life. Find totals and give change I have 14. I am given another 9. How much do I have now? A pear costs 15p more than an apple. An apple costs. 12p. What does a pear cost? Rhian spent 24p. She spent 8p more than Amy. How much did Amy spend? Patrick bought three choc bars at 15p each. How much change did he get from 50p? Solve simple word problems involving measures and explain how the problem was solved. As outcomes, Year 2 pupils should, for example: Solve problems such as: Length, mass, capacity My cat is 30cm tall. My dog is 25cm taller. How tall is my dog?

31 Time There are 5kg of pears in 1 box. How many kilograms of pears are in 3 boxes? I have 50L of water. How many 10L buckets can I fill? Sue got on the bus at 9 o clock. Her journey lasted half an hour. What time did she get off the bus? Mary went into a shop at 10:30. She came out at 10:45. How long was she in the shop? James walked from 9:45 until 10:15. How many minutes did he walk? Measures Length, mass and capacity Understand and use the vocabulary related to length, mass and capacity; begin to know the relationship between standard metric units. Measure and compare: by direct (side-by-side) comparison; using uniform non-standard units; using standard units. As outcomes, Year 2 pupils should, for example: Understand and use in practical contexts: Length and distance: long, short, tall, high, low, wide, narrow, deep, shallow, thick, thin, far, near, close. Mass: weight, weighs, heavy, light, balances Capacity: full, empty, holds And comparative words such as longer, longest.. Know that: 1 metre = 100 centimetres 1 kilogram = 1000 grams 1 litre = 1000 millilitres Suggest things that are: Longer or shorter than 1 metre, or 1 centimetre, or 10 centimetres; Heavier or lighter than 1 kilogram; Holding more or less than 1 litre. Suggest suitable units to estimate or measure length, mass or capacity. As outcomes, Year 2 pupils should, for example: Understand and use in practical contexts: Guess, estimate Roughly, nearly, about, close to. Suggest things that could be measured using Metres, centimetres Kilograms, grams

32 Litres Suggest a unit you could use to measure: The width of the classroom The height of a plant How much water will fill a bowl Respond to questions such as: What is about 1cm, 10cm, 100cm long/wide/tall/deep? What will balance about 1kg? 5kg? What holds about 1 litre? About 5 litres? Read and interpret number scales with some accuracy. As outcomes, Year 2 pupils should, for example: Read a simple scale to the nearest labelled division Time Understand and use the vocabulary related to time; know and use units of time and the relationships between them; read the time from clocks; solve problems involving time. As outcomes, Year 2 pupils should, for example: Know that: 1 week = 7 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds Know in order the months and seasons of the year. Read the time to the half or quarter hour on a digital clock or an analogue clock, knowing, for example, that the time is a quarter to 5 or 15 minutes to 5 Suggest a suitable unit of time to measure the time needed to walk home, to sleep each night, etc Know what takes about 10 seconds, 1 minute, 1 hour. Shape and space Properties of 3-D and 2-D shapes Describe and classify common 3-D and 2-D shapes according to their properties As outcomes, Year 2 pupils should, for example: Understand, use and begin to read the vocabulary from the previous year, and extend to:

33 Circular, triangular, rectangular surface Use mathematical vocabulary to name, classify and describe some features of 3-D and 2-D shapes, extending the shapes used to: Pyramid Pentagon, hexagon, octagon Sort 3-D shapes in different ways according to the properties of their faces such as whether they: Have six faces Have a triangular face, a rectangular face Using a set of solid shapes, choose an example to match given properties and name the shape Sort a set of flat shapes according to properties such as: The number of corners The number of sides Whether the sides are straight or curved Make models, shapes and patterns with increasing accuracy, and describe their features. As outcomes, Year 2 pupils should, for example: Relate 3-D shapes to pictures of them Use 2-D shapes to make and describe pictures and patterns Line symmetry Recognise line symmetry in simple cases. As outcomes, Year 2 pupils should, for example: Understand, use and begin to read: Line of symmetry Fold, match, mirror line, reflection, symmetrical Begin to recognise a line of symmetry Complete a symmetrical pattern by drawing or making the other half : for example, using a pegboard Position and direction Describe positions and directions. As outcomes, Year 2 pupils should, for example: Extend vocabulary from previous year to include clockwise, anti-clockwise Describe positions. Respond to questions or instructions by describing, placing or ticking objects which are in a given position.

34 Use a squared area and a counter to move in a described path, for example: Three squares along and two squares down; or three squares to the left and two squares up. Describe directions Movement and angle Describe movements (in a straight line and turning) and understand angle as a measure of turn. As outcomes, Year 2 pupils should, for example: Understand, use and begin to read the vocabulary from the previous year, and extend to: Quarter turn Right angle, straight line Recognise whole, half and quarter turns Know that a quarter turn is called a right angle Recognise that the corners of doors, windows, books, tables. Are right angles Recognise that a square and a rectangle have right angles at each corner/vertex Use two geo-strips to make and draw half and quarter turns from the same starting position Handling data Organising and using data Solve a given problem by collecting, sorting and organising information in simple ways. As outcomes, Year 2 pupils should, for example: Understand, use and begin to read: Sort, set, represent, graph, table, list, count, label Most/least common or popular Classify numbers and organise them in lists and simple tables. For example, make a list of: All the multiples of 10 between 0 and 100 Five different numbers that are more than 70 All the odd numbers from 15 to 35 Organise a table and use it to respond to questions such as: What is the most common number of letters in a name? How many names have more than 5 letters? How many names have fewer than 5 letters? Make a simple block graph and use it to respond to questions such as:

35 What do most children like to drink? How many children were asked? Make a simple pictogram, where one symbol represents one unit, and use it to respond to questions such as: How many children are in bed by 7:30? Are more children in bed before 7:30 than after 7:30? How many children altogether?

36 YEAR 3 Numbers and the Number System Counting Pupils should be taught to: Know the number names and recite them in order, from and back to zero Count reliably a set of objects Describe and extend number sequences: count on or back in steps of 1, 10 or 100 from any number As outcomes, Year 3 pupils should, for example: Counting in tens Respond to questions such as: Count on and back in tens, crossing 100 Count on 40 in tens: from 30, from 27, from 480, from 652 Count back 40 in tens: from 80, from 72, from 590, from 724 Count on in tens from 36 to 76. How many tens did you count? Count back in tens from 84 to 34. How many tens did you count? Counting in hundreds Respond to questions such as: Count on or back in 400 in hundreds: from 500, from 520, from 570 Count on in hundreds from 460 to 960. How many hundreds did you count? Describe these sequences: 256, 356, 456, , 431, 441, 451. Write the next three numbers in each sequence. Describe and extend number sequences: count on or back in twos, and recognise odd and even numbers As outcomes, Year 3 pupils should, for example: Use, read and begin to write: odd, even, sequence, predict, continue, rule, relationship Count from 0 or 1 in steps of two to about 50. Count back again. Respond to questions such as: Is 74 odd or even? How do you know? Select the odd numbers: Continue these sequences: 35, 37, 39, 41 68, 66, 64 What odd number comes before 91? After 69? Make general statements about odd or even numbers such as: An even number ends in 0, 2, 4, 6 or 8. An odd number ends in 1,3,5,7 or 9. If you add two even numbers, the answer is even. If you add two odd numbers, the answer is even.

37 Describe and extend number sequences: count on or back in steps of any size As outcomes, Year 3 pupils should, for example: Respond to questions such as: Count on from any small number in step of 2, 3, 4, 5, 10 or 100 and then back 2, 7, 12, 17 78, 76, 74, 72 Describe each pattern. What is the rule? What are the next three numbers in each sequence? Fill in the missing numbers in this sequence: 5, 9, *, 17, 21, *, * Recognise familiar multiples As outcomes, Year 3 pupils should, for example: Use, read and begin to write: multiple Recognise that multiples of 100 end in 00; 51 end in 00 or 50; 10 end in 0; 5 end in 0 or 5; 2 end in 0, 2, 4, 6, 8. Respond to questions such as: Select the numbers which are multiples of 5: Count in 50s to Write three different multiples of 50. What is the multiple of 10 before 140? What is the multiple of 100 after 500? What si the next multiple of 5 after 195? Place value and ordering Read and write numbers in figures and words As outcomes, Year 3 pupils should, for example: Read and write numbers to at least 1000 Respond to questions such as: Write numbers to 1000 in figures and words Know what each digit in a number represents, and partition a number into a multiple of 10 and ones (TU), or a multiple of 100, a multiple of 10 and ones (HTU) As outcomes, Year 3 pupils should, for example: Know what each digit in a three-digit number represents Recognise 0 as a place holder in three-digit numbers such as 430, 506 Respond to questions such as: Say what the digit 3 in 364 represents. And the 6? And the 4? Say which number is equivalent to: four hundreds, five tens and six ones; nine hundreds and two ones

38 In one step make 478 into 978; make 326 into 396; change 707 to 507; change 263 to 203 What number needs to go in each box? 364 = * = * + 2 Make the biggest/smallest number you can with these digits: 2, 5, 3 Understand and use the vocabulary of comparing and ordering numbers, including ordinal numbers; use the = sign to represent equality; compare two given numbers, say which is more or less, and give a number lying between them As outcomes, Year 3 pupils should, for example: Understand and use in practical contexts ordinal numbers: first, second, third How many As many as equal to more than les than fewer than Most, least, smallest, largest. Order, first, last, before, after, next, between Use the = sign to represent equality Respond to questions such as: What colour would the 19th bead be in this pattern of beads? Which is more: 216 or 261? Which is shorter: 157cm or 517cm? Which is lighter: 3.5kg or 5.5kg? Jo has 47 stamps. Ny has 92 stamps. Who has fewer stamps? How many more stamps has Ny than Jo? What is the number halfway between 40 and 60? Between 300 and 400? Between 7 and 8? Say the number that is 1, 10, or 100 more or less than any given number As outcomes, Year 3 pupils should, for example: Use and apply knowledge of adding and subtracting 1 or 10 in a variety of contexts Respond to questions such as: What is 1 more than 485? Than 569? Than 299? What is 1 less than 756? Than 340? Than 500? What number is 10 after 437? 10 less? 100 more? 100 less? 1 more? 1 less? Jack walks 645 metres to school. Suzy walks 100 metres less. How far does Suzy walk?

39 Order a set of familiar numbers and position them on a number line and, where appropriate, a 100 square As outcomes, Year 3 pupils should, for example: Order numbers in real contexts in science, design and technology, geography, history, physical education Respond to questions such as: Put these in order, largest/smallest first: 27, 16, 85, 72, 52 Fill in the missing numbers on this number line: * * 259 * Write a number in each blank space so that the five numbers are in order: 697 * 701 * 706 Estimation and rounding Round a number to the nearest 10 or 100 As outcomes, Year 3 pupils should, for example: Round numbers less than 100 to the nearest ten. For example: 33 is 30 rounded to the nearest ten. 37 is 40 rounded to the nearest ten. 35 is 40 rounded to the nearest ten. Begin to approximate by rounding any three-digit number to the nearest hundred. Fractions Recognise and find simple fractions; recognise the equivalence between them; compare two simple fractions in practical contexts As outcomes, Year 3 pupils should, for example: Understand, use and begin to read: part, fraction. One half, one whole, one quarter Three quarters, one third, two thirds, one tenth Recognise 1/10 as one tenth, and know that it means one whole divided into ten equal parts. Respond to questions such as: What fraction of this shape is shaded? What fraction of this set of objects is circled? What fraction is not circled? Find half of each of the numbers up to 30. What is one tenth of 20? What is three quarters of 20? Find one quarter of 12 biscuits, of 8 pencils. Say what fraction of a cake each person will get when: 1 cake is divided equally among 10 people;

40 5 cakes are divided equally between 2 people. Recognise what is not ½, ¼. 1/3, 1/10. Recognise and find simple fractions; recognise the equivalence between them; compare two simple fractions in practical contexts (continued) As outcomes, Year 3 pupils should, for example: Know that: Two quarters are the same as one half; One half is equivalent to five tenths; Ten tenths make one whole; One whole is three quarters plus one quarter; One whole is three tenths plus seven tenths.; One quarter is half of one half. Answer questions such as: What number is halfway between three and four? Between 2½ and 3? Estimate a fraction Calculations Addition Understand the operation of addition and the related vocabulary, and recognise that addition can be done in any order As outcomes, Year 3 pupils should, for example: Understand and use in practical contexts: More, add, sum, total, altogether, equals, sign And read and write the plus (+) and equals (=) signs Continue to develop understanding of addition as: Combining sets to make a total; Counting on steps along a number line Understand that adding zero leaves a number unchanged Begin to understand that addition reverses subtraction (addition is the inverse of subtraction) Respond rapidly to oral questions phrased in a variety of ways such as: 94 add 10 Add 60 to plus 50 What is the sum/total of 26 and 9? How many are 11 and 35 altogether? What must be added to 4 to make 23? I think of a number. I add 45. The answer is 90. What is my number?

41 Record simple mental additions in a number sentence using the + and = signs Recognise the use of symbols such as or to stand for unknown numbers, and complete, for example: = Understand, for example, that: = , but that is not the same as = ( ) or ( ) and use these properties when appropriate Understand that more than two numbers can be added together As outcomes, Year 3 pupils should, for example: With the aid of apparatus Find the missing number in = 150 Using coins if necessary, total a shopping bill such as: Mentally Add mentally three or more small numbers, within the range of 1 to about 50 Respond to oral/written questions such as: Add 15, 6, 15 and 1. 7 plus 5 plus 9 What is the sum/total of 13, 12 and 3? How many altogether are 11, 17 and 6? Work mentally to complete written questions such as: = = 37 Subtraction Understand the operation of subtraction and the related vocabulary As outcomes, Year 3 pupils should, for example: Understand and use in practical contexts: Take away, subtract, how many are left, how much less is than.., Difference between, how much more is than, how many more to make. and read and write the minus (-) sign Continue to develop understanding of subtraction as: Taking away; Finding the difference between; Complementary addition Understand that:

42 Subtracting zero leaves a number unchanged 4-2 (for example) is different from 2-4 Begin to understand the principle that subtraction reverses addition (subtraction is the inverse of addition). Respond rapidly to oral questions phrased in a variety of ways such as: 7 take away 3 Take 30 from subtract 2 Subtract 30 from 70 3 less than 7 What number must I take from 20 to leave 3? What is the difference between 10 and 18? How many more is 11 than 3? How many less is 7 than 18? 5 taken from a number leaves 11. What is the number? 8 added to a number is 18. What is the number? Find pairs of numbers with a difference of 10 with a difference of 9 Record simple mental subtractions in a number sentence using the - and = signs Recognise the use of symbols such as or to stand for unknown numbers, and complete, for example: 7-3 = - 6 = 2 Using rods, counters or cubes, coins or a number line, then mental strategies: 25 8 = 25 - = = - 40 = 28 Rapid recall of addition and subtraction facts Know by heart addition and subtraction facts As outcomes, Year 3 pupils should, for example: Know by heart all addition and subtraction facts for all numbers up to and including 20 Derive quickly these addition doubles: Doubles of numbers from to Doubles of multiples of 5 from to (e.g., ) Know by heart all pairs of numbers that total 20 Know by heart all pairs of multiples of 5 that total 100. For example: Put numbers in the boxes to make 100: + 20 = 100 Know by heart all pairs of multiples of 100 that total For example: Put numbers in the boxes to make 1000: = 1000

43 Use knowledge that addition can be done in any order Find a small difference by counting up Identify near doubles Add or subtract 9, 19, 29 or 11, 21, 31 by adding or subtracting 10, 20, 30 and adjusting by 1 Use patterns of similar calculations Use the relationship between addition and subtraction As outcomes, Year 3 pupils should, for example: Mentally add or subtract 9 or 11 to/from any three-digit number Mentally add or subtract 9, 19, 29 or 11, 21, 31 to/from any two-digit number without crossing 100 Develop and recognise a pattern such as: = = = = = = 43 Recognise and use the pattern in, for example: = = = 1200 Say and write the subtraction fact corresponding to a given addition fact, and vice versa (2-digit, including crossing 10s). For example: = 83 implies that = 56, etc You know that = 16 so: What is ?, or 16 4? Or 16 12? Given three 2-digit numbers, say or write four different sentences relating these numbers. Use known number facts and place value to add or subtract a pair of numbers mentally As outcomes, Year 3 pupils should, for example: Add/subtract a single digit to or from any three-digit number, without crossing the tens boundary ( ; 345 3) Add a two-digit number to a multiple of 100 ( ) Subtract a single digit from a multiple of 100 (400 8) Add a two-digit number to a multiple of 10, crossing 100 ( ) Add/subtract a pair of two-digit numbers, without crossing the tens boundary or 100 ( ; 87 62)

44 Use known number facts to add or subtract a pair of numbers mentally (continued) As outcomes, Year 3 pupils should, for example: Add/subtract 10 to or from any two- or three-digit number, including crossing the hundreds boundary ( ; ; ; ) Begin to add/subtract a pair of multiples of ten, crossing 100 ( , ) Find what must be added to a three-digit multiple of ten to make the next highest multiple of 100 (540 + = 600) Add/subtract a multiple of ten to/from a two-digit number, crossing 100 ( ; ) Add/subtract a pair of hundred multiples, crossing 1000 ( ; ) Add/subtract 100 to/from any three-digit number, without crossing 1000 ( ; = 447, ; = 513) Add or subtract a pair of numbers mentally (continued) by bridging through 10 or 100, or a multiple of 10 or 100, and adjusting As outcomes, Year 3 pupils should, for example: Consolidate subtracting a single digit from a teens number, crossing 10 Use two steps and cross 10 as a middle stage Add/subtract single digit to/from a two-digit number, crossing the tens boundary Use two steps, crossing a multiple of ten as a middle stage Find a small difference between a pair of numbers lying either side of a multiple of 100 from 100 to 1000 (604 7; = 502) Begin to add/subtract any pair of two-digit numbers Understanding multiplication Understand the operation of multiplication and the associated vocabulary, and that multiplication can be carried out in any order As outcomes, Year 3 pupils should, for example: Understand, use and begin to read: Double, times, multiply, multiplied by, product, multiple of Lots of, groups of. Times as (big, long, wide ) Read and write the x sign Understand multiplication as: Repeated addition Describing an array Scaling ( a number of times as wide, tall )

45 Understand that multiplication can be done in any order: for example, 5 x 8 = 8 x 5 but that 16 2 is not the same as 2 16 and use this property appropriately Understand the principle that multiplication reverses division (multiplication is the inverse of division) Respond rapidly to oral or written questions such as: Two tens; Double 2; 3 times 4; 9 multiplied by 2; Multiply 5 by 8 Is 20 a multiple of 5? Recognise the use of symbols such as to stand for unknown numbers and complete with rapid mental recall, for example: 6 x 2 = x 2 = 14 Begin to interpret situations as multiplication calculations. For example: A baker puts 6 buns in each of 4 rows. How many buns does she bake? Sue has 10 stamps. Tim has 3 stamps for every one of Sue s. How many stamps has Tim? Alex has 4 stickers. Jo has 3 times as many stickers as Alex. How many stickers does Jo have? Understanding division Understand the operation of division and the associated vocabulary As outcomes, Year 3 pupils should, for example: Understand, use and begin to read: Share, halve, divide, divided by, equal groups of, Read and write the sign Understand that ½ means one divided into two equal parts Understand the operation of division as: Grouping or repeated subtraction, including interpreting, for example, 35 5 as how many 5s make 35? Sharing Know that dividing a whole number by 1 leaves the number unchanged: for example 12 1 = 12 Understand that that 16 2 does not equal 2 16 Understand division reverses multiplication (division is the inverse of multiplication) Respond rapidly to oral or written questions phrased in a variety of ways such as: Share 18 between 2. Divide 25 by 5. How many fives make 45? How many 5p coins do you get for 35p? How many lengths of 10m can you cut from 80m of rope? Is 35 a multiple of 5? Recognise the use of symbols such as to stand for unknown numbers and complete with rapid mental recall, for example: 16 2 = 2 = 14 Interpret in every situations as division calculations. For example:

46 A baker bakes 24 buns. She puts 6 buns in every box. How many boxes can she fill? William has made a pattern using 12 tiles. One tile in every 4 is red. How many tiles are red? Understand the idea of a remainder Make sensible decisions about rounding up or down after division in the context of a problem As outcomes, Year 3 pupils should, for example: Use, read and begin to write: Left over, remainder Give a whole-number remainder when one number is divided by another. Make sensible decisions about rounding down or up after division, depending on the context of the problem. For example: 46 5 is 9 remainder 1, but whether the answer should be rounded up to 10 or rounded down to 9 depends on the context. Examples of rounding down: I have 46. Tickets cost 5 each. I can only afford 9 tickets. I have 46 cakes. One box holds 5 cakes. I can fill only 9 boxes of cakes. Examples of rounding up: I have 46 cakes. One box holds 5 cakes. I need 10 boxes to hold all 46 cakes. There are 46 children. Each table seats tables are needed to seat all the children. Know simple multiplication and division facts by heart Derive doubles and halves quickly As outcomes, Year 3 pupils should, for example: Know by heart multiplication facts for 2 up to 2 x 10 5 up to 5 x up to 10 x 10 and derive quickly the corresponding division facts Begin to know multiplication facts for 3 up to 3 x 10 4 up to 4 x 10 and derive quickly the corresponding division facts Use known facts to derive quickly: Doubles of all numbers 1 to 20 Doubles of 5, 15, 25 up to 100

47 Doubles of 50, 100, 150, 200 up to 500 and the corresponding halves Shift the digits of a number one place to the left/right to multiply/divide by 10 Use knowledge of doubles/halves to multiply/divide Say or write a division statement corresponding to a given multiplication statement Use knowledge of number facts and place value to multiply or divide mentally As outcomes, Year 3 pupils should, for example: Multiply a single digit by 1, 10 or 100 Divide a three-digit multiple of 100 by 10 or 100 Double any multiple of 5 up to 50 Halve any multiple of 10 up to 100 Multiply 10, or 50 by 2, 3, 4, 5, or 10 Multiply a two-digit number by 2, 3, 4, or 5 without crossing the tens boundary (24 x 2, 21 x 4, but not 21x5 or 32x4) Solving Problems Making decisions Choose and use appropriate number operations and ways of calculating to solve problems As outcomes, Year 3 pupils should, for example: Understand and use in practical contexts: Operation, sign, symbol, number sentence Make up number stories to reflect statements such as: = 280 A burger cost 1.35 and a large fries cost Together they cost What operation sign does each stand for? = = 90

48 Reasoning about numbers or shapes Solve mathematical problems or puzzles, recognise simple patterns or relationships, generalise and predict. As outcomes, Year 3 pupils should, for example: Solve puzzles and problems such as: Find a pair of numbers with: a sum of 7 and a product of 10 Solve simple word problems set in real life contexts and explain how the problem was solved. As outcomes, Year 3 pupils should, for example: Use any of the four operations to solve story problems about numbers in real life. Solve puzzles and problems such as: One-step operations I think of a numbers, then subtract 12. The answer is 26. What was my number? A spider has 8 legs. How many legs do 5 spiders have? A box holds 35 nuts. How many are left if you eat 17? How many people can have 5 each? How many are in 3 boxes? How many boxes are needed for 70? For 80? Two-step operations There are 19 books on the top shelf and 32 books on the bottom shelf. 24 are removed. How many are left? I think of a number, double it and add 5. The answer is 35. What was my number? Solve simple word problems involving money and explain how the problem was solved. As outcomes, Year 3 pupils should, for example: Use any of the four operations to solve money problems. Find totals and give change It costs 75p for a child to swim. How much does it cost for two children? Anna has a 50p coin and 3 20p coins. She pays 90p for a Big Dipper ride. How much does she have left? A set of paints costs 3. Parveen saves 20p each week. How many weeks must she save to buy the paints? Dad bought three packets of cornflakes at 70p each. What was his change from 3?

49 Solve simple word problems involving measures and explain how the problem was solved. As outcomes, Year 3 pupils should, for example: Solve problems such as: Length, mass, capacity Two rolls of tape are 35cm and 41cm long. What is their total length? What is the difference in their lengths? An egg weighs about 50 grams. About how much do 6 eggs weigh? A bottle of medicine holds 35mL. A teaspoon holds 5mL. How many teaspoons of medicine are in the bottle? Time Mark got into the pool at 3:30pm. He swam for 40 minutes. What time did he get out? The cake went into the oven at 10:20. It came out at 10:45. How long was it in the oven? Lunch break lasts 50 minutes. It ends at 1:00pm. What time does it start? Measures Length, mass and capacity Understand and use the vocabulary related to length, mass and capacity; begin to know the relationship between standard metric units. Measure and compare: by direct (side-by-side) comparison; using uniform non-standard units; using standard units. As outcomes, Year 3 pupils should, for example: Understand and use in practical contexts: Length and distance: long, short, tall, high, low, wide, narrow, deep, shallow, thick, thin, far, near, close. Mass: weight, weighs, heavy, light, balances Capacity: full, empty, holds And comparative words such as longer, longest.. Know that: 1 kilometre = 1000 metres 1 metre = 100 centimetres 1 kilogram = 1000 grams 1 litre = 1000 millilitres Begin to recognise that 3.5m represents three-and-a-half metres, and that 3.05m is 3 metres and 5 centimetres. Solve problems involving length, mass and capacity in a variety of contexts, using standard units such as: Miles, kilometres, metres, centimetres

50 Kilograms, half-kilograms, units of 100 grams. Litres, half-litres, units of 100mL. Suggest suitable units to estimate or measure length, mass or capacity. As outcomes, Year 3 pupils should, for example: Understand and use in practical contexts: Guess, estimate Roughly, nearly, about, approximately. Suggest things that could be measured using Miles or kilometres, metres, centimetres Kilograms, grams Litres, millilitres Suggest a unit you could use to measure: How far it is to London The height of a door The length and width of a greetings card The capacity of a kitchen bucket Respond to questions such as: Would you expect: A front door to be 1, 2 or 5 metres tall? A hand span to be 5, 15 or 20cm wide? A new-born baby to be 3kg or 30kg? A teapot to hold 1 litre, 10 litres or 100 litres? Read and interpret number scales with some accuracy. As outcomes, Year 3 pupils should, for example: Read a scale to the nearest marked division Time Understand and use the vocabulary related to time; know and use units of time and the relationships between them; read the time from clocks; solve problems involving time. As outcomes, Year 3 pupils should, for example: Know that: 1 year = 365 days or 52 weeks or 12 months 1 week = 7 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds Use a calendar and write the date correctly.

51 Read the time to five minutes on a digital clock or an analogue clock, knowing, for example, that the time is 8:35, 35 minutes past 8 or 25 minutes to 5 Use am and pm Suggest a suitable unit of time to measure the time to the end of the month, to boil an egg, etc Know what takes about 30 minutes, 5 hours, 4 weeks. Shape and space Properties of 3-D and 2-D shapes Describe and classify common 3-D and 2-D shapes according to their properties As outcomes, Year 3 pupils should, for example: Understand, use and begin to read the vocabulary from the previous year, and extend to: Pentagonal, hexagonal, octagonal Right-angled vertex, vertices layer Diagram Name, classify and describe some features of 3-D and 2-D shapes, extending the shapes used to: Prism, hemi-sphere Quadrilateral, semi-circle Know that a prism has the same cross-section along its length, and that its two end faces are identical. Sort 3-D shapes in different ways according to properties such as: Whether or not they are prisms The number of faces, edges and vertices Name and describe solids Know that a quadrilateral is any flat shape with four straight sides Using a collection of flat shapes, choose an example to match given properties Sort a set of flat shapes according to properties such as: The number of vertices or sides Whether the sides are the same length Whether or not at least one angle is a right angle Whether or not a shape has a line of symmetry Make models, shapes and patterns with increasing accuracy, and describe their features. As outcomes, Year 3 pupils should, for example: Recognise that two or more shapes can be put together in different ways to make new shapes Relate 3-D shapes to pictures of them

52 Use 2-D shapes to make and describe pictures and patterns Line symmetry Recognise line symmetry in simple cases. As outcomes, Year 3 pupils should, for example: Recognise more than one line of symmetry Recognise the reflection of a simple 2-D shape in a mirror line along one edge Position and direction Describe positions and directions. As outcomes, Year 3 pupils should, for example: Extend vocabulary from previous year to include Grid, row, column map, plan Compass point, north, south, east, west Horizontal, vertical, diagonal Descend, ascend. Describe and find the position of a square on a grid of squares with rows and columns labelled Make and use simple maps or plans on squared paper and describe the position of a feature Describe directions: Recognise the four compass directions N, S, E, W Use a squared field and a counter to move, for example, from A3 to C1, describing the route as two squares east and two squares south Movement and angle Describe movements (in a straight line and turning) and understand angle as a measure of turn. As outcomes, Year 3 pupils should, for example: Use, read and begin to write the vocabulary from the previous year, and extend to: Angle Is a greater/smaller angle than Recognise whole, half and quarter turns Know that a quarter turn is called a right angle and that a straight line is two right angles. Know that after turning through a half a turn, or two quarter turns in the same direction, you are facing the opposite direction Sort 2-D shapes according to whether they have all, some or no right angles

53 Use a template to measure right angles In a shape, mark the smallest and largest angles Handling data Organising and using data Solve a given problem by collecting, sorting and organising information in simple ways. As outcomes, Year 3 pupils should, for example: Use, read and begin to write: Sort, set, represent, graph, chart, pictogram, diagram, table, list, count, tally, axis, label, title Most/least common or popular Classify objects, numbers or shapes according to one criterion, progressing to two criteria, and display on a Carroll of Venn diagram. Examples might include: Children who are 8 years old and not 8 years old Shapes that are squares and not squares Use a frequency table to respond to questions such as: Which is the most/least popular? Who voted either for this or for this? Which colour had fewer than 5 votes? Make a simple bar chart, with the vertical axis labelled in ones, then twos. Use it to respond to questions such as: Which day had most/least packed lunches? How many packed lunches in the whole week? Make a simple pictogram, where one symbol represents 2 units, and use it to respond to questions such as: Do most children walk to school? More children walk than come by bike. How many more?

54 YEAR 4 Numbers and the Number System Place value (whole numbers) Pupils should be taught to: Read and write whole numbers, know what each digit in a number represents, and partition numbers into thousands, hundreds, tens and ones. As outcomes, Year 4 pupils should, for example: Use, read and write: Units or ones, tens. Hundreds, thousands, ten thousand, hundred thousand, million, digit, one-digit number, two digit number, three-digit number, four digit number, numeral.. place value Respond to written questions such as: Read these: 785, 1179, 4601, 3002, Find the card with: two thousand, three hundred and sixty on it. five thousand and seven on it What number needs to go in each box? 3642 = = What does the digit 3 in 3642 represent? The 6? The 4? The 2? (They represent 3000 and 600 and 40 and 2.) What is the figure worth in the number 7451? Write the number that is equivalent to: seven thousands, four hundred, five tens and six ones. Write in the figures: four thousand, one hundred and sixty-seven. Write in words: 7001, 5090,8300 Which is less: 4 hundreds or 41 tens? What needs to be added/subtracted to change: 4782 to 9782; 3261 to 3961; Make the biggest/smallest number you can with these digits: 3,2,5,4,0. Add or subtract 1, 10, 100 or 1000 to/from whole numbers and count on or back in tens, hundreds or thousands from any whole number up to As outcomes, Year 4 pupils should, for example: From any three- or four digit number count on or back in ones, tens, hundreds or thousands, including crossing boundaries.

