NAME:... FORM:... MATHS TOOLKIT 017 018 Year 6
Click on the topic of choice Contents Number... 4 Revision of Four Operations... 4 Multiplication and Division of Decimals... 5 Long Multiplication... 7 Long Division/Division by a double digit divisor... 8 Addition and Subtraction of Fractions... 9 Revision of Number Properties... 10 Product of Prime Factors... 11 Highest Common Factor... 1 Lowest Common Multiple... 13 Negative Numbers... 14 Order of operations - BIDMAS... 15 Ratio... 16 Fractions and Percentages of Quantities... 17 Fraction, Decimal and Percentage Conversions... 18 Measures, Shape and Space... 0 Shape properties... 0 Perimeter and Area... Surface Area and Volume of Cubes and Cuboids... 4 Metric and Imperial Measurements... 5 Data Handling... 7 Pie Charts... 7 Algebra... 8 Collecting Like Terms... 8 Forming expressions... 9 Simple Substitution... 30 Number... 31 Multiplication and Division of Fractions... 31 Measures, Shape and Space... 3 Plotting Shapes... 3 Translation... 33 Reflection... 34 Copyright 017 Dulwich Prep London
Rotation... 35 Enlargement... 36 Data Handling... 37 Revision of Averages... 37 Algebra... 38 One stage and two stage equations... 38 Expanding brackets and simplifying... 39 Further solving equations (ext.)... 40 Using the nth term to derive sequences... 41 Number... 4 Percentage Change Profit and Loss... 4 Measures, Shape and Space: Constructing Triangles... 43 Speed Distance Time... 44
Number Revision of Four Operations Remember to set out your calculations properly using column calculations. Most mistakes are made when trying to work out answers in your head! Make sure your calculations are aligned properly ensure your units columns are underneath each other when adding or subtracting. 7459 + 395 7 4 5 9 + 3 9 5 7 8 5 4 1 1 034 917 1 1 0 3 1 4 9 1 7 1 1 1 7 836 x 8 8 3 6 x 8 6 6 8 8 6 4 871 9 0 3 1 9 9 8 1 7 8 1
Number Multiplication and Division of Decimals Multiplying or Dividing decimals by a whole number: These are very easy as long as we remember to line up the decimal point in our answer with the decimal point in the question. 8.567 x 5 8 5 6 7 x 5 4 8 3 5 4 3 3 5.65 9 0 8 5 9 5 7 6 4 5 Multiplying a Decimal by a Decimal: If both the numbers are decimals, multiply them by powers of ten until you have whole numbers. Work out the calculation with the whole numbers, then divide the answer. 0.08 x 0.6 = 0 048 8 (x 100) x 6 (x 10) 4 8 ( 1000) Remember you can check whether your decimal place is in the correct place in your answer by counting up the number of numbers after the decimal points in the question are they the same as in your answer? Here there are three decimal places in the question so three decimal places in the answer.
Extension topic: Dividing a Decimal by a Decimal: Think of the calculation as a fraction. Multiplying the top and bottom of a fraction by the same power of ten won t change the value of the fraction (think about it: 1 = 10 = 100 ) so we can 0 00 use this fact to help us solve the calculation. 14.4 0.1 14.4 x 100 1440 = 0.1 x 100 1 Choose a power of 10 that will make the denominator into a whole number! Now do the calculation as usual: 0 1 0 1 1 1 4 4 0 So the answer is 10. This might seem surprising as we are used to division calculations having smaller answers than the questions but it makes sense if you rephrase the question as how many times does 0.1 go into 14.4?.
