Home NAME:... FORM:... MATHS TOOLKIT Year 5. Copyright 2017 Dulwich Prep London

Transcription

1 NAME:... FORM:... MATHS TOOLKIT Year 5 Copyright 2017 Dulwich Prep London

2 YEAR 5 OVERVIEW Home Contents Number... 5 Addition and Subtraction... 5 Multiplication and Division... 6 Place Value and Rounding whole numbers... 7 Place Value and Rounding decimals... 8 Place Value and Rounding rounding to 1 or 2 Decimal Places... 9 Number Properties Divisibility Rules Long Multiplication Equivalent fractions Improper fractions and mixed numbers Introduction to percentage Fractions and Percentages of Quantities Ratio Measures, Shape and Space Properties of 2D shapes Length and Measurement Perimeter and Area Data Handling Probability Number Addition and Subtraction of Decimals Negative Numbers Long Division/Division by a double digit divisor Multiplying or Dividing Decimals by a Whole Number Measures, Shape and Space Plotting coordinates in all four quadrants Drawing and measuring angles Calculating Angles Time Metric units and scales Surface Area and Volume of Cubes and Cuboids Number... 37

3 Addition and Subtraction of Fractions Order of operations - BIDMAS Averages Bar graphs and line graphs Algebra Introduction to Algebra: Collecting Like Terms Solving Equations... 43

4 Autumn Term Number: Addition and subtraction Multiplication and division Place value and rounding Number properties Long Multiplication Equivalent fractions Improper fractions/mixed Numbers Introduction to percentage Fractions and percentages of quantities Ratio Measures, Shape and Space: Properties of 2D and 3D shapes Length and Measurement Perimeter and Area Data Handling: Probability Spring Term Number: Decimal addition and subtraction Negative numbers Long division/division by a double digit divisor Multiplication and division of decimals Measures, Shape and Space: Plotting coordinates in all four quadrants Drawing and measuring angles Calculating angles (triangles / quadrilaterals / straight lines / around a point) Time Metric units and scales Surface Area and Volume Summer Term Number: Addition and subtraction of fractions Introduction to BIDMAS Data Handling: Averages: Mean, median, mode, range Interpreting graphs Algebra: Introduction to algebra: collecting like terms Solving one-stage equations

5 AUTUMN TERM Addition and Subtraction Number Remember to set out your calculations properly using column sums. Most mistakes are made when trying to work out answers in your head! Make sure your sums are aligned properly ensure your units columns are underneath each other when adding or subtracting

6 Multiplication and Division Number Remember to set out your calculations properly using column sums. Most mistakes are made when trying to work out answers in your head! Make sure your sums are aligned properly eg ensure your units columns are underneath each other when adding or subtracting. 836 x x

7 Number Place Value and Rounding whole numbers Remember the names of the columns Hundreds of Thousands Tens of Thousands Thousands Hundreds Tens Units HTh TTh Th H T U Four hundred and six thousand, two hundred and twenty eight: Hundreds of Thousands Tens of Thousands Thousands Hundreds Tens Units HTh TTh Th H T U Moving a digit one column to the left is the same as MULTIPLYING by 10 Moving a digit one column to the right is the same as DIVIDING by 10 When rounding, we need to look at the next digit to the right to decide whether or not to round up. If the digit to the right is less than five, we don t round up. If it is five or greater, we do round up. Round 3876 to the nearest 10: Find the tens column. The next digit to the right is greater than five so we round up and the answer is Round 82,393 to the nearest 1000: Find the thousands column. The next digit to the right is less than five so we don t round up and the answer is 82,000.

8 Place Value and Rounding decimals Number We use decimals as a way of showing parts of a whole number, like fractions. You need to know the different column names for decimals: Hundreds Tens Units Tenths Hundredths Thousandths H T U The decimal point does not have its own column; it sits between the units and tenths columns. Seven of the ten parts are coloured in, so we say this is 0.7 or Hundreds Tens Units Tenths Hundredths Thousandths H T U of the 100 parts are coloured in, so we say this is 0.23 or (We can t fit 23 in the hundredths column; the 3 goes in the hundredths column and the 2 in the tenths column. This is because 23 hundredths is the same as 2 tenths and 3 hundredths - you can see this from the diagram!) Hundreds Tens Units Tenths Hundredths Thousandths H T U

9 Number Place Value and Rounding rounding to 1 or 2 Decimal Places Any number rounded to 1 decimal place (dp) will have one digit after the decimal point. Any number rounded to 2 decimal places (dp) will have two digits after the decimal point. We need to look at the next digit to the right to decide whether or not to round up. If the digit to the right is less than five, we don t round up. If it is five or greater, we do round up. Round these numbers to 1 dp: The next digit to the right is less than five so we don t round up = 28 2 to 1dp The next digit to the right is greater than five so we round up = 7 7 to 1dp Round these numbers to 2 dp: The next digit to the right is greater than five so we round up = to 2dp The next digit to the right is less than five so we don t round up = 7.68 to 2dp