55 Respond to oral questions such as: Count on, for example: 6 in ones from in tens from 569 Count back for example: 6 in ones from in hundreds from 732 Starting with 23, how many tens do you need to add to get more than 100? Starting with 374, how many hundreds do you need to add to get more than 1000? Answer written questions such as: What is 1 more than: ? What is 1 less than: ? What is 10, 100, 1000 more/less than the numbers above? What is 10p, 100p, 1000p more or less than 1005p? (ml/g/m) Write the correct numbers in the boxes more is 6500 Pupils should be taught to: Multiply and divide whole numbers, then decimals, by 10, 100 or As outcomes, Year 4 pupils should, for example: Demonstrate understanding of multiplying or dividing a whole number by 10. Understand that : When you multiply a number by 10, the digits move one place to the left. When you divide a number by 10, the digits move one place to the right. Extend to multiplying integers less than 1000 by 100. Respond to written questions, such as: Tins of dog food are put in packs of 10. One tin costs 42p. How much does one pack cost? 10 packs? Work out mentally the answers to written questions such as: 6 x 10 =, 329 x 10 =, = Ordering (whole numbers) Pupils should be taught to: Use the vocabulary of comparing and ordering numbers, and the symbols <,>, =; give a number lying between two numbers and order a set of numbers

56 As outcomes, Year 4 pupils should, for example: Use, read and write: How many, as many as, the same number as, equal to More than, fewer than, greater than, less than, smaller than, larger than most, least, smallest, largest Order, first, last, before, after, next, between, half way between Ordinal numbers: first, second, third, fourth. 1 st, 2 nd, 3 rd, 4 th and the < and > signs Respond to questions such as: Which is greater: 7216 or 7261? Which is longer: 3157m or 3517m? Put these numbers in order, largest/smallest first: 4521, 2451, 5124, 5241 Rounding (whole numbers) Round whole numbers to the nearest 10, 100 or 1000 As outcomes, Year 4 pupils should, for example: Round any two- or three-digit number to the nearest 10 or 100 Round measurements in seconds, minutes, hours, metres, kilometres, miles, kilograms, litres to the nearest 10 or 100 units Estimate calculations by approximating. Negative Numbers Recognise and order negative numbers As outcomes, Year 4 pupils should, for example: Use, read and write in context: Integer, positive, negative, minus, above/below zero Use negative numbers in the context of temperature. For example: Which temperature is lower: -4 C or -2 C? Put these temperatures in order, lowest first:2 C, -8 C, -1 C, -6 C, Properties of numbers and number sequences Recognise and extend number sequences formed by counting on and back in steps of any size, extending beyond zero when counting back As outcomes, Year 4 pupils should, for example: Use, read and write: next, consecutive, sequence, predict, continue, rule, relationship sort, classify, property Count on and back. For example:

57 From any number, count on in 2s, 3s, 4s, 5s to about 100 and then back Count back in 4s from 40 Count in 25s to 500, then back Describe, extend and explain number sequences and patterns Count on or back from any number in steps of any single-digit number Recognise odd and even numbers and make general statements about them Recognise multiples and know some tests of divisibility As outcomes, Year 4 pupils should, for example: Use, read and write: multiple, digit Recognise multiples in the 2, 3, 4, 5 and 10 times-tables Fractions and decimals Use fraction notation and recognise the equivalence between fractions As outcomes, Year 4 pupils should, for example: Use, read and write: fraction. Half, quarter, eighth third, sixth Fifth, tenth, twentieth Use fraction notation: for example, read and write 1 10 as one tenth, 3 10 as three tenths Recognise that one whole is equivalent to two halves, three thirds, four quarters. Begin to know the equivalence between: Halves, quarters and eighths: 2 8 equals ¼; 4 8 equals 2 4 or ½; 6 8 equals ¾ Tenths and fifths: 2 10 equals 1 5 Thirds and sixths: 2 6 equals 1 3, 4 6 equals 2 3 Order familiar fractions As outcomes, Year 4 pupils should, for example: Recognise from practical work that: One half is more than one quarter and less than three quarters Which of these fractions are greater than one half: ¾, 1 3, 5 8, 1 8, 2 3, 3 10 Find fractions of numbers or quantities As outcomes, Year 4 pupils should, for example: Begin to relate fractions to division. For example:

58 Understand that finding one half is equivalent to dividing by 2, so that ½ of 16 is equivalent to 16 2 Recognise that when 1 whole cake is divided equally into 4, each person gets one quarter, or 1 4 = ¼ Find fractions of numbers and quantities. For example, answer questions such as: What is one tenth of: 100, ? What is one fifth of: 15, 10 35? What is ¼ of: 8, 16, 40? What is 1/10 of: 50, 10, 80? What is one tenth, one quarter, one fifth of 1? Of 1 metre? What fraction of 1 is 10p? What fraction of 1 metre is 25cm? What fraction of the larger shape is the smaller shape? Solve simple problems involving ratio and proportion As outcomes, Year 4 pupils should, for example: Use, read and write: in every, for every For example, discuss statements such as: In every week I spend 5 days at school, so in every 3 weeks I spend 15 days at school. For every two bags of crisps you buy, you get 1 sticker. To get 3 stickers you must buy 6 bags of crisps. Use decimal notation, know what each digit in a decimal fraction represents and order a set of decimal fractions As outcomes, Year 4 pupils should, for example: Use, read and write: decimal fraction, decimal, decimal point, decimal place Respond to questions such as: What does the digit 6 in 3.6 represent? What is the figure 4 worth in the number 17.4? Write the decimal fraction equivalent to four tenths; fifty-seven and nine tenths Start at 5.1 and count on or back in steps of 0.1 Put in order, largest/smallest first: 6.2, 5.7, 4.5, 7.6, 5.2 Write centimetres in metres. For example: write 125cm in metres. In the context of word problems, work out calculations involving mixed units of pounds and pence, or metres and centimetres, such as: I cut 65cm off 4 metres of rope. How much is left? Recognise the equivalence between decimals and fractions

59 As outcomes, Year 4 pupils should Know that, for example: 0.5 is equivalent to ½ 0.25 is equivalent to ¼ 0.75 is equivalent to ¾ 0.1 is equivalent to 1 10 particularly in the context of money and measurement. Calculations Addition Understand the operation of addition and the associated vocabulary, and its relationship to subtraction As outcomes, Year 4 pupils should, for example: Understand and use in practical contexts: More, add, sum, total, altogether, increase, equals, sign, inverse and read and write the plus (+) and equals (=) signs Understand and use when appropriate the principles (but not the names) of the commutative and associative laws as they apply to addition Understand that addition is the inverse of subtraction Respond rapidly to questions phrased in a variety of ways such as: 654 add 50 Add 68 to 74 7 add 12 add 9. Add 15, 6, 4, 15 and 1 What is the sum/total of 26 and 39? And of 13, 62 and 3? How many altogether are 121 and 35? And 61, 37 and 6? Increase 48 by 22. Use mental or written methods to: Find the missing number in: = 250 Subtraction Understand the operation of subtraction and the associated vocabulary, and its relationship to addition As outcomes, Year 4 pupils should, for example: Use, read and write: Take away, subtract, how many are left, how much less is than.., Difference between, how much more is than, how many more to make.decrease, inverse, and read and write the minus (-) sign

60 Consolidate understanding of subtraction as: Taking away; Finding the difference between; Complementary addition Understand that: Subtraction is non-commutative: that is, 7-5 is not the same as 5 7 When a larger number is subtracted from a smaller number, the answer is negative: for example, 3 8 = -5 Understand that subtracting a (positive) number makes a number less Understand that subtracting zero leaves a number unchanged Understand that subtraction is the inverse of addition Respond rapidly to questions phrased in a variety of ways such as: 93 take away 8 Take 7 from subtract 46 Subtract 120 from less than less than 5437 What must I take from 84 to leave 26? What is the difference between 28 and 65? How many more than 234 is 249? How many less than 68 is 42? What must I add to 54 to make 93? Decrease 72 by added to a number is 43. What is the number? Find pairs of numbers with a difference of 79 Use mental or written methods to: Find the missing number in: 91 - = 48 Rapid recall of addition and subtraction facts Know, with rapid recall, addition and subtraction facts As outcomes, Year 4 pupils should, for example: Know by heart all addition and subtraction facts for all numbers up to and including 20 Derive quickly related facts such as: = = 1600 Derive quickly number pairs that total 100 Derive quickly pairs of multiples of 50 that total 1000 Derive quickly addition doubles: Doubles of numbers from to Doubles of multiples of 10 from to (e.g., )

61 Doubles of multiples of 100 from to (e.g., ) Mental calculation strategies (+ and -) Find a difference by counting up through the next multiple of 10, 100 or 1000 Count on or back in repeated steps of 1, 10, 100, 1000 Partition into hundreds, tens and ones Identify near doubles Add or subtract the nearest multiple of 10, 100 or 100 and adjust As outcomes, Year 4 pupils should, for example: Work out mentally that: = 1995 by counting back in ones from = 693 by counting on in tens from = 337 by counting back in tens from = 960 by counting on in hundreds from 460 Work out mentally that: = 82 because it is ( ) + (4 + 8) or ( ) + 8 Work out mentally that: = 74 because it is double 40 subtract 2, subtract 4, or double 37 Add 9, 19, 29 or 11, to any two-digit number Subtract 9, 19, 29 or 11, 21, 31 from any two- or three-digit number Use the relationship between addition and subtraction As outcomes, Year 4 pupils should, for example: Continue to recognise that knowing one addition/subtraction fact means that you also know three others Add several numbers As outcomes, Year 4 pupils should, for example: Add mentally several small numbers: for example, Work mentally to complete questions like: = 37 and = using strategies such as Looking for pairs that make 10 or 100 and doing these first; Starting with the largest number; Looking for pairs that make 9 or 11, and adding these to the total by adding 10 and then adjusting by 1 Add a set of numbers such as recognising that this is equivalent to 4 x 6 Use known number facts and place value to add or subtract a pair of numbers mentally

62 As outcomes, Year 4 pupils should, for example: Continue to add/subtract two-digit multiples of 10 Add or subtract a pair of multiples of 100 crossing 1000 Revise adding/subtracting a multiple of 10 to/from a two- or three-digit number, without crossing the hundreds boundary Revise adding a two- or three-digit number to a multiple of 10, 100 or 1000 Find what to add to a two-digit number to make 100, or a three-digit number to make the next 100 multiple Find what to add to a four-digit multiple of 100 to make the next highest multiple of 1000 Add a single digit to any three- or four-digit number, crossing the tens boundary Subtract a single digit from a multiple of 100 or 1000 Subtract a single digit from a three- or four-digit number, crossing the tens boundary Find a small difference between a pair of numbers lying either side of a 1000 multiple Add or subtract any pair of two-digit numbers, including crossing the tens boundary Pencil and paper procedures (addition) Develop and refine written methods for addition, building on mental methods Standard written methods Develop an efficient stand method that can be applied generally. For example: Adding the least significant digits, preparing for carrying Leading to basic carrying Using similar methods, add several numbers with different numbers of digits. For example, find the total of: 83, 256, 4, 57 Extending to decimals Using methods similar to those above, begin to add two or more threedigit sums of money, with or without adjustment from the pence to pounds. Know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts such as 3.59 ± 78p.

63 Pencil and paper procedures (subtraction) Develop and refine written methods for subtraction, building on mental methods Standard written methods Develop an efficient standard method that can be applied generally. For example: Subtraction is taught by decomposition of hundreds, tens and units which will then lead to direct adjustment from the tens to the units, and then from the hundreds to the tens. 754 = leading to = (adjust from T to U) = (adjust from H to T) = Subtract with numbers of digits. For example, find difference between: 671 and 58, 46 and 518. Extend to decimals Using methods similar to those above, begin to find the difference between two three-digit sums of money, with or without adjustment from the pence to the pounds. Know that decimal points should line up under each other. For example: Understanding multiplication Understand the operation of multiplication and the associated vocabulary, and its relationship to addition and division As outcomes, Year 4 pupils should, for example: Use, read and write: Times, multiply, multiplied by, product, multiple, inverse Read and write the x sign Understand and use when appropriate the principles (but not the names) of the commutative, associative and distributive laws as they apply to multiplication Understand that is equivalent to 86 x 3 or 3 x 86 Understand that multiplication by 1 leaves a number unchanged Multiplication of zero results in zero

64 Understand that multiplication is the inverse of division Understanding division Understand the operation of division and the associated vocabulary, and its relation to subtraction and multiplication As outcomes, Year 4 pupils should, for example: Use, read and write: Share, group, divide, divided by, divided into, divisible by, factor, quotient, remainder, inverse Read and write the division signs and / Understand the operation of division as: Grouping or repeated subtraction Sharing equally Understand that division by 1 leaves the number unchanged Understand that division is the inverse of multiplication Respond to questions such as: Share 44 between (sic) 4 Divide 69 by divided by 3. Divide 3 into 69. How many groups of 6 can be made from 48? How many lengths of 10cm can be cut from 183cm? What are the factors of 12? Is 72 divisible by 3? Begin to relate division and fractions Understand that ½ of 10 is the same as 10 2; ¼ of 3 is the same as 3 4 Complete written questions, for example: 36 4 = 60 = 6 3= = 240 = = 7 And (25 ) + 2 = 7 ( 5) 2 = 3 Progressing to = 156 = 26 9 = 460 Understand the idea of a remainder and when to round up or down after division As outcomes, Year 4 pupils should, for example: Give a remainder as a whole number. For example, 41 4 is 10 remainder 1 28 = (5 x 5) is 14 remainder 2 97 = (9 x 10) is 7 remainder = (3x100) + Answer questions such as: There are 64 children in Year 5. How many teams of 6 children can be made? How many children will be left over? Divide a whole number of pounds by 2, 4, 5 or 10. For example: Four children collected 19 for charity. They each collected the same amount. How much did each one collect? ( 4.75)

65 Make sensible decisions about rounding down or up after division. For example: 62 8 is 7 remainder 6, but whether the answer should be rounded up to 8 or rounded down to 7 depends on the context. Know multiplication facts by heart and derive quickly the corresponding division facts Know by heart or derive rapidly doubles and halves Use related facts and doubling or halving As outcomes, Year 4 pupils should, for example: Know by heart multiplication facts for the 2, 3, 4, 5 and 10 times-tables, up to 10x, including multiplication by 0 and 1, and begin to know them for the 6, 7,8 and 9 times-tables. Derive quickly the corresponding division facts Know by heart or derive quickly: Doubles of all numbers 1 to 50 Doubles of 10 multiples up to 500 Doubles of 100 multiples up to 5000 and all the corresponding halves Use related facts and doubling or halving. For example: To multiply by 4, double and double again To multiply by 5, multiply by 10 and halve To multiply by 20, multiply by 10, then double Work out the 8 times-table facts by doubling the 4 times-table facts Work out quarters and eighths by halving Use closely related facts already known Partition and use the distributive law Use the relationship between multiplication and addition, or multiplication and division Use known number facts and place value to multiply or divide mentally As outcomes, Year 4 pupils should, for example: Work out the 6 times-table by adding the 2 times-table facts to the 4 timestable facts To multiply a number by 9 or 11, multiply by 10 and then add or subtract the number Begin to multiply a two digit number by a single-digit number, multiplying the tens first. For example: 32 x 3 = (30 x 3) + (2 x 3) = = 96 Continue to recognise that knowing one of: 12 x 9 = x 12 = = = 9 means that you also know the other three. Recognise and use for example, 25 x 4 = Answer written questions

66 Given 14 x 6 = 84 what is 6 x 14, or 84 6, or 84 14? Multiply a two- or three- digit number by 10 or 100 For example: 327 x x 100 Work mentally to complete written questions like: 96 x 100 = 82 x = 8200 Divide a four-digit multiple of 1000 by 10 or 100 Double any multiple of 5 up to 100 Halve any multiple of 10 to 200 Consolidate multiplying a two-digit multiple of 10 by 2, 3, 4, 5 or 10 and begin to multiply by 6, 7, 8 or 9 Multiply a two- digit number by 2, 3, 4 or 5, crossing the tens boundary Solving Problems Making decisions Choose and use appropriate number operations and ways of calculating to solve problems As outcomes, Year 4 pupils should, for example: Understand and use in practical contexts: Operation, sign, symbol, number sentence Make decisions: Choose the appropriate operation to solve word problems and number puzzles. Explain and record how the problem was solved. From number stories recognise/determine the number statement that reflects the problem, for example: Four portions of fries at 90p each cost 360p 3.60 altogether. That gives the statement = 680. What operation sign does each stand for? = 161: 8 5 = 90 Reasoning about numbers or shapes Methods and reasoning about numbers in writing. Complete calculations that have wholly or partly been done mentally, beginning to use conventional notation and vocabulary to record the calculation. For Example: : Add 17 and to get 20, then 20 more to get. Extend to calculations that cannot entirely be done mentally. For example:

67 : = 612 Solve mathematical problems or puzzles, recognise simple patterns or relationships, generalise and predict. As outcomes, Year 4 pupils should, for example: Solve puzzles and problems such as: Find a pair of numbers with: a sum of 11 and a product of 24. Find three consecutive numbers which add up to 39. Arrange the numbers 1, 2, 3.. to 9 in the circles so that each side of the square adds up to12. Make and investigate a general statement about familiar numbers or shapes by finding examples that satisfy it As outcomes, Year 4 pupils should, for example: Find examples that match a general statement. For example: explain and start to make general statements like: If I multiply a whole number by 10, every digit moves one place to the left. Examples: 63 x 10 = 630: 5 x 10 = 50 The perimeter of a rectangle is twice the length plus twice the breadth. Example: 5 cm + 3 cm + 5 cm + 3 cm = 16 cm Which is the same as 5 cm x 2 add 3 cm x 2. Use all four operations to solve word problems involving numbers in real life As outcomes, Year 4 pupils should, for example: Solve story problems about numbers in real life, choosing the appropriate operation and method of calculation. Record using numbers, signs and symbols how the problem was solved. Examples of problems: I think of number, then subtract 18. The answer was 26. What was my number? A beetle has 6 legs. How many legs have 9 beetles?

68 Multi-step operations There are 129 books on the top shelf. There are 87 books on the bottom shelf. I remove 60 of the books. How many books are left on the shelves? Use all four operations to solve word problems involving money. As outcomes, Year 4 pupils should, for example: Use read and write: Money, coin, pound,, pence, note, price, cost, cheaper, more expensive, pay, change, total, value, amount Solve problems involving money, choosing the appropriate operation. For example: Shopping problems What is the total cost of a 4.70 book and a 6.10 game? A CD costs 4. Parveen saves 40p a week. How many weeks must she save to buy the CD? A chocolate bar costs 19p. How many bars can be bought for 5? Calculating Fractions Harry spent one quarter of his savings on a book. What did the book cost if he saved: ? Use all four operations to solve word problems involving length mass or capacity. As outcomes, year 4 pupils should for example: Solve story problems involving : kilometres, metres, centimetres, millimetres kilograms, half kilograms, grams litres, half litres, millimetres For Example: A family sets off to drive 524 miles. After 267 miles, how much further do they have to go? A potato weighs about 250g. Roughly how much do 10 potatoes weigh? Use all four operations to solve word problems involving time. As outcomes, Year 4 pupils should, for example: Solve story problems involving units of time. For example: Lunch takes 40 minutes. It ends at 1:10pm. What time did it start? Jan went swimming on Wednesday, 14 th January. She went swimming again 4 weeks later. On what date did she go swimming the second time.

69 Measures Length, mass and capacity Understand and use the vocabulary related to measures. As outcomes, Year 4 pupils should, for example: Use, read and write: unit, standard unit, metric unit, names of standard metric units: kilometre, metre, centimetre, millimetre kilogram, gram litre, millilitre.. length and distance: long, short, tall, high, low, wide narrow, deep, shallow, thick, thin.. far, near, close, distance, perimeter, circumference.. mass: big, bigger, small, smaller, balances.. weight: heavy, light, weighs.. capacity: full, empty, holds.. and comparative words such as: longer, longest.. Use correctly the abbreviations: mm ( millimetre), cm ( centimetre), m ( metre), km (kilometre), g ( gram), kg (kilogram), ml( millilitre), l (litre).. and cm 2 (square centimetre) and m 2 (square metre). Know and use relationships between familiar units. Know that: 1 kilometre = 1000 metres 1 metre = 100 centimetres 1 kilogram = 1000 grams 1 litre = 1000 millilitres Know the equivalent of one half, one quarter, three quarters and one tenth of 1 km, 1m, 1kg, 1 litre in m, cm g and ml respectively. For example know that: 500g is half of 1 kg. Begin to write, for example: 1.6 m in centimetres (160 centimetres) 5 litres in millilitres (5000 millilitres) and visa versa

70 Suggest suitable units to estimate or measure length, mass or capacity. As outcomes, Year 4 pupils should, for example: Use, read and write: estimate roughly, nearly, about, approximately Respond to questions like: Would you expect: a big potato to be about 20 g, 200 g, 2 000g? a small bottle of lemonade to hold about 250ml or 1250ml? Suggest things that would be measured in: kilometres, metres, centimetres, millimetres kilograms, grams litres, millilitres Suggest a metric unit to measure, for example: how far it is from London to Birmingham; the height of a telegraph pole. Suggest suitable measuring equipment record estimates and readings from scales to a suitable degree of accuracy. Choose a suitable measuring instrument to measure, for example: the length of the classroom a small library book the weight of a person. Read measuring scales to the nearest division. Record measured and estimated lengths in metres and centimetres in decimal form: 1.35 metres. Record other estimates and measurements using a mix of units: For example, write 4125 grams as 4 kilograms and 125 grams Round measurements to the nearest ten or hundred units. For example: I am about 157 cm tall, or 160 cm to the nearest 10 cm; Time Use the vocabulary related to time; suggest suitable units of time to estimate or measure Use, read and write: names of days of the week, months and seasons day, week, fortnight, month, season, year, leap year, century,

71 millennium, morning, afternoon, evening, night, midnight, noon, hour, minute, second, today, yesterday, tomorrow, weekend am and pm how long ago, how long will it be to arrive, depart, faster, fastest, slower, slowest, takes longer, takes less time, earliest, latest. Know and use: 1 millennium = 1000 years 1 century = 100 years 1 year = 12 months 1 week = 7 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds Estimate or measure, suggesting suitable units Suggest things you would estimate or measure in; hours, minutes, seconds, days, weeks, months, years Suggest a unit to estimate or measure, for example: The time it will take to eat lunch. What measuring instrument would you use to time, for example: Running100 metres. Cooking a cake.? Estimate, using standard units, for example: how long it takes to run across a school field. Respond to questions like: Would you expect to cook a soft boiled egg in 3 minutes or in 30 minutes? Read the time from clocks, calendars and timetables. Read clocks and calendars Read the time to the minute on a 12 hour digital clock and an analogue clock. Know, for example, that 4:37, or 37 minutes past 4, or 23 minutes to 5 are all equivalent. Use am and pm. Use the calendar given to work out, for example: the number of days, between, the 27 th of February and the 10 th of March. Use timetables For Example: The bus takes 20 minutes between each stop. Complete the table.

72 High street 11:05 1:45 Church Post Office 1:05 Sports centre Area and perimeter Measure and calculate the perimeter and area of simple shapes. Use, read and write: Area, covers, surface, perimeter, distance, edge and use symbols for square centimetres (cm 2 ). Perimeter Respond to questions such as: How long is the perimeter of a 5cm x 5 cm square and a 4cm x 7cm rectangle.. a triangle whose sides are 10m, 20m, 24m. Find a short way to work out the perimeter of a rectangle. Area Find out which of two or more things has the greatest area by covering with squares. Finding areas by counting squares. For example: Each square is 1 square centimetre. What area is shaded? Choose a suitable unit and estimate the area of, for example: a postcard. Suggest areas to measure in square centimetres. Shape and space Properties of 3-D and 2-D shapes Describe and visualise 3-D and 2-D shapes; classify them according to their properties As outcomes, Year 4 pupils should, for example: Use, read and understand words: Pattern, shape, 2-D, two-dimensional, 3-D, three-dimensional line, side, edge, face, surface, base, point, angle, vertex, vertices centre, radius, diameter net make, build, construct, draw, sketch And adjectives such as: curved, straight regular, irregular concave, convex closed open circular, triangular, hexagonal, cylindrical, spherical square-based, right-angled. Name, classify and describe 2-D and 3-D shapes:

73 circle, semi-circle, triangle, equilateral triangle, isosceles triangle, quadrilateral, rectangle, oblong, square, pentagon, hexagon, heptagon, octagon, polygon.cube cuboid, pyramid, sphere, hemi-sphere, cylinder, cone, prism, tetrahedron, polyhedron. For example: 3-D shapes Know that in a polyhedron: each face is a flat surface and is a polygon; an edge is the straight line where two faces meet; a vertex is the point where three or more edges meet. Know that a prism has two identical end faces and the same cross section throughout its length. Collect, name and describe examples. 2-D shapes Know that a polygon is a closed, flat shape with three or more straight sides, and that regular polygons have all their sides and all their angles equal. Know the angle and side properties of isosceles and equilateral triangles, and use them: for example, to make triangular patterns. Name and classify polygons: Triangles Quadrilaterals Regular Irregular Know some of their properties. For example: all heptagons have seven sides; a quadrilateral is any shape with four straight sides; the square and the equilateral triangle are examples of regular polygon; an isosceles triangle is an example of an irregular polygon; a polygon can be concave or convex. Identify a particular shape from a mixed set. For example, which of these are hexagons? Reflective symmetry, reflection and translation Recognise reflective symmetry in 2-D shapes, reflections and translations

74 As outcomes, Year 4 pupils should, for example: Use, read and understand: Mirror line, line of symmetry, line symmetry, symmetrical, reflect, reflection, translation Identify and sketch two or more lines of reflective symmetry, and recognise shapes with no lines of symmetry. For example: Identify whether designs have a line of symmetry. Classify 2-D shapes according to their lines of symmetry. ( For example: At least one line of symmetry or no lines of symmetry.) Select the correct reflection of a simple shape in a mirror line parallel to one edge, where the edges of the shape of the lines of the pattern are parallel or perpendicular to the mirror line. Know that equivalent points are the same (shortest) distance from the mirror line. Make patterns by repeating a translating shape. Use this program to answer questions: By completing patterns, reflecting it in a horizontal or vertical line. Identifying translated tile patterns. Position and direction Recognise positions and directions, and use coordinates As outcomes, Year 4 pupils should, for example: Use, read and understand: Prepositions and everyday words to describe position and direction. position, direction ascend, descend journey, route, map, plan grid, row, column, origin, co-ordinates compass point, north, south, east, west, north-east, north-west, south-east, south-west horizontal, vertical, diagonal Describe and find the position of a point on a grid of squares where the lines are numbered. Begin to understand the convention that (3,2) describes a point found by starting from the origin (0,0) and moving three lines across and two lines and two lines up. Recognise that (4,1) and (1,4) describe different points. Recognise and identify simple examples of horizontal or vertical lines or edges in the environment. For example: the edge of the table is horizontal;

75 the edge of the door is vertical. Know that rows on a grid are described as horizontal, columns as vertical and lines joining corners as diagonal. Use the eight compass directions N, S, E, W, NW, NE, SW, SE For example: Describe a south-east route from (1, 4) as going through the points (2,3), (3,2), (4,1) and (5,0). Angle and rotation Make turns; estimate, draw and measure angles; recognise rotations As outcomes, Year 4 pupils should, for example: Use, read and understand: Turn, rotate, whole turn, half turn, quarter turn angle, right angle, straight line degree, ruler, set square, angle measurer Know that angles are measured in degrees and that: One whole turn is 360 O or four right angles; a quarter turn is 90 O or one right angle; half a right angle is 45 O. Know that the angles at the corners of rectangles and squares are 90 O, and that the angles of an equilateral triangle are 60 O. Recognise which of two angles is greater: For example, that an angle of 60 O is greater than an angle of 45 O. Handling data Organising and interpreting data Solve a given problem by extracting and interpreting data in tables, graphs and charts. As outcomes, Year 4 pupils should, for example: Use, read and begin to write: Vote, survey, questionnaire, data, count tally, sort, set, represent table, list, graph, chart, diagram, axes, label, title.. most common or popular Find the answer to a question by interpreting a tally chart. Find the answer to a question by interpreting a pictogram where the symbol represents several units. For example: Which student practises the most on their musical instrument? How long does this student practise for?

76 Annabell Length of time for each music practice Simon Christine Each symbol represents 15 minutes music practice. Answer a question or solve a problem by interpreting a bar chart with the vertical axis marked in multiples of 2, 5, 10, 20, noting that the graph has a title and axes are labelled. Use sorting diagrams such as two-way Venn and Carroll diagrams to display information about shapes or numbers. For example: Venn Diagram: Put these numbers on the diagram: 8, 33, 36, 41, 37, 50 Numbers with 3 tens Even numbers 33, , This Carroll diagram records how the whole numbers from 20 to 39 were sorted.

77 Numbers that have 3 tens Numbers that do not have 3 tens. Odd Even

78 YEAR 5 Numbers and the Number System Place value (whole numbers) Pupils should be taught to: Read and write whole numbers, know what each digit in a number represents, and partition numbers into thousands, hundreds, tens and ones As outcomes, Year 5 pupils should, for example: Use, read, write and spell correctly: Units or ones, tens, hundreds, thousands ten thousand, hundred thousand, million digit, one-digit number, two digit number, three-digit number, four digit number, numeral.. place value Respond to written questions such as: Read these: , , , Find the card with: sixty-two thousand, six hundred and twenty on it. six hundred and forty-five thousand and nine on it What does the digit 3 in represent? And the 6? The 4? The 5? What is the value of the digit 7 in the number ? And the 9? Write the number that is equivalent to: five hundred and forty-seven thousands, four hundreds, nine tens and two ones (units). Write in figures: two hundred and ninety-four thousand, one hundred and sixty-one. Write in words: , , Which is less: 4 thousands or 41 hundreds? What needs to be added/subtracted to change: to ; to ; Make the biggest/ smallest integer you can with these digits: 8, 3, 0, 7, 6, 0, 2. Multiply and divide whole numbers, then decimals, by 10, 100 or As outcomes, Year 5 pupils should, for example: Demonstrate understanding of multiplying or dividing a whole number by 10 or 100. Understand that :

79 When you multiply a number by 10/100, the digits move one/two places to the left. When you divide a number by 10/100, the digits move one/two places to the right. Understand that multiplying by 100 is equivalent to multiplying by 10, and again by 10. Respond to written questions, such as: How many times larger is 2600 than 26? Work out mentally the answers to written questions such as: 329 x 100 =, 56 x = 5600, = Ordering (whole numbers) Use the vocabulary of comparing and ordering numbers, and the symbols <,>, =; give a number lying between two numbers and order a set of numbers As outcomes, Year 5 pupils should, for example: Use, read and write, spelling correctly, the vocabulary from the previous year, and extend to ascending and descending order and the < and > signs. Respond to questions such as: Which is greater: or ? Which is longer: m or m? What is halfway between: and A journey takes about 2 hours, give of take 10 minutes. What is the longest time the journey might take? Put these numbers in ascending/descending order: , , , Rounding (whole numbers) Round whole numbers to the nearest 10, 100 or 1000 As outcomes, Year 5 pupils should, for example: Round any two- or three- or four-digit number to the nearest 10, 100 or For example: 5633 is 5630 rounded to the nearest ten is 8000 rounded to the nearest hundred. Round measurements in days, metres, kilometres, miles, kilograms, litres to the nearest 10, 100 or 1000 units. Estimate calculations. For example: Which is the best approximation of ?

80 , , , Negative Numbers Recognise and order negative numbers As outcomes, Year 5 pupils should, for example: Use, read and write, spelling correctly: Integer, positive, negative, minus, above/below zero Count back through zero, for example: Seven, three, negative one, negative five Respond to questions such as: Put these numbers in order, least first: -2, -8, -1, -6, -4 Use negative numbers in the context of temperature. For example: The temperature falls from 11 o C to -2 o C. How many degrees does the temperature fall? Use negative numbers in other contexts such as: A diver is below the surface of the water at -30 m. He goes up 12 metres, then down 4 metres. Where is he now? Properties of numbers and number sequences Recognise and extend number sequences formed by counting on and back in steps of any size, extending beyond zero when counting back As outcomes, Year 5 pupils should, for example: Use and read: next, consecutive, sequence, predict, continue, rule, relationship formula, classify, property Count on and back. For example: From zero, count on in 6s, 7s, 8s, 9s to about 100 and then back. Count in 11s from 132, then count back. Count in 25s to 1000, then back Count in steps of 0.1 to 5.0, then back. Describe, extend and explain number sequences and patterns. For example, respond to questions like. Describe and extend this sequence: -40, -37, -34. Explain the rule in writing. Fill in the missing numbers in these sequences. Explain the rule in writing. 38, 49,,, 82,,, 71, 62, 44 Recognise odd and even numbers and make general statements about

81 them Recognise multiples and know some tests of divisibility As outcomes, Year 5 pupils should, for example: Make general statements about odd or even numbers and/or give examples that match them. For example, explore and give some examples to satisfy these general statements: The sum of three even numbers is even. The sum of three odd numbers is odd. The difference between one odd number and one even number is odd. Use, read and write, spelling correctly: multiple, digit, divisible, divisibility, factor Recognise multiples in the 6, 7, 8, 9 times-tables and in the 11 times tables to 99. Respond to questions such as: Click the numbers in the box that are divisible by 7 (or have a factor of 7). Recognise multiples of more than one number: for example, multiples of both 2 and 3. multiples of , 2 9, 21, 24 8, 20 15, 3 18, 12 10, 16 Multiples of neither 2 nor 3. multiples of 3 multiples of 6 Recognise that a whole number is divisible by: 100 if the last two digits are 00; 10 if the last digit is 0; 2 if the last digit is 0, 2, 4, 6, 8; 4 if the last two digits are divisible by 4; 5 if the last digit is 0 or 5. 41, 19, 17, 13 Use this knowledge to work out, for example, that the year 2004 is a leap year because 2004 is divisible by 4. Recognise square numbers Recognise prime numbers and identify factors Use, read and write, spelling correctly: Square number

82 Begin to recognise: 6 2 as six squared. Recognise 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 as square numbers. Relate to drawings of squares. Respond to questions such as: What is 4 squared? What is the square of 6? What is 8 2? Which number multiplied by itself gives 36? What is the area of a square whose side is 6 cm in length? Use, read and write, spelling correctly: Factor, divisible by Find all the pairs of factors of any number to 100. For example, the pairs of factors of 36 are: 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6. Use factors, when appropriate, for finding products mentally: for example, 16 x 12 = 16 x 3 x 2 x 2 = 48 x 2 x 2 = 96 x 2 = 192 Fractions and decimals Use fraction notation and recognise the equivalence between fractions As outcomes, Year 5 pupils should, for example: Use and read: fraction, proper/improper fraction, mixed number half, quarter, eighth, third, sixth, ninth, twelfth fifth, tenth, twentieth, hundredth equivalent, reduced to, cancel Convert improper fractions to mixed numbers and visa versa: for example, change to Recognise from simple graphics relationships between fractions. For example:

83 one quarter is half of one half; one eighth is half of one quarter; one sixth is half of one third; one tenth is half of one fifth; one twentieth is half of one tenth. Recognise patterns in equivalent fractions, such as: 1 2 = 2 4 = 3 6 = 4 8 = 5 10 = 6 12 = And similar patterns for 1 4, 1 5, 1 10 and 1 3 Start to recognise that: is equivalent to is equivalent to is equivalent to 5 10 or ½ is equivalent to ¼ is equivalent to 3 4 Order familiar fractions As outcomes, Year 5 pupils should, for example: Recognise from simple graphics that, for example: two thirds is less than three quarters. Answer questions such as: Which of these fractions are less than one half? 1 10, 1 20, 2 5, ½, 7 10, 4 5, Place these in order, smallest first: ½, 1 ½, 2, 1 4, 1 ¾ Find fractions of numbers or quantities As outcomes, Year 5 pupils should, for example: Relate fractions to division. For example: Understand that finding one third is equivalent to dividing by 3, so that 1 3 of 15 is equivalent to 15 3; When 3 whole cakes are divided equally into 4, each person gets three quarters, or 3 4 = 3 4 ; Recognise that 12 3 is another way of writing Find fractions of numbers and quantities. For example, answer questions such as: What is one tenth of: 80, 240, 1000? What is one hundredth of: 100, 800, 1000? What is 3 10 of: 50, 20, 100? What is ¾ of: 16, 40, 100? What is 23/100 of 1 in pence? What fraction of 1km is 250 m? 200? What fraction of 1 day is 1 hour, 8 hours, 24 hours? What fraction of the smaller shape is the larger shape?