Long Multiplication Number Long multiplication means both of the numbers you are multiplying will have at least two digits. We add two lines to our working to help us break the calculation down. 43 x 7 4 3 x 7 3 0 1 (7 x 43) 3 (0 x 43) 8 6 0 1 1 6 1 Add the rows together 1 In the first row under the line, calculate 7 x 43 as you would in a normal multiplication calculation. In the second row, calculate 0 x 43. To make it easy, start by putting a zero in the units column. Then you can work out x 43 because the extra zero has the effect of multiplying your answer by 10. Finally, add together the two rows, making sure that you don t add in any of your carrying numbers by mistake (the little ones above!) 36 x 86 8 6 x 3 6 1 7 1 6 (6 x 86) 1 5 3 (30 x 86) 8 5 8 0 1 1 0 9 6 Add the rows together 1 1 Make sure you put the 3 digit number above the digit number to make it easier. It is good to get into the habit of filling in the extra zero before you even start completing the sum so that you don t forget about it!
Number Long Division/Division by a double digit divisor Long division involves dividing by a number with two or more digits. With some straightforward numbers we can use our normal short division method. 6456 1 0 5 3 8 1 6 6 4 4 5 9 6 With larger numbers we can use the method below: 5110 35 35 goes into 51 once. Write 1 above the line and take 35 away from 51 to leave you with 16. Then pull the 1 down from the tens column of the sum. 35 goes into 161 four times. Four 35s are 140, so take 140 away from 161, which gives you 1. Pull the final zero down from the sum to give you 10. 35 goes into 10 6 times. 10 10 = 0. You have finished! 0 1 4 6 3 5 5 1 1 0 3 5 1 6 1 1 4 0 1 0 1 0 0 or consider dividing by factors 5110 35 Take any factor pair of the divisor (but not 1 and the number itself). In this case 7 and 5 are the only factor pair. Now divide successively by each factor. It doesn t matter what order you do this in. 0 7 3 0 7 5 5 1 1 0 1 4 6 5 7 3 3 0
Number Addition and Subtraction of Fractions It is very easy to add or subtract fractions if their denominators are the same. 7 + 3 7 = 5 7 Sometimes you may need to simplify your answer or to convert it to a mixed number 7 8 3 8 eg 6 7 + 4 7 = 4 8 = 10 7 = 1 = 1 3 7 If the denominators are not the same, then you should use your knowledge of equivalent fractions to make them the same. 3 4 1 3 = 9 1 4 1 Think of the smallest number which is a multiple of both 4 and 3. In this case it is 1. Convert both fractions to twelfths. Here, we multiply both numbers in the first fraction by 3 and in the second fraction by 4. = 5 1 Add or subtract any whole numbers before you start. 4 5 + 3 3 4 = 7 8 0 + 15 0 = 7 3 0 = 8 3 0 In this case, the lowest common denominator is 0. Multiply both numbers in the first fraction by 4 and in the second fraction by 5. In the third stage, we need to convert the improper fraction ( 3 0 ) into a mixed number (1 3 ) and add the 1 onto the 7 to make 8. 0
Revision of Number Properties Number Factors: Factors are numbers which multiply together to make a number. and 3 are factors of 6 because x 3 = 6. Most numbers have an even number of factors because they come in pairs. The factors of 1 are 1,, 3, 4, 6, 1 because 1 x 1 = 1, x 6 = 1 and 3 x 4 = 1 Square numbers have an odd number of factors because one factor multiplies by itself The factors of 16 are 1,, 4, 8, 16 because 1 x 16 = 16, x 8 = 16, 4 x 4 = 16 Multiples: The multiples of a number divide by that number without leaving a remainder. multiples of 7 include 7, 14, 1, 8, 35 because all these numbers divide by 7 without leaving a remainder Prime Numbers: Prime numbers have exactly two factors; themselves and 1. 