84 Ratio and Proportion Solve simple problems involving ratio and proportion As outcomes, Year 5 pupils should, for example: Use and read, vocabulary to express simple ratios and proportions: for every to every in every as many as Solve simple problems involving in every or for every. For example: Chicken must be cooked 50 minutes for every kg. How long does it take to cook a 3 kg chicken? Zara uses 3 tomatoes for every ½ litre of sauce. How much sauce can she make from 15 tomatoes? For every 50p coin Mum gives Dad, he gives her five 10p coins. Dad gave Mum twenty-five 10p coins. How many coins did Mum give him? Use decimal notation, know what each digit in a decimal fraction represents and order a set of decimal fractions As outcomes, Year 5 pupils should, for example: Use and read: decimal fraction, decimal, decimal point, decimal place Respond to questions such as: What does the digit 6 in 3.64 represent? The 4? What is the 4 worth in the number 7.45? The 5? Write the decimal fraction equivalent to: two tenths and five hundreds. Continue the pattern: 1.2, 1.4, 1.6, 1.8. Put these in order, largest/smallest first: 5.51, 3.75, 7.35, 5.73, 3.77 Begin to convert halves of a metric unit to a smaller unit and visa versa. For example, write: 7.5 m in centimetres (750 centimetres) 3.5 kg in grams (3500 grams) In the context of word problems, work out calculations involving mixed units such as 3 kilograms ± 150 grams Round decimal fractions to the nearest whole number or the nearest tenth Round decimals with one decimal place to the nearest whole number. For example: Round these to the nearest whole number: Round these costs to the nearest :

85 Recognise the equivalence between decimals and fractions Recognise that for example: (particularly in the context of money and measurement) 0.07 is equivalent to is equivalent to Respond to questions such as: Which of these decimals is equal to 19/100? Write each of these as a decimal fraction: Fractions, decimals and percentages Understanding percentage as the number of parts in every 100, recognise the equivalence between percentages and fractions and decimals, and find simple percentages of numbers or quantities. Understand and read percentage, per cent, %... Recognise the % sign on clothes labels, in sales, on food packets Recognise what percentage of 100 Multilink cubes are red, yellow, blue, green Know that: one whole = 100% one quarter = 25% one half = 50% one tenth = 10% Know that: 10% = 0.1 = % = 0.2 = % = 0.25 = ¼ 1% = 0.01= % = 0.5 = ½ 75% = 0.75 = ¾ Identify a percentage of a shape: for example, What percentage of these shapes are shaded? Calculate questions such as: Find: 25% of % of 1 70% of 100cm Richard got 40 marks out of 80 in his maths test. Sarah got 45%. Who did better: Richard or Sarah?

86 Find percentages by using halving and quartering. For example, to find 75% of 300: 50% is one half = % is one half = 75 75% is three quarters = 225 Calculations Addition Understand the operation of addition and the associated vocabulary, and its relationship to subtraction As outcomes, Year 5 pupils should, for example: Respond rapidly to written questions, for example: 3754 add 30 Add 700 to add 30 add 29.. Add 250, 60, 40, 150 and 3 What is the sum/total of 226 and 39? Increase 190 by 37. Complete written questions, for example: Working rapidly, using know facts: + 62 = 189, = Using a standard written method: = Total a shopping bill or set of measurements. Subtraction Understand the operation of subtraction and the associated vocabulary, and its relationship to addition As outcomes, Year 5 pupils should, for example: Respond rapidly to written questions, 127 take away 35 Take 80 from subtract 105 Subtract 50 from less than 720. What must I take from 220 to leave 55? What is the difference between 155 and 390? How many more than 952 is 1050? How many less than 305 is 94? What must I add to 720 to make 908? Decrease 92 by add a number is 620. What is the number? Find pairs of numbers with a difference of Complete written questions, for example: Working rapidly using know facts:

87 - 62 = = Using a standard written method: = Rapid recall of addition and subtraction facts Know, with rapid recall, addition and subtraction facts As outcomes, Year 5 pupils should, for example: Derive quickly related facts such as: = = = = = = 0.7 Derive quickly, or continue to derive quickly: Two-digit number pairs that total 100 Multiples of 50 that total 1000 Decimal tenths with a total of 1: Decimal (ones and tenths) with a total of 10. Derive quickly addition doubles: Doubles of numbers from to Multiples of 10 from to (eg ) Multiples of 100 from to Mental calculation strategies (+ and -) Find a difference by counting up through the next multiple of 10, 100 or 1000 Count on or back in repeated steps of 1, 10, 100, 1000 Partition into hundreds, tens and ones Identify near doubles Add or subtract the nearest multiple of 10, 100 or 100 and adjust As outcomes, Year 5 pupils should, for example: Work out mentally that: , by counting up from the smaller number to the larger number. Work out mentally that: = 382 because it is = 370 and = 12, or = 382, or it is = = 382. And = = = 385, or it is = = = 385 Work out mentally that: = 3.1 (double 1.5 plus 0.1). Continue to add/subtract 9, 19, 29, or 11, 21,31.. by adding/subtracting 10, 20,30 then adjusting by 1.

88 Use the relationship between addition and subtraction As outcomes, Year 5 pupils should, for example: Recognise that knowing a fact such as = 455 makes it possible to find: or Work out mentally one fact such as or , and then state three other related facts. Working mentally, answer oral questions like: You know that = 700. What is: , , or ? You know that = 810. What is: , or , or Given the numbers 135, 288 and 363, write four different sentences relating these numbers. For example: 228 add 135 equals 363, = 363; 135 add 228 equals 363, = 363; 363 subtract 228 equals 135, = 135; 363 subtract 135 equals 228, = 228. Add several numbers As outcomes, Year 5 pupils should, for example: Add mentally several small numbers: for example: ; And three multiples of 10, such as: Work mentally to complete questions like: = using strategies such as Looking for pairs that make 10 and doing these first; Starting with the largest number; Add a set of numbers such as , recognising that this is equivalent to 30 x 5 Use known number facts and place value to add or subtract a pair of numbers mentally As outcomes, Year 5 pupils should, for example: Add or subtract three-digit multiples of 10 Work mentally to complete written questions like: = = 240 then explain the method in writing. Add three or more three-digit multiples of 100 Work mentally to complete written questions like: = then explain the method in writing. Add/subtract a single-digit multiple of 100 to/from a three- or four- digit number, crossing 1000

89 Work mentally to complete written questions like: = = = 982 then explain the method in writing. Add/subtract a three-digit multiple of 10 to/from a three-digit number, without crossing the hundreds boundary. Work mentally to complete written questions like: = 538 = = 532 Continue to find what to add to a three-digit number to make the next higher multiple of 100. Work mentally to complete written questions like: = = 300 then explain in writing. Find what to add to a decimal with units and tenths to make the next higher whole number Work mentally to complete written questions like: = = 8 Then explain the method in writing. Find the difference between a pair of numbers lying either side of a multiple of 1000 For example, work out mentally that: = 104 By counting up 1 from 6899 to 6900, then 100 to 7000, then 3 to Work mentally to complete written questions like: = = = 19 Add or subtract a pair of decimal fractions each with units and tenths, or with tenths and hundredths, including crossing the units boundary or the tenths boundary. Work mentally to complete written questions like: = = 0.78 then explain method in writing. Pencil and paper procedures (addition) Develop and refine written methods for addition, building on mental methods Standard written methods Continue to develop an efficient standard method that can be applied generally. For example: Using carrying

90 Extend methods to numbers with at least four digits. Using similar methods, add several numbers with different numbers of digits. For example, find the total of: 58, 671, 9, 468, 2187 Extend to decimals Using the chosen method, add two or more decimal fractions with up to three digits and the same number of decimal places. Know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts such as 3.2 m ± 350 cm. For example: Pencil and paper procedures (subtraction) Develop and refine written methods for subtraction, building on mental methods Standard written methods Continue to develop an efficient standard method that can be applied generally. For example: Subtraction is taught by decomposition of hundreds, tens and units which will then lead to direct adjustment from the tens to the units, and then from the hundreds to the tens. 754 = leading to = = Subtract numbers with different numbers of digits. For example: 764 and 5821 Extend to decimals Using the chosen method, find the difference between two decimal fractions with up to three digits and the same number of decimal places. Know that decimal points should line up under each other. For example: Understanding multiplication

91 Understand the operation of multiplication and the associated vocabulary, and its relationship to addition and division As outcomes, Year 5 pupils should, for example: Use and read: Times, multiply, multiplied by, product, multiple, inverse and the x sign. Understand and use as appropriate the principles (but not the names) of the commutative, associative and distributive laws as they apply to multiplication. Understand that with positive whole numbers, multiplying makes a number larger. Understand that multiplication is the inverse of division and use this to check results. Start to use brackets: know that they determine the order of operations, and that their contents are worked out first. For example: 3 + ( 6 x 5 ) = 33, whereas (3 + 6) x 5 = 45 Respond rapidly to written questions, explaining the strategy used. For example: Two twelves. Double times 8. Multiply 31 by 8 by zero by 1. Is 81 a multiple of 3? How do you know? What is the product of 25 and 4? Find all the different products you can make by using three of these: 6, 7, 8, 9, 11. Complete the written questions, for example: Working rapidly, using pencil and paper jottings and/or mental strategies: 70 x 6 = 6 x 8.4 = x 7 = 0.49 Using informal and standard written methods: 46 x 28 = 14 x + 8 = 50 Using written methods work out: 132 x 46 = (14 x 60) + = 850 Understanding division Understand the operation of division and the associated vocabulary, and its relation to subtraction and multiplication As outcomes, Year 5 pupils should, for example: Use and read: Share, group, divide, divided by, divided into, divisible by, factor, quotient, remainder, inverse

92 And the division signs and /. Understand the operation of division as either sharing equally or repeated subtraction (grouping): o sharing is better for dividing by small numbers; o grouping is better for dividing by larger numbers. Understand that: o With positive whole numbers, division makes a number smaller; o Division is non-commutative that is, 72 9 is not the same as 9 72; o A number cannot be divided by zero. Understand that division is the inverse of multiplication and use this to check results. Respond to questions, explaining the strategy used. For example: Share 48 between 8 Divide 56 by 7. Divide 3 into 72. How many groups of 8 can be made from 73? How many lengths of 20cm can be cut from 270cm? What are the factors of 36? Tell me two numbers with a quotient of 100. Relate division and fractions. Understand that: 1 3 of 24 is equivalent to 24 3 or is equivalent to 16 5 or Complete written questions, for example: With rapid mental recall: 63 7 = 56 = 8 9 = 8 Using a pencil and paper jottings and/ or mental strategies: = 54 = = 90 Use written methods to work out: (125 ) + 2 = = 120 Understand the idea of a remainder and when to round up or down after division As outcomes, Year 5 pupils should, for example: Begin to give a quotient as a fraction when dividing by a whole number. For example: 43 9 = Begin to give a quotient as a decimal fraction: When dividing by 10, 5, 4, or 2, for example: = = When dividing pounds and pence by a small whole number, for example: It cost 4 children a total of 5.40 to swim. What did it cost each child? ( 1.35) Decide what to do after division, and round up or down accordingly.

93 Make sensible decisions about rounding down or up after division. For example, is 4 remainder 32, but whether the answer should be rounded up to 5 or rounded down to 4 depends on the context. Example of rounding down: I have saved 240. A train ticket to Durham is is on my calculator. I can buy 4 tickets. Example of rounding up: I have 240 cakes. One box holds 52 cakes. I will need 5 boxes to hold all the cakes. Mental Calculation strategies (x and ) Know multiplication facts by heart and derive quickly the corresponding division facts Know by heart or derive rapidly doubles and halves Use related facts and doubling or halving Use factors As outcomes, Year 5 pupils should, for example: Know by heart multiplication up to 10 x 10, including multiplication by 0 and 1. Derive quickly the corresponding division facts Know by heart the squares of all numbers from 1 x 1 to 10 x 10. Use and read: double, twice, half, halve, whole, divide by 2, divide into 2.. and ½ as one half. Understand that halving is the inverse of doubling: for example, if half of 72 is 36, then double 36 is 72. Know by heart or quickly derive: Doubles of all numbers 1 to 100 Doubles of 10 multiples up to 1000 Doubles of 100 multiples up to and all the corresponding halves Respond to written questions like: What is half of 71.30? How many millimetres are there in half a metre? Complete written questions, for example: 160 x 2 = /2 = 65 Use related facts and doubling/halving. For example: Double 78 = double 70 + double 8 = =156 For example: Doubling a number ending in 5, and halve the other number. For example: 16 x 5 is equivalent to 8 x 10 = 80 Halving an even number in the calculation, find the product, then double it. For example:

94 13 x x 7 = x 2 = 182 To multiply by 50, multiply by 100, then halve. For example: 36 x x 100 = = 1800 Work out: 1 x 25 2 x 25 4 x 25 8 x x 25 Use combinations of these facts to work out, say, 25 x 25 = (16x 25) + (8 x 25) + (1 x 25) = 625 Explain how to find sixths by halving thirds, or twentieths by halving tenths. For example, work out mentally that: One sixth of 300 is 50 (one third of 300 is 100, half of that is 50) Use factors. For example: 15 x 6 = 90 since 15 x 3 = 45, 45 x 2 = = 30, 30 2 = 15, 90 6 = 15 To multiply a number by 19 or 21, multiply it by 20 and add or subtract the number. For example: 13 x 21 = (13 x 20) + 13 = = 273 Use closely related facts already known Partition and use the distributive law Use the relationship between multiplication and addition, or multiplication and division Use known number facts and place value to multiply or divide mentally As outcomes, Year 5 pupils should, for example: Work out the 12 times-table by adding 2 times-table facts to the 10 timestable facts. To multiply a number by 19 or 212, multiply it by 20 and add or subtract the number. Multiplying a two digit number by a single digit number, multiplying the tens first. Continue to recognise that knowing one form of a number sentence, for example: 23 x 3 = 69, means that you also know the other 3 forms. Recognise, for example, that: if 12 x 6 = 72, then 1 6 of 72= 12, 1 12 of 72= 6 Multiply a two- or three-digit number by 10 or 100 Divide a four-digit multiple of 1000 by 10 or 100 Double any multiple of 5 up to 100 Halve any multiple of 10 to 200 Consolidate multiplying a two digit multiple of 10 by 2, 3, 4, 5 or 10 and begin to multiply by 6, 7, 8 or 9 Multiplying a two-digit number by 2, 3, 4, or 5, crossing tens boundary.

95 Solving Problems Making decisions Choose and use appropriate number operations and ways of calculating to solve problems As outcomes, Year 5 pupils should, for example: Understand and use in practical contexts: Operation, sign, symbol, number sentence, equation Make decisions: Choose the appropriate operation(s) to solve word problems and number puzzles. Explain and record how the problem was solved. From number stories recognise/determine the statements that reflect them such as: If 8 equal pieces are cut from 564mm of string, each piece is 70.5mm long. The number statement is = 70.5 What operation sign does each stand for? = 593: = 218 Reasoning about numbers or shapes Methods and reasoning about numbers in writing. Complete calculations that have wholly or partly been done mentally, and develop use conventional notation and vocabulary to record the calculation. For Example: x 5 = 105, plus 4 more is 109. Answer: of 424 ½ of 424 = 212, and ½ of 212 = 106, and ½ of 106 = 53, so 1 8 of 424 = 53. Extend to calculations that cannot entirely be done mentally. For example: : = = 8081 Compare ways of recording and understanding that different ways of recording are equivalent: for example, that is equivalent to and 28)176. Work towards more efficient methods of recording to support and/or explain calculations that are too difficult to do mentally. Solve mathematical problems or puzzles, recognise simple patterns or

96 relationships, generalise and predict. As outcomes, Year 5 pupils should, for example: Solve puzzles and problems such as: Find two consecutive numbers with a product of 182; Find three consecutive numbers with a total of 333. Choose any four numbers from the grid. Add them up. Find as many ways as possible of making Write a number in each circle so that the number in each square box equals the sum of the two numbers on either side of it With 12 squares you can make 3 different rectangles. Find how many squares can be rearranged to make 5 different rectangles. A two-digit number is an odd multiple of 9. When its digits are multiplied, the result is also a multiple of 9. What is the number? Find ways to complete: + + =1 Make and investigate a general statement about familiar numbers or shapes by finding examples that satisfy it As outcomes, Year 5 pupils should, for example:

97 Find examples that match a general statement. For example, explain and start to make general statements like: A multiple of 6 is both a multiple of 2 and a multiple of 3. A multiple of 6 is twice the multiple of 3. A number is not a multiple of 9 if its digits do not add up to a multiple of 9. The product of two consecutive numbers is even. If you divide two different numbers the other way round, the answer is not the same. The perimeter of a regular polygon is length of side x number of sides. Angles on a straight line add up to 180 o. Express a relationship in words in writing. For example: for finding the number of months in any number of years; for finding the change from 50p for a number of chews at 4p. The rule is double the previous number, add 1. Start with 1. Write the next six numbers in the sequence. A sequence starts 1, 4, 9, 16, 25,. Explain the rule. Use all four operations to solve word problems involving numbers in real life As outcomes, Year 5 pupils should, for example: Solve story problems about numbers in real life, choosing the appropriate operation and method of calculation. Record using numbers, signs and symbols how the problem was solved. Examples of problems: Three children play Tiddywinks. What was each child s score? Yasmin Steven Micky Multi-step operations I have read 134 of the 513 pages of my book. How many more pages must I read to reach the middle? Use all four operations to solve word problems involving money. As outcomes, Year 5 pupils should, for example: Use, understand and read: vocabulary of the previous year, and extended to: discount Solve problems involving money, choosing the appropriate operation. Record how the problem was solved. For example: Shopping problems Find the total of: 9.63, and 3.72;

98 66p, 98p 48 p and You have four 35p and four 25p stamps. Find all the different amounts you could stick on a parcel. Calculating Fractions The deposit on a 230 chair is 50%. How much is the deposit? Use all four operations to solve word problems involving length mass or capacity. As outcomes, year 5 pupils should for example: Solve story problems involving : kilometres, metres, centimetres, millimetres kilograms, half kilograms, grams litres, half litres, millimetres For Example: Greg uses 5 tomatoes to make ½ a litre of sauce. How much sauce can he make from 15 tomatoes? Change this pancake recipe for 4 people to a recipe for six people. 240g flour 300ml milk 2 eggs Use all four operations to solve word problems involving time. As outcomes, Year 5 pupils should, for example: Solve story problems involving units of time. For example: The car race began at 08:45 and finished at 14:35. How long did the race last? A train leaves at 09:45h and arrives at 15:46 h. How long does the journey last? Measures Length, mass and capacity Understand and use the vocabulary related to measures. As outcomes, Year 5 pupils should, for example: Understand, use and read vocabulary from the previous year. Use correctly the abbreviations: mm, cm, m, km, g, kg, ml, l... and mm 2 (square millimetre), cm 2 and m 2. Know the relationships between units from the previous year.

99 Recognise that: A mile is a unit of distance, and is a bit more than 1.5km (about 1600metres); Know the equivalent of one half, one quarter, three quarters, one tenth and one hundredth of a metre, 1 kilometre, 1 kilogram and 1 litre in m, cm, g and ml respectively. For example, know that: 10g is one hundredth of 1 kg. Know and use relationships between familiar units. Know the relationships between units from the previous year. Recognise that: A mile is a unit of distance, and is a bit more than 1.5km (about 1600metres); Know the equivalent of one half, one quarter, three quarters, one tenth and one hundredth of a metre, 1 kilometre, 1 kilogram and 1 litre in m, cm, g and ml respectively. For example, know that: 10g is one hundredth of 1 kg. Write, for example: 1.6 m in centimetres (160 centimetres) 5 litres in millilitres (5000 millilitres) and visa versa Suggest suitable units to estimate or measure length, mass or capacity. As outcomes, Year 5 pupils should, for example: Use, read and understand: estimate roughly, nearly, about, approximately Respond to questions like: Estimate the measurement: Is the classroom; 3 metres, 6 metres, 9 metres or 12 metres high. Is this crayon about 5 mm, 55 mm, or 555mm long? Suggest things that would be measured in: kilometres, metres, centimetres, millimetres kilograms, grams litres, millilitres Suggest a metric unit to measure, for example: The weight of a daffodil bulb; The amount held by a teaspoon.

100 Suggest suitable measuring equipment record estimates and readings from scales to a suitable degree of accuracy. Choose a suitable measuring instrument to measure, for example: the height of the classroom the depth of a swimming pool the weight of a ball of wool. Read measuring scales between divisions. Begin to record measurements involving halves, quarters or tenths of 1km, 1kg or 1 litre in decimal form: for example, record 1500ml as 1.5 litres, and 600m as 0.6km. Begin to round decimal measurements to the nearest whole unit. For example: A saucepan holding 4275 millimetres holds 4 litres to the nearest litre. Time Use the vocabulary related to time; suggest suitable units of time to estimate or measure Use, read and write, spelling correctly, the vocabulary from the previous year, and extend to decade Digital/analogue clock, 24-hour clock, 12 hour clock. Know and use: 1 millennium = 1000 years 1 century = 100 years 1 decade = 10 years 1 year = 12 months or 52 weeks or 365 days 1 leap year = 366 days 1 week = 7 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds Estimate or measure, suggesting suitable units Suggest things you would estimate or measure in; weeks, months, years, decades, centuries Suggest a unit to estimate or measure, for example: How long it takes from planting a daffodil bulb to when it flowers. Suggest a unit to measure, for example: How long until year birthday Estimate, using standard units, for example: The hours of darkness in December in June

101 Respond to questions like: Have you lived more or less than 3650 days? Than days? Read the time from clocks, calendars and timetables. Read clocks and calendars Read the time to the minute on a 24-hour digital clock and an analogue clock. Understand 8:48am, 8:48pm, 08:48 and 20:48. For example: Fill in the gaps: seven o/clock in the evening 19:00h 7:00pm quarter to ten in the morning 14:20h 22:15h Midnight 17 minutes past 4 in the afternoon Use timetables Use a train timetable. For Example: Birmingham 09:40 10:05 11:05 12:35 New Street Birmingham 09:50 10:15 11:15 12:45 International Coventry 10:10 10:30 11:30 13:00 Lamington Spa 10:25 11:45 13:15 Banbury 10:45 12:05 Oxford 11:05 11:20 12:25 13:55 Reading 11:30 11:55 12:50 14:25 Area and perimeter Measure and calculate the perimeter and area of simple shapes. Use, read and write, spelling correctly: Area, covers, surface, perimeter, distance, edge and use symbols for: square centimetres (cm 2 ), square metres (m 2 ), square millimetres (mm 2 ). Perimeter Express the formula for the perimeter of a rectangle first in words, then in letter. Work out and express in words a formula for finding the perimeter of a regular polygon. Respond to questions such as:

102 The perimeter of a rectangle is 72cm. The shortest side is 9 cm. What is the length of the longest side? Area Express the formula for the area of a rectangle first in words, then in letters. Choose a suitable unit to estimate the area of, for example: A sheet of newspaper the top of a desk Respond to questions like: Would the area of a standard playing card to be 5cm, 50cm 2 or 100cm 2? Shape and space Properties of 3-D and 2-D shapes Describe and classify common 3-D and 2-D shapes according to their properties As outcomes, Year 5 pupils should, for example: Understand, use and begin to read the vocabulary from the previous year, and extend to: congruent Continue to name and describe shapes, extending the shapes used to: Scalene triangle.. octahedron For example: 3-D shapes Classify solids according to properties such as: the shapes of the faces; the number of faces, edges, vertices; whether or not any face is right-angled; whether the number of edges meeting at each vertex is the same or different. 2-D shapes Recognise properties of rectangles such as: all four angles are right angles; opposite sides are equal and parallel; the diagonals bisect one another. Name and classify triangles. Know some of their properties. For example: In an equilateral triangle all three sides are equal in length and all three angles are equal in size; An isosceles triangle has two equal sides and two equal angles; In a scalene triangle no two sides or angles are equal; In a right-angled triangle one of the angles is a right angle. Make models, shapes and patterns with increasing accuracy, and describe their features.

103 As outcomes, Year 5 pupils should, for example: Visualise and comprehend models, shapes and patterns with increasing accuracy. For example: Identify the different nets for an open cube. Work out the least number of unit cubes needed to change this shape into a cuboid: Reflective symmetry, reflection and translation Reco gnise reflective symmetry in 2-D shapes, reflections and translations As outcomes, Year 5 pupils should, for example: Use read and understand the vocabulary from the previous year and extend to: Axis of symmetry, reflective symmetry. Recognise the number of axes of reflection symmetry in regular polygons. For example: Investigate the lines of symmetry in regular polygons. Know that the number of lines of symmetry in a regular polygon is equal to the number of sides, so a square has four lines of symmetry and an equilateral triangle has three. Sketch the reflection of a simple shape in a mirror line parallel to one edge, where the edges of the shape are not parallel or perpendicular to the mirror line. Complete symmetrical patterns on squared paper with two lines of symmetry at right angles. Show the new position of a shape on a grid after it has been translated, say, 2 units to the left. Identify reflection and translation patterns of individual tiles.

104 Position and direction Recognise positions and directions, and use co-ordinates As outcomes, Year 5 pupils should, for example: Use read and understand, the vocabulary from the previous year and extend to: x-axis, y-axis quadrant parallel, perpendicular. Read and plot points using co-ordinates in the first quadrant. Know the convention that (3, 2) describes a point found by starting from the origin (0, 0) and moving three lines across and two lines up. Respond to questions such as: Three of the vertices of a square are (2, 1), (2, 4), and (5, 4). What are the co-ordinates of the fourth vertex? Know that perpendicular lines are at right angles to each other; Know that parallel lines are the same distance apart. Recognise and identify parallel and perpendicular lines in the environment and in regular polygons such as the square, hexagon and octagon. Know that a diagonal is a straight line drawn from a vertex of a polygon to a non-adjacent vertex. Angle and rotation Make turns; estimate and recognise rotation A s outcomes, Year 5 pupils should, for example: Use, read and understand, the vocabulary from the previous year, and extend to: Rotation.. acute, obtuse, protractor Begin to identify, estimate, order obtuse, acute and right angles. Know that: An angle less than 90 o is acute; An angle between 90 o and 180 o is obtuse; An angle of 180 o is a straight line. For example: Identify acute, obtuse and right angles in a picture of, for example: a classroom. Calculate angles in a straight line.

105 Handling data Organ ising and interpreting data Sol ve a given problem by extracting and interpreting data in tables, graphs and charts. As outcomes, Year 5 pupils should, for example: Use, read and understand, the vocabulary from the previous year and extend to: Classify, mode, maximum/minimum value, range outcome Begin to interpret information in bar and bar line charts. Answer simple questions that relate to the data in the graph. Answer questions related to frequency. Know that it is not appropriate to join the tops of bars when the values in between have no meaning: for example, a dice does not show the number 2.5. Develop understanding of the mode (most common item) and the range (difference between the greatest and least) of a set of data. Interpret a line graph. Understand that intermediate points may or may not have meaning.

106 YEAR 6 Numbers and the Number System Place value (whole numbers) Pupils should be taught to: Multiply and divide whole numbers, then decimals, by 10, 100 or As outcomes, Year 6 pupils should, for example: Demonstrate understanding of multiplying or dividing a whole number by 10, 100, Understand that : When you multiply a number by 10/100/100, the digits move one/two/three places to the left. When you divide a number by 10/100/1000, the digits move one/two/three places to the right. Understand that multiplying by 1000 is equivalent to multiplying by 10, then by 10, then by 10 or is equivalent to multiplying by 10 then by 100. Respond to written questions, such as: How many times larger is than 26? How many 100 notes are in 1300, , ? How many 10 notes, 1 coins, 10p coins, 1p coins? Work out mentally the answers to written questions such as: 0.8 x 10 =, 56 x = , 4 = 0.4 Rounding (whole numbers) Round whole numbers to the nearest 10, 100 or 1000 As outcomes, Year 6 pupils should, for example: Round any whole number to the nearest 10, 100 or Round to the nearest 10,100, 1000 units measurements such as: The capacity of a larger saucepan in millimetres. Estimate calculations. For example: Which is the best approximation of ? , 40-29, 41-30, Negative Numbers Recognise and order negative numbers As outcomes, Year 6 pupils should, for example:

107 Use, read and understand: Integer, positive, negative, minus, above/below zero Respond to questions such as: Put these numbers in order, least first: -37, 4, 29, -4, -28. In this equation, and represent whole numbers. + = 17, select possible values for this sentence. Plot these points on a co-ordinate grid: (5, 4), (5, 8), (-3, 4), (-3, 8) What shape do they make? What is the length of its perimeter? Use negative numbers in the context of temperature. For example: The temperature is -5 o C. It falls by 6 degrees. What is the temperature now? The temperature is -11 o C. It rises by 2 degrees. What is the temperature now? The temperature at the North Pole is -20 o C. How much must it rise to reach -5 o C? Use negative numbers in other contexts such as: Lena set herself a target of 1 metre for her high jump. She recorded each attempt in centimetres above and below her target What was her highest (best) jump? What was her average jump? Properties of numbers and number sequences Recognise and extend number sequences formed by counting on and back in steps of any size, extending beyond zero when counting back As outcomes, Year 6 pupils should, for example: Use and read: next, consecutive, sequence, predict, continue, rule, relationship formula, classify, property Count on and back. For example: From any number, count on in 6s, 7s, 8s, 9s to about 100, and then back. Count in 11s, 15s, 19s 21s and 25s, then back. Can you go on past zero? Count in steps of 0.1, 0.5, 0.25 to 10, then back. Describe, extend and explain number sequences and patterns. For example, respond to questions like. Extend this sequence: 1, 3, 6, 10, 15, 21 (triangular numbers) Fill in the missing numbers in these sequences. Explain the rule in writing. 10, 25,,, 70

108 1, 4,,, 25, 36,,,, -61, -42, -23. Recognise odd and even numbers and make general statements about them Recognise multiples and know some tests of divisibility As outcomes, Year 6 pupils should, for example: Make general statements about odd or even numbers and/or give examples that match them. For example, explore and give some examples to satisfy these general statements: The sum of two even numbers is even. The product of two odd numbers is odd; the product of one odd and one even number is even; an odd number can be written as twice a number plus one (an example is 21, which is 2 x ). Use, read and write, spelling correctly: multiple, digit, divisible, divisibility, factor Recognise multiples to at least 10 x 10. Respond to questions such as: Ring the numbers in the box that are divisible by 12 (or have a factor of 12) A line of counters is set our in a pattern: Five white, four blue, five white, four blue What colour is the 65 th counter? What position in the line is the 17 th blue counter? Ring the numbers that are divisible by Find the smallest number that is a common multiple of two numbers such as: 8 and 12, 12 and 16, 6 and 15 Recognise that a whole number is divisible by: 3 if the sum of its digits is divisible by 3; 6 if it is even and is also divisible by 3; 8 if half of it is divisible by 4, or 8 if the last three digits are divisible by 8; 9 if the sum of its digits is divisible by 9; 25 if the last two digits are 00, 25, 50, or 75.

109 Recognise square numbers Recognise prime numbers and identify factors Use, read and write, spelling correctly: Square number Recognise: 6 2 as six squared. Recognise squares up to 12 x 12, and calculate the values of larger squares: for example, 15 2, Identify two-digit numbers which are the sum of two squares: for example, 34 = Use a calculator to respond to questions such as: Find which number, when multiplied by itself, gives Find two consecutive numbers with a product of The area of a square is 265cm 2, what is the length of the side? Use, read and write spelling correctly: factor, divisible by, prime, prime factor factorise Find all the prime factors of any number to 100. For example: the prime factors of 60 are 2, 2, 3, and 5, since 60 = 2 x 30 = 2 x 2 x 15 = 2 x 2 x 3 x 5. Recognise, for example, that since 60 is a multiple of 12, it is also a multiple of all the factors of 12. Use factors, when appropriate, for finding products mentally: for example, 32 x 24 = 32 x 3 x 8 = 96 x 8 = 800 ( 4 x 8) = 768 Identify numbers with an odd number of factors (squares). Identify two digit numbers with only two factors (primes). For example: Which of these are prime numbers? Recognise prime numbers to at least 20. Fractions and decimals Use fraction notation and recognise the equivalence between fractions As outcomes, Year 6 pupils should, for example: Use and read, vocabulary from the previous year and extend to thousandth Continue to convert improper fractions to mixed numbers and visa versa: for example, change 49 8 to Recognise from simple graphics relationships between fractions. For example: one half is twice as much as one quarter, and three times as much as one sixth; one quarter is twice as much as one eighth;

110 one tenth is ten times as much as one hundredth. Recognise that: A fraction such as 5/ 20 can be reduced to an equivalent fraction ¼ by dividing both numerator and denominator by the same number (cancel); A fraction such as 3 / 10 can be changed to an equivalent fraction 30 /100 by multiplying both numerator and denominator by the same number. Recognise patterns in equivalent fractions, such as: 1 3 = 2 6 = 3 9 = 4 12 = 5 15 = 6 18 = And similar patterns for other unit fractions, relating them to ratios: 1 in every 7, 2 in every 14, and so on. Order familiar fractions As outcomes, Year 6 pupils should, for example: Compare or order simple fractions by converting them to a common denominator. For Example: Suggest a fraction that is greater than one quarter and less than one third. Answer questions such as: Place these numbers in order, smallest first: 2 1/ 10, 1 3/ 10, 2 ½, 1 1 /5, 1 3 /4 Find fractions of numbers or quantities As outcomes, Year 6 pupils should, for example: Relate fractions to division. For example: Understand that finding one tenth is equivalent to dividing by 10, so that 1 10 of 95 is equivalent to 95 10; When 9 whole cakes are divided equally into 4, each person gets nine quarters, or 9 4 = 9 4 ; Recognise that 60 8 is another way of writing Find fractions of numbers and quantities. For example, answer questions such as: What is three tenths of: 80, 10, 100? What is seven hundredths of: 50, 20, 200? What is 4 5 of: 50, 20, 100? 2 litres, 5 metres? What is 5 / 6 of: 16, 40, 100? 12 km, 30 kg? What is 23/ 100 of 4 kilograms in grams. What fraction of 1km is 253 m? What fraction of 1 litre is 413ml? Relate fractions to simple proportions.

111 Ratio and proportion Solve simple problems involving ratio and proportion As outcomes, Year 6 pupils should, for example: Appreciate that two to every three compares part to part; it is equivalent to two in every five, which compares a part to the whole. For example: How many blue tiles to white tiles? (1 to every 2) What is the proportion of black tiles in the whole line? ( 1 /3) What proportion of the small squares are coloured? Solve simple problems involving in every or for every. For example: Kate shares out 12 sweets. She gives Jim 1 sweet for every 3 sweets she takes. How many sweets does Jim get? At the gym club there are 2 boys for every 3 girls. There are 30 children at the club. How many boys are there? Use decimal notation, know what each digit in a decimal fraction represents and order a set of decimal fractions As outcomes, Year 6 pupils should, for example: Use and read: decimal fraction, decimal, decimal point, decimal place Respond to questions such as: What does the digit 5 in represent? The 4? The 6? Write the decimal fraction equivalent to: two tenths, five hundredths and nine thousandths; eight and seven thousandths; sixteen and twenty-nine thousandths. Continue the pattern: 1.92, 1.94, 1.96, Put these in order, largest/smallest first: 7.765, 7.675, 6.765, 7.756, 6.776; and other sets involving measures. Convert a larger metric unit to a smaller. For example, write:

112 3.215 km in metres (3125 metres) 1.25 litres in millimetres (1259 millimetres) Begin to convert halves, quarters, tenths hundredths to a larger unit. For example, write: 300ml in litres (0.3 millilitres) 750g in kg (0.75 kg) In the context of word problems, work out calculations involving mixed units such as 1.3 litres ± 300 millilitres Round decimal fractions to the nearest whole number or the nearest tenth Round decimals with one or two decimal places to the nearest whole number. For example: Round these to the nearest whole number: Round decimals with two or more decimal places to the nearest tenth. For example: What is 5.28 to the nearest tenth? What is 3.82 to one decimal place? Recognise the equivalence between decimals and fractions Recognise that for example: is equivalent to ; is equivalent to particularly in the context of measurement. Respond to questions such as: Which of these decimals is equal to 193/100? Write each of these as a fraction: Fractions, decimals and percentages Understanding percentage as the number of parts in every 100, recognise the equivalence between percentages and fractions and decimals, and find simple percentages of numbers or quantities. Understand and read percentage, per cent, %...Work out, for example, What proportion of the pupils in the class are girls, aged 11, have brown eyes Know that for example: 43% = 0.43= 43/ 100 Answer questions such as:

113 Which of these percentages is equivalent to 0.26? 0.26% 2.6% 26% 260% Calculate questions such as: Find: 25% of % of 5 70% of 300cm A school party of 50 is at the tower of London. 52% are girls. 10% are adults. How many are boys? Find percentages by using halving and quartering. For example, to find 12.5% of : 50% is one half = % is one half = % is one eighth = 4500 Calculations Addition Understand the operation of addition and the associated vocabulary, and its relationship to subtraction As outcomes, Year 6 pupils should, for example: Respond rapidly to written questions, for example: 4250 add add 314 add 750 Add 1200, 400, and 15. What is the sum/total of 753 and 227? Increase 250 by 420. Complete written questions, for example: Working rapidly, using know facts: = 5.38, = 250 Using a standard written method: = Find the average (mean): for example price of some goods, the average of a set of measurements or a set of numbers Subtraction Understand the operation of subtraction and the associated vocabulary, and its relationship to addition As outcomes, Year 6 pupils should, for example: Respond rapidly to written questions, 750 take away 255 Take 300 from subtract 2050 Subtract 2250 from less than less than What must I take from 8.4 to leave 2.6? What is the difference between 2.2 and 6.5? How many more than 23.4 is 24.9?

114 How many less than 6.8 is 4.2? What must I add to 5.4 to make 9.3? Decrease 5.6 by add a number is 4.2. What is the number? Find pairs of numbers with a difference of Complete written questions, for example: Working rapidly using know facts: = = Using a standard written method: = Use mental methods to find the missing number in: = 4128 Mental calculation strategies (+ and -) Find a difference by counting up through the next multiple of 10, 100 or 1000 Identify near doubles Add or subtract the nearest multiple of 10, 100 or 100 and adjust As outcomes, Year 6 pupils should, for example: Work out mentally by counting up from the smaller number to the larger number: is = 5215 Work out mentally that: = 808 (double 400 minus 13). Add/subtract 0.9, 1.9, 2.9, or 1.1, 2.1,3.1.. by adding/subtracting 1, 2, 3 then adjusting by 0.1. Use the relationship between addition and subtraction As outcomes, Year 6 pupils should, for example: Continue to make use of the relationship between addition and subtraction. For example: Work out mentally one fact such as or and then state three other related facts. Add several numbers As outcomes, Year 6 pupils should, for example: Add mentally: And three or more multiples of 10, such as: Work mentally to complete questions like: = 87 using strategies such as: Looking for pairs that make 10 and doing these first;

115 Starting with the largest number; Add a set of numbers such as , recognising this as equivalent to (70 x 4) + ( ). Use known number facts and place value to add or subtract a pair of numbers mentally As outcomes, Year 6 pupils should, for example: Add or subtract four-digit multiples of 100 Work mentally to complete written questions like: = = Find what to add to a decimal with units, 10ths and 100ths to make the next higher whole number or 10 th. Respond to questions: What must be added to 6.45 to make 7? And to 2.78 to make 2.8? Work mentally to complete written questions like: = = 8 Then explain the method in writing. Add or subtract a pair of decimal fractions each less than 1 and with up to two decimal places Respond to questions like: Work mentally to complete written questions like: = = 0.19 Pencil and paper procedures (addition) Develop and refine written methods for addition, building on mental methods Standard written methods Continue to develop an efficient standard method that can be applied generally. For example: Using carrying Extend methods to numbers with any number of digits. Using similar methods, add several numbers with different numbers of digits. For example, find the total of: 42, 6432, 786, 3, 4681.

116 Extend to decimals Using the chosen method, add two or more decimal fractions with up to four digits and with either one or two decimal places. Know that decimal points should line up under each other, particularly when adding or subtracting mixed amounts such as 14.5 kg ± 750g. For example: Pencil and paper procedures (subtraction) Develop and refine written methods for subtraction, building on mental methods Standard written methods Continue to develop an efficient standard method that can be applied generally. For example: decomposition Subtract numbers with different numbers of digits. For example: and 4387 Extend to decimals Using the chosen method, subtract two or more decimal fractions with up to three digits and either one or two decimal places. Know that decimal points should line up under each other. For example: Understanding multiplication Understand the operation of multiplication and the associated vocabulary, and its relationship to addition and division As outcomes, Year 6 pupils should, for example: Use and read: Times, multiply, multiplied by, product, multiple, inverse and the x sign. Understand and use as appropriate the principles (but not the names) of the commutative, associative and distributive laws as they apply to multiplication.

117 Understand that multiplication is the inverse of division and use this to check results. Use brackets: know that they determine the order of operations, and that their contents are worked out first. Respond rapidly to written questions, explaining the strategy used. For example: Two nineteens. Double times 8. 9 multiplied by 8. Multiply 25 by 8 by zero by 1. Is 210 a multiple of 6? How do you know? What is the product of 125 and 4? Find all the different products you can make by using three of these: 0.2, 1.4, 0.03, 1.5, 0.5. Complete the written questions, for example: Working rapidly, using pencil and paper jottings and/or mental strategies: 0.7 x 20 = 20 x = 8000 x 5 = 3.5 Using standard written methods: 132 x 46 = x 9 = 18.9 Understanding division Understand the operation of division and the associated vocabulary, and its relation to subtraction and multiplication As outcomes, Year 6 pupils should, for example: Use and read: Share, group, divide, divided by, divided into, divisible by, factor, quotient, remainder, inverse And the division signs and /. Continue to understand the operation of division as either sharing equally or repeated subtraction (grouping): o sharing is better for dividing by small numbers; o grouping is better for dividing by larger numbers. Understand that division is the inverse of multiplication and use this to check results. Respond to questions, explaining the strategy used. For example: Share 108 between 9 Divide 112 by 7. Divide 15 into 225. How many groups of 16 can be made from 100? How many lengths of 25cm can be cut from 625cm? What are the factors of 98? Tell me two numbers with a quotient of 0.5.

118 Relate division and fractions. Understand that: 1 8 of 72 is equivalent to 72 8 or is equivalent to 4 7 Complete written questions, for example: With rapid mental recall: = 9.9 = = 0.8 Using a pencil and paper jottings and/ or mental strategies: = 25 = 39 Understand the idea of a remainder and when to round up or down after division As outcomes, Year 6 pupils should, for example: Give a quotient as a fraction when dividing by a whole number. For example: 90 7 = Begin to give a quotient as a decimal fraction: When dividing by a whole number, for example: = = 6.12 When dividing pounds and pence, for example: It cost 15 people a total of for a theatre trip. What did it cost each one? ( 5.25) Decide what to do after division, and round up or down accordingly. Make sensible decisions about rounding down or up after division. For example, is 3.8, but whether the answer should be rounded up to 4 or rounded down to 3 depends on the context. Example of rounding down: Dad have saved An air fare to Sydney is is on my calculator. He can buy 5 tickets. Example of rounding up: I have 5000 sheets of paper. A box holds 865 sheets. I will need 6 boxes to hold all 5000 sheets. Mental Calculation strategies (x and ) Know multiplication facts by heart and derive quickly the corresponding division facts Know by heart or derive rapidly doubles and halves Use related facts and doubling or halving Use factors As outcomes, Year 6 pupils should, for example: Continue to know by heart multiplication up to 10 x 10, including multiplication by 0 and 1. Derive quickly the corresponding division facts Know by heart the squares of all numbers from 1 x 1 to 12 x 12.

119 Derive quickly squares of multiples of 10 to 100, such as 20 2, Respond rapidly to written questions like: Nine eighths. How many sevens in 35? 8 times 8. 6 multiplied by 7. Multiply 11 by 8. Respond quickly to questions like: 7 multiplied by 0.8 by 0. Multiply 0.9 by 0.6 by 0. Divide 3.6 by 9 by 1. What is 88 shared between 8? Divide 6 into divided by times 7 times 2. One twentieth of 360. Use and read: double, twice, half, halve, whole, divide by 2, divide into 2.. and ½ as one half. Understand that halving is the inverse of doubling: for example, if half of 0.3 is 0.15, then double 0.15 is 0.3. Know by heart or quickly derive: Doubles of two digit whole numbers or decimals; Doubles of 10 multiples up to 1000 Doubles of 100 multiples up to and all the corresponding halves Respond to written questions like: Double 37 ½ What is half of 581? What fraction of 1 cm is half a millimetre? Complete written questions, for example: 370 x 2 = /2 = 165 Use related facts and doubling or halving. For example: Double 176 = = 352 For example: Doubling a number ending in 5, and halve the other number. Halve/double one number in the calculation, find the product, then double/halve it. To multiply by 15, multiply by 10, then halve. The result, then add the two parts together. For example: 14 x x 10 = = x 15 = 210 Alternatively multiply by 30 and divide by 2. To multiply by 25, multiply by 100, then divide by 4. For example: 39 x x 100 = = 975 Work out 1 x 32 = 32 and so deduce that 2 x 32 = 64

120 4 x 32 = x 32 = x 32 = 512. Use combinations of these facts to work out other multiples of 32. Explain how to find sixths and twelfths by halving thirds, or twentieths by halving tenths. For example, work out mentally that: One twelfth of 300 is 25 Since 1 / 3 of 300 is 100, 1 / 6 is half 100, 1 / 12 is half 50 Use factors. For example: 35 x x 6 = 210 and 210 x 3 = 630, 35 x 18 = = 126 and = 18, = 18 Use closely related facts already known Partition and use the distributive law Use the relationship between multiplication and addition, or multiplication and division Use known number facts and place value to multiply or divide mentally As outcomes, Year 6 pupils should, for example: To multiply a number by 49 or 51, multiply it by 50 and add or subtract the number. To multiply a number by 99 or 101, multiply it by 100 and add or subtract the number. Continue to multiply a two digit number by a single digit number, multiplying the tens first. Multiply a whole number and tenths by a single digit number, multiplying the units first. For example: 8.6 x 7 = ( 8 x 7 ) + ( 0.6 x 7 ) = = 60.2 Continue to recognise that knowing one form of a number sentence, for example: 0.75 x 4 = 3, means that you also know the other 3 forms. Recognise, for example, that: if 5 x 60 = 300, then 1 5 of 300= 60, 1 6 of 300 = 50 and ¾ of 4 = 3, then 4 x ¾ = 3 Answer written questions Use the numbers 0.2, 0.3 and 0.06 Say or write four different multiplication or division statements relating the numbers. Multiply a decimal fraction with one or two decimal places by 10 or 100 Divide a one- or two digit whole number by 100 or 10 For example: Double a decimal fraction less than 1 with one or two decimal places Complete written questions like: 0.65 x 2 =

121 Halve a decimal fraction less than 1 with one or two decimal places Complete written questions like: = Multiply a decimal fraction such as 0.6 by a single- digit number Work mentally to complete written questions like: 0.7 x 5 = 0.2 x = 1.8 Multiply a two-digit whole number or decimal fraction by any single-digit number Work mentally to complete written questions like 3.7 x 5 = 4.2 x = 16.8 Solving Problems Making decisions Choose and use appropriate number operations and ways of calculating to solve problems As outcomes, Year 6 pupils should, for example: Use, read and understand: Operation, sign, symbol, number sentence, equation Make decisions: Choose the appropriate operation(s) to solve word problems and number puzzles. Explain and record how the problem was solved. From number stories recognise/determine the statements that reflect them such as: = compact discs at 6.83 each will cost What operation sign does each stand for? = 435; = Reasoning about numbers or shapes Methods and reasoning about numbers in writing. Complete calculations that have wholly or partly been done mentally, and develop use conventional notation and vocabulary to record the calculation. For Example: 17.5 % of % = % = % = % = = 129, = of 400 1/ 10 of 400 = 40, and ½ of 40 = 20, and so 1/ 20 of 400 = 20.

122 Extend to calculations that cannot entirely be done mentally. For example: : x x Answer: 22 18/ 27 = 22 2 / 3 Develop efficient mehtods of recording calculations, including generally applicable or standard written methods for; Addition and subtraction of whole numbers ( three or more digits, including decimals with up to two decimal places); Solve mathematical problems or puzzles, recognise simple patterns or relationships, generalise and predict. As outcomes, Year 6 pupils should, for example: Solve puzzles and problems such as: Find two consecutive numbers with a product of 1332; Find two numbers with a product of 899. Complete this table of multiplication: x Each represents one of the digits 1 to 6. Use each of the digits 1 to 6 once. Replace each to make a correct product. x = A number sequence is made from counters. There are 7 counters in the third number. (1) (2) (3) How many counters in the 6 th number? The 20 th? Write a formula for the number of counters in the nth number in the sequence.

123 For how many three-digit numbers does the sum of the digits equal 25? Each letter from A to G is a code for one of these digits: 1, 3, 4, 8, 9. Crack the code. A + A = B A x A = DF A + C = DE C + C = DB C x C = BD A x C = EF Make and investigate a general statement about familiar numbers or shapes by finding examples that satisfy it As outcomes, Year 6 pupils should, for example: Find examples that match a general statement. For example, explain and start to make general statements like: If 0.24 < < 0.27, then any number between 0.24 and 0.27 can go in the box. Examples: 0.25, 0.26, 0.251, If you add three consecutive numbers, the sum is three times the middle number. To multiply by 25, multiply by 100 and divide by 4. Example: 12 x 25 = 12 x = 300 Any square number is the sum of two consecutive triangular numbers. Examples: 4 = 1 + 3, 25 = , Dividing a whole number by one half makes the answer twice as big. Example: = 68 = 2 x 34 If I multiply a decimal number by 10, every digit moves one place to the left. A trapezium is a quadrilateral with one pair of parallel sides. The sum of the angles of a triangle is 180 o. Express a relationship in words in writing. For example: Use symbols and identify a formula for the number of months m in y years. Select a formula for the cost of c chews at 4p each. Identify a formula for the nth term of this sequence: 3, 6, 9, 12, 15. The perimeter of a rectangle is 2 x (l +b), where l is the length and b is the breadth of the rectangle. What is the perimeter if l = 8 cm and b = 5cm? The number of bean stick needed for a row which is m metres long is 2m + 1. How many bean sticks do you need for a row which is 60 metres long? Plot the points which show pairs of numbers with a sum or 9. Use all four operations to solve word problems involving numbers in real life

124 As outcomes, Year 6 pupils should, for example: Solve story problems about numbers in real life, choosing the appropriate operation and method of calculation. Record using numbers, signs and symbols how the problem was solved. Examples of problems: Single-step operations people visited the museum this year. This is 2568 more than last year. How many people visited the museum last year? Multi-step operations There is space in the multi-storey car park for 17 rows of 30 cars on each of 4 floors. How many cars can park? Use all four operations to solve word problems involving money. As outcomes, Year 6 pupils should, for example: Use, understand and read, the vocabulary from the previous year. Solve problems involving money, choosing the appropriate operation. Record how the problem was solved. For example: Shopping problems Find the cost of 145 bottles of lemonade at 21p each. What change do you get from 50? Converting to European or foreign currency There are 1.43 euros to 1. What is the price in pounds of a car costing euros? Calculating Fractions The agent s fee for selling a house is 5%. Calculate the fee on a house sold for Use all four operations to solve word problems involving length mass or capacity. As outcomes, year 6 pupils should for example: Solve story problems involving : kilometres, metres, centimetres, millimetres kilograms, grams litres, millimetres, centilitres miles For Example: Sarah travelled km by car, 2.7 km by bus and 1000m on foot. How many kilometres did she travel? How many metres?

125 I cut 65 cm off 3.5 metres of rope. How much is left? How many grams of carrots must be added to 2.76 kg to make 5 kg of carrots altogether? Solve, for example: A pin is made from 14 mm of wire. How many pins can be made from 1 m of wire? Use all four operations to solve word problems involving time. As outcomes, Year 6 pupils should, for example: Solve story problems involving units of time. For example: Lamb must be cooked for 60 minutes for every kg. Chick must be cooked for 50 minutes for every kg. Complete this table of cooking times. Kilograms Cooking time in minutes (lamb) Cooking time in minutes (chicken) Measures Length, mass and capacity Understand and use the vocabulary related to measures. As outcomes, Year 6 pupils should, for example: Understand, use and read vocabulary from the previous year, and extend to: Names of standard metric unit: tonne Names of commonly used imperial units: pound yard, foot, inch Use correctly the abbreviations: km, m, cm, mm, kg, g, l, ml... cm 2, mm 2, m 2. Know and use relationships between familiar units. Know the relationships between units from the previous year, and extend to:

126 1 tonne = 1000 kilograms Know the approximate equivalence between commonly used imperial units and metric units: 1 litre 2 pints (more accurately, 1 ¾ pints) 4.5 litres 1 gallon or 8 pints 1 kilogram 2 lb (more accurately, 2,2 lb) 30 grams 1 oz 8 kilometres 5 miles Know the equivalent of one thousandth of 1km, 1kg, 1litre in m, g and ml respectively. Convert a larger metric unit to a smaller. For example: Write km in metres (3125 metres) Begin to convert a smaller unit to a larger. For example: Write 750grams in kilograms (0.75 kilograms) Suggest suitable units to estimate or measure length, mass or capacity. As outcomes, Year 6 pupils should, for example: Use, read and understand: estimate roughly, nearly, about, approximately Respond to questions like: Estimate the measurement in metric units: The weight of an egg. The perimeter of the classroom? Suggest how you could measure: The weight of a grain of rice; The quantity of water in a raindrop. Suggest things that would be measured in: kilometres, metres, centimetres, millimetres tonnes, kilograms, grams litres, millilitres Suggest a metric unit to measure, for example: The distance from the earth to the moon; The amount of milk in a milk jug in a milk tanker. What units of measurement might you see for example, in: TV weather forecast;. Suggest suitable measuring equipment record estimates and readings from scales to a suitable degree of accuracy. Choose a suitable measuring instrument to measure, for example: The width of a manhole The weight of a lorry..

127 Read measuring scales, converting to an equivalent metric unit.. For example: How many grams of flour are there on the scales? Time Record measurements of lengths, weights or capacities in decimal form: for example, write 4125 grams as kg. Round a measurement to the nearest whole unit or tenth of a unit. For example: 3879 grams of potatoes weigh 3.9 kg to the nearest tenth of a kilogram, or 4 kg to the nearest kilogram. Use the vocabulary related to time; suggest suitable units of time to estimate or measure Use, read and write, spelling correctly, the vocabulary from the previous year, and extend to: Greenwich mean time, British summer time. Know and use: 1 millennium = 1000 years 1 century = 100 years 1 decade = 10 years 1 year = 12 months or 52 weeks or 365 days 1 leap year = 366 days 1 week = 7 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds Read the time from clocks, calendars and timetables. Read world time charts Understand different times around the world. Use a world time chart to answer questions such as:

128 Area and perimeter It is 12:00 noon in London. What is it in Delhi, Tokyo, Hawaii, San Francisco..? It is 4:36 am Sydney. What is it in New York? Measure and calculate the perimeter and area of simple shapes. Use, read and write, spelling correctly: Area, covers, surface, perimeter, distance, edge and use symbols for: square centimetres (cm 2 ), square metres (m 2 ), square millimetres (mm 2 ). Perimeter Calculate the perimeters of compound shapes that can be split into rectangles. For example, find the perimeter of this shape. Area Know the formula for finding the area of a rectangle. Begin to find the areas of compound shapes that can be split into rectangles, such as the shape above. Respond to questions like: Find the length, breadth and height of this box. Calculate it s surface area. Shape and space Properties of 3-D and 2-D shapes Describe and visualise 3-D and 2-D shapes; classify them according to

129 their properties As outcomes, Year 6 pupils should, for example: Understand, use and begin to read the vocabulary from the previous year, and extend to: Concentric tangram circumference Continue to name and describe shapes, extending to: Parallelogram, rhombus, kite, trapezium, dodecahedron For example: 3-D shapes Describe properties of 3-D shapes, such as parallel or perpendicular faces or edges. 2-D shapes Name and begin to classify quadrilaterals, using criteria such as parallel sides, equal angles, equal sides, lines of symmetry. Know properties. For example: A parallelogram has its opposite sides equal and parallel; A rhombus is a parallelogram with four equal sides; A rectangle has four right angles and its opposite sides are equal; A square is a rectangle with four equal sides; A trapezium has one pair of opposite parallel sides; A kite has two pairs of adjacent sides equal. Begin to know properties such as: The diagonals of any square, rhombus or kite intersect at right angles; The diagonals of any square rectangle, rhombus or parallelogram bisect one another. Visualise models, shapes and patterns with increasing accuracy, and describe their features. As outcomes, Year 6 pupils should, for example: Visualise and comprehend models, shapes and patterns with increasing accuracy. For example: Identify the different nets for a closed cube. Visualise 3-D shapes from 2-D drawings Work out the least number of cubes needed to cover and join the blackfaces.

130 Reflective symmetry, reflection and translation Recognise reflective symmetry in 2-D shapes, reflections and translations As outcomes, Year 6 pupils should, for example: Use read and understand the vocabulary from the previous year. Sketch the reflection of a simple shape in a mirror line touching it at one point, where the edges of the shape are not necessarily parallel or perpendicular to the mirror line. Show the reflection of a simple shape in two mirror lines at right angles, where the sides of the shape are parallel or perpendicular to the mirror line Show the position of a simple shape after it has been translated, say, 3 units to the right, then 2 units down

131 Position and direction Recognise positions and directions, and use co-ordinates As outcomes, Year 6 pupils should, for example: Use read and understand, the vocabulary from the previous year and extend to: Intersecting, intersection Plane.. Read and plot points using co-ordinates beyond the first quadrant. Respond to questions such as: The points (-1, 1), (2, 5) and (6, 2) are three of the four vertices of a square. What are the co-ordinates of the fourth vertex? Draw a polygon with each vertex lying in the first quadrant. Plot its reflection in the y-axis, and name the coordinates of the reflected shape. Recognise parallel and perpendicular lines in quadrilaterals. Know that two lines that cross each other are called intersecting lines, and the point at which they cross is an intersection Angle and rotation Make turns; estimate and recognise rotation As outcomes, Year 6 pupils should, for example: Use, read and understand the vocabulary from the previous year, and extend to: reflex Sketch the position of a simple shape after a rotation of 90 o or 180 o about a vertex Identify and calculate acute, obtuse and right angles, for example: Calculate the third angle in a triangle given the other two. Calculate angles at a point.

132 Handling data Organising and interpreting data Solve a given problem by extracting and interpreting data in tables, graphs and charts. As outcomes, Year 6 pupils should, for example: Use, read and understand, the vocabulary from the previous year and extend to: statistics, average, distribution, median, mean Begin to interpret information in bar and bar line charts. Answer simple questions that relate to the data in the graph. Answer questions related to frequency. Begin to interpret simple pie charts and answer simple questions relating to the fraction or percentage of the whole. Know the formula for mode and range for a simple set of data. Begin to understand the meaning of the mean and median of a data set.

133 YEAR 7 Using and applying mathematics to solve problems Applying mathematics and solving problems Pupils should be taught to: Solve word problems and investigate in a range of contexts As outcomes, Year 7 pupils should, for example: Solve word problems and investigate in context of number, algebra, shape, space and measures, and handling data; compare and evaluate solutions. Problems involving money For example: o A drink and a box of popcorn together cost 90p. Two drinks and a box of popcorn together cost What does a box of popcorn cost? Problems involving percentages For example: o The value of a flat in increased by 12% in June. Its new value increased by a further 10% in October. What was its value in November? Problems involving ratio and proportion: For example: o In a country dance there are 7 boys and 6 girls in every line. 42 boys take part in the dance. How many girls take part? o Some children voted between a safari park and a zoo for a school visit. The result was 10: 3 in favour of the safari park. 130 children voted in favour of the safari park. How many children voted in favour of the zoo? o A boy was sent 35 in birthday money. He spent some of the money. He saved four times as much as he spent. How much did he save? o 10 bags of crisps cost What is the cost of 6 bags of crisps? o 1 = Danish kroner. How much is 1.50 Danish kroner? Problems involving number and algebra For example: o I think of a number, add 3, 7, then multiply by 5. The answer is What is the number? o Two prime numbers are added. the answer is 45. What are the numbers? o What operation is represented by each? a = 543 c = 393 b = 6.24 d =

134 o Find two consecutive numbers with the product 702. o A function machine changes the number n to the number 3n + 1. What does it do to these numbers? 2, 5, 9, 21, 0 What numbers must be input to get these numbers? 10, 37, 100 More problems involving number and algebra For example: Here is a sequence of five numbers The rule is to start with 2 then add the same amount each time. Write in the missing numbers. Does the line y = x 5 pass through the point (10, 10)? This is a series of patterns with white and blue tiles. How many while tiles and blue tiles will there be in pattern number 8? Will one of the patterns have 25 white tiles? A girl at a fair tries the hoopla. She pays 2 coin for 4 goes and is given change. The cost of each go is c pence. Which of these expressions gives her change in pence? 4c 2 4c c 200 4c Problems involving shape and space For example: o Here is a net with a tab to make a cuboid. These is a dot ( ) in a corner of one of the faces. Imagine folding the net up. Write T on the edge that the tab will be stuck to. Draw a dot on each of the two corners that will meet the corner with the ( ). o Use the diagram to calculate angle w.

135 From a selection, choose the sequence of facts that will lead the concluding the size of angle w. - The angles in the quadrilateral add to 360 o. Therefore w = 360 o (90 o + 90 o o ) = 99.5 o o Draw and label axes on squared paper, choosing a suitable scale. Plot these points. A (-1, 1) B (2, -1) C (3, -1) D (5, -1) E (2, 2) F (1, -2) G (5, 2) H (2, 1) a. Name the four point that are the vertices of: i. a square, ii. a non-square parallelogram, iii. a non-square trapezium. b. Name the three points that are the vertices of: i. a right angled triangle, ii. a non-right angled isosceles triangle. Problems involving perimeter and area For example: Twelve rectangles, all the same size, are arranged to make a square outline, as shown in the diagram. Calculate the area of one rectangle. Problems involving measures For example: o A pupil paced the length and width of a corridor. She found it was 5 paces wide by 142 paces long. She measured one pace. It was 73cm. What was the approximate length of the corridor in metres?

136 o A map has a scale of 1 cm to 6 km. A road measured on the map is 6.6cm long. What is the length of the road in kilometres? o A box of figs costs 2.80 per kilogram. A fig from the box weighs 150g. Find the cost of the fig. o A number of bars of soap are packed in a box that weighs 850g. Each bar of soap weighs 54g. When it is full, the total weight of the box and the soap is 7.6kg. How many bars of soap are in the box? o The maximum load in a small service lift is 50kg. 60 tins of food must go up in the lift. Each tin weighs 840g. Is it safe to load all of these tins into the lift? Problems involving probability For example: o There are six balls in a bag. The probability of taking a red ball is 0.5. A red ball is taken out of the bag and put to one side. What is the probability of taking another red ball out of the bag? o Some children choose six tickets numbered from 1 to 200. Kay chooses numbers 1, 2, 3, 4, 5 and 6 Zach chooses numbers 14, 45, 76, 120, 137 and 182. Mary then picks a number at random from 1 to Is Kay or Zach more likely to have Mary s number? Problems involving handling data For example o Jeff has played two of the three games in a competition. Game A Game B Game C Score To win Jeff needs a mean score of 60. How many points does he need to score in game C? o James has four number cards. Their mean is 4. James takes another card. The mean of his five cards is now 5. What is the number on the new card? Identify the information necessary to solve a problem; represent problems mathematically in a variety of forms As outcomes, Year 7 pupils should, for example: Use, read and understand: investigate, explore, solve, explain true, false. problem, solution, method, answer, results, reasons, evidence Identify the necessary information; represent problems mathematically, making correct use of symbols, words, diagrams, tables and graphs. For example, solve: Birthday candles Mrs Sargent is 71 years old.

137 Every year since she was born she has blown out the corresponding number of candles on her birthday cake. How many birthday candles has she blown out altogether? Link to rapid recall of number facts including complements to 100 and multiplication and division facts using the order of operations, and brackets. Tiles A T- shape is made from 5 square tiles. Three tiles horizontally for the hat of the T, and two vertically for the trunk. The length of a tile is w cm. Select an expression for the are of the T-shape. If the area is 720cm 2 what is the length of its perimeter? Break problems into smaller steps or tasks; choose and use efficient operations, methods and resources Break a complex calculation into smaller steps, choosing and using appropriate and efficient operations, methods and resources. Link to generating and describing simple integer sequences. Select which number statement reflects a given word problem. Numbers and the Number System Place value, ordering and rounding Understand and use decimal notation and place value; multiply and divide integers and decimals by powers of 10 As outcomes, Year 7 pupils should, for example: Use, read and understand: place value, zero place holder, tenth, hundredth, thousandth, equivalent, equivalence. Understand and use decimal notation and place value. Read and write any number from and , knowing what each digit represents. For example, know that: In the digit 9 represents nine thousandths, which is written as The number in words is five point two three nine, not five point two hundred and thirty-nine. The fraction is read as five and two hundred and thirtynine thousandths. Know the significance of 0 in 0.35, 3.05, 3.50 and so on. Know that decimals used in context may be spoken in different ways. For example: 1.56 is written in mathematics as one point five six is written as one pound fifty-six.

138 1.06 is written as one pound and six pence is written as fifty pence. 1.56km is sometimes written as one kilometre, five hundred and sixty metres. 3.5 hours can be written as three and a half hours or three hours and thirty minutes. Answer questions such as: Match the figures to these words: Four hundred and three thousand, and seventeen. Match the words with these figures: 4.236, 0.5, Match these words to the decimal the fraction six, and two hundred and forty-three thousandths. Make the largest and smallest number you can using: The digits 2, 0, 3, 4; The digits 2, 0, 3, 4, and a decimal point. Add or subtract 0.1 and 0.01 to or from any number. Count forwards or backwards from any number. For example: Count on in 0.1s from 4.5 Count back from 23.5 in 0.1s. Count on in 0.01s from Answer questions such as: What is 0.1 less than 2.0? What is 0.01 more than 2.09? What needs to be added or subtracted to change: to 37.48, to 27.38, to 5.302? Multiply and divide numbers by 10, 100 and Understand the effects of multiplying or dividing a number by 10, 100, In particular, recognise that: Multiplying a positive number by 10, 100, has the effect of increasing the value of that number. Dividing a positive number by 10, 100, 1000 has the effect of decreasing the value of that number. When a number is multiplied by 10, the digits move one place to the left: multiplied by 10 = When a number is divided by 10, the digits move one place to the right divided by 10 = Complete statements such as: 4 x 10 = 4 x = = 4 = x 10 = 0.4 x = 400

139 = 0.4 = = = 40 x 1000 = x 10 = 400 Link to converting mm to cm and m, cm to m, m to km Compare and order decimals As outcomes, Year 7 pupils should, for example: Use, read and understand: decimal number, decimal fraction, less than, greater than, between order, compare, digit, most/least significant digit And use accurately these symbols: =,, <, >,,. Know that to order decimals, digits in the same position must be compared, working from the left, beginning with the first non-zero digit. In these examples, the order is determined by: < the second decimal place; 3.18 km >3.172 the second decimal place; 0.42 < 0.54 the first decimal place; 5.4 < 5.6 < 5.65 the first decimal place initially, then the second decimal place. Know that when comparing measures it is necessary to convert all measures into the same units. For example: Order these measurements, starting with the smallest: 5kg, 500g, 0.55kg 45cm, 1.23m, 0.96m 3.67, 3.71, 39p Identify and estimate decimal fractions on a number line and find a number between two others by looking at the next decimal place. For example: Find the number that is half way between two: 3 and 4, 0.3 and 0.4, -3 and 4 and -4 and 3. Use accurately the symbols <, >,,. For example: Place < or > between these: o C -7 o C Round numbers, including to a given number of decimal places Use and read and understand Round, nearest, to one decimal (1 d.p.) approximately Round positive whole numbers to the nearest 10, 100 or 1000, What is the volume of the liquid in the measuring cylinder to the nearest 10ml? What is the mass of the flour to the nearest 100 grams?

140 Round decimals to the nearest whole number or to one decimal place. When rounding a decimal to a whole number, know that: If these are 5 or more tenths, then the number is rounded up to the next whole number; otherwise, the whole number is left unchanged; Decimals with more than one decimal place are not first rounded to one decimal place, eg to 7, not to 7.5 then rounds to 8. When rounding a decimal such as 3.96 to one decimal place, know that the answer is 4.0, not 4, because the zero in the first decimal place is significant. For example: 4.48 rounded to the nearest whole number is rounded to the nearest whole number is 5, and rounded to one decimal place is rounded to the nearest whole number is rounded to one decimal place is 5.0. Answer questions such as Round 5.28: a. To the nearest whole number; b. To one decimal place. Integers, powers and roots Order, add, subtract, multiply and divide positive and negative numbers As outcomes, Year 7 pupils should, for example: Use, read and understand: Integer, positive, negative, minus And know that -6 is read as negative 6. Order integers and position them on a number line. For example: Put a > or < sign between these pairs of temperatures: -6 o C 4 o C -6 o C -4 o C Link to plotting coordinates in all four quadrants.