5 is a prime number because its only factors are 1 and 5. NB 1 is not a prime number because it only has one factor. The first ten prime numbers are:, 3, 5, 7, 11, 13, 17, 19, 3, 9 Square Numbers: Square numbers are produced by multiplying whole numbers by themselves 16 is a square number because 4 x 4 = 16 The first ten square numbers are: 1, 4, 9, 16, 5, 36, 49, 64, 81, 100
Product of Prime Factors Number The word Product means to multiply. So the Product of Prime Factors is which prime numbers can be multiplied together to make a number. The Product of Prime Factors for 6 are: 3 To find the product of prime factors we use a Factor Tree. This involves splitting (dividing) a number by the factors that go into it. For example: 18 9 3 3 18 = 3 3 OR 60 30 15 3 5 60 = 3 5 = 3 5
Highest Common Factor Number The Highest Common Factor (HCF) is the largest number that can go into or more numbers. The HCF of: 16 and 1 is 4 30 and 18 is 6 4 and 35 is 7 We find the HCF by using the Product of Prime Factors. 18 4 9 1 3 3 6 18 = 3 3 3 4 = 3 18 and 4 both have one and one 3 in common. The HCF is found by multiplying these common factors together. HCF = 3 = 6
Lowest Common Multiple Number The Lowest Common Multiple (LCM) is the smallest number that can or more numbers go into. It will be the lowest number that is in both (all) the times tables of that number. The LCM of: 1 and 10 is 60 15 and 6 is 30 5 and 0 is 100 We find the LCM by using the Product of Prime Factors. 18 4 9 1 3 3 6 18 = 3 3 3 4 = 3 We take all of the prime factors from the first number, then take those from the second number not already used and multiply them altogether. LCM = 3 3 = 7 Check 18, 36, 54, 7, 90, 108 4, 48, 7, 96, 10
Negative Numbers Number It is often useful to think of a number line when working with negative numbers What is -8 + 6? +6 So -8 + 6 = - Subtracting negative numbers is the same as adding positive numbers 8 - - 9 = 8 + 9 = 17 Multiplying and Dividing Negative Numbers: Remind yourself of the rules: x = 4 4 = x = 4 4 = x = 4 4 = X = 4 4 =
Order of operations - BIDMAS Number BIDMAS tells us the order in which to do a calculation Brackets Indices Division and Multiplication (left to right across the page) Addition and Subtraction (left to right across the page) (a) 16-7 x + 4 (b) 4 x 3 = 16 14 + 4 = 4 x 9 = + 4 = 36 = 6 (c) 1 x 4 1 = 1 x 16 1 = 8 1 = - 4
Ratio Number Simplifying ratios: Ratios can be simplified in a similar way to fractions, by dividing both sides of the ratio by the same number. eg There are 60 cats and 7 dogs at a rescue centre. Express the ratio of cats to dogs in its simplest form. ( 1) 60 : 7 ( 1) 5 : 6 Missing numbers: Make sure you multiply both sides of the ratio by the same number. eg The ratio of boys to girls in a school is 3:. If there are 180 boys, how many girls are there in the school? 3 : x 60 x 60 180 : 10 Sharing out: There are three stages to remember: 1. Add the ratio numbers together to find the total number of parts. Divide the quantity you are sharing by the number of parts to find out how much is in each part 3. Multiply each number in the ratio by the quantity in each part eg 80 was shared between James and Emma in the ratio 3:5. How much money did they each receive? 1. 3 + 5 = 8. 80 8 = 10 3. James: 3 x 10 = 30 Emma: 5 x 10 = 50