141 Begin to add and subtract integers. Extend patterns such as: = = = = = = = = = = 0 Use negative number cards to help answer questions such as: = = = = = = = = Sole simple puzzles or problems involving addition and subtraction of positive and negative numbers, such as: Complete the magic square Link to substituting positive and negative numbers in expressions and formulae. Use positive and negative numbers in context. For example, find: The final position of an object after moves forwards and backwards along a line; The mean of a set of temperatures above and below zero Recognise and use multiples, factors and primes; use tests of divisibility Use, read and understand: multiple, lowest common multiple (LCM), factor, common factor, highest common factor (HCF), divisor, divisible, divisibility, prime, prime factor, factorise. Know that a prime number has two and only two distinct factors (and hence that 1 is not a prime number). Know the prime numbers up to 30 and test whether two-digit numbers are prime by using simple tests of divisibility, such as: 2 the last digit is 0, 3, 4, 6, or 8; 3 the sum of the digits is divisible by 3; 4 the last two digits are divisible by 4; 5 the last digit is 0 or 5; 6 it is divisible by both 2 and 3; 8 half of it is divisible by 4;

142 9 the sum of the digits is divisible by 9. Find the factors of a number. Find the factors of a number by checking for divisibility by primes. For example, to find the factors of 123, check mentally or otherwise if the number divides by 2, then 3, 5, 7, 11. Find all the pairs of factors of non-prime numbers. Use factors when appropriate to calculate mentally, as in: 35 x 12 = 35 x 2 x6 = 70 x 6 = 420 Find the lowest common multiple (LCM) of two numbers, such as: 6 and 8; 25 and times table: 6, 12, 18, 24, 30 8 times table: 8, 16, 24, 32 The lowest common multiple of 6 and 8 is 24. Find the highest common factor (HCF) of two numbers, such as: 18 and 24; 40 and 65. The factors of 18 are The factors of 34 are , 2, 3 and 6 are common factors of 18 and 24, So 6 is the highest common factor of 18 and 24. Link to cancelling fractions Investigate problems such as: Write a number in each circle so that the number in each square is the product of the two numbers on either side of it Recognise squares and cubes, and the corresponding roots; use index notation and simple instances of the index laws Use, read and understand: property, consecutive, classify;;; square number, squared, square root triangular number. The notation 6 2 as six squared and the square root sign. Use index notation to write squares such as 2 2, 3 2, 4 2, Recognise: Squares of numbers

143 1 to 12 and the corresponding roots; Triangular numbers: 1, 3, 6, 10, 15.. Work out the values of squares 15 2, Square roots Recognise that squaring and finding the square root are inverse of each other. Fractions, decimals. percentages, ratio and proportion Use fraction notation; recognise and use the equivalence between fractions and decimals As outcomes, Year 7 pupils should, for example: Use and read, Numerator, denominator, mixed number, proper fraction, improper fraction decimal fraction, percentage equivalent, cancel, simplify, convert.. lowest terms, simplest form Understand a fraction as part of a whole. Understand a fraction as part of a whole. Use fraction notation to describe a proportion of a shape. For example: Estimate the fraction of each shape that is shaded. Relate fractions to division. Know that 4 8 is another way of writing 4 8, which is the same as ½. Express a smaller number as a fraction of a larger one. For example: What fraction of: 1 metre is 35cm? 1 hour is 33 minutes? What fraction of the big shape is the small one? ( 3 8 ) Know the meaning of numerator and denominator. Simplify fractions by cancellation and recognise equivalent fractions. Understand how equivalent fractions can be shown in diagrammatic form, with shapes sectioned into equal parts. Find equivalent fractions by multiplying or dividing the numerator and denominator by the same number.

144 Know that if the numerator and the denominator have no common factors, the fraction is expressed in its lowest terms. Answer questions such as: Find the unknown numerator or denominator in ¼ = 7 48, 12 = 35 Link to finding the highest common factor. Continue to convert improper fractions to mixed numbers and visa versa: for example, change 34 5 to Answer questions such as: Convert 36/5 to a mixed number. Which fraction is greater, or 29 7? How many fifths are there in 7 1 5? Convert terminating decimals to fractions. Recognise that each terminating decimal is a fraction for example, 0.27 = Convert decimals (up to two decimal places) to fractions. For example: Convert 0.4 to 4 10 and then cancel to 2 5. Convert 0.32 to and then cancel to Convert 3.25 to = 3 ¼. Convert fractions to decimals. Convert a fraction to a decimal by using a known equivalent fraction. For example: 2 8 = ¼ = = 6 10 = = = 0.15 Convert a fraction to a decimal by using a known equivalent decimal. For example: Because 1/5 = 0.2 3/5 = 0.2 x 3 = 0.6 Compare two or more simple fractions. Deduce from a model or diagram that ½ > 1 3 > ¼ > 1 5 > and that, for example, 2 3 < 3 4. Answer questions such as: Insert a > or < symbol between each pair of fractions: ½ Place these fractions in order, smallest first: ¾, 2 3 and 5 6 Calculate fractions of quantities; add, subtract, multiply and divide fractions As outcomes, Year 7 pupils should, for example: Know addition facts for simple fractions, such as ¼ = ¾ = 1 4 and derive other totals from these results, such as:

145 Begin to add and subtract simple fractions by writing them with a common denominator. Calculate fractions of numbers, quantities or measurements. Know that, for example: 1/5 of 35 has the same value as 35 5 = 7; 2/3 of 15 has the same value as 15 3 x 2 = 10; 0.5 of 18 has the same value as ½ of 18 = 9. Use mental methods to answer short questions with whole number answers, such as: Find: one fifth of 40; two thirds of 150g. Find: 0.5 of 50; 1.25 of 40. Calculate: 7 10 of 420; 1 ¼ of 2.4. Know that ¼ of 12, ¼ x 12, 12 x ¼ and 12 4 are all equivalent. Multiply a simple fraction by an integer. For example: 1 5 x 3 = x 4 = 8 5 Simplify the product of a simple fraction and an integer. For example: 1 5 x 15 = x 15= 2 x 1 5 x 15 = 2 x 3 = 6 Answer questions such as: Find: 1 9 x x 90 1 ¼ x 10 Understand percentage as the number of parts per 100; recognise the equivalence of fractions, decimals and percentages; calculate percentages and use them to solve problems As outcomes, Year 7 pupils should, for example: Understand percentage as the number of parts in every hundred, and express a percentage as an equivalent fraction or decimal. For example: Convert percentages to fractions by writing them as the number of parts per 100, then cancelling. For example: 60% is equivalent to = 3 5 ; 150% is equivalent to = 3 2. Convert percentages to decimals by writing them as the number of parts per 100, then using knowledge of place value to write the fraction as a decimal. For example: 135% is equivalent to = 1.35 Recognise the equivalence of fractions, decimals and percentages. Know decimal and percentage equivalents of simple fractions, For example, know that 1 100%. Use this to show that: 1 10 = 0.1 which is equivalent to 10% = 0.01 which is equivalent to 1% 1 8 = which is equivalent to 12.5% 1 ¾ = 1.75 which is equivalent to 175%

146 1 3 = which is equivalent to % Express simple fractions and decimals as equivalent percentages by using equivalent fractions. For example: 3 5 = which is equivalent to 60%; 7 20 = which is equivalent to 35%; = which is equivalent to 275%; 0.48 = which is equivalent to 48%; 0.3 = which is equivalent to 30%. Calculate percentages of numbers, quantities and measurements. Know that 10% of equivalent to 1/10 = 0.1, and 5% is half of 10%. Use mental methods. For example, find: 10% of 20 by dividing by 10; 10% of 37g by dividing by 10; 5% of 5 by finding 10% and then halving; 100% of 5 litres by knowing that 100% represents the whole; 15% of 40 by finding 10% then 5% and adding the results together. Use informal written methods. For example, find: 11% of 2800 by calculating 10% and 1% as jottings, and adding the results together. 70% of 130g by calculating 10% and multiplying this by 7 as jottings; or by calculating 50% and 20% as jottings and adding the results. Use, read and understand: change, total, value amount, sale price, discount, decrease, increase, convert. Begin to recognise percentages as a way to analyse data from everyday contexts. Answer questions such as: 12% of a 125g pot of yoghurt is whole fruit. How many grams are not whole fruit? 48% of the pupils at a school are girls. 25% of the girls and of the boys travel to school by bus. What is the percentage of the whole school travels by bus? Use proportions to interpret pie charts. For example: Some people were asked which fruit they liked best. This chart shows the results.

147 Estimate: a. the percentages of the people that liked oranges best; b. the proportion that liked apples best; c. the percentage that did hot chose pears. Understand the relationship between ratio and proportion, and use ratio and proportion to solve simple problems As outcomes, Year 7 pupils should, for example: Use read and understand ratio, proportion And the notation 3 : 2 Proportion compares part to whole, and is usually expressed as a fraction, decimal or percentage. Solve problems such as: Tina and Fred each have some Smarties in a jar. The table shows how many Smarties they have, and how many of these Smarties are red. Number of Smarties Number of red Smarties Tina Fred Who has the great proportion of red smarties? Use direct proportion in simple contexts. Three bars of chocolate costs 90p. How much will six bars cost? And twelve bars? Understand the idea of a ratio and use ratio notation. Ratio compares part to part. For example:

148 If Lee and Ann divide 100 in the ratio 2 : 3, Lee gets 2 parts and Ann gets 3 parts. 1 part is = 20. So Lee gets 20 x 2 = 40 and Ann gets 20 x 3 = 60. Know that the ratio 3 : 2 is not the same as 2 : 3. Simplify a (two part) ratio to an equivalent ratio by cancelling, e.g. Which of these ratios is equivalent to 3 : 12? A. 3:1 B. 9:36 C: 3:6 D. 1:3 Understand the relationship between ratio and proportion, and relate them both to everyday situations. Divide a quantity into two parts in a given ratio and solve simple problems using informal strategies. 28 pupils are going on a visit. They are in the ratio of 3 girls to 4 boys. How many boys are there? Calculations Number operations and the relationship between them Consolidate understanding of the operations of multiplication and division, their relationship to each other and to addition and subtraction; know how to use the laws of arithmetic As outcomes, Year 7 pupils should, for example: Use, read and understand: operation, commutative, inverse, add, subtract, multiply, divide, sum, total, difference, product, multiply, divide, sum, total, difference, product, multiple, factor, quotient, divisor, remainder.. Understand addition, subtraction, multiplication and division as they apply to whole numbers and decimals. Multiplication Understand that: Multiplication is equivalent to and is more efficient than repeated addition. Because multiplication involves fewer calculations than addition, it is likely to be carried out more accurately. Understand the effect of multiplying by 0 or 1. Division Recognise that: can be interpreted as How many 13s in 910?, and calculate by repeatedly subtracting 13 from 910, or convenient multiples of 13. Division by zero is not allowed.

149 A quotient (the result obtained after division) can be expressed as a remainder, a fraction or as a decimal, e.g = 6 R12 Or = Or = 6.92 (2 d.p.) Know how to use the laws of arithmetic to support efficient and accurate mental and written calculations. (commutative law, associative law, and the distributive law) Inverses Understand that addition is the inverse of subtraction and multiplication is the inverse of division. For example: Fill in the missing number: ( x 4) 8 = 5. Know and use the order of operations, including brackets As outcomes, Year 7 pupils should, for example: Use, read an understand: order of operations, brackets Know the conventions that apply when evaluating expressions: Contents of brackets are evaluated first. In the absence of brackets, multiplication and division take precedence over subtraction and addition. A horizontal line acts as a bracket in expressions such as: or a + b 2 5 Brackets Powers or indices Multiplication (including of ) and division Addition and subtraction With strings of multiplication and divisions, or strings of addition and subtractions, and no brackets, the convention is to work from left to right. Calculate with mixed operations. For example: Find mentally: a = 12 b = 18 c. 14 x x 11 = 186 d. 100 = 5 4 x 5 e x 5 = 97 f. ( ) 2 =625 g. (5 2 7)/ (2 2-1) = 6 In algebra recognise that, for example, when a = 4,

150 3a 2 = 3 x 4 2 = 3 x 16 = 48 Mental methods and rapid recall of number facts Consolidate the rapid recall of number facts and use known facts to derive unknown facts As outcomes, Year 7 pupils should, for example: Use read and understand: increase, decrease, double, halve, complement, partition Addition and subtraction facts Know with rapid recall addition and subtraction facts to 20. Complements Derive quickly: Whole number complements in 100 and 50, e.g. 100 = = Decimal complements in 1 (one or two decimal places) e.g. 1 = , 1 = Doubles and halves Derive quickly: Doubles in two digit numbers including decimals, Doubles of multiples of 10 to 1000, Doubles of multiples of 100 to , and all corresponding halves. Multiplication and division facts Know with rapid recall multiplication facts up to 10 x 10, and squares to at least 12 x 12. Derive quickly the associated division facts, e.g. 56 7, 81. Use knowledge of place value to multiply and divide mentally any number by 10, 100, 1000 or by a small multiple of ten. Use knowledge of multiplication facts and place value to multiply mentally examples such as: 0.2 x 8 8 x 0.5 x 0.2 = 10 Factors, powers and roots Know or derive quickly: Prime numbers less than 30; Squares of numbers 0.1 to 0.9, and of multiples of 10 to 100, and corresponding roots; Pairs of factors of numbers to 100. Calculate mentally: (4 + 3) (9 +7) ( ) What is the fourth square number? Solve mentally: 3a = 15 x 2 = 49 n(n +1) = 12 Measurements

151 Recall and use formulae for: the perimeter and area of a rectangle. Calculate simple examples mentally. Recall: Relationships between units of time; Relationships between metric units of length, mass and capacity (eg. Between km, m, cm and mm). Convert between units of measurement. For example: Convert 38cm into mm. Convert 348p into pounds. Convert 45 minutes into seconds. Consolidate and extend mental methods of calculation, accompanied where appropriate by suitable jottings As outcomes, Year 7 pupils should, for example: Strategies for mental addition and subtraction Count forwards and backwards from any number. For example: Count on in 0.1s from 4.5. Count back from 4.05 in 0.01s. Count on from and back to zero in steps of ¾. Identify positions of 0.1s and 0.01s on a number line. Add and subtract several small numbers. For example: Extend to adding and subtracting several small multiples of 10: Continue to add and subtract any pair of two-digit whole numbers, such as , Extend to: Adding and subtracting a two digit whole number to or from a three-digit whole number; Adding and subtracting decimals such as: 8.6 ± ± ± 1.5 By considering 86 ± ± ± 150 Partition and deal with the most significant digits first. For example: = ( ) + 5 = = = = =8.8 Find a difference by counting up from the smaller to the larger number. For example: = = 3138

152 Use compensation, by adding or subtracting too much, and then compensating. For example: = = = = = = 254 Recognise special cases. For example: Near doubles = 16.7 (double 8.2 plus 0.3) = 793 (double 400 plus 27 minus 34) Nearly numbers Add and subtract near 10s and near 100s by adding or subtracting a multiple of 10 or 100 and adjusting. For example: Use the relationship between addition and subtraction. For example, recognise that knowing one of: = = = = 5.8 means that you also know the other three. Strategies for multiplication and division Use factors. For example 3.2 x x 10 = x 3 = 96 Use partitioning. For example: For multiplication, partition either part of the product: 7.3 x 11 = (7.3 x 10) = = 80.3 For division, partition the dividend (the number that is to be divided by another): = 30 R = 2 R = 32 R 14 = 33 R 1 Recognise special cases where doubling or halving can be used. For example: To multiply by 50, first multiply by 100 and then divide by 2. Double one number and halve the other. For example: 6 x x 9 = 27 Use the relationship between multiplication and division. For example, knowing one of these facts means you also know the other three. Recall of fraction, decimal and percentage facts Know and derive quickly: Simple decimal/fraction/percentage equivalents, such as:

153 ¼ = 0.25 or 25% 0.23 is equivalent to 23% Simple addition facts for the fractions, such as: ¼ + ¼ = ½ ¼ + ½ = ¾ Some simple equivalent fractions for ¼ and ½, such as: ½ = 2 4 = 3 6 = 4 8 = 5 10 = ¼ = 2 8 = 3 12 = 4 16 = 5 20 = Strategies for finding equivalent fractions, decimals and percentages For example: Convert 1 8 into a decimal. (Know that ¼ = 0.25 so 1 8 is = ) Express 3 5 as a percentage. (Know that 3 5 = 6 10 or , so it is equivalent to 60%.) Express 23% as a fraction and as a decimal. (Know that 23% is equivalent to or 0.23.) Express 70% as a fraction in its lowest terms. (Know that 70% is equivalent to , and cancel this to 7 10.) Use known facts such as 1/5 = 0.2 to convert fractions to decimals mentally. For example: 3 5 = 0.2 x 3 = 0.6 Find simple equivalent fractions. For example: State three fractions equivalent to 3 5, such as: 6 10, 30 50, Fill in the boxes: ¾ = 8 = 12 = 16 = 20 7 = Strategies for calculating fractions and percentages of whole numbers and quantities. For example: 1/8 of 20 = 2.5 (e.g. find one quarter, halve it) 75% of 24 = 18 (e.g. find 50% then 25% and add the results) 15% of 40 (e.g. find 10% then %% and add the results) 40% of 400kg (e.g. find 10% them multiply by 4) 60 pupils go to the gym club. 25% of them are girls. How many are boys. Word problems and puzzles (all four operations) Apply mental skills to solving simple problems, using jottings if appropriate. For example: Pencils cost 37p each. How many can you buy with 3.70? A 55g bag of crisps has 20% less fat. How much fat is that? A boy saved 215. He bought a Walkman for 69. How much money did he have left? A girl used 2 metres of wood to make 5 identical shelves. How long was each shelf?

154 Sandy and Michael dug a neighbour s garden. They were paid 32 to share for their hours of work. Sandy worked for 6 hours. Michael worked for 2 hours. How much should Sandy get paid? The mean of a, b, and c is 6. a is 5 and b is 11. What is c? Tony, David and Estelle are playing a team game. They need to get a mean of 75 points to win. Tony scores 63 points, Estelle scores 77 points and David scores 77 points. Have they scored enough points to win? What is the value of 6n + 3 when n = 2.5? Solve problems or puzzles such as: Choose from 1, 2, 3, 4, and 5 to place in the boxes. In any question, you cannot use a number more than once. a. - + = 5 b. + - = 4 c. x - = 3 Written methods Use efficient column methods for addition and subtraction of whole numbers, and extend to decimals As outcomes, Year 7 pupils should, for example: Continue to use and refine efficient methods for column addition and subtraction, while maintaining accuracy and understanding. Extend to decimals with up to two decimal places, including: sums and differences with different numbers of digits; totals of more than two numbers. For example: Algebra Equation, formulae and identities Use the letter symbols and distinguish their different roles in algebra As outcomes, Year 7 pupils should, for example: Use, read and understand algebra, unknown, symbol, variable.. equals brackets evaluate, simplify, substitute, solve term, expression, equation squared commutative Reinforce the idea of an unknown. Answer questions such as:

155 5 + = 17 3 x - 5 = 7 Know that letters are used to stand for numbers in algebra. Begin to distinguish between different uses of letters. For example: In the equation 3n + 2 = 11, n is a particular unknown number, but in the equation a + b =12, a and b can take many different values. Recognise algebraic conventions, such as: 3 x n or n x 3 can be thought of as 3 lots of n or n + n + n, and can be shortened to 3n. In the expression 3n, n can take any value, but when the value of an expression is known, an equation is formed, i.e. If 3n is 18, then the equation is written as 3n = 18. Understand the meaning of and begin to use simple expressions with brackets, e.g. 3(n + 2) meaning 3 x (n + 2), where the addition operation is to be performed first and the result of this is then multiplied by 3. Use the equals sign appropriately and correctly. Recognise that if a = b then b = a, and that a + b = c can also be written as c = a + b. Avoid errors arising on from misuse of the sign when setting out the steps in a calculation, e.g. in correctly writing = = 68 1 = 67 Use letter symbols to write expressions in meaningful contexts. For example: Add 7 to a number n + 7 Subtract 4 from a number n 4 4 minus a number 4 n A number multiplied by (n x 2) + 5 or 2n and then 5 added A number divided by 2 n 2 or n/2 A number plus 7 and (n + 7) x 10 or 10(n + 7) then multiplied by 10 a number multiplied by itself n x n or n 2 Understand the difference between expressions such as: 2n and n + 2 3(c + 5) and 3c + 5 n 2 and 2n 2n 2 and (2n) 2 Know that algebraic operations follow the same conventions and order as arithmetic operations; use index notation and the index laws As outcomes, Year 7 pupils should, for example: Know that algebraic operations follow the same conventions and order as arithmetic operations. Begin to generalise from arithmetic that multiplication and division have precedence over addition and subtraction. For example:

156 In the expression 2 + 5a, the multiplication is to be performed first. Know that the commutative and associative laws apply to algebraic expressions as the do to arithmetic expressions, so: = a + b = b + a 2 x 3 = 3 x 2 a x b = b x a 2 x (3 x 4) = (2 x 3) x 4 a + (b + c) = (a + b) + c 2 x (3 x 4) = (2 x 3) x 4 a x (b x c) = (a x b) x c Or a(bc) = (ab)c Inverses Understand addition and subtraction as the inverse of each other, and multiplication and division as the inverse of each other. Generalise from arithmetic that: a + b = 5 implies b + a = 5, using the commutative law, and the corresponding inverse relationships 5 b = a and 5 a = b. Similarly, a x b = 24 implies that b x a = 24, b = 24/a and a = 24/b. Verify by substituting suitable sets of numbers. Begin to apply inverse operations when two successive operations are involved. For example: The inverse of multiplying by 6 and add on 4. The answer is 34. What was the original number? Alternatively: I think of a number, multiply by 6 and add on 4. The answer is 34. What was the original number? Simplify or transform algebraic expressions As outcomes, Year 7 pupils should, for example: Simplify linear expressions by collecting like terms; begin to multiply a single term over a bracket. Understand that partitioning a number helps to break a multiplication into a series of steps. For example: By partitioning 38, 38 x 7 becomes (30 + 8) x 7 = 30 x x 7 Generalise, from this and similar examples, to: (a + b) x c = (a x c) + (b x c) or ac +bc Recognise that letters stand for numbers in problems. For example: Simplify expressions such as: a. a + a + a = 3a b. b + 2b + b = 4b c. x x = 3x +6 d. 3n + 2n = 5n e. 3(n + 2) = 3n + 6

157 And a/a = 1, 2a/a = 2,. And 4a/2 = 2a, 6a/2 = 3a, etc. The number in each cell is the result of adding the numbers in the two cells beneath it. Write an expression for the number in the top cell. Write you expression as simply as possible. Construct and solve linear equations, selecting an appropriate method As outcomes, Year 7 pupils should, for example: Use, read and understand: equation, solution, unknown, solve, verify, prove, therefore ( ). Construct and solve simple linear equations with integer coefficients, the unknown on one side only. Choose a suitable unknown and form expressions leading to an equation. Solve the equation by using inverse operations or other mental or written methods. For example: I think of a number, subtract 7 and the answer is 16. What is my number? Let n be the number. n 7 = 16 n = = 23 In this diagram, the number in each cell is formed by adding the two numbers above it. What is n? 31 n 45 n + 31 n I think of a number, multiply it by 6 and add 1. The answer is 37. What is my number? Solve the equations: a. a + 5 = 12 b. 3m = 18 c. 7h 3 =20 d. 7 = 5 + 2z Explore ways of constructing simple equations to express relationships, and begin to recognise equivalent statements. For example: Recognise that statements such as B C = 3 and B = C + 3 express the same relationship in different ways. Use formulae from mathematics and other subjects As outcomes, Year 7 pupils should, for example: Substitute positive integers into simple linear expressions.

158 For example: Substitute positive integer values into: x + y z 3(x + y) 20/x 9y x 2(8 x) x/2 6 The expression 3s + 1 gives the number of matches needed to make a row of s squares. How many matches are needed to make a row of 13 squares? Explain the meaning of and substitute integers into formulae expressed in words, or partly in words, such as: Number of days = 7 times the number of weeks Cost = price of one item x number of items Pence = number of pounds x 100 Cost of petrol for a journey = cost per litre x number of litres used. Progress to substituting into formulae such as: Conversion of centimetres c to metres m: m = c 100 The perimeter P of a rectangle length l and width w: A = lw Derive simple algebraic expressions and formulae. Check for correctness by substituting particular values. For example: You have p pencils. a.rashida has twice as many pencils as you have. How many pencils does Rashida have? b. You give away 2 pencils. How many pencils do you have left? c. Rashida shares her pencils equally between herself and 4 other friends. How many pencils do they each get? Derive formulae such as: The number of 1-metre square concrete slabs that will surround a rectangular ornamental pond that is 1 metre wide and m metres long: s = 2m + 6 Sequences and functions Describe sequences As outcomes, Year7 pupils should, for example: Use read and understand: Sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite. Know that:

159 o A number sequence is a set of numbers in a given order. o Each number in a sequence is called a term. o Terms next to each other are called consecutive and are often separated by commas, e.g. 6, 8, 10 and 12 are consecutive terms in a sequence of even numbers. Know that a sequence can have a finite of an infinite number of terms. Describe simple integer sequences and relate them to geometrical patterns. For example: Odds and evens, multiples of 3, square numbers, triangular numbers. Find the nth term, justifying its form by referring to the context in which it was generated As outcomes, Year 7 pupils should, for example: Generate sequences from simple practical contexts. For example: Find the first few terms of the sequence. Describe how it continues by reference to the context. Begin to describe the general term, first using words, then symbols; justify the generalisation by referring to the context. For example: Growing matchstick squares Numbers of squares Numbers of matchsticks Justify the pattern by explaining that the first square needs 4 matches, then 3 matches for each additional square, or you need 3 matches for every square plus an extra one for the first square. In the nth arrangement there are 3n + 1 matches. Begin to find a simple rule tor the nth term of some simple sequences. For example: 6, 12, 18, 24, 30 nth term is six times n. 9, 19, 29, 39, 49 nth term is 10 times n minus 1. Express functions and represent mappings As outcomes, Year 7 pupils should, for example: Use, read and understand: input, output, rule, function, function machine, mapping Express simple functions at first in words then using symbols. For example:

160 Explore simple function machines by: finding outputs (y) for different inputs (x); finding inputs for different outputs. For example: x multiply add 8 y by 2 What input gives an output of 40? Describe the effect of this fraction machine as x x = y. Given inputs and outputs, find the function. For example: Find the rule (single machine) 2, 1, 7, 4? 8, 4, 28, 16 Explore inverse operations to find the input given the output. Given the output, find the input for a particular machine:? subtract 6 multiply by 6 9, 3, 15, 6 Begin to recognise some properties of simple functions. A function can be sometimes expressed in more than one way. A function can sometimes be expressed more simply. A function can often be inverted. Graphs and functions Generate points and plot graphs of functions As outcomes, Year 7 pupils should, for example: Use, read and understand: coordinates, coordinate pair/point, x-coordinate grid, origin, axis, axes, x-axis variable, straight line graph, equation (of a graph) Generate and plot pairs of coordinates that satisfy a simple linear relationship. For example: y = x + 1 (0, 1), (1, 2), (2, 3), (3, 4). Complete the table of values that satisfy the rule y = x + 2: x y = x Plot the points on a coordinate grid. Begin to consider the features of graphs of simple linear functions, where y is given explicitly in terms of x.

161 Recognise and interpret graphs such as: y = x, y = 2x, y = 3x, y = 4x, y = 5x Note that graphs that are of the form y = mx: - are all straight lines which pass through the origin; - vary in steepness, depending on the function; - match the graphs of multiples, but are continuous lines rather than discrete points. Identify graphs such as: y = x + 1, y = 10 x Note the positive or negative slope of the graph and the intercept points with the axes. Make connections with the value of the constant term. Interpret linear functions arising from real-life problems and plot and interpret their corresponding graphs As outcomes, Year 7 pupils should, for example: Begin to interpret graphs of linear functions. In interpreting graphs of functions: Read values from a graph; Say whether intermediate points have any practical significance; Say how the variables are related, e.g. they increase together. For example: Some pupils put a lighted candle under jars of different sizes. The jars varied from 200cm 3 to 500cm 3 in volume. They timed how long the candle took to go out. Answer:

162 About how long would it take for a candle in a 450cm 3 jar to go out? a. 15 seconds b. 22 seconds c. 27 seconds d. 25 seconds Which of the four sentences below best describes the relationship between the volume of the jar and the time it takes for the candle to go out? A. The greater the volume, the shorter the time for the candle to go out. B. The biggest jar kept the candle going longest. C. As the volume of the jar increases, so the time the candle burns gets longer. D. The candle went out most quickly under the smallest jar. Shape, Space and measures Geometric reasoning: lines, angles and shapes Use accurately the vocabulary, notation and labelling conventions for lines, angles and shapes; distinguish between conventions, facts, definitions and derived properties As outcomes, Year 7 pupils should, for example: Use, read and understand: line segment, line parallel, perpendicular. plane horizontal, vertical, diagonal adjacent, opposite point, intersect, intersection. vertex, vertices side angle, degree ( o ) acute, obtuse, reflex vertically opposite angles base angles Use accurately the notation and labelling conventions for lines, angles and shapes. Understand that a straight line can be considered to have infinite length and no measurable width, and that a line segment AB has end-points A and B. Know that: Two straight lines in a plane (a flat surface) can cross once or are parallel; if they cross, they are said to intersect, and the point at which they cross is an intersection. When two line segments meet at a point, the angle formed is the measure of rotation of one of the line segments to the other. The angle can be described as DEF or DÊF or E.

163 A polygon is a 2D or a plane shape constructed from line segments enclosing a region. The line segments are called sides; the ends are called vertices. The polygon is named according to the number of its sides, vertices or angles: triangle, quadrilateral, pentagon Know the labelling convention for: Triangles- capital letters for the vertices (going round in order, clockwise or anti-clockwise) and corresponding lower-case letters for each opposite side, the triangle them being described as ABC; Equal sides and parallel sides in diagrams. AB = AC AB is parallel to DC, AB//DC AD is parallel to BC, AD//BC. Identify properties of angles and parallel and perpendicular lines, and use these properties to solve problems As outcomes, Year 7 pupils should, for example: Identify parallel and perpendicular lines. Recognise parallel and perpendicular lines in the environment, and in 2D and 3D shapes: for example, rail tracks, side edges of doors, ruled lines on a page, double yellow lines Know the sum of angles at a point, on a straight line and in a triangle, and recognise vertically opposite angles and angles on a straight line.

164 Give sufficient information, calculate: -angles in a straight line and at a point; -the third angle of a triangle; -the base angles of an isosceles triangle. Identify and use the geometric properties of triangles, quadrilaterals and other polygons to solve problems; explain and justify inferences and deductions using mathematical reasoning As outcomes, Year 7 pupils should, for example: Use, read and understand: polygon, regular, irregular, convex, concave circle, triangle, isosceles, equilateral, scalene, right-angled, quadrilateral, square, rectangle, parallelogram, rhombus, trapezium, kite, delta. and names of other polygons. Use 2-D representations, including plans and elevations, to visualise 3-D shapes and deduce some of their properties As outcomes, Year 7 pupils should, for example: Use read and understand: 2-D, 3-D, cube, cuboid, pyramid, tetrahedron, prism, cylinder, sphere, hemisphere face, vertex, vertices, edge net.. Use 2-D representations and oral descriptions to visualise 3-D shapes and deduce some of their properties. For example: -Imagine you have two identical cubes. Place them together, matching face to face. Name and describe the new solid. How many faces, edges, vertices.? -Imagine cutting off a corner of a cube. Describe the new face. How many faces, edges and vertices does the new solid have?

165 -On a six-sided dice, the faces are numbered from 1 to 6, and the opposite faces should add up to 7. Here is a net for a cube. Choose a face and write 5 on it. Now write numbers on the other faces so that when the cube is folded up, opposite faces add up to 7. A piece of card is viewed from different angles against the light. Which of the following are possible views of the square card? Which are impossible? Transformations Understand and use language and notation associated with reflections, translations and rotations Recognise and visualise transformations and symmetries of shapes Use read and understand; transformation image, object, congruent reflection, mirror line, line of symmetry, line of symmetry, reflection symmetry, symmetrical.. translation rotate, rotation, rotation symmetry, order of rotation symmetry, centre of rotation. Reflection Understand reflection in two dimensions as a transformation of a plane in which points are mapped to images in a mirror line or axes of reflection, such that: The mirror line is the perpendicular bisector of the line joining point A to the image A; The image is the same distance behind the mirror as the original is in front of it. Know that a reflection has these properties: Points on the mirror line do not change their position after reflection, i.e. reflection which maps A to A also maps A to A, i.e. reflection is a self-inverse transformation. Reflect a shape in a line along one side. For example:

166 Reflect each shape in the dotted line. What is the name of the resulting quadrilateral? Which angles and which sides are equal? Construct the reflections of shapes in mirror lines placed at different angles relative to the shape. For example: F A N S F A N S F A N S Which shape appears not to have changed after a reflection? In this diagram below describe why the rectangle R is not the reflection of R in the line L. R R Reflection symmetry Know that if a line can be found such that one half of a shape reflects to the other, then the shape has the reflection symmetry, and the line is called a line of symmetry. Recognise: reflection symmetry in familiar shapes such as isosceles triangle, a rectangle, a rhombus, a regular hexagon lines of symmetry in 2-D shapes; shapes with no lines of symmetry. For example: Identify the lines of symmetry in this pattern. L

167 Rotation Understand rotation in two dimensions as a transformation of a plane in which points (such as A) are mapped to images (A ) by turning about a fixed point in the plane, called the centre of rotation. A A A A centre of rotation A rotation is specified by a centre of rotation and an (anticlockwise) angle of rotation. Know that a rotation has these properties: The centre of rotation can be inside or outside the shape and its position remains fixed throughout the rotation. The inverse of any rotation is either: a. an equal rotation about the same point in the opposite direction, or b. a rotation about the same point in the same direction, same that the two rotations have a sum of 360 o. Rotate shapes anticlockwise about (0, 0) through right angles and simple fractions of a turn. Rotate shapes about points other than (0, 0)

168 Know that: A 2-D shape has rotation symmetry of order n when n is the largest positive integer for which a rotation of 360 o n produces an identical looking shape in the same position. The order of rotation symmetry is the number of ways the shape will map on to itself in a rotation of 360 o. For example: This shape has rotation symmetry of order 8 because it maps onto itself eight distinct positions under rotations of 45 o about the centre point. Solve problems such as: Identify the centre of rotation in a shape with rotation symmetry, such as an equilateral triangle. Translations Understand translation as a transformation of a plane in which points (such as A and B) are mapped on to images (A and B ) by moving a specified direction. Know that when describing a translation, it is essential to state either the direction and distance or, with reference to a coordinate grid, the moves parallel to the x-axis and parallel to the y-axis. Know that a translation has these properties: The orientations of the original and the image are the same. The inverse of any translation is an equal move in the opposite direction. Translate shapes on a coordinate grid, e.g. 4 units to the right, 2 units down, then 3 units to the left. Determine which two instructions are equivalent to the three used.