Number Fractions and Percentages of Quantities Fractions of Quantities: To find a fraction of a quantity, divide by the denominator then multiply by the numerator Find 1 of 80 5 80 5 = 16 16 x 1 = 16 Find 3 of 48 4 48 4 = 1 1 x 3 = 36 Percentages of Quantities: It is often easiest to start by finding a simple percentage of a quantity o To find 10%, divide the quantity by 10 (because 10% = 1 10 ) o To find 1%, divide the quantity by 100 (because 1% = 1 100 ) o To find 50%, divide the quantity by (because 50% = 1 ) We can use these simple percentages to find more complex percentages eg Find 40% of 700kg 10% = 70 (700 10 = 70) 40% = 80kg (70 x 4 = 80) Find 13% of 900m 10% = 90 (900 10 = 90) 1% = 9 (900 100 = 9) 3% = 7 (9 x 3 = 7) 13% = 117m (90 + 7 = 117)
Number Fraction, Decimal and Percentage Conversions Decimal to Percentage: Percent means out of 100 so simply multiply by 100. 0 3 x 100 = 30% 0 73 x 100 = 73% Percentage to Decimal: Reverse the process above; divide by 100. 43% 100 = 0 43 9% 100 = 0 09 Fraction to Decimal: 1. One easy way to change a fraction into a decimal is to find an equivalent fraction in terms often tenths or hundredths 5 3 = 75 4 5 100 = 0.75 5 = 4 10 = 0.4. Some fractions cannot be easily converted into a fraction in terms of tenths or hundredths. In this case we can divide the numerator by the denominator 3 8 = 0.375 0 3 7 5 8 3 3 0 6 0 4 0 Decimal to Fraction: Put the numbers over the appropriate power of ten (usually 10, 100 or 1000) and simplify. 0.8 = 8 10 = 4 5 0.68 = 68 4 100 4 = 17 5
Percentage to Fraction: Put the numbers over 100 and simplify. 4% = 4 4 100 4 = 6 5 80% = 80 0 100 0 = 4 5 Fraction to Percentage: Convert the fraction to a decimal and then multiply by 100. 5 13 0 5 = 65 100 = 0.65 0 65 x 100 = 65% 5 8 = 0.65 0 65 x 100 = 6.5% 0 6 5 8 5 5 0 0 4 0 Make sure you learn the common conversions: Fraction Decimal Percentage 1 100 0.01 1% 1 10 0.1 10% 1 5 0. 0% 1 4 0.5 5% 1 3 0. 3 33 1 % 3 1 0.5 50% 3 0. 6 66 % 3 3 4 0.75 75%
Measures, Shape and Space Shape properties You need to learn the following D (two dimensional flat) shapes: TRIANGLES (Three sided shapes): EQUILATERAL (3 sides equal) ISOSCELES ( sides equal) SCALENE (all sides different) RIGHT ANGLED QUADRILATERALS (Four sided shapes) SQUARE RECTANGLE PARALLELOGRAM RHOMBUS (all sides equal don t call it a diamond!) TRAPEZIUM KITE OTHERS: PENTAGON (5 sides) HEXAGON (6 sides) HEPTAGON (7 sides) OCTAGON (8 sides) NONAGON (9 sides) DECAGON (10 sides) CIRCLE ELLIPSE (NOT Oval)
You also need to know these 3D (three dimensional solid) shapes: CUBE CUBOID CONE CYLINDER SPHERE SQUARE BASED PYRAMID TRIANGULAR PRISM Line symmetry and rotational symmetry If you can put a mirror on a shape and its reflection would look the same, it has line symmetry. A square has four lines of symmetry: A rectangle has two lines of symmetry: The rotational symmetry of a shape is how many times it would fit onto itself in one full turn. Every shape has rotational symmetry of order 1 or above A square has rotational symmetry of order 4, a parallelogram has rotational symmetry of order
Perimeter and Area Measures, Shape and Space The perimeter of any shape is the distance around the edge of it. For a rectangle, this will mean four straight lines. (not to scale) 10cm 6cm Perimeter = 10 + 6 + 10 + 6 = 3cm You can find the area of a rectangle by counting the number of square centimetres that fit inside it, but this could take a very long time! A much quicker way is to multiply the length (number of columns of squares) by the width (number of rows of squares) 5cm 4cm Area = length x width = 5 x 4 = 0cm² Area of a triangle = base x height 5cm 7cm Area = 5 x 7 = 35 = 17.5cm²