169 Coordinates Use coordinates in all four quadrants As outcomes, Year 7 pupils should, for example: Use read and understand.. row, column, coordinates, origin, x-axis, y-axis position, direction intersecting, intersection Read and plot points using coordinates in all four quadrants. Given a set of points discover the shape they form. Plot points determined by geometric information. The points (-3, 1) and (2, 1) are two points of the four vertices of a rectangle. Suggest coordinates of the other two vertices. Find the perimeter and area of the rectangle.. Plot these three points: (1, 3), (-2, 2), (-1, 4). What fourth point will make: a. a kite b. a parallelogram c. an arrowhead? Measures and mensuration Use units of measurement to measure estimate, calculate and solve problems in a range of contexts; convert between metric units. As outcomes, Year 7 pupils should, for example: Use and read names and abbreviations of: Standard metric units Millimetre (mm), centimetre (cm), metre (m), kilometre (km) Gram (g), kilogram (kg) Millilitre (ml), centilitre (cl), litre (l) Square millimetre (mm 2 ), square centimetre (cm 2 ), square metre (m 2 ), and square kilometre (km 2 ).. Units of temperature, time and angle Degree Celsius ( o C) Seconds (s), minute (min), hour (h), day, week, month, year, Degree ( o ) Know relationships between units of a particular measure, e.g. 1kg = 1000g Convert one metric unit to another. Know the relationship between metric units in common use and how they are derived from the decimal system. For example: km - - m - cm mm

170 km = 8000m 4m = 4000mm 37cm = 0.37m 230mm=0.23m Understand that for the same measurement in two different units; If the unit is smaller, the number of units will be greater; If the unit is bigger, the number of units will be smaller. Change a larger unit to a smaller one. For example: Change 36 centilitres into millilitres. Change 0.89km into metres. Change 0.56 litres into millilitres. Change a smaller unit to a larger one. For example: Change 750g into kilograms. Change 237 ml into litres. Change 3 cm into metres. Change 4 mm into centimetres. Read and interpret scales on a range of measuring instruments, with appropriate accuracy, including: Vertical scales, e.g. thermometer, tape measure, ruler, measuring cylinder Scales around a circle or semicircle, e.g. for measuring time, mass, angle. Solve problems involving length, area, capacity, mass time and angle, rounding measurements to an appropriate degree of accuracy. Extend the range of measures used to angle measure and bearings, and compound measures Angle measure Use and read: angle, degree (o), right angle, acute angle, obtuse angle, reflex angle Use angle measure; distinguish between and estimate the size of acute, obtuse and reflex angles. Know that: An angle less than 90 o is an acute angle. An angle between 90 o and 180 o is an obtuse angle. An angle greater than 360 o involves at least one complete turn. Estimate acute, obtuse and reflex angles. For example: Decide whether these angles are acute, obtuse or reflex, estimate their size, then measure each of them to the nearest degree.

171 Deduce and use formulae to calculate lengths, perimeters, area and volumes in 2-D and 3-D shapes As outcomes, Year 7 pupils should for example: Use, read and understand: area, surface area, perimeter, distance, edge and use the units: square centimetre (cm 2 ), square metre (m 2 ), square millimetre (mm 2 ) Deduce and use formulae for the perimeter and area of a rectangle. Derive and use a formulae for the area of a right-angled triangle, thinking of it as half a rectangle: area = ½ x base length x height area = ½ bh Calculate the perimeter and area of shapes made from rectangles. For example: Find the area of this shape. Here is a flag. Calculate the area of the shaded cross. Find the area of cuboids and shapes made from cuboids. Check by measurement and calculation. Derive and use the a formula for the surface area S of a cuboid with length l, width w and height h: S = 2(length x width) + 2(length x height) + 2(height x width) S = 2lw + 2lh +2hw

172 Solve simple problems such as: How many unit cubes are there in this shape? What is the surface area? Investigate the different cuboids you can make with 24 cubes. Do they all have the same surface area? Calculate the surface area in cm 2 of this girder. Handling data Processing and representing data Calculate statistics from data, finding the mode, mean, median and range As outcomes, Year 7 pupils should, for example: Use, read and understand: statistic, interval range, mean, median, mode, modal class/group, average Know that:

173 The mode is the only statistic appropriate for data based on nonnumeric categories, e.g. the most common way of travelling to school. The mean is often referred to as the average. Find the mode of a small set of discrete data. Know that the mode of a set of numbers is the number that occurs most often in the set. For example: For 1, 2, 3, 3, 4, 6, 9 the mode is 3. For 3, 4, 4, 4, 7, 7, 8 the mode is 4. For 2, 2, 3, 5, 6, 9, 9 there are two modes, 2 and 9. In a grouped frequency distribution, the group that contains the most members is called the modal class or modal group. Calculate the mean for a small set of discrete data. The mean of a set of numbers is the sum of all the numbers divided by the number in the set. For example: The mean of 2, 6, 8, 9 and 12 is: = 37 = Find and use the range of a small set of discrete data The range of a set of values is the difference between the largest and the smallest numbers in the set. For example, for 2, 3, 4, 7, 9, 10, 12, 15, the range is 15 2 = 13. Find the median of a small set of discrete data. The median of a set of numbers is the value of the middle number when they are arranged in ascending order. For example, 2, 5, 8, 3, 1, 7, 6 becomes 1, 2, 3, 5, 6, 7, 8 and the median is 5. If there is no single middle number, the mean of the two middle numbers is taken. For example, the set 1, 5, 7, 8, 9, 10 has a median of (7 + 8)/2 = 7.5. Calculate statistics. For example: A competition has three different games. Jane has played two of the games. Game A Game B Game C Score To win, Jane needs a mean score of 60. How many points does she need to score in game C? Interpreting and discussing results Interpret diagrams and graphs, and draw inferences Interpret diagrams, graphs and charts, and draw inferences based on the shape of the graphs and simple statistics for a single distribution. Relate these to the initial problem. For example: Interpret data in a pie chart from a newspaper, or generated by a computer. For example:

174 a. Which species of trees grow best in the local wood? 14% 24 % How many of each species of tree would there be in the wood if it had 600 trees? Interpret data in a simple compound bar chart. Interpret a bar chart (discrete data). For example: This chart shows the lengths of 100 words in two different newspaper passages. Compare the two distributions. 22% 30% 10% Observe that the differences are not great, but these may be slightly greater word length and variety of word length in the broadsheet. Express these observations by selecting the statement that best represents the inferences made from this graph. A school has five year groups. Eighty pupils took part in a sponsored swim. Lara drew this graph.

175 Look at the graph. Did Year 10 have fewer pupils taking part than Year 7? Tick the correct box. Yes No Cannot tell Compare two simple distributions using the range, mode, mean or median Compare the distributions of two sets of data, and the relationships between them, using range and one of the mode, mean or median. For example: Which newspaper is easiest to read? In a newspaper survey of the numbers of letters in 100-word samples, compare the mean and the range. Newspaper type Mean Range Tabloid Broadsheet Probability Use the vocabulary of probability Use, read and understand: fair, unfair, unlikely, equally likely, certain, uncertain, probable, possible, impossible, chance, good chance, poor chance, no chance, fifty-fifty chance, good chance, even chance, likelihood, probability, risk, doubt, random, outcome. Use vocabulary and ideas of probability, drawing on experience. For example: A class is going to play three games. In each game some cards are put into a bag. Each card has a square or a circle on it. One card will be taken out, then put back. If it is a circle, the girls will get a point. If it is a square, the boys will get a point.

176 a. Which game are the girls most likely to win? b. Which game are the boys least likely to win? c. Which game are the boys certain to win? d. Which game is it equally likely that the boys or girls win? e. Are any of the games unfair? Use the probability scale; find and justify theoretical probabilities Understand and use the probability scale from 0 to 1; find and justify probabilities based on equally likely outcomes in simple contexts. Recognise that, for a finite number of possible outcomes, probability is a way of measuring the chance or likelihood of a particular outcome on a scale from 0 to 1, with the lowest probability outcome on a scale from 0 to 1, with the lowest probability at zero (impossible) and the highest probability at 1 (certain). For example: What fraction would you use to describe: a. the chance of picking a red card at random from a pack of 52 cards? b. The chance of picking a club card? Position the fractions on this probability scale. Know that the probability is related to proportion and can be represented as a fraction, decimal or percentage, e.g. What is meant by weather forecast of a 20% chance of rain? Select a statement. Know that if several equally likely outcomes are possible, the probability of a particular outcome chosen at random can be measured by: number of events favourable to the outcome total number of possible events For example: The letters in the word RABBIT are placed in a tub, and a letter taken at random. What is the probability of taking out: a. a letter T? (one in six, or 1 /6) b. a letter B? ( 2 /6 or 1 /3) Identify all the possible outcomes of a single event. For example: What are the possible outcomes a. when a fair coin is tossed? There are two outcomes: heads or tails. The probability of each is ½. b. when a letter from the word HIPPOPOTAMUS is picked at random? There are nine outcomes: H, I, P, O, T, A, M, U, S

177 The probability of O is 2/ 12 or 1 / 6 The probability of P is 3/ 12 or ¼.

178 YEAR 8 Using and applying mathematics to solve problems Applying mathematics and solving problems Pupils should be taught to: Solve word problems and investigate in a range of contexts As outcomes, Year 8 pupils should, for example: Solve more demanding problems and investigate in a ranged of contexts; compare and evaluate solutions. Problems involving money For example: o At Alan s sports shop. GoFast trainers usually cost 40.95, but there is 1/3 off in the sale. At Irene s sports shop, GoFast trainers normally cost 40, but there is a discount of 30% in the sale. Which shop sells the trainers for less in the sales? o A supermarket sells biscuits in these packets: 15 biscuits for 56p 24 biscuits for 88p 36 biscuits for 1.33 Which packet is the best value for money? Why do you think this is the case? o Tina and Jill each gave some money to a charity. Jill gave twice as much as Tina, and added 4 more. Between them, the two girls gave 25 to the charity. How much did Tina give? Problems involving percentages For example: o Chloe and Denise each bought identical T-shirts from the same shop. Chloe bought hers on Monday when there was 15% off the original price. Denise bought hers on Friday when there was 20% off the original price. Chloe paid 35p more than Denise. What was the original price of the T-shirt? Problems involving ratio and proportion: For example: o To make orange paint, you mix 13 litres of yellow paint to 6 litres of red paint to 1 litre of white paint. How many litres of each colour do you need to make 10 litres of orange paint? o Flaky pastry is made by using flour, margarine and lard in the ratio 8 : 3 : 2 by weight. How many grams of margarine and lard are needed to mix with 200 grams of flour? o A square and a rectangle have the same area. The sides of the rectangle are in the ratio 9:1 its perimeter is 200cm. What is the length of a side of the square?

179 o A 365g packet of coffee costs How much per 100g of coffee is this? o The cost of a take-away pizza is 6. A pie chart is drawn to show how the total cost is made up. The cost of labour is represented by a sector of 252 o on the pie chart. What is the cost of labour? Problems involving number and algebra For example: o What fraction is half way between ¾ and 5 /7? o Fine the missing digits. The need not be the same digit in each case. a. + = 3_ 8 b. ( ) 3 = 7 o Find four consecutive numbers with a total of 80. Find three consecutive numbers with a product of Find two prime numbers with a product of o Each inside edge of two cube-shaped tins is 2s cm and s cm respectively. The larger tin is full of water; the smaller tin is empty. How much water is left in the larger tin when the smaller tin has been filled from it? o A boy gets 2 marks for each sum that he gets right and -1 mark for each that he gets wrong. He did 18 sums and got 15 marks. How many sums did he get right? More problems involving number and algebra For example: o The next number in the sequence is the sum of the two previous numbers. Fill in the missing numbers o Here is a sequence: The sequence continues in the same way. a. Write the next three terms of the sequence. b. Explain the rule for finding the next term. c. What is the 30 th term of the sequence? d. What is the nth term in the sequence? o Elaine played a number game. She said: Multiplying my number by 4 then subtracting 5 gives the same answer as multiplying my number by 2 then adding 1. Work out the value of Elaine s number. o A line with the equation y = mx + 9 passes through the point (10, 10). What is the value of m? Problems involving shape and space For example: o The diagram shows a model with nine cubes, five blue and four white.

180 o The diagram shows the positions of three points A, B, and C. The distances AB and AC are equal. Calculate the sizes of the angles marked x, y, and z. Problems involving perimeter and area For example: o What is the smallest perimeter for a shape made of 8 regular hexagons each of side a? 9 regular hexagons? 10 regular hexagons? o The length of a rectangle is 4 cm more than its width. Its area is 96 cm 2. What is its perimeter? o Boxes measure 2.5cm by 4.5cm by 6.2cm. Work out the largest number of boxes that can lie flat in a 9 cm by 31 cm tray. Problems involving measures For example:

181 o A school expects between 240 and 280 parents for a concert. Chairs are to be put out in the hall. Each chair is 45cm wide, and no more than 8 chairs must be put in a row together. There must be a corridor 1 m wide down the middle of the hall and 0.5 m of space between the rows for people to get to their seats. What is the minimum space needed to set out the chairs? o The Wiro Company makes wire coat-hangers each day. Each coat hanger uses 87.4 cm of wire. How many km of wire are used each day? What is the greatest number of coat-hangers that can be made from 100m wire? Problems involving probability For example: o Here is a spinner with five equal sections. Jane and Sam play many times. If it stops on an odd number, Jane gets 2 points. If it stops on an even number, Sam gets 3 points. Is this a fair game? Explain your answer. o The names of all the pupils, all the teachers and all the canteen staff of a school are put tin a box. One name is taken out at random. A pupil says: There are only three choices. It could be a pupil, a teacher or one of the canteen staff. The probability of it being a pupil is 1 / 3. Is this statement true or false? Select a reason. Problems involving handling data For example o Imran and Nia play three different games in a competition. Their scores have the same mean. The range of Imran s scores is twice the range of Nia s scores. Fill in the missing scores in the table below. Imran s scores 40 Nia s scores o The cost of exporting 350 wide-screen TV sets was: Item Cost Freight charge 371 Insurance 49 Packing 280 Port charges 140 a. Find the mean cost of exporting a wide screen TV. b. Construct a pie chart showing how the export costs in the table make up the total cost. Identify the information necessary to solve a problem; represent problems mathematically in a variety of forms As outcomes, Year 8 pupils should, for example:

182 Use, read and understand: best estimate, degree of accuracy justify, prove, deduce conclude, conclusion counter-example, exceptional case Identify the necessary information; represent problems in algebraic, using correct notation. For example, solve: Crosses How many crosses are there in each pattern? a. Find the value of: i ii iii b. What is the value of ? From the next pattern in the sequence, determine if it is equal to: (n +1) 2 n n 2 ( ) n x 4 Where n is the numbers of rows of x in the next sequence. Link to comparing two distributions using the range and one more of the measures of average interpreting tables, graphs and diagrams for both discrete and continuous data and drawing inferences; relating summarised data to questions being explored. Break problems into smaller steps or tasks; choose and use efficient operations, methods and resources Solve more complex problems by breaking them into smaller steps, choosing efficient operations for calculation, algebraic representation, methods and resources. For example solve: Missing digits Find the missing digits represented by the in the examples such as: ( 5) 2 = (3 x ) 2 = Link to factors, powers and roots. Numbers and the Number System Place value, ordering and rounding Understand and use decimal notation and place value; multiply and divide integers and decimals by powers of 10

183 As outcomes, Year 8 pupils should, for example: Use vocabulary from previous year and extend to: Billion, power, index. Read and write positive integer powers of 10. Know that: 1 hundred is 10 x 10 = thousand is 10 x 10 x 10 = thousand is 10 x 10 x 10 x 10 = 10 4, etc. 1 million is billion is 10 9, one thousand millions (In the past, 1 billion was 10 12, one million millions, in the UK.) Recognise that successive powers of 10 (i.e. 10, 10 2, 10 3, )underpin decimal (base 10) notation. Read numbers in standard form, e.g. read 7.2 x 10 3 as seven point two times ten to the power three. Link to using index notation Add or subtract to or from any number. Answer questions such as: What is more than 3.009? What is more than 3.299? What is less than 5? What is less than 10? What needs to be added or subtracted to change: to 4.277? to 6.139? to 5.053? to 4.111? Multiply and divide numbers by 0.01 and Investigate, describe the effects of, and explain multiplying and dividing a number by 0.1 and 0.01, In particular, recognise how numbers are increased or decreased by these operations. 0.1 is equivalent to 1/ 10 and 0.01 is equivalent to 1 /100, so: o Multiplying by 0.1 has the same effect as multiplying by 1/ 10 or dividing by 10. For example, 3 x 0.1 has the same value as 3 x 1/ 10, which has the same value as 3 10 = 0.3, and 0.3 x 0.1 has the same value as 3/ 10 x 1/ 10 = 3/ 100 = o Multiplying by 0.01 has the same effect as multiplying by 1/ 100 or dividing by 100. For example, 3 x 0.01 has the same value as 3 x 1/ 100, which has the same value as = 0.03, and 0.3 x 0.01 has the same value as 3/ 10 x 1/ 100 = o Dividing by 0.1 has the same effect as dividing by 1/ 10 or multiplying by 10. For example, has the same value as 3 1/ 10. (How many tenths in 3? 3 x 10 = 30) has the same value as 3/ 10 1/ 100 (How many hundredths in three tenths? 0.3 x 100 = 30)

184 o Dividing by 0.01 has the same value as dividing by 1/100 or multiplying by 100. For example, has the same value as 3 1/100. (How many hundredths in three? 3 x 100 = 300) has the same value as 3/10 1/100. (How many hundredths in three tenths? 0.3 x 100 = 30) Complete statements such as: 0.5 x 0.1 = 0.8 x = = 0.6 = 6 Dividing a positive number by 10, 100, 1000 has the effect of decreasing the value of that number. Discuss the effects of multiplying and dividing by a number less than 1. Does division always make a number smaller? Does multiplication always make a number larger? Compare and order decimals As outcomes, Year 8 pupils should, for example: Use, read and understand vocabulary from the previous year and extend to: ascending and descending Know that to order decimals, digits in the same position must be compared, working from the left, beginning with the first non-zero digit. In these examples, the order is determined by: < the fourth decimal place < < the first decimal place initially, then the second. Extend to negative numbers, e.g > Use accurately the symbols <, >,,, for example: Place < or > between: metres 330 millimetres Round numbers, including to a given number of decimal places Use and read and understand vocabulary from the previous year and extend to: recurring decimal Round positive whole numbers to a given power of 10, for example: There are people out of work. Politician A says: We have just over 1 million people out of work. Politician B says: We have nearly one and a half million people out of work. Who is more accurate? Recognise recurring decimals

185 Recurring decimals contain an infinitely repeating block of one or more decimal digits. For example: Fractions with denominators containing prime factors other than 2 or 5 will recur if written in decimal form. Round decimals to the nearest whole number or to one or two decimal places. For example, know that: rounded to the nearest whole number is 4 to one decimal place is 3.7, and to two decimal places is rounded to the nearest whole number is 2, to one decimal place 2.2, and to two decimal places is rounded to two decimal places is When substituting numbers into expressions and formulae, know that rounding should not be done until the final answer has been computed. Answer questions such as: Round to one decimal place. Round decimals in context in context, selecting an appropriate number of decimal places to use when, for example: Using decimal measurements for work on perimeter, area and volume; Calculating summary statistics, such as the mean; Investigating recurring decimals; Dividing. Integers, powers and roots Order, add, subtract, multiply and divide positive and negative numbers As outcomes, Year 8 pupils should, for example: Use, read and understand vocabulary from the previous year. Order positive and negative decimals. See ordering decimals. Add and subtract integers. Understand that 1 add -1 is zero, and use this to calculate, for example: = = 3 + =3 = -3 + = = = Recognise that: has the same value as ( ) = 1; 0-1 has the same value as ( ) -1 = -1 Use this to calculate, for example:

186 = = -2 - = 7 Solve puzzles such as: Complete this table. a b a b Multiply and divide positive and negative numbers. Link known multiplication tables to negative number multiplication tables. For example: -2 x 1 = -2-2 x 2 = -4-2 x 3 = -6 and so on.. Write the tables, continuing the pattern: 2 x 2 = 4 2 x -2 = -4 1 x 2 = 2 1 x -2 = -2 0 x 2 = 0 0 x -2 = 0-1 x 2 = -2-1 x -2 = 2-2 x 2 = -4-2 x -2 = 4-3 x 2 = -6-3 x -2 = 6 Recognise that division by a negative number is the inverse of multiplication by a negative number. Use this and the negative number multiplication tables, to show, for example, that -4-2 = 2, and relate this to the question How many -2s in -4? For a fact such as -3 x 2 = -6, write three other facts, i.e. 2 x -3 = -6, -6 2 = -3, -6-3 = 2 Answer questions such as: How many negative twos make negative four? (Two) Extend to the distributive law. For example: -1 x ( ) = -1 x 7 = -7-1 x ( ) = ( -1 x 3 ) + ( -1 x 4 ) = = -7 Recognise and use multiples, factors and primes; use tests of divisibility Use, read and understand vocabulary from the previous year. Use factors when appropriate to calculate, as in: 64 x 75 = 64 x 25 x = (3 x 3 x 8 x 8) = 1600 x 3 = 3 x 8 = 4800 = 24 Recognise squares and cubes, and the corresponding roots; use index notation and simple instances of the index laws Use, read and understand vocabulary from the previous year extended to: Cube number, cubed, cube root power The notation 6 3 as six cubed, and 6 4 as six to the power 4 and the cube root sign.

187 Fractions, decimals. percentages, ratio and proportion Use fraction notation; recognise and use the equivalence between fractions and decimals As outcomes, Year 8 pupils should, for example: Use and read vocabulary from the previous year and extend to; Terminating decimal, recurring decimal, unit fraction Use fraction notation to describe a proportion of a shape. For example: Draw a 3 by 4 rectangle. Divide it into four parts that are ½, ¼, 1 / 6 and 1/ 12 of the whole rectangle. Parts must not overlap. Now draw a 4 by 5 rectangle. Divide it into parts. Each part must be a unit fraction of the whole rectangle, i.e. with numerator 1. Try a 5 by 6 rectangle. And a 3 by 7 rectangle? The pie chart shows the proportions of components in soil. Estimate the fraction of the soil that is: a. water; b. air. Relate fractions to division. Know that 43 7 is another way of writing 43 / 7, which is the same as 6 1 / 7. Express a number as a fraction (in its lowest terms) of another. For example: What fraction of 180 is 120? Convert decimals to fractions. Continue to recognise that each terminating decimal is a fraction. For example, = Recognise that a recurring decimal is a fraction. Convert decimals (up to three decimal places) to fractions. For example: Convert to and then cancel to 5 / 8. Link to percentages. Convert fractions to decimals. Use division to convert a fraction to a decimal, without a calculator. For example: Use short division to work out that: 1 5 = = = Investigate fractions such as 1/3, 1/6, 2/3, 1/9, 1/11, convert to decimals. Order Fractions Compare and order fractions by converting them to fractions with a common denominator or by converting them to decimals. For example, find the larger of 7 / 8 and 4 / 5 ; Using common denominators: 7 / 8 is 35 / 40 and 4 / 5 is 32 / 40, so 7 / 8 is larger. Using decimals:

188 7 / 8 is and 4 / 5 is 0.8, so 7 / 8 is larger. Answer questions such as: Which is greater, 0.23 or 3/ 16? Which fraction is exactly half way between 3 / 5 and 5 / 7? Calculate fractions of quantities; add, subtract, multiply and divide fractions As outcomes, Year 8 pupils should, for example: Using diagrams to illustrate adding and subtracting fractions, showing equivalence. Know that fractions can only be added and subtracted if they have the same denominator. Answer questions such as: ¼ + 5/ 12 3 / 5 + ¾ 5 / 6 ¾ Calculate fractions of numbers, quantities or measurements. Develop written methods to answer short questions with fraction answers, such as: Find : three fifths of 17; Two fifths of 140g; 6/ 25 of 34. Multiply an integer by a fraction. Know that 2 / 3 of 12, 2 / 3 x 12 and 12 x 2 / 3 are all equivalent. Think of multiplication by 1/8 as division by 8, so 6 x 1 / 8 = 6 8, and 6 x 3 / 8 = 6 x 3 x 1 / 8 = 18 8 Use cancellation to simplify the product of a fraction and an integer. For example: 5 7 x 15 = 7 x 15 = Answer such questions as: 3/ Find: 12 x 30 5 / 9 x / 8 x 10 Understand that when multiplying a positive number by a fraction less than one, the result will be a smaller number. For example: 24 x 1 / 4 = 6 Divide an integer by a fraction. Know that a statement such as 24 ¼ can be interpreted as: How many quarters are there in 24? 24 = x ¼ or 24 = ¼ x For example: Look at one whole circle (or rectangle prism ) How many sevenths can you see? (seven) Look at 1. How many fifths do you see? (5)

189 Look at 4 circles. How many fifths are there now? (20) Understand that when dividing a positive number by a fraction less than one, the result will be a larger number. For example: 24 ¼ = 96 Understand percentage as the number of parts per 100; recognise the equivalence of fractions, decimals and percentages; calculate percentages and use them to solve problems As outcomes, Year 8 pupils should, for example: Understand percentage as the operator so many hundredths of. For example, know that 15% means 15 parts per hundred, so 15% of Z means 15/ 100 x z. Convert fraction and decimal operators to percentage operators by multiplying by 100. For example: x 100% = 45% 7/ 12 (7 12) x 100% = 58.3 % (to 1 d.p.) Link the equivalence of fractions, decimals and percentages to the probability scale and to interpretation of data in pie charts and bar charts. Calculate percentages of numbers, quantities and measurements. Continue to use mental methods. For example. Find: 65% of 40 by finding 50% then 10% then 5%; 35% of 70ml by finding 10% trebling the result and then adding 5%; 125% of 240 by finding 25% then adding this to 240. Use written methods. For example; Use an equivalent fraction, as in: 13/ 13% of x 48 = 624/ 100 = 6.24 Use an equivalent decimal, as in: 13% of x 48 = 6.24 Use a unitary method, as in: 13% of 48 1% of 48 = % of 48 = 0.48 x 13 = 6.24 Use vocabulary from the previous year and extend to: profit, loss, interest, service charge, tax. unitary method Use the equivalence of fractions, decimals and percentages to compare simple proportions and solve simple problems. Answer questions such as: There is 20% orange juice in every litre of a fruit drink. How much orange juice is there in 2.5 litres of fruit drink? How much fruit drink can be made from 1 litre of orange juice? 6 out of every 300 paper clips produced by a machine are rejected. What is this as a percentage? Find the outcome of a percentage increase or decrease. Understand that:

190 If something increases by 100%, it doubles. If something increases by 500%, it increases by 5. A 100% decrease leaves zero. An increase of 15% will result in 85%, and 85% is equivalent to An increase of 10% followed by a further increase of 10% is not equivalent to an increase of 20%. For example: An increase of 15% on an original cost of 12 gives a new price of 12 x 1.15 = Investigate problems such as: I can buy a bicycle for one cash payment of 119, or pay a deposit of 20% and then six equal monthly payments of 17. Understand the relationship between ratio and proportion, and use ratio and proportion to solve simple problems As outcomes, Year 8 pupils should, for example: Use read and understand vocabulary from the previous year and extend to: direct proportion Solve simple problems involving direct proportion. For example: Pizzas cost 16. What will 6 pizzas cost? 6 stuffed peppers cost 9. What will 9 stuffed peppers cost. Simplify a (three-part) ratio to an equivalent ratio by cancelling. For example: Write the ratio 12 : 9 : 3 in its simplest form. Link to fraction notation Simplify a ratio expressed in different units. For example: 2m : 50cm 450g : 5 kg Link to converting between measures Consolidate understanding of the relationship between ratio and proportion. For example: In a game, Tom scored 6, Sunil scored 8 and Amy scored 10. The ratio of their scores was 6 : 8 : 10, or 3 : 4 : 5. Tom scored a proportion of 3/ 12 or ¼ or 25% of the total score. Divide a quantity into two or more parts in a given ratio. Solve simple problems using a unitary method. Potting compost is made from loam, peat and sand, in the ratio 7 : 3 : 2 respectively. A gardener used 1 ½ litres of peat to make compost.

191 How much loam did she use? How much sand? The angles in a triangle are in the ratio 6 : 5 : 7. Find the sizes of the three angles. Lottery winnings were divided in the ratio 2 : 5. Dermot got the smaller amount of How much in total were the lottery winnings? 2 parts = part = parts = 2500 Total = = 3500 Link to problems involving ratio Calculations Number operations and the relationship between them Consolidate understanding of the operations of multiplication and division, their relationship to each other and to addition and subtraction; know how to use the laws of arithmetic As outcomes, Year 8 pupils should, for example: Use, read and understand vocabulary from the previous year and extend to: associative, distributive partition. Understand the operations of addition, subtraction, multiplication and division as they apply to positive and negative numbers. Link to integers Understand the operations of addition and subtraction as they apply to fractions. Link to fractions Understand that multiplying does not always make a number larger and that division does not always make a number smaller. Recognise that: can be interpreted as How many 0.1s (or tenths) in 9.1? can be interpreted as How many 0.01s (or hundredths) in 9.1? Link to multiplying and dividing by 0.1 and For example, use mental or informal written methods to calculate: 484 x 25 = 484 x = = x 25 = 3.15 x = = Recognise the application of the distributive law when multiplying a single term over a bracket in number and in algebra. Link to algebraic operations and mental calculations Inverses Use inverse operations. For example:

192 Fill in the missing number: 2 4 = 16 Use inverses to check results. For example: o = 355 appears to be about right, because 350 x 20 = 7000 Link to inverse operations in algebra and checking results. Know and use the order of operations, including brackets As outcomes, Year 8 pupils should, for example: Use vocabulary from the previous year. Recognise that, for example: 100 = = 5 4 x 5 Calculate with more complex mixed operations. For example: Find the value of: a. 2.1 ( ) + ( ) = 5 b. (2 + 3) 2 = 5 2 = 1 (14 9) Find to two decimal places, the value of: 25 6 x 93 Evaluate expressions using nested brackets, such as: 120 { 30 (2 7)} Understand that the position of the brackets is important. For example; Make as many different answers as possible by putting brackets into the expression: 3 x x For example: a. 3 x (5 + 3) (2 x 7) + 1 = 11 b. 3 x (5 + 3) 2 x ( 7 + 1) = 8 Mental methods and rapid recall of number facts Consolidate the rapid recall of number facts and use known facts to derive unknown facts As outcomes, Year 8 pupils should, for example: Use vocabulary from the previous year. Use known facts to derive unknown facts For example, generate constant-step sequences, such as: Start at 108, the rule is add 8. The start number is 5, target is 33. What is the rule? Complements Derive quickly:

193 Complements in 1, 10, 50, 100, Solve mentally equations such as: 100 = x = x 50 x = = 1000 x Doubles and halves Using doubling and halving methods to multiply and divide by powers of 2. For example: 18 x 16 = 18 x 2 x 2 x 2 x = Link to using the laws of arithmetic Multiplication and division facts Derive the product and quotient of multiples of 10 and 100 (wholenumber answers). For example: 30 x x Use knowledge of place value to multiply and divides whole numbers by 0.1 and For example: 47 x x x Extend to decimals, such as: 0.5 x x = = 3 Use knowledge of multiplication and division facts and place value to: derive products involving numbers such as 0.4 and For example: 4 x 0.6 = 4 x 6 10 = = 2.4 Divide mentally by 2, 4, and 5. Factors, powers and roots Know or derive quickly: - cubes of numbers from 1 to 5, and 10, and the corresponding roots. Calculate mentally: (24 +12) (7 + 4) 2 (89-25) ( ) 2 Solve mentally: 3a 2 = 31 n(n -1) = 56 Measurements Recall and use formulae for: the perimeter and area of a rectangle; the area of a triangle; the volume of a cuboid. Calculate simple examples mentally. Convert between units of time. For example:

194 How many hours in 5 ¼ days? How many days in 36 hours? Consolidate and extend mental methods of calculation, accompanied where appropriate by suitable jottings As outcomes, Year 8 pupils should, for example: Strategies for mental addition and subtraction Consolidate and use addition and subtraction strategies from previous years. For example: Add and subtract mentally pairs of integers. Use strategies for addition and subtraction to add and subtract pairs of integers. For example: = = The result of subtracting one integer from another is -29. What could the integers be? Add mentally several positive or negative numbers. Including larger multiples of 10. For example: Calculate a mean using an assured mean. For example: Find the mean of 18.7, 18.4, 19.1, 18.3 and 19.5 Use 19.0 as the assumed mean. The differences are -0.3, -0. 6, 0. 1, -0.7 and 0.5, giving a total difference of The actual mean is 19.0 ( ) = 18.8 Link to integers Add and subtract pairs of numbers of the same order (both with two significant figures). For example: Strategies for multiplication and division Use factors. For example 22 x x 0.01 = x 2 = 0.44 Use partitioning. For example: For multiplication, partition either part of the product: 13 x 1.4 = (10 x 1.4) + (3 x 1.4) = = 18.2 Use knowledge of place value to multiply and divide mentally any number by 0.1 or For example: o 3.6 x Recognise special cases where doubling or halving can be used.

195 Extend doubling and halving methods to include decimals and negative numbers. For example: 3.4 x 4.5 = 1.7 x 9 = 15.3 Multiply be near 10s. For example: 23 x 11 = 23 x x -19 = 8 x ( ) = = -152 Recall of fraction, decimal and percentage facts Know and derive quickly: Decimal/fraction/percentage equivalents, such as: 1 / 8 = or 12.5% Simplified form of fractions such as: 3/ 15 = 1 / 5 Know that 1 / 3 is and 2 / 3 is. Know that 0.03 is 3/100 or 3%. Strategies for finding equivalent fractions, decimals and percentages For example: Express 136% as a decimal. (Know that 136% is equivalent to or 1.36.) Express 55% as a fraction in its lowest terms. (Know that 55% is equivalent to , cancel to /20.) Convert 4 25 into a decimal. (Work out that 4/ 25 = 16 / 100, so it is equivalent to 0.16.) Use known facts such as 1 / 8 = to convert fractions to decimals mentally. For example: 5 8 = x 5 = Convert between improper fractions and mixed numbers. For example: Convert 7 1 / 3 into an improper fraction. Convert 36 / 5 into a mixed number. Strategies for calculating fractions and percentages of whole numbers and quantities. For example: o 3 / 5 of 20 = 12 (e.g. find one fifth, and multiply by 3.) o 1½ of 16 = 24 (e.g. find half, and add it to 16) o 125% of 240 (e.g. find 25% then add it to 240) o 35% of 40 (e.g. find 10%, then 30% and 5%, and add.) o A 5% discount on a 45 coat in a sale. By how much is the coat s price reduced? (Find 1% and then %5.) Word problems and puzzles (all four operations) Apply mental skills to solving simple problems, using jottings if appropriate. For example: o 14 x 39 = 546. What is 14 x 3.9? o Four sunflowers have heights of 225cm, 199cm, 185cm and 239cm. What is their mean height? o The sum of p and q is 12. The product of p and q is 27. Calculate their values. o Find 25% of 10% of 400.

196 o Calculate the area of this triangle. (With height 6.5cm and length 10cm.) Written methods Use efficient column methods for addition and subtraction of whole numbers, and extend to decimals As outcomes, Year 8 pupils should, for example: Consolidate the methods learned and used in previous years, and extend to harder examples of sums and differences with different numbers of digits. For example: Algebra Equation, formulae and identities Use the letter symbols and distinguish their different roles in algebra As outcomes, Year 8 pupils should, for example: Use, read and understand vocabulary from the previous year and extend to: algebraic expressions, formula, function partition, multiply out.. cubed, to the power of Know that an algebraic expression is formed from letter symbols and numbers, combined with operation signs such as +,-, x, (,), and /. Use letter symbols in different ways and begin to distinguish their different roles. For example: o In the equation 4x + 3 = 47, x is a particular unknown number. o In a formula V = IR, V, I and R are variable quantities related by the formula. In the formula F = 9C / , once C is known, the value of F can be calculated. In the function y = 3x - 4, for any chosen value of x, the related value of y can be calculated. Know how multiplication and division are represented in algebraic expressions. For example: 2 x n is written as 2n. p x q is written a pq. a x (b + c) is written as a(b + c)

197 (x + y) z is written as x + y z Use the equals sign appropriately and correctly. Know that the symbol = denotes equality. Avoid misuse of the equals sign when working through a sequence of steps, e.g. incorrectly writing = = = 93. Avoid interpreting the equals sign as makes, which suggests it means merely the answer to a calculation, as in 3 x = 13. Begin to interpret the equals sign more broadly, including in equations with expressions on each side. For example: Recognise equalities in different forms, such as: a + b = c + d 8 = 15 3x And know that they can be written as: a + b = c + d or c + d = a + b 8 = 15 3x or 15 3x = 8 Know that expressions such as 2a + 2 and 2 ( a + 1) always have the same value for any value of a. Know that algebraic operations follow the same conventions and order as arithmetic operations; use index notation and the index laws As outcomes, Year 8 pupils should, for example: Recognise that algebraic operations follow the same conventions and order as arithmetic operations. Know that contents of brackets are evaluated first, and that multiplication and division are carried out before addition and subtraction. For example: In 7 5s, the multiplication is performed first. In 6 s 2 the square is evaluated first. In 3(x 2), the expression in the brackets is evaluated first. Use index notation for small positive integer powers. Know that expressions involving repeated multiplication of the same number, such as: n x n n x n x n n x n x n x n are written as n 2, n 3, and n 4 and are referred to as n squared, n cubed n to the power 4 etc. Know why the terms squared and cubed are used for to the power of 2 to the power of 3. Understand the different meanings of expressions, such as: 2n x n 2 3n and n 3 Simplify expressions such as: 2x 2 + 3x 2 n 2 x n 3 p 3 p 2 Understand and use inverse operations. Recognise that any one of the equations: a + b = c, b + a = c, c a = b and c b = a ab = c, ba = c, b = c/a and a = c/b

198 implies each of the other three in the same set, as can be verified by substituting suitable sets of numbers into the equations. Apply inverse operations when two successive operations are involved. For example: The inverse of dividing by 4 and subtracting 7 is adding 7 and multiplying by 4, i.e. m/4 7 = n, then m = 4(n + 7). Simplify or transform algebraic expressions As outcomes, Year 8 pupils should, for example: Simplify or transform linear expressions by collecting like terms; multiply a single term over a bracket. Understand the application of the distributive law to arithmetic calculations such as: o 7 x 36 = 7 ( ) = 7 x x 6 o 7 x 49 = 7 (50-1) = 7 x 50 7 x 1 Know and use the distributive law for multiplication: o Over addition, namely a( b + c) = ab + ac o Over subtraction, namely a ( b c) = ab ac Recognise that the letters stand for numbers in problems. For example: o Simplify these expressions: a. 3a + 2b + 2a b b. 4x x 3 x c. 3(x + 5) d. 12 (n 3) e. M(n p) f. 4(a + 2b) 2(2a + b) o Write different equivalent expressions for the total length of the lines in this diagram. Simplify each expression as far as possible. Construct and solve linear equations, selecting an appropriate method As outcomes, Year 8 pupils should, for example: Use, read and understand: Linear equation Consolidate forming and solving linear equations with an unknown on one side. Choose a suitable unknown and form expressions leading to an equation. Solve the equation by removing brackets, where appropriate, collecting like terms and using inverse operations.

199 For example: o There are 376 stones in three piles. The second pile has 24 more stones than the first pile. The third pile has twice as many stones as the second. How many stones are there in each pile? o Let s stand for the number in the first pile. Pile 1 Pile 2 Pile 3 Total s s (s + 24) 376 s + ( s + 24) + 2( s + 24) = 376 s + s s + 48 = 376 4s + 72 = 376 4s = 304 s = 76 o On Dwain s next birthday, half of his age will be 16. How old is Dwain now? o Solve the equations: 5x = 7 2( p+ 5) = 24 3 = z = 14.3 n Explore alternative ways of solving simple equations, e.g. deciding whether or not to remove brackets first. For example: o 2(x + 5) = 36 or 2(x + 5) = 36 x + 5 = 18 2x + 10 = 36 x = 13 2x = 26 x = 13 Begin to understand that an equation can be thought of as a balance where, provided the same operation is performed on both sides, the resulting equation remains true. For example: o Start with a true statement, such as: 52 7 = Make the same change to both sides, e.g. subtract = 41 Check that this statement is true. Then start with a simple equation, such as: y = x + 4 add 3 y + 3 = x + 7 double 2(y + 3) = 2( x + 7) subtract d 2(y + 3) d = 2(x + 7) d Check that the result equation is true by substituting numbers which fit the original, e.g. x = 1, y = 5. Form linear equations (unknown on both sides) and solve them by transforming both sides in the same way. Begin to recognise what transformations are needed and in what order. For example: o Jill and Ben each have the same number of pens. Jill has 3 full boxes of pens and 2 loose pens.

200 Ben has 2 full boxes of pens and 14 loose pens. How many pens are there in a full box? 3n + 2 = 2n + 14 o Solve these equations: 3x + 2 = 2x + 5 5x 7 = 13 3z Set up and use equations to solve word and other problems involving direct proportion Begin to use graphs and set up equations to solve simple problems involving direct proportion. Discuss practical examples of direct proportion, such as: Discuss the distance travelled in a given time, assuming constant speed. For example: Consider a walking speed of 200 metres every 2 minutes. Generate (distance, time) pairs. Observe that distance/time is constant. Use this relationship to find the distance walked in: a. 12 minutes b. 7 minutes Use formulae from mathematics and other subjects As outcomes, Year 8 pupils should, for example: Substitute positive and negative numbers into linear expressions and positive integers into simple expressions involving powers. For example: Find the value of these expressions when a = 4. 3a a 3 Find the value of these expressions when x = x x 7(x - 1) Find the value of y when expressions when x = 2.5. y = 2x + 3 y = x 1

201 x x + 1 Explain the meaning of and substitute numbers into formulae such as: The volume V of a cuboid of length l and breadth b and height h: V = lbh The surface area S of a cuboid with width w, depth d and height h: S = 2dw + 2dh + 2hw Answer questions such as: The voltage V in an electrical circuit, with current I and resistance R, is given by the formula: V= IR What is V when l = 5 and R = 7? What is R when V= 42 and l = 3? In simple cases, find an unknown where it is not the subject of the formula and where an equation must be solved. For example: The formula for the change C from 50 for d compact discs at 7 each is C = 50 7d, If C = 15, what is d? The formula for the perimeter P of a rectangle l by w is P = 2 ( l + w ), If P = 20 and l = 7, what is w? Derive algebraic expressions and formulae. Check by substituting particular values. For example: o Mr Varma bought n apples and some oranges. a. He had 4 times as many oranges as apples. How many oranges did he have? b. He had 3 oranges left after making a pudding. How many oranges did he use? c. He used half the apples in a pie and his son ate one. How many apples were left? Derive formulae such as: The number f of square faces that can be seen by f = 4n + 2 The sum S of the interior angles of a polygon with n sides: S = ( n 2) x 180 o The area A of a parallelogram with base b and height h: A = b x h The number n half way between two numbers, n 1 and n 2 : n = n 1 + n 2 2 Sequences and functions Generate and describe sequences As outcomes, Year7 pupils should, for example: Use read and understand vocabulary from the previous year and extend to:

202 difference pattern, general term, T(n) flow chart linear sequence, arithmetic sequence Generate sequences from flow charts. For example: Continue familiar sequences. For example: Square numbers: 1, 4, 9, 16 Powers of 10 10, 100, 1000, Powers of 2 64, 32, 16 Generate sequences by multiplying and dividing by a constant factor. For example: 1, -2, 4, -8, 16, -32. Generate sequences by counting forwards or backwards in increasing or decreasing steps. For example: Start at 30 and count forwards by 1, 2, 3, to give 31, 32, 36 Begin to appreciate that: o Seeing a pattern in results enables predictions to be made, but any prediction must always be checked for correctness. o A satisfactory conclusion is reached only when a general explanation of either a term- to- term or a position to term rule is given and the pattern can be justified. Generate and describe integer sequences, relating them to geometrical patterns. For example: o Powers of 2 Generate terms of a sequence using term-to-term and position-to-term definitions of the sequence.

203 As outcomes, Year 8 pupils should, for example: Generate terms of a sequence given a rule for finding each term from the previous term, Generate these sequences: 1 st term Term to term rule 8 subtract 4 1 add consecutive odd numbers, starting with halve divide by 10 1, 2, add the two previous terms. Generate a sequence given a rule for finding each term from its position in the sequence, referring to terms as T(1) = first term, T(2) = second term,, T(n) = nth term. For example: o The nth term of a sequence is 2n, i.e. T(n) = 2n. Write the first five terms. o Write the first five terms of a sequence whose nth term or T(n) is: a. 5n + 4 b n c. 99 9n Find the nth term, justifying its form by referring to the context in which it was generated As outcomes, Year 8 pupils should, for example: Generate sequences from simple practical contexts. For example: Find the first terms of the sequence; describe how it continues using a term-to-term rule. Describe the general (nth) term. Begin to use linear expressions to describe the nth term of an arithmetic sequence, justifying its form by referring to the activity from which it was generated. Develop an expression for the nth term of sequences, such as: 7, 12, 17, 22, n Express functions and represent mappings As outcomes, Year 8 pupils should, for example: Use vocab from the previous year and extend to: Linear function Express simple functions in symbols. Extend mapping to include fractional values. Given inputs and outputs, find the function. Given a linear function use difference patterns to help find the function. For example, find the rules:

204 2, 3, 5, 1?? 5, 7, 11, 3 Know some properties of functions produced by combining number operations. Graphs and functions Generate points and plot graphs of functions As outcomes, Year 8 pupils should, for example: Use vocabulary from the previous year and extend to: Linear relationship Intercept, steepness, slope, gradient Generate coordinates pairs and plot graphs of simple linear functions, using all four quadrants. For example: y = 2x 3 (-3, -9), (-2, -7), (-1, -5), (0, -3), (1, -1), (2, 1),. Plot the points. Observe that the points lie in a straight line. Read other coordinate pairs from the line (including fractional values) and confirm that they also fit the function. Recognise that a graph of the form y = mx + b; o Corresponds to a straight line, whereas the graph of a linear sequence consists of set of discrete points lying on an imagined straight line ; o Represents an infinite set of points, and that: The values of the coordinates of each point satisfy the equation represented by the graph; Any coordinate pair which represents a point not on the graph does not satisfy the equation. Identify the graphs of linear functions in the form y = mx + b. Identify and interpret graphs such as: y = 2x, y = 2x + 1, y = 2x + 4, y = 2x 2, y = 2x -5 Describe similarities and differences. Notice that: -the lines are all parallel to y = 2x. -the lines all have the same gradient; -the number (constant) tells you where the line cuts the y-axis (the intercept). Recognise that equations of the form y = mx + b corresponds to a straight-line graph. Know these properties of functions of the form y = mx + b: o They are straight lines; o For a given value of b, all lines pass through the point (0, c) on the y-axis;

205 o All lines with the same given value of m are parallel. o Use knowledge of these properties to find the equation of straight line graphs. Interpret linear functions arising from real-life problems and plot and interpret their corresponding graphs As outcomes, Year 8 pupils should, for example: Interpret graphs of functions from a range of sources. For example: Give plausible explanations for the shape of graphs such as: In interpreting graphs of functions: Read values from a graph; Select accurate statements that reflect trends. Discuss and interpret line graphs from other subjects. For example: o Some pupils poured different volumes of water on to a small towelling flannel. Each time they found its mass. The water was always completely absorbed or soaked up by the flannel. Which would be the most likely line for their graph? Shape, Space and measures Geometric reasoning: lines, angles and shapes Use accurately the vocabulary, notation and labelling conventions for lines,

206 angles and shapes; distinguish between conventions, facts, definitions and derived properties As outcomes, Year 8 pupils should, for example: Use vocabulary from the previous year and extend to: corresponding angles, alternate angles.. supplementary, complementary interior angle, exterior angle.. equidistant prove, proof Continue to use accurately the notation and labelling conventions for lines, angles and shapes. Know that <DEF is an interior angle of DEF and that <GDF is an exterior angle of DEF. Know that: o A pair of complementing angles have a sum of 90 o. A pair of supplementary angles have a sum of 180 o. Identify properties of angles and parallel and perpendicular lines, and use these properties to solve problems As outcomes, Year 8 pupils should, for example: Identify alternate and corresponding angles. Understand that the sum of the angles of a triangle is 180 o and of a quadrilateral is 360 o, and that the exterior angle of a triangle equals the sum of the two interior opposite angles. Consider relationships between three lines meeting at a point and a fourth line parallel to one of them. Given sufficient information, calculate: o Interior and exterior angles of triangles o Interior angles of quadrilaterals For example:

207 Identify and use the geometric properties of triangles, quadrilaterals and other polygons to solve problems; explain and justify inferences and deductions using mathematical reasoning As outcomes, Year 8 pupils should, for example: Use vocabulary from the previous year and extend to: Bisect, bisector, midpoint Congruent tessellate, tessellation Know and use side, angle and symmetry properties of equilateral, isosceles and right-angled triangles. Classify quadrilaterals by their geometric properties (equal and/or parallel sides, equal angles, right angles, diagonals bisected and/or at right angles, reflection and rotation symmetry ) Know properties such as: o An isosceles trapezium is a trapezium in which the two opposite non-parallel sides are the same length. It has one line of symmetry and both diagonals are the same length. o A parallelogram has its opposite sides equal and parallel. Its diagonals bisect each other. It has rotation symmetry of order 2. o A rhombus is a parallelogram with four equal sides. Its diagonals bisect each other at right angles. Both diagonals are lines of symmetry. It has rotational symmetry of order 2. o A kite is a quadrilateral that has two pairs of adjacent sides of equal length, and no interior angle larger than 180 o. It has one line of symmetry and its diagonals cross at right angles. o An arrowhead or delta has two pairs of adjacent edges of equal length and one interior angle larger than 180 o. It has one line of symmetry. Its diagonals cross at right angles outside the shape. Use angle and side properties of triangles, and angle properties of parallel and intersecting lines, to solve problems. Explain reasoning. For example: o The angle at the vertex of a regular pentagon is 108 o. Two diagonals are drawn to the same vertex to make three triangles in the pentagon. Calculate the sizes of the angles in each triangle. Understand congruence and similarity As outcomes, Year 8 pupils should, for example:

208 Congruence Know that if two 2-D shapes are congruent, they have the same shape and size, and corresponding sides and angles are equal. For example: o From a collection of different triangles or quadrilaterals, identify those that are congruent to each other. Realise that corresponding sides and angles are equal. o Two congruent scalene triangles without right angles are joined with two equal edges fitted together. What shapes can result? What if the two triangles are right angled, isosceles or equilateral? Use 2-D representations, including plans and elevations, to visualise 3-D shapes and deduce some of their properties As outcomes, Year 8 pupils should, for example: Use vocabulary from the previous year and extend to: View, plan, elevation isometric Know and use geometric properties of cuboids and shapes made from cuboids; begin to use plans and elevations. For example; o Identify a net to make this model. Construct the shape. o Here are three views of the same cube. Which letters are opposite each other? Identify the position of hidden lines in an isometric drawing. Begin to use plans and elevations. For example: o The diagrams below are of solids viewed from directly above.

209 Select the names of these solids from a list. For example square pyramid; cylinder; triangular pyramid.. Recognise the front elevation, side elevation and plan of this shape. Transformations Understand and use language and notation associated with reflections, translations and rotations Recognise and visualise transformations and symmetries of shapes Use vocabulary from the previous year Combinations of two transformations Transform 2 D shapes by repeated reflections, rotations and translations. Explore the effect of repeated reflections in parallel or perpendicular lines. For example: o Reflect a shape in one coordinate axis and then the other. For example, reflect the shape below first in the x-axis and then in the y axis. What happens? What is the equivalent transformation? Now reflect it first in the y-axis and then in the x-axis. What happens? What is the equivalent transformation?

210 Explore the effect of repeated rotations, such as half turns about different points. Understand and demonstrate some general results about repeated transformations. For example: Reflection in two parallel lines is equivalent to a translation. Reflection in two perpendicular lines is equivalent to a half turn rotation. Two rotations about the same centre are equivalent to a single rotation. Two translations are equivalent to a single rotation. Two translations are equivalent to a single translation. Explore the effect of combining transformations. o Triangles A, B, C and D are drawn on a grid. a. Find a single transformation that sill map: i. A on to C ii. C on to D b. Find a combination of two transformations that will map: i. B on to C ii. A on to D c. Find other examples of combined transformations, such as: A to C: with centre (0, 0), rotation of 90 o, followed by a further rotation of 90 o ;

211 A to C: reflection in the y-axis followed by reflection in the x- axis; B to C: rotation of 90 o centre (-2, 2), followed by translation (-4, 0); C to D: reflection in the y-axis followed by reflection in the line x = 4; C to D: rotation of 270 o centre (0, 0), followed by a rotation of 90 o centre (4, 4). Rotation Rotate shapes, and deduce properties of the new shapes formed, using knowledge that the images are congruent to the original and identifying equal angles and sides. For example: o Rotate a right-angled triangle through 180 o about the mid point of its shortest side. Name the shape formed by the object and image. Identify the equal angles and equal sides. Explain why they are equal. What happens when you rotate the triangle about the mid-point of its longest side? Reflection symmetry and rotation symmetry Recognise all the symmetries of 2-D shapes. Identify and describe the reflection and rotation symmetries of: o Regular polygons (including equilateral triangles and squares); o Isosceles triangles; o Parallelograms, rhombuses, rectangles, trapeziums and kites. Sort quadrilaterals, based on questions relating to their symmetries. Relate symmetries of triangles and quadrilaterals to their side angle and diagonal properties. For example: o An isosceles triangle has reflection symmetry. Use this to confirm known properties, such as: - The line of symmetry passes through the vertex which is the intersection of the two equal sides, and is the perpendicular bisector of the third side. - The base angles on the unequal side are equal. Use and interpret maps and scale drawings As outcomes, Year 8 pupils should, for example: Use scales. For example: Identify the scale of each of these: a. the plan of a room with 2cm representing 1m b. the plan of the school field with 1 inch representing 50 yards c. a map of the area surrounding the school, with 4cm representing 1km.

212 Coordinates Use coordinates in all four quadrants As outcomes, Year 8 pupils should, for example: Use vocabulary from the previous year and extend to.. midpoint Read and plot points in all four quadrants. For example: The points (-5, -3), (-1, 2) and (3, -1) are the vertices of a triangle. Identify where the vertices lie after: a. translation of 3 units parallel to the x-axis; b. reflection in the x-axis; c. rotation of 180 o about the origin. Link to transformations Given the coordinates of points A and B, find the mid point of the line segment AB. Construction and loci Find simple loci, by reasoning As outcomes, Year 8 pupils should, for example: Know that: o The perpendicular bisector of a line segment is the locus of points that are equidistant from the two end points of the line segment. o The bisector of an angel is the locus of points that are equidistant from the two lines. Visualise a simple path. For example: o Imagine a robot moving so that it is always the same distance from a fixed point. Describe the shape of the path that the robot makes. o Imagine two trees. Imagine walking so that you are always an equal distance from each tree. Describe the shape of the path you would walk. (The perpendicular bisector of the line segment joining the two trees.) Measures and mensuration Use units of measurement to measure estimate, calculate and solve problems in a range of contexts; convert between metric units. As outcomes, Year 8 pupils should, for example: Continue to use familiar units of measurement from previous year and extend to: Standard metric units Tonne (not usually abbreviate) Hectare(ha)

213 Cubic millimetre (mm 3 ), cubic centimetre (cm 3 ), cubic metre (m 3 ).. Know the relationship between the units of a particular measure, e.g. o 1 hectare = m 2 o 1 tonne = 1000kg o 1 litre = 1000cm 3 o 1 millilitre = 1 cm 3 o 1000 litres = 1 m 3 Convert between one unit and another. Convert between area measures in simple cases. For example: o A rectangular field measures 250m by 200m What is its area in hectares (ha)? o Each side of a square tablecloth measures 120cm. Write its area in square metres (m 2 ). Convert between units of time. For example: At what time of what day of what year will it be: a minutes b weeks after the start of the year 2000? Consolidate changing a smaller unit to a larger one. For example: o Change 760g into kilograms. o Change 400ml into litres. Suggest appropriate units and methods to estimate or measure volume. Solve problems involving length, area, volume, capacity, mass, time, angle and bearings, rounding measurements to an appropriate degree of accuracy. Extend the range of measures used to angle measure and bearings, and compound measures Bearings Use and read: bearing, three-figure bearing and compass directions and compass directions. Use bearings to specify direction and solve problems, including making simple scale drawings. Know that the bearing of a point A from an observer O is the angle between the line OA and the north line through O, measured in a clockwise direction. In the diagram the bearing of A from O is 210 o. The bearing of O from A is 30 o.

214 For example: o If the bearing of P from Q is 45 o, what is the bearing of Q from P? If the bearing of X from Y is 120 o, what is the bearing from Y to X. Deduce and use formulae to calculate lengths, perimeters, area and volumes in 2-D and 3-D shapes As outcomes, Year 8 pupils should for example: Use vocabulary from the previous year and extend to: Volume, space, displacement and use the units: hectare (ha), cubic centimetre (cm3), cubic metre (m3), cubic millimetre (mm3) Deduce formulae for the area of a parallelogram and triangle. For example, explain why: The area of a triangle is given by A = ½ bh, where b is the base and h is the height of the triangle. Calculate areas of triangles, parallelograms and trapezia, and of shapes made from rectangles and triangles. For example: A right-angled triangle lies inside a circle. The circle has a radius of 5 cm. Calculate the area of the triangle. The diagram shows a shaded square inside a larger square. Calculate the area of the shaded square. o The area of the parallelogram is 10cm 2. Calculate the height of the parallelogram.

215 o The area of the trapezium is 10cm2. What might be the value of h, a and b (a < b)? Know the formula for the volume of a cuboid and use it to solve problems involving cuboids. Understand the formula for the volume of a cuboid by considering how to count unit cubes. Estimate volumes. For example: Estimate the volume of everyday objects such as a rectangular chopping board, a bar of soap and a shoe box. Suggest volumes to be measured in cm 3, m 3. Solve problems such as: o Find the volume of a 3 cm by 4 cm by 5 cm box. o Find the volume in cm3 of this H- shaped girder by splitting it into cuboids.

216 Handling data Processing and representing data Calculate statistics from data, finding the mode, mean, median and range As outcomes, Year 8 pupils should, for example: Use vocabulary from the previous year and extend to: Distribution Stem-and leaf diagram Know when it is appropriate to use the mode (or the modal class), mean, median and range: o The median is useful for comparing with a middle value, e.g. half the class swam more than 500 m. o The range gives a simple measure of spread. o The mode indicates the item or class that occurs most often and is useful in reporting opinion polls. o The mean gives an idea of what would happen if there were equal shares. Use a stem-and-leaf diagram to help find the median, range and mode. For example: Hours of sunshine for UK weather stations 10/05/00 (stem = hours, leaves = tenths) There are 58 items of data. The median is the half way between the 29 th and 30 th item. Which is 4.8 which is 4 hours and 48 minutes of sunshine. The range is = 7.4h. The mode is 3.0 hours. Calculate statistics. For example: o Imran and Nia play three games. Their scores have the same mean. The range of Imran s score is twice the range of Nia s scores. Write the missing scores in the table below. Imran s score 40 Nia s

217 o John has three darts scores with a mean of 30 and a range of 20. His first dart scored 26. What were his other two scores? Interpreting and discussing results Interpret diagrams and graphs, and draw inferences Interpret diagrams, graphs and charts, and draw inferences related to the problem; relate summarised data to the questions being explored. For example: Interpret a compound bar chart. For example: Select a statement that accurately answers the following question. How has the method of travel changed over the last 20 years? Interpret line graphs, e.g. weather data. Compare two simple distributions using the range, mode, mean or median Compare two distributions using the range and one or more of the mode, mean or median. For example: Which type of battery last longer? Use data from an experiment to calculate the range, median and mean of each type. Conclude, for example, that one brand is generally of higher quality and one has less consistent manufacturing standards, as evidenced by a greater range. Probability Use the vocabulary of probability

218 Use vocabulary from the previous year and extend to: Event, theory, sample, sample space, biased. Use the vocabulary of probability when interpreting the results of an experiment; appreciate that random processes are unpredictable. For example: o Two boxes of sweets contain different numbers of hard and soft centred sweets. o Box 1 has 8 hard-centred sweets and 10 with soft centres. Box 2 has 6 hard-centred sweets and 12 with soft centres. Kate only likes hard-centred sweets. She can pick a sweet at random from either box. Which box should she pick from? Why? Kate is given a third box of sweets 5 hard centred sweets and 6 soft centres. Which box should Kate choose from now? Use the probability scale; find and justify theoretical probabilities Know that if the probability of an event occurring is p, then the probability of it not occurring is 1 - p. Use this to solve problems. For example: o Consider a pack of 52 cards (no jokers). If the probability of drawing a club from the pack of cards is ¼ then the probability of drawing a card that is not a club is 1 ¼ = 3 / 4. Calculate the probability that a card chosen at random will be: a. a red card; b. a heart: c. not a picture card; d. not an ace; e. either a club or a diamond; f. an even numbered red card. o There are 25 cars parked in a garage. 12 are red, 7 are blue, 3 are white and rest black. Calculate the probability that the next car to leave the garage will be: a. red; b. blue; c. neither red nor blue; d. black or white. Find and record all possible outcomes for single events and two successive events in a systematic way, using diagrams and tables. o What are the possible outcomes when: A mother gives birth to twins? A glazier puts red, green or blue in each of two windows? You can choose two pizza toppings from onion, mushroom and sweet corn? o 200 raffle tickets are numbered from 1 to 200. They have all been sold. One ticket will be drawn at random to win first prize.

219 a. Karen has number 125. What is the probability that she will win? b. Andrew buys tickets with numbers 81, 82, 83, 84. Sue buys tickets with numbers 30, 60, 90, 120. Who has the better chance of winning? c. Rob buys several tickets. He has 5% chance of winning. How many tickets has he bought? d. Three people have each lost a ticket and do not play. What is the chance that nobody wins?

220 YEAR 9 Using and applying mathematics to solve problems Applying mathematics and solving problems Pupils should be taught to: Solve word problems and investigate in a range of contexts As outcomes, Year 9 pupils should, for example: Solve increasingly demanding problems; explore connections in mathematics across a range of contexts; generate fuller solutions. Problems involving money For example: o Two families went to the cinema. The Smith family bought tickets for one adult and four children and paid 19. The Jones family bought tickets for two adults and two children and paid 17. What was the cost of one child s ticket? o Thirty years ago the money used in Great Britain was pounds, shillings and pence. There were 20 shilling in 1. o A gallon of petrol cost 7 shillings thirty years ago. Today it costs about How much has the cost of petrol risen in the last thirty years? Problems involving percentages For example: o After an advertising campaign costing 950, a firm found that its profits rose by 15% to Did they recover the cost of the campaign from this initial rise in profits? Problems involving ratio and proportion: For example: o 2 parts of red paint mixed with 3 parts of blue paint make purple paint. What is the maximum amount of purple paint that can be made from 50ml of red paint and 100 ml of blue? o A recipe for jam uses 55g of fruit for every 100g of jam. I want to make ten 454 g jars of jam. How many grams of fruit do I need? Problems involving number and algebra For example: Two satellites circle round the Earth. Their distance from the centre of the earth is: Satellite A 1.53 x 10 7 miles Satellite B 9.48 x 10 6 miles What is: a. the maximum distance apart.

221 b. the minimum distance apart, the satellites could be? More problems involving number and algebra o For example: o This bottle is being steadily filled with water. Which graph shows the relationship between time and the height of water in the bottle? Problems involving shape and space For example: o Triangle T has vertices at (1, 2), (2, 4) an d(3, 4). a. Place crosses on these coordinates, in the provided grid. b. Triangle R is obtained by reflecting T in the x-axis. What are the coordinates of its vertices? c. Triangle S is obtained by reflecting R in the y-axis. What are its coordinates? d. There is a transformation that takes triangle T directly to triangle S. Describe transformation as precisely as you can. More problems involving shape and space. For example:

222 o In the diagram: PG = PS, QR = QS, PQ is parallel to SR, angle SQR is 62 o. Calculate the sizes of the other angles. o Is the angle a 90 o. o BC is a diameter of a circle centre O. A is any point on the circumference. Prove that angle BAC is a right angle. Problems involving circumference, area and volume For example: o A letter B is made out of a piece of wire. It has a straight side and two equal semicircles. Calculate the total length of the wire. Problems involving measures

223 For example: o A plank of wood weighed 1.4 kg. 25 centimetres of the plank were cut off its length. The plank then weighed 0.8 kg. What was the length of the original plank? o The diameter of a red blood cell is cm and the diameter of a red cell and the diameter of a white cell is cm. Work out the difference between the diameter of a red cell and the diameter of a white cell. Give the answer in millimetres. Problems involving probability For example: o Karen and Huw each have three cards, numbered 2, 3 and 4. They each take one of their own cards. They then add together the numbers of the four remaining cards. What is the probability that their answer is an even number? o John makes clay pots. Each pot is fired independently. The probability that a pot cracks while being fired is a. John fires two pots. Calculate the probability that both pot cracks, and secondly that only one of them cracks. b. John has enough clay for 80 pots. He gets an order for 75 pots. Does he have enough clay to make 75 pots without cracks? Explain your answer. Problems involving handling data For example o Here are some data about the population of the regions of the world in 1950 and Regions of the world Population in millions in Populations in millions in Africa Asia Europe Latin America North America Oceania a. In 1990, for every person who lived in North America, how many people lived in Asia? b. A pupil thinks that from 1950 to 1990 the population of Oceania went up by 100%. Is the pupil right? Identify the information necessary to solve a problem; represent the problems mathematically in a variety of forms. As outcomes, Year 9 pupils should, for example: Use vocabulary from the previous years and extend to:

224 Trail and improvement Generalise Identify the information necessary to solve a problem. For example: A cube of side is made up of 27 individual centimetre cubes. The cube is dipped into a pot of paint, so that all the exterior sides are covered in the paint. The cube is them broken up into the individual 72 centimetre cubes. How many of the cm cubes have 3 sides painted? 2 sides painted? 1 side painted? 0 sides painted? Identify and select representations for problems by synthesising information in algebraic, geometric and graphical form. For example: On Dave s next birthday he will be twice as old as his little brother Danny. He ll also be one third the age of his Dad. Dave is now n years old. How old will Dad be on Dave s next birthday? Click to choose. 3(n + 1) 3 x n n Link to generating sequences using term- to-term and position-toterm definitions. Break problems into smaller steps or tasks; choose and use efficient operations, methods and resources Solve substantial problems by breaking them into simpler tasks, using a range of efficient techniques and methods. Use trial and improvement where a more efficient method is not obvious. For example: The inside of a running track is a rectangular shape with identical semicircles on each end. The total perimeter of this inside area of the track must be 400m and the straight sides must be at least 80m. What is the greatest area the running track can enclose? Link to using the formulae for the circumference and area of a circle. Link to simplifying algebraic expressions; expanding the product of two linear expressions; using systematic trial and improvement methods. Numbers and the Number System Place value, ordering and rounding Understand and use decimal notation and place value; multiply and divide integers and decimals by powers of 10

225 As outcomes, Year 9 pupils should, for example: Use vocabulary from previous year and extend to: Standard (index) form exponent. Extend knowledge of integer powers of 10. Know that: 10 o = = = 1/10 1 = 1/ = 1/10 2 = 1/100 Know the prefixes associated with powers of 10. Relate to commonly used units. For example: 10 9 giga 10 6 mega 10 3 kilo 10-2 centi 10-3 milli 10-6 micro 10-9 nano pico Know the term standard (index) form and read numbers such as 7.2 x Link to using index notation and writing numbers in standard form Multiply and divide by any integer power of ten. For example: Calculate: 7.34 x x x x Link to converting mm 2 to cm 2, cm 2 to m 2, mm 3 to cm 3 and cm 3 to m 3. Begin to write numbers in standard form, expressing them as A x 10n where 1 A < 10 and n is an integer. For example: = x = 6.3 x 10-3 Answer questions such as: o Complete these. The first is done for you. 3 x 10 n = 300 x 10 n x 10 n = x 3 10 n = x o Put these numbers in ascending order: 2 x 10-2, 3 x 10-1, 2.5 x 10-3, 2.9x10-2, 3.2x10-1 Round numbers, including to a given number of decimal places or significant figures.