To find the area of compound shapes you will need to split the shape up into simpler shapes 3cm Area of A = 4 x 3 A 4cm = 1 cm 7cm B Area of B = 6 x 3 = 18cm² Total area = 1 + 18 6cm = 30 cm To find the area of a compound shape, add up all of the side lengths. Write in any missing lengths onto the shape. 3cm 4cm 3cm Work out the missing lengths, labelled here with the arrows. Then add up all of the sides: 7cm 7 + 3 + 4 + 3 + 3 + 6 = 6cm 6cm 3cm
Measures, Shape and Space Surface Area and Volume of Cubes and Cuboids Surface area remember that cubes and cuboids have six faces: top, bottom, left, right front, back. 8cm 3cm 5cm Top: 8 x 3 = 4 Bottom: 8 x 3 = 4 Left: 3 x 5 = 15 Right: 3 x 5 = 15 Front: 5 x 8 = 40 Back: 5 x 8 = 40 SURFACE AREA = 158cm² Volume is the amount of space that a 3 dimensional (3D) object occupies. We may also consider volume as the amount of space inside a 3D object. Volume is measured in cubic units, such as cm 3, mm 3, m 3. To calculate the volume of a cube or cuboid we need to know the LENGTH, WIDTH and HEIGHT. Formula: Example: Volume = Length Width Height Or V = L W H l = 3 cm Volume = l w h h = cm w = 1 cm Volume = 3 1 Volume = 6 cm 3
Measures, Shape and Space Metric and Imperial Measurements USEFUL METRIC/IMPERIAL CONVERSIONS: 8 km 5 miles/1 mile 1.6km 1 litre 1.75 pints/1 pint 570ml 1 kg.lbs/1 lb 0.45kg 1 m 3 feet 3 inches
For metric units it might help to remember that kilo always means 1000 kilogram is 1000 grams and kilometre is 1000 metres. Similarly, milli means one thousandth so there are 1000 millilitres in a litre, 1000 millimetres in a metre and 1000 milligrams in a gram. Length and Measurement Metric Units: o 1 km = 1000 m o 1 m = 0.001 km o 1 m = 100 cm o 1 cm = 0.01 m o 1 cm = 10 mm o 1 mm = 0.1 cm Imperial Units o 1 ft = 1 in o 1 yard = 3ft o 1760yards = 1 mile [ yards (a chain) = distance between cricket wickets; 1 acre = area of 10 square chains] Mass Metric Units kg = kilograms g = grams mg = milligrams t = tonne o 1kg = 1000g o 1g = 0.001kg o 1g = 1000mg o 1mg = 0.001g o 1kg = 1 000 000mg o 1mg = 0.000 001kg o 1000kg = 1 tonne Imperial Units 1 lb = 16 oz 14lb = 1 stone Capacity Metric Units l = litre cl = centilitre ml = millilitre o 1litre (l) = 1000ml o 1 litre (l) = 100cl o 1cl = 10ml Imperial Units 1 gallon = 8 pints
Data Handling Pie Charts Pie charts show information by splitting a circle into slices or sectors of different sizes. Pie charts are often used to show the results of a survey. The sectors of the circle show what fraction of the total is in each group. 4 people were asked where they went on holiday last year. Their answers are summarised in this frequency table. Country Frequency Spain 7 France 5 Greece 3 Portugal 1 Britain 8 TOTAL: 4 To draw a pie chart 1. Find the angle that each person (or unit) will take up, by dividing 360 0 by the total number. 360 0 4 = 15 0 per person.. I can then find out the angle for each group of data by multiplying the frequency by the degrees per person (or unit). Country Number of Multiply Angle people Spain 7 15 o x 7 105 o France 5 15 o x 5 75 o Greece 3 15 o x 3 45 o Portugal 1 15 o x 1 15 o Britain 8 15 o x 8 10 o Total 4 360 o Remember: The total of the angles adds up to 360 0. Plot the pie chart using a protractor. Remember to include: (a) a title, (b) a label for each sector; and (c) the angle for each sector.