226 Use and read and understand vocabulary from the previous year and extend to: Significant figures, upper and lower bounds Read and write the approximately equal to sign ( ) Round decimals to the nearest whole number on to one, two and three decimal places. For example, know that: o rounded to the nearest whole number is 3, rounded to 1 d.p. is 3.1, to 2 d.p. is 3.06, and to 3 d.p is o rounded to the nearest whole number is 10, to 1 d.p. is 10.0, and to 2 d.p. is o 22/7 is an approximation to π and can be given as 3.14 to 2 d.p. or correct to 3 d.p. Know that rounding should not be done until a final result has been computed. Understand upper and lower bounds. For example: o For discrete data such as: The population p of Sweden is to the nearest million is 15 million. Know that the least population could be and the greatest population could be o For continuous data such as measurements of distance; The distance d km from Exeter to Plymouth is 62 km to the nearest km. Know that the shortest possible distance is 61.5km and the longest possible distance is 62.5km. Round numbers to a given number of significant figures. Know, for example that: o 5.78 to 5.8 to two significant figures ( 2 s.f). o is 35 to 2 s.f., and 34.7 to 3 s.f. Know when to insert zeros as place holders to indicate the degree of significance of the number. For example, is 1.40 to 3 s.f. Use numbers to a given number of significant figures to work out an approximate answer. For example: o The area of a circle with radius 7 cm is approximately 3 x 50 cm 2. Give answers to calculations to an appropriate number of significant figures. Integers, powers and roots Recognise and use multiples, factors and primes; use tests of divisibility Find the common factors of algebraic expressions. For example: o 2x 2 yz and 3wxy have common factor xy. o (x 1)(2x + 3) 3 and (x 1) 2 (2x 3) have the common factor (x 1).

227 Recognise squares and cubes, and the corresponding roots; use index notation and simple instances of the index laws Use, read and understand vocabulary from the previous year extended to: Index, indices, index notation, index law. Use index notation for small integer powers. For example: o 19 3 = = = Know that x o = 1, for all values of x. Know that: 10-1 = 1/ 10 = = 1/ 10 = 0.01 Square roots and cube roots. Know that: a + b (a + b) Know that: There are two square roots of a positive integer, one positive and one negative number, written as ± ; The cube root of a positive number is positive and the cube root of a negative number is negative. Investigate problems such as: o The outside of a cube made from smaller cubes is painted blue. How many small cubes have 0,1, 2 or 3 faces painted blue? Investigate. o Three integers, each less than 100, fit the equation a 2 + b 2 = c 2 What could the integers be? Use simple instances of the index laws and start to multiply and divide numbers in index form. o Recognise that: Indices are added when multiplying, e.g. 4 3 x 4 2 = (4 x 4 x 4 ) x (4 x 4) (3 + 2) = 4 x 4 x 4 x 4 x 4 = 4 5 = 4 Indices are subtracted when dividing, e.g = (4 x 4 x 4 x 4 x 4) ( 4 x 4 ) = 4 x 4 x 4 = 4 3 = 4 (5-2) = 4 (2-5) = = 7 0 = 1 Generalise to algebra. Apply simple instances of the index laws (small integral powers), as in: n 2 x n 3 = n = n 5 p 3 p 2 = p 3-2 = p Know and use the general forms of the index laws for multiplication and division of integer powers. p a x p b = p a +b, p a p b = p a-b, (p a ) b = p ab Begin to extend understanding of index notation to negative and fractional powers; recognise that he index laws can be applied to these as well.

228 / 4 2 = 1/ 16 1 / 3 2 = 1/ 8 1 / 2 2 = 1/ 4 1 / 1 2 = 1/ 2 1 / 0 2 = Know the notation 5 1/2 = 5 and 5 1/3 = 3 5. Extend to simple surds (unresolved roots): 3 x 3 = 3 3 x 3 x 3 = = (2 x 16) = 2 x 16 = 4 2 Fractions, decimals. percentages, ratio and proportion Use fraction notation; recognise and use the equivalence between fractions and decimals As outcomes, Year 9 pupils should, for example: Use fraction notation to describe a proportion of a shape. For example: The curves of this shape are semicircles or quarter circles. Express the shaded shape as a fraction of the large dashed square. Express a number as a fraction (in its lowest terms)of another. For example: What fraction of 120 is 180? ( 3 / 2 or 1 1 / 2 ) Understand the equivalence of algebraic fractions. For example: o For example: ab b ac c ab 2 = abc a 1 x b 1 x b = b a 1 x b 1 x c c 3ab = a 6bc 2c Simplify algebraic fractions by finding common factors. For example:

229 o Simplify 3a + 2ab 4a 2 Recognise when cancelling is inappropriate. For example, recognise that: o a + b is not equivalent to a + 1; b o a + b is not equivalent to a; b o ab 1 is not equivalent to a 1. b Link to adding algebraic fractions. Know that a recurring decimal is an exact fraction. Know and use simple conversions for recurring decimals to fractions. For example: o = 1 / 3 o = 2 / 3 o = 1 / 9 Order fractions. Answer questions such as: o The numbers 1 / 2, a, b, ¾ are in increasing order of size. The differences between successive numbers are all the same. What is the value of b? Calculate fractions of quantities; add, subtract, multiply and divide fractions Add and subtract fractions. Add and subtract more complex fractions. For example: = = o Begin to add and subtract algebraic fractions, linking to number examples. Calculate fraction of numbers, quantities or measurements. Understand the multiplicative nature of fractions as operators. For example: o Which is the greater: ¾ of 24 or 2 / 3 of 21? o In a survey of 24 pupils, ½ liked football best, ¼ liked basketball, 3/8 liked athletics. The rest liked swimming. How many liked swimming. Multiply a fraction by a fraction. Multiply fractions and simplify: 3 x 2 = Divide a fraction by a fraction. 1 3 = Understand percentage as the number of parts per 100; recognise the equivalence of fractions, decimals and percentages; calculate percentages

230 and use them to solve problems As outcomes, Year 9 pupils should, for example: Use vocabulary from previous years and extend to: cost, price, selling price... Recognise when fractions or percentages are needed to compare proportions and solve problems. Answer questions such as: o Which is the better buy: A 400g pack of biscuits at 52p, or A pack of biscuits with 400g 25% + extra, at 57p? o In a phone bill, VAT at 17.5% is added to the total cost of calls and line rental. What percentage of the total bill is VAT? Use percentage changes to solve problems, choosing the correct numbers to take as 100%, or as a whole. For example: o There was a 25% discount in a sale. A boy paid 30 for a pair of jeans in the sale. What was the original price of the jeans? Solve problems such as: o I bought a fridge freezer in a sale and saved 49. The label said that it a 20% reduction. What was the original price of the fridge freezer? o people visited a museum in This was an increase of 25% on How many visitors were there in 1999? Understand the relationship between ratio and proportion, and use ratio and proportion to solve simple problems As outcomes, Year 9 pupils should, for example: Use vocabulary from the previous year and extend to: Proportionality, proportional to Identify when proportional reasoning is needed to solve a problem. For example: o A recipe for fruit squash for six people is: 300g chopped oranges 1500ml lemonade 750ml orange juice Trina made fruit squash for ten people. How many millilitres of lemonade did she use? Jim used 2 litres of orange juice for the same recipe. How many people was this enough for? Simplify a ratio expressed in fractions or decimals. For example: Write the ratio 0.5 : 2 in whole number form. Compare ratios by changing them to the form m : 1 or 1 : m. For example:

231 o The ratios of Lycra to other materials in two stretch fabrics are 2 : 25 and 3 : 40. By changing each ratio to the form 1 : m, say which fabric has the greater proportion of Lycra. Interpret and use ratio in a range of contexts. For example: o Shortcrust pastry is made from flour and fat in a mass ratio 2 : 1. How much flour will make 450g of pastry? o An alloy is made from iron, copper, nickel and aluminium in the ratio 5 : 4 : 4 : 1. Find how much copper is needed to mix with 85 g of iron. o On 1 st June the height of a sunflower was 1 m. By 1 st July, the height had increased by 40%. What was the ratio of the height of the sunflower on 1 st June to its height on 1 st July? Calculations Number operations and the relationship between them Consolidate understanding of the operations of multiplication and division, their relationship to each other and to addition and subtraction; know how to use the laws of arithmetic As outcomes, Year 9 pupils should, for example: Use, read and understand vocabulary from the previous year and extend to: reciprocal. Understand the effect of multiplying and dividing by numbers between 0 and 1. Understand the operations of multiplication and division as they apply to fractions. Understand that multiplying a positive number by a number between 0 and 1 makes it smaller and that dividing it by a number between 0 and 1 makes it larger. Know that division by zero has no meaning. Continue to use the laws of arithmetic to support efficient and accurate mental and written calculations. Recognise the application of the distributive law when expanding the product of two linear expressions in algebra. Know and use the order of operations, including brackets As outcomes, Year 9 pupils should, for example: Use vocabulary from the previous year. Understand the effect of powers when evaluating an expression. For example: o Find the value: o 36 ( 3 + 9) x ( 8 2) 3 = 188

232 o = -44 o ( 4 / 3 ) 2 = = 16 / 9 = 1 7 /9 Recognise that -a 2 -a 2. Mental methods and rapid recall of number facts Consolidate the rapid recall of number facts and use known facts to derive unknown facts As outcomes, Year 9 pupils should, for example: Use vocabulary from the previous year. Use known facts to derive unknown facts For example, Derive 36 x 14 from 36 x 25? Multiplication and division facts Derive the product and quotient of multiples of 10, 100, and For example: 600 x Use knowledge of place value to multiply and divide decimals by 0.1 and For example: o 0.47 x o 9.6 x o x Consolidate knowledge of multiplication and division facts and place value to multiply and divide mentally. For example: o 0.24 x 0.4 = 24 x = = x 0.7 = 80 X 7 = 56 x 10 = 560 Factors, powers and roots Find mentally: - The HCF and LCM of pairs of numbers such as 36 and 48, 27 and 36; - Products of small integer powers, such as 3 3 x 4 2 = 27 x 16 = 432;. - Factor pairs for a given number. Calculate mentally: o ( ) 3 Solve mentally: (3 + x) 2 = 25 (12 - x) 2 = 56 Identify numbers from property questions, such as: o This number is a multiple of 5. It leaves remainder 1 when divided by 4. What could it be? Know simple Pythagorean triples such as 3, 4, 5, or 5, 12, 13, and their multiples. o Apply Pythagoras theorem. Measurements

233 Recall and use formulae for: the perimeter of a rectangle and circumference of a circle; the area of a rectangle, triangle, parallelogram, trapezium, circle; the volume of a cuboid and a prism. Calculate simple examples mentally. Know that; speed = distance/ time. Use this to derive facts from statements such as: o A girl takes 20 minutes to walk to school, a distance of 1.5km. Find her average speed in km/h. Solve problems such as: o 1 is equivalent to 1.65 euros. 1 is also equivalent to 1.5 US dollars ($1.5). How many euros are equivalent to $6? o A car travels 450 km on 50 litres of fuel. How many litres will it use to travel 81 km? Consolidate and extend mental methods of calculation, accompanied where appropriate by suitable jottings As outcomes, Year 9 pupils should, for example: Strategies for mental addition and subtraction Consolidate and use addition and subtraction strategies from previous years. For example: o Find the length of wire in this framework. 4(6.7) + 4(5.2) +4(4.1) = 4 x 16 = 64cm Strategies for multiplication and division Consolidate and use multiplication and division strategies from previous years. For example: o Find the volume of this square-based cuboid.

234 4.5 x 4.5 x 12 = 9 / 2 x 9 / 2 x 12 = 9 x 9 x 3 = 243 cm 3 Recall of fraction, decimal and percentage facts Know that is half of one per cent, so that o 37.5% = 37% + 0.5% o = Strategies for finding equivalent fractions, decimals and percentages For example: o Express as an equivalent percentage. (Recognise this as 62%, plus half of one percent, or 62.5%) o Express 10.5 as an equivalent percentage (recognise this as 1000% plus 50% or 1050%.) Simplify fractions by cancelling highest common factors mentally For example: o Simplify 85/ / 720 Strategies for calculating fractions and percentages of numbers and quantities. For example: o 2/5 of 20.5 = 8.2 (e.g. find one fifth, multiply by 2) o 3/8 of 6400 = 2400 (e.g. find one eighth, multiply by 3) o Find 20% of Word problems and puzzles (all four operations) Apply mental skills to solving simple problems, using jottings if appropriate. For example: o The probability that a train will be late is Of 50 trains, how many would you expect to be late? o Some girls and boys have 32 between them. Each boy has 4 and each girl has 5. How many boys are there? o Find two numbers: Whose sum is -11 and whose product is 28; Whose difference is 4 and whose quotient is 3. Make estimates and approximations Recognise the effects of rounding up and down on a calculation. Discuss questions such as: Why is 6 2 a better approximation for than 7 2? Recognise when approximations to the nearest 10, 100, 1000 are good enough, and when they are not. Estimating the answer, e.g x x 0.1 = = 5

235 Written methods Use efficient column methods for addition and subtraction of whole numbers, and extend to decimals As outcomes, Year 9 pupils should, for example: Use a standard column procedure for addition and subtraction of numbers of any size, including a mixture of large and small numbers with differing numbers of decimal places Algebra Equation, formulae and identities Use the letter symbols and distinguish their different roles in algebra As outcomes, Year 9 pupils should, for example: Use, read and understand vocabulary from the previous year and extend to: Identity, identically equal, inequality... subject of the formula Common factor, factorise, index law Linear, quadratic, cubic.. Know the distinction between equations, formulae and functions. For example: o In the equation 5x + 4 = 2x + 31, x is a particular unknown. o In the formula v = u + at, v, u, a and t are variable quantities, related by the formula. Once the values of three of the variables are known, the fourth value can be calculated. o In the function y = 8x + 11, for any chose value of x, the related value of y can be calculated. Know that an inequality or ordering is a statement that one expression is greater or less than another. For example: o x 1 means that x is greater than or equal to 1. o y 2 means that y is less than or equal to 2. An inequality remains true if the same number is added to or subtracted from each side, or if both sides are multiplied or divided by the same positive number. Multiplying or dividing by a negative number reverses the sense of the inequality. Know that algebraic operations follow the same conventions and order as

236 arithmetic operations; use index notation and the index laws As outcomes, Year 9 pupils should, for example: Apply simple instances of the index laws for multiplication and division of small integer powers. For example: n 2 x n 3 = n 2+3 = n 5 p 3 p 2 - p 3-2 = p Know and use general forms of the index laws for multiplication and division of positive inter powers. p a x p b = p a+b p a p b = p a-b (p a ) b = p ab Simplify or transform algebraic expressions As outcomes, Year 9 pupils should, for example: Simplify or transform linear expressions by taking out single-term common factors. Continue to use the distributive law to multiply a single term over a bracket. Extend to taking out single term common factors. Recognise that letters stand for numbers in problems. For example: o Simplify these expressions: 3(x 2) 2(4 3x) (n + 1) 2 (n + 1) + 1 o Factorise: 3a + 6b = 3(a + 2b) x 3 + x 2 + 2x = x(x 2 + x + 2) o Write an expression for each missing length in this rectangle. Write each expression as simply as possible. 2 o The area of a rectangle is 2x + 4x. Suggest suitable lengths for its sides. What if the perimeter of a rectangle is 2x 2 + 4x? Add simple algebraic fractions. Generalise from arithmetic that: a ± c = ad ± bc b d bd Square a linear expression, and expand and simplify the product of two linear expressions of the form x ± n.

237 Derive and use identities for the product of two linear expressions of the form (a ± b)(c ± d) = ac ± ad ± bc + bd Solve problems such as: In this diagram h, j, and k can be any integers. The missing number in each cell is found by adding the two numbers beneath it. Will the number in the top cell well always be even? Click the correct answer and explanation. Solve linear equations, selecting an appropriate method As outcomes, Year 9 pupils should, for example: Use, read and understand: inequality, region and, or Solve linear equations with negative signs anywhere in the equation, negative solution. Solve linear equations using inverse operations, by transforming both sides in the same way or by other methods. For example: o Compare different ways of solving think of a number problems and decide which would be more efficient retaining brackets and using inverse operations, or removing brackets first. For example: I think of a number, add 3, multiply by 4, add 7, divide by 9, then multiply by 15. The final answer is 105. What was the number that I thought of? o Jack, Jo and Jim are sailors. They were shipwrecked on an island with a monkey and a crate of 185 bananas. Jack ate 5 more bananas than Jim. Jo ate 3 more bananas than Jim. The monkey ate 6 bananas. How many bananas did each sailor eat? o The length of a rectangle is three times its width. Its perimeter is 24 centimetres. Find its area. Click on the correct expression for its area. o Solve these equations: 3c 7 = -13 4(b 1) - 5 (b + 1) = 0 4( z + 5) = 8

238 Form linear equations (unknown on both sides) and solve them by transforming both sides in the same way. Begin to recognise what transformations are needed and in what order. For example: o Multiplying a number by 2 and then adding 5 gives the same answer as subtracting the number from 23. What is the number? o The sum of the ages of a mother and her daughter is 46. In three years; time the mother will be three times as old as her daughter is then. How old is the daughter now? o Solve these equations: 7(s + 3)= 45 3(12 s) 3(2a 1) = 5(4a 1) 4( 3a 2) Solve a pair of simultaneous linear equations Solve a pair of simultaneous equations by eliminating one variable. Know that simultaneous equations are true at the same time and are satisfied by the same values of the unknowns involved, and that the linear simultaneous equations may be solved in a variety of ways. Substitute from one equation into another. For example: x and y satisfy the equation 5x + y = 49. They also satisfy the equation y = 2x. Find x and y. Write down another equation satisfied by x and y. Method: From the second equation, y = 2. Substituting into the first equation gives: 5x + 2x = 49 So x = 7 and y = 14. Other equations might be x + y = 21, y = x + 7. o Solve the equations: 3x 5y = 22 x = 3y + 2 Extend the substitution method to examples where one equation must be rearranged before the substitution can be made. For example: o Solve the equations: x 2y = 5 2x + 5y = 100 From the first equation, x = 5 +2y. Substituting into the second equation gives. 2(5 + 2y) + 5y = 100 Add or subtract equations. For example: o Solve the equations: x + 3y = 56 x + 6y = 101 Method x + 3y = 56 x + 6y = 101

239 Compare the two equations and deduce that 3y = 45 so y = 15. Substituting into the first equation gives x = 11. o Solve the equations: 4x + y = 44 x + y = 20 Extend to adding or subtracting equations in order to eliminate one variable. For example: 2x + y = 17 Multiply by 2: 4x + 2y = 34 3x + 2y = 28 Subtract: 3x + 2y = 28 x = 6 Form and solve linear simultaneous equations to solve problems. For example: o In five years time, Ravi s father will be twice as old as Ravi. In 13 years; time, the sum of their ages will be 100. How old is Ravi now? Use algebraic methods to solve simple non-linear equations. For example: o Solve these equations exactly. Each has two solutions. a. c = 60 c. x 2 5= 220 b. 9 = y + 2 d. 3 = 12 y + 2 x 2 Set up and use equations to solve word and other problems involving direct proportion Solve problems involving proportion using algebraic methods, relating algebraic solutions to graphical representations of the equations. For example: If variables x and y take these values: x y o The relationship between the variables is expressed by y = 3.5x o The graph of y against x is a straight line through the origin. Use algebraic methods to solve problems such as: o Green paint is made by mixing 11 parts of blue paint with 4 parts of yellow paint. How many litres of blue paint would be needed to mix with 70 litres of yellow paint? Algebraic method: Let b be the number of litres of blue paint needed: Blue b = 11 Yellow 70 4 How many litres of blue paint would be needed to make up 100 tins of green paint? Blue b = 11 Green Use formulae from mathematics and other subjects

240 As outcomes, Year 9 pupils should, for example: Substitute positive and negative numbers into linear expressions and expressions involving powers. For example: o Find the value of these expressions 3x x 3-2x When x = -3 and when x = 0.1. o Find the values of a and b when p = 10. a = 3p 3 b = 2p 2 (p 3) 2 7p o Here are two formulae. P = s + t + 5 (s 2 + t 2 ) A = 1 / 2 st + (s 2 + t 2 ) 3 9 Work out the values of P and A when s = 1.7 and t = 0.9. Explain the meaning of and substitute numbers into formulae such as: o A formula to convert C degrees Celsius to F degrees Fahrenheit: F = 9C + 32 or F = 9(C + 40) Find an unknown where it is not the subject of the formula and where an equation must be solved. For example: o The surface area S of a cuboid of length l, width w and height h is S = 2lw + 2lh + 2hw What is h if S = 410, l = 10 and w = 5? In simple cases, change the subject of a formula, using inverse operations. For example: o Make I or R the subject of the formula V = IR o Make l or w the subject of the formula P = 2(l + w) o Make b or h the subject of the formulae A = ½ bh V = lbh o Make u or v the subject of the formula = 1 v u f Sequences and functions Generate terms of a sequence using term-to-term and position-to-term definitions of the sequence. As outcomes, Year 9 pupils should, for example: Generate terms of a sequence given a rule for finding each term from the previous term.

241 Review general properties of linear sequences of the form an + b, where a and b take particular values, e.g. 2n + 5, 3n 7, 10 4n. o The sequence can also be defined by 1 st term, T(1) = a + b Term to term rule, add a to the previous term so T(2) = a + T(1), etc. o The nth term of a sequence is 2n, i.e. T(n) = 2n. Write the first five terms. Find the nth term As outcomes, Year 9 pupils should, for example: Analyse sequences from practical contexts. For example: Find the first terms of the sequence; describe how it continues using a term-to-term rule. Use algebraic expressions to describe the nth term, justifying them by referring to the context. When appropriate, compare different ways of arriving at the generalisation. Begin to use linear expressions to describe the nth term of an arithmetic sequence, justifying its form by referring to the activity from which it was generated. Develop an expression for the nth term of sequences, such as: 7, 12, 17, 22, n Express functions and represent mappings As outcomes, Year 9 pupils should, for example: Use vocab from the previous year and extend to: Identity function, inverse function, quadratic function Identify the graph of a linear function. Know some properties of a quadratic functions and features of their graphs. For example: o The graph is a curve, symmetrical about the vertical line through its turning point. o The value of the y-coordinate at the turning point is either a maximum or a minimum value of the function. Graphs and functions

242 Generate points and plot graphs of functions As outcomes, Year 9 pupils should, for example: Use vocabulary from the previous year and extend to: Quadratic function and cubic function Recognise the graphs of linear functions in the form: ay + bx + c = 0 and consider its features. Given values of m and c, find the gradient of lines given by equations of the form y = mx + c. Compare changes in y with corresponding changes in x, and relate to changes to a graph of the function. For example: Recognise that a graph of the form y = mx + b: o y = 3x + 1 x y The difference is 3. Change in y = 4 1 = = 3 Change in x Know that for the straight line y = mx + c. m = change in y ; change in x m is called the gradient of the line and is a measure of the steepness of the line; if y decreases as x increases then m will be negative. Interpret linear functions arising from real-life problems and plot and interpret their corresponding graphs As outcomes, Year 9 pupils should, for example: Interpret a range of graphs arising from real situations. For each of the situations below, suggest which sketch graph has a shape that most accurately describes it:

243 The distance (y) travelled by a car moving at constant speed on a motorway, plotted against time (x); The number (y) of litres of fuel left in the tank of a car moving at constant speed, plotted against time (x). The distance (y) travelled by an accelerating racing car, plotted against time (x); The number (y) of dollars you can purchase for a given amount in pounds sterling (x); The temperature (y) of a cup of tea left to cool to room temperature, plotted against time (x); The distance (y) a runner runs, plotted against time (x), if he starts by running flat out and gradually slowing down until he collapses with exhaustion. The amount (y) of an infection left in the body as it responds to treatment, slowly at first, then more rapidly, plotted against time (x). Shape, Space and measures Geometric reasoning: lines, angles and shapes Use accurately the vocabulary, notation and labelling conventions for lines, angles and shapes; distinguish between conventions, facts, definitions and derived properties As outcomes, Year 9 pupils should, for example: Use vocabulary from the previous year. Identify properties of angles and parallel and perpendicular lines, and use these properties to solve problems As outcomes, Year 9 pupils should, for example: Explain how to find, calculate and use properties of the interior and exterior angles of regular and irregular polygons. Explain how to find the interior angle sum and the exterior angle sum in (irregular) quadrilaterals, pentagons and hexagons. For example: o A polygon with n sides can be split into n 2 triangles, each with an angle sum of 180 o. So the interior angle sum is (n 2) x 180o, giving 360o for a quadrilateral, 540o for a pentagon and 720o for a hexagon. Find calculate and use the interior and exterior angles of a regular polygon with n sides. For example: o The interior angle sum S for a polygon with n sides is S = (n 2) x 180 o. In a regular polygon all the angles are equal, so each interior

244 angle equals S divided by n. Since the interior and exterior angles are on a straight line, the exterior angle can be found by subtracting the interior angle from 180 o. Recall that the interior angles if an equilateral triangle, a square and a regular hexagon are 60 o, 90 o, 120 o respectively. Identify and use the geometric properties of triangles, quadrilaterals and other polygons to solve problems; explain and justify inferences and deductions using mathematical reasoning As outcomes, Year 9 pupils should, for example: Use vocabulary from the previous year and extend to: Hypotenuse, Pythagoras theorem Know and use angle and symmetry properties of polygons, and angle properties of parallel and intersecting lines, to solve problems and explain reasoning. For example: Know and use properties of triangles, including Pythagoras theorem. Know that: o In any triangle, the largest angle is opposite the longest side and the smallest side. o In a right-angle triangle, the side opposite the right angle is the longest and is called the hypotenuse. Understand and recall Pythagoras theorem: o As a property of areas: in a right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. o As a property of lengths; a 2 + b 2 = c 2 o Appreciate that: If a 2 > b 2 + c 2, then A is an obtuse angle. If a 2 < b 2 + c 2, then A is an acute angle. Use Pythagoras theorem to solve simple problems in two dimensions. For example: o You walk due north for 5 miles, then due east for 3 miles. What is the shortest distance you are from your starting point? o A ladder leans against a wall with its foot 1.5 m away from the wall. How far up the wall does the ladder reach? Understand congruence and similarity As outcomes, Year 9 pupils should, for example: Congruence Appreciate that when two shapes are congruent one can be mapped on the other by a translation, reflection or rotation, or some combination of these transformations.

245 Know from experience of constructing them that triangles satisfy SSS, SAS, ASA or RHS are unique, but that triangles satisfying SSA or AAA are not. Appreciate that two triangles will be congruent if they satisfy the same conditions: o Three sides are equal (SSS); o Two sides and the included angle are equal (SAS); o Two angles and a corresponding side are equal (ASA) o A right angle and a corresponding side are equal (RHS). Use these conditions to deduce the properties of triangles and quadrilaterals. For example: o Draw triangle ABC with AB= AC. Draw triangle ABC, AB = AC. Draw the perpendicular from A to BC to meet BC at point D. o Show that triangles ABD and ACD are congruent. Hence show that the two base angles of an isosceles triangle are equal. o Use congruence to prove that the diagonals of a rhombus bisect each other at right angles. o Consider a parallelogram with a diagonal drawn. By using alternate angle property, use congruence to prove that the opposite sides of a parallelogram are equal. Identify and use the properties of circles As outcomes, Year 9 pupils should, for example: Circles Know the parts of a circle including: Centre, radius, diameter, circumference, chord, arc, segment, sector, tangent.. Know that: A circle is a set of points equidistant from its centre. The circumference is the distance around the circle. The radius is the distance from the centre to the circumference. An arc is part of the circumference.

246 A sector is the region bounded an arc and two radii. Know that when the line; Touches the circle at a point P, it is called a tangent to the circle at that point; Intersects the circle at two points A and B, the line divides the area enclosed by the circle into two regions called segments; Passes through the centre of the circle, the line segment AB becomes a diameter, which is twice the radius and divides the area by the circle into two semicircles. Recognise that a tangent is perpendicular to the radius at the point of contact P. Use 2-D representations, including plans and elevations, to visualise 3-D shapes and deduce some of their properties As outcomes, Year 9 pupils should, for example: Use vocabulary from the previous year and extend to: Cross-section, projection plane Analyse 3 D shapes through 2-D projections and cross-sections, including plans and elevations. Visualise and describe sections obtained by slicing in different planes. For example: o Is it possible to slice a cube so that the cross-section is: a. a rectangle? b. a pentagon? c. a hexagon? Imagine you have a cube. Put a dot in the centre of each face. Join the dots on adjacent sides by straight lines. What shape is generated by these lines? Transformations Understand and use language and notation associated with reflections, translations and rotations Recognise and visualise transformations and symmetries of shapes Use vocabulary from the previous years and extend to: plane symmetry, plane of symmetry axis of rotation symmetry Combinations of transformations Transform 2 D shapes by combining reflections, rotations and translations. Know that reflections, rotations and translations preserve length and angle, and map objects on to congruent images.

247 Use mental imagery to consider a combination of transformations and relate the results to symmetry and other properties of the shapes. For example: o Say what shape the combined object and image(s) form when: a. a right-angled triangle is reflected along its hypotenuse; b. a square is rotated three times through a quarter turn about a corner; c. a scalene triangle is rotated through 180o about the mid-point of one of its sides. Work practically when appropriate, solve problems such as: o Reflect this quadrilateral in the y- axis. Then reflect in both shapes in the x-axis. In the resulting pattern, which lines and which angles are equal in size? Some congruent L-shapes are placed on a grid in this formation. Describe transformations from shape C to each of the other shapes. Use and interpret maps and scale drawings

248 As outcomes, Year 9 pupils should, for example: Use and interpret maps and scale drawings. Understand different ways in which the scale of a map can be represented and convert between them, e.g. 1 : or 2 cm to 1 km. Measure from a real map or scale drawing. For example; o Use the scale of a map to convert a measured map distance to an actual distance on the ground. o Measure dimensions in a scale drawing and convert them into actual dimensions. Coordinates Use coordinates in all four quadrants As outcomes, Year 9 pupils should, for example: Use vocabulary from the previous year and extend to.. Pythagoras theorem Find points that divide a line in a given ratio, using the properties of similar triangles. Given the coordinates of points A and B, calculate the length of AB. Construction and loci Find simple loci, by reasoning As outcomes, Year 9 pupils should, for example: Find loci, by reasoning: o A spider is dangling motionless on a single web. I move a finger so that its tip is always 10 cm from the spider. What is the locus of my finger tip? (the surface of a sphere) Measures and mensuration Use units of measurement to measure estimate, calculate and solve problems in a range of contexts; convert between metric units. As outcomes, Year 9 pupils should, for example: Continue to use familiar units of measurement from the previous years and extend to: Compound measures Average speed (distance/time) Density (mass/volume) Convert between metric units, including area, volume and capacity measures. For example:

249 o Change square centimetres (cm 2 ) into square metres (m 2 ). o Change square metres (m 2 ) into hectares; o Change 5.5 cubic centimetres (cm 3 ) into cubic millimetres (mm 3 ); o Change 3.5 litres into cubic centimetres (cm 3 ) Suggest appropriate units to estimate or measure speed. For example, estimate or suggest units to measure the average speed of: o An aeroplane flying to New York from London, o A swimmer in a race, a snail in motion Extend the range of measures used to angle measure and bearings, and compound measures Compound measures Use and read: Speed, density.. and units such as miles per hour (mph) and meters per second (m/s). Understand that: o Rate is a way of comparing how one quantity changes with anther, e.g. a car s fuel consumption is litres per 100km. o The two quantities are usually measured in different units and per, the abbreviation p or an oblique / is used to mean for every or in every. Know that if a rate is constant(uniform), then the two variables are connected by a simple formula. For example: o Speed = o Density = distance travelled time taken mass of object volume of object Know that if a rate varies, the same formula can be used to calculate the average rate. Solve problems involving average rates of change. For example: o The distance from London to Leeds is 190 miles. An intercity train takes about 2 ¼ hours to travel from London to Leeds. What is its average speed? Deduce and use formulae to calculate lengths, perimeters, area and volumes in 2-D and 3-D shapes As outcomes, Year 9 pupils should for example: Use vocabulary from the previous year and extend to: Circumference, π And the names of the parts of a circle. Know and use the formula for the circumference of a circle. For example:

250 Know that the formula for the circumference of a circle is C = πd, or C = 2πr, and that different approximations to π are 3, 22 / 7, or 3.14 correct to 2 d.p. Calculate the circumference of circles and arcs of circles. For example: o The diameter of King Arthur s Round Table is 5.5m. A book claims that 50 people sat round the table. Assume each person needs 45 cm round the circumference of the table. Is it possible to sit 50 people around it? o The large wheel on Wyn s wheelchair has a diameter of 60 cm. Wyn pushes the wheel round exactly once. Calculate how far Wyn has moved. The large wheel on Jay s wheelchair has a diameter of 52 cm. Jay moves her wheelchair forward 950 cm. How many times does the large wheel go round? o A Ferris wheel has a diameter of 40 metres. How far do you travel in one revolution of the wheel? Know and use the formula for the area of a circle: A = πd 2 /4 or A = πr 2 For example: o A circle has a radius of 15 cm. What is its area? o Calculate the area of the shaded shape. o The inside lane of a running track is 400m long, 100m on each straight and 100m on each semicircular end. What area in the middle is free for field sports? Surface area and volume of a cylinder Know that the total surface area A of a cylinder of height h and radius r is given by the formula A = 2πr 2 + 2πrh And that the volume of the cylinder is given by the formula V = πr 2 h Solve problems such as: o This door wedge is in the shape of a prism. The shaded face is a trapezium. Calculate its area. Calculate the volume of the door wedge.

251 Handling data Processing and representing data Calculate statistics from data, finding the mode, mean, median and range As outcomes, Year 9 pupils should, for example: Use vocabulary from the previous year. Calculate statistics. For example. o Three people have a median age of 30 and a mean age of 36. The range of their ages is 20. How old is each person? o Three children have a mean age of 10. The range of their ages is 6. What is the lowest possible age: a. of the youngest child? b. Of the oldest child? o Amrita has five cards numbered in the range 0 to 20. She says: The range of my cards is 4, the mode is 6 and the mean is 5. Is this possible? Interpreting and discussing results Interpret diagrams and graphs, and draw inferences Interpret diagrams, graphs and charts, and draw inferences from data representations to support and to cast doubt on initial conjectures. For example: o Interpret pie charts, e.g. showing how British adults spend their time.

252 Consider this graph records the average time occupations of adults. If only every second adult had paid work, about how many hours would the paid workers be working each week? 20 hours 30 hours 40 hours 54 hours o Interpret frequency diagrams. Probability Use the probability scale; find and justify theoretical probabilities Know that the sum of probabilities of all mutually exclusive outcomes is 1. Use this to solve problems. For example: o A number of discs are placed in a bag. Most are marked with a number 1, 2, 3, 4, or 5. The rest are unmarked. The probabilities of drawing out a disc marked with a particular number are: P(1) = 0.15 P(2) = 0.1 P(3) = 0.05 P(4) = 0.35 P(5) = 0.2 What is the probability of drawing a disc: a. marked 1, 2 or 3 b. not marked with a number? o In an arcade game only one of four possible symbols can be seen in the final window. The probability of each occurring is: Symbol Probability Jackpot 1 / 16 Moon 1 / 4 Star?