Algebra Collecting Like Terms Adding and Subtracting: We refer to the simplification of terms that are added or subtracted as collecting like terms. This is because we can only add or subtract algebraic terms which are the same. Expressions we can simplify 5a + 8a + a = 15a 3b²c + 9b²c 5b²c = 4b²c d 3 + 9d = 11d 3 6g² + 9g g = 6g² + 7g 5ef + 7fe + ef = 14ef All these letters are the same All these terms involve b²c The numbers and letters can be collected separately The g² and g terms can be collected separately ef and fe are the same as e x f is the same as f x e Expressions we cannot simplify a + 3b + 3c 3b²c + 9bc² d 3 6g² + 9g 5ef + 7EF Different letters cannot be combined b²c means b x b x c bc² means b x c x c Numbers and letters cannot be combined g² and g are not the same and cannot be combined Capital letters are not the same as lower case letters! Multiplying: Unlike adding and subtracting, any algebraic terms can be combined through multiplication. Make sure your simplified expression has no multiplication signs as they are not used in simplified algebra. Expression Simplification Explanation a a a² Use indices where possible 7 d 6 c 4cd Multiply the numbers together and put them at the beginning. Put the letters in alphabetical order. 5 e e² 8 40e³ Multiply numbers then letters. e x e x e = e³
Forming expressions Algebra Decide what the letters will represent in your expression. Often, they represent the number you are trying to discover. Read the instructions very carefully you might wish to underline or highlight key words. Take care to distinguish between multiplying 3 times bigger, 3 times the size and adding 3 years older, 3 metres longer Check that your algebraic expression works by substituting numbers into it. My brother is twice my age. My sister is three years younger than me. Write algebraic expressions for our ages. Let my age be x My brother s age is x My sister s age is x 3 Check with numbers if I am 8, my brother is 8 = 16 and my sister is 8 3 = 5 years old. This makes sense, so the expressions are correct. There are twice as many chocolates as toffees in a packet of sweets, and seven more caramels than toffees. Altogether there are 43 sweets in the packet. Form an equation and solve it to find out how many chocolates there are in the packet. Let the number of toffees be t The number of chocolates is t (twice as many chocolates as toffees) The number of caramels is t + 7 (seven more caramels than toffees) Add these together: t + t + t + 7 = 43 4t + 7 = 43 (-7) 4t = 36 ( 4) t = 9 There are 9 = 18 chocolates in the packet. Check it works: 9 toffees + 18 chocolates + 16 caramels = 43 sweets altogether
Simple Substitution Algebra Remember to show all your working, not just a solution Remember to follow the rules of BIDMAS Be especially careful with negatives! If a = 3, b = - and c = 1, find: a) ab + c = 3 x + 1 = 6 + 1 = 5 BIDMAS tells us to do the multiplication before the addition. Remember that a negative multiplied by a positive is a negative. Then add the two numbers together. b) b² + c² = ( )² + 1² = 4 + 1 = 8 + 1 = 9 BIDMAS tells us to do the indices first (the squaring). Remember that - x - = +4. Any negative number squared gives a positive answer. Next you do the multiplication. Then add the two numbers together. c) (a)² b² = ( 3)² ( )² = 6² ( )² = 36 4 = 3 BIDMAS tells us to deal with the brackets first. Then we do the indices (the squaring). Remember that - x - = +4. Any negative number squared gives a positive answer. Then do the subtraction. d) ac b = 3 1 = 3 Always work out the numerator and denominator of a fraction before dividing. A positive divided by a negative is a negative. Change the improper fraction to a mixed number. = 1 1
Number Multiplication and Division of Fractions SPRING TERM Multiplying Fractions: First cancel any numbers you can. Remember you can only cancel a number on the top with a number on the bottom you cannot cancel two numbers on the top or two numbers on the bottom. Then multiply the numerators together and the denominators together. = 3 3 5 1 5 1 Dividing Fractions: Flip the second fraction and then cancel and multiply as above. = = 4 3 5 1 5 3 4 5 6 Extension topic - Multiplying and Dividing Mixed Numbers: Before you start, make any mixed numbers into improper (top heavy) fractions. 1 4 5 3 = 3 9 8 5 3 1 = 4 5 = 4 4 5
Measures, Shape and Space Plotting Shapes (5,4) (-6,-) When plotting coordinates, always start at the point (0,0) which is in the middle of the grid above. The first number in the coordinate tells you how far to move to the right (positive) or left (negative). The second number tells you how far to move up (positive) or down (negative). When you find the correct point, draw a small neat cross at the intersection of the two lines. You may be asked to label the point with a letter or to join the points up with a ruler.
Translation Measures, Shape and Space Translation means moving a shape or point. We normally use vectors to describe a translation. The top number indicates how far right (+) or left (-) to move ( 3 4 ) The bottom number indicates how far up (+) or down (-) to move So, this vector tells us to move a shape or point 3 squares to the right and 4 squares down. Make sure you check more than one corner of a shape when translating. Translate shape A by the vector ( 7 6 ) A B
Reflection Measures, Shape and Space Shapes are reflected in a mirror line Every point on the reflected shape should be the same distance away from the mirror line as the equivalent point on the original shape Remember that the reflected shape will be the opposite way round from the original shape Reflect shape A in the x-axis and label it B. A B
Rotation Measures, Shape and Space To rotate a shape, we need a centre of rotation, an angle and a direction (clockwise or anticlockwise). You should be able to rotate a shape using tracing paper; try to do it without as well! It may help you to count squares or to draw lines between the centre of rotation and the corners of the shape. NB The origin is the point (0,0). Rotate shape A through 90 clockwise about the centre of rotation marked and label it B. B A Rotate shape A through 180 about the origin and label it B. A B
Enlargement Measures, Shape and Space To enlarge a shape, we need a centre of enlargement and a scale factor. Use the number of squares between the centre of enlargement and the corners of the shape to help you enlarge effectively. Enlarge shape A by a scale factor 3 with a centre of enlargement (-6, 4) Q P R A S Point P is one square to the right and one square up from the centre of enlargement. Multiply these distances by 3 for the enlargement. Point Q is three squares to the right and three squares up from the centre of enlargement. Point R is one square to the right and two squares down from the centre of enlargement. Multiply these distances by 3 for the enlargement. Point Q is three squares to the right and six squares down from the centre of enlargement.
Data Handling Revision of Averages There are three types of average that we study in Year 5. MEAN: Found by adding all the quantities and dividing by the number of quantities MEDIAN: Found by putting all the quantities in order and finding the number in the middle MODE: The most common quantity We also look at the RANGE of a set of numbers. This is the difference between the biggest and smallest quantities., 7, 4, 6, 6 13, 9, 11, 9, 9, 15 MEAN + 7 + 4 + 6 + 6 5 = 5 5 = 5 13 + 9 + 11 + 9 + 9 + 15 6 = 66 6 = 11 MEDIAN, 4, 6, 6, 7 Median = 6 9, 9, 9, 11, 13, 15 (There are two numbers in the middle, so find the half way point between them.) Median = 10 MODE 6 9 RANGE 7 = 5 15 9 = 6
Algebra One stage and two stage equations Remember that you are trying to end up with the unknown term (often x) on its own. Remember that you must keep your equations balanced whatever you do to one side, you must also do to the other side. Remember that you must show what you are doing to each side, either by writing it in brackets or by writing it underneath each side. x + 4 = 7 (-4) x = 3 Subtract 4 from both sides to leave x on its own. x 7 = 6 (+7) x = 13 Add 7 to both sides to leave x on its own. 6x = 30 ( 6) x = 5 Divide both sides by 6 to leave x on its own. x 4 = 9 (x4) x = 36 Multiply both sides by 4 to leave x on its own. x + 3 = 9 (-3) x = 6 ( ) x = 3 8 x = 11 (+ x) 8 = 11 + x (- 11) 3 = x x = 3 NEGATIVE x terms can cause confusion so add x to both sides to make it positive. Remember that your answer can be a negative number! 3(x 4) = 3 3x 1 = 3 (+ 1) 3x = 15 ( 3) x = 5 Start by multiplying out the brackets. Then continue as normal.
Algebra Expanding brackets and simplifying Remember to multiply everything inside the bracket by the numbers and letters outside the bracket. Arrows may help you remember! 3(a + 4) = 3a + 1 c(d 3) = cd 3c After you multiply out (expand) the brackets, you may need to collect like terms to simplify the expression. 4(b 3) + 5(4 + b) = 4b 1 + 0 + 5b = 9b + 8 4a 6(a + 3) = 4a 6a 18 = a 18 t(t + 6) t(7 + t) = t² + 6t 7t t² Remember that, t t = t² = t
Further solving equations (ext.) Algebra Remember that you are trying to end up with the unknown term (often an x) on its own. Remember that you must keep your equations balanced whatever you do to one side, you must also do to the other side. Remember that you must show what you are doing to each side, either by writing it in brackets or by writing it underneath each side. 5x 8 = 9x + 8 (- 5x) 8 = 4x + 8 (- 8) 16 = 4x 4 4 4 = x x = 4 ( 4) If you have x terms on both sides, subtract the smaller number of x s to eliminate them from one side. Then proceed as normal. x 3 = 5 (x ) x 3 = 10 (+ 3) x = 13 The left hand side has been divided by so multiply both sides by. Then add 3 to both sides. 4(x + 3) x = 17 4x + 1 x = 17 3x + 1 = 17 (- 1) 3x = 5 3 3 ( 3) Start by expanding the brackets. Then collect like terms. Then proceed as normal. There is no reason why your answer can t be a fraction or mixed number but avoid using decimals. x = 1 3
Algebra Using the nth term to derive sequences Make sure you know how to find missing terms in a sequence. It helps to look at the gaps between the numbers. 15 3 31 39 The sequence is going up by 7 each time, so the missing number is 46. 30 38 4 46 The sequence is going up by 4 each time, so the missing number is 34. The nth term is an algebraic expression used to describe a sequence. By substituting the term number into the nth term expression we can find out what the sequence is. Find the first 4 terms and the 100 th term of the sequence 4n + 1 1 st term: 4 x 1 + 1 = 5 nd term: 4 x + 1 = 9 3 rd term: 4 x 3 + 1 = 13 4 th term: 4 x 4 + 1 = 17 100 th term: 4 x 100 + 1 = 401 Find the first 4 terms and the 100 th term of the sequence 6 3n 1 st term: 6 3 x 1 = 3 nd term: 6 3 x = 0 3 rd term: 6 3 x 3 = -3 4 th term: 6 3 x 4 = -6 100 th term: 6 3 x 100 = -94
Number Percentage Change Profit and Loss SUMMER TERM The percentage change is the % by which something has increased or decreased. When dealing in currency (money) we refer to this as Profit and Loss. The percentage change is calculated by finding the difference between to the original value and the end value, then we divide it by the original value and multiply it by 100 to turn it in to a %. Formula: % increase = change original 100 Examples: In January a cow weighed 400 lbs. By June the cow weighed 600 lbs. The cow s weight has risen so it is a % increase. % increase = change original 100 = 600 400 400 = 00 400 100 100 The cow has increased its weight by 50%. = 50% A man buys a car for 3500. A year later he sells it for 100. The car has gone down in value, so it is a % decrease or loss. % Loss = change original 100 = 3500 100 3500 = 1400 3500 100 = 40% 100 The value of the car has fallen by 40%.
Measures, Shape and Space: Constructing Triangles Home
Measures Shape and Space Speed Distance Time Distance To calculate the distance travelled we need to know the average speed (velocity) and the time taken. We calculate the distance travelled by multiplying together the average speed and the time taken. For example: If a car travels at 60 mph for hours it would go 10 miles. We set it out as follows: Speed D = s t D = 60 D = 10 miles To calculate the average speed we need to know the distance travelled and the time taken. We calculate the average speed by dividing the distance travelled by the time taken. For example: If a mouse travels at 90 metres in 30 seconds it would have an average speed of 3 m/s. We set it out as follows: s = D t s = 90 30 s = 3 m/s
Time To calculate the time taken we need to know the distance travelled and the average speed. We calculate the time taken by dividing the distance travelled by the average speed. For example: If a train travels at 900 km at 90 km/h it would take of 10 hours. We set it out as follows: t = D s t = 900 90 t = 10 hours NB: We must remember that time uses base 60. This means that 0.5 is 30 minutes not 50 minutes. To calculate time use the your calculator or make the time into a fraction. 14 minutes converts to 14 = 7 60 30 hours. button on