10 Watch out for nines! We need to round the 9 up, but we can t write the number 10 in a single column, so the answer is: = to 2dp We need to round the 9 up, but we can t write the number 10 in a single column and each 9 is going to round the next to 10, so the answer is: = to 2dp

11 Number Properties Number Factors: Factors are numbers which multiply together to make a number. 2 and 3 are factors of 6 because 2 x 3 = 6. Most numbers have an even number of factors because they come in pairs. The factors of 12 are 1, 2, 3, 4, 6, 12 because 1 x 12 = 12, 2 x 6 = 12 and 3 x 4 = 12 Square numbers have an odd number of factors because one factor multiplies by itself The factors of 16 are 1, 2, 4, 8, 16 because 1 x 16 = 16, 2 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, 21, 28, 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: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 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, 25, 36, 49, 64, 81, 100

12 Divisibility Rules Number Divisible by 2 Look at the last digit. It if is even, the number is divisible by is divisible by is not divisible by 2 Divisible by 3 Add up the digits. If you get a multiple of 3, the number is divisible by : = is divisible by 3 647: = is not divisible by 3 Divisible by 4 Halve the numbers first. If it is even then it is divisible by 4. 36: 36 2 = 18 (which is even) 36 is divisible by 4 74: 74 2 = 37 (which is odd) 74 is not divisible by 4 Divisible by 5 Look at the last digit. It must be a 5 or a zero is divisible by is not divisible by 5

13 Divisible by 6 Two things to check: (a) it must be even; and (b) it must be a multiple of 3 264: even and = : even and = is a multiple of 3, so 264 is divisible by 6 13 is not a multiple of 3, so 1264 is not divisible by 6 Divisible by 9 If the sum of its digits is a multiple of 9, it is divisible by : = 18, which is divisible by 9, so is divisible by : = 24, which is not divisible by 9, so 897 is not divisible by 9 Divisible by 10 The number must end in a zero is divisible by is not divisible by 10 There are rules of divisibility for 7, 8 and 11 but they are more complicated. It is better just to be good at these times tables and to be able to divide by 7, 8 and 11.

14 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 sum down. 43 x x (7 x 43) 3 2 (20 x 43) Add the rows together 1 In the first row under the line, calculate 7 x 43 as you would in a normal multiplication sum. In the second row, calculate 20 x 43. To make it easy, start by putting a zero in the units column. Then you can work out 2 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 x (6 x 286) (30 x 286) Add the rows together 1 1 Make sure you put the 3 digit number above the 2 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!

15 Equivalent fractions Number There are lots of different ways of describing the same fraction of a whole. 1 2 = 2 4 = 3 6 We can form equivalent fractions by multiplying the top and bottom of the fraction by the same number. 1 3 x 3 = x Fill in the gap to make the fractions equivalent: 2 7 =? 28 The denominator of the fraction has been multiplied by 4 to give 28, so multiply the top of the fraction (numerator) by 4 to make x 4 = x

16 Improper fractions and mixed numbers Number Improper (sometimes called top heavy) fractions and mixed numbers are two different ways of representing the same information. Look at the boxes below which are divided into sixths. We could say that I have shaded 19 because I have shaded 19 of the sections. This is called 6 an improper fraction because the numerator (top number) is larger than the denominator (bottom number). Or, we could say that I have shaded 3 1 because I have shaded 3 whole boxes and one sixth 6 of another. This is called a mixed number because it is a mixture of a whole number and a fraction. Converting improper fractions to mixed numbers: Divide the numerator by the denominator to find out how many whole ones there are. The remainder forms the numerator of the fraction = divided by 5 is 2 remainder 3. Therefore, there are 2 whole ones and three fifths. Imagine colouring in boxes like the ones above to help you picture this. Converting mixed numbers to improper fractions: Multiply the whole number by the denominator and add the answer to the numerator of the fraction = 23 7 Multiply 3 by 7. This tells you how many sevenths there are in three whole ones there are 21. Then add on the extra 2 sevenths from the fraction to end up with 23 sevenths.

17 Introduction to percentage Number Per cent means out of a hundred. 38% of the shape below is shaded because 38 out of 100 equal parts are shaded. We can use our knowledge of equivalent fractions to work out how percentages and fractions are related. eg = 1 2 so 50% is the same as one half = 1 4 so 25% is the same as one quarter = 1 10 so 10% is the same as one tenth

18 Fractions and Percentages of Quantities Number Fractions of Quantities: To find a fraction of a quantity, divide by the denominator then multiply by the numerator Find 1 of = x 1 = 16 Find 3 of = 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% = ) o To find 50%, divide the quantity by 2 (because 50% = 1 2 ) We can use these simple percentages to find more complex percentages Find 40% of 700kg 10% = 70 ( = 70) 40% = 280kg (70 x 4 = 280)

19 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. There are 60 cats and 72 dogs at a rescue centre. Express the ratio of cats to dogs in its simplest form. ( 12) 60 : 72 ( 12) 5 : 6 Missing numbers: Make sure you multiply both sides of the ratio by the same number. The ratio of boys to girls in a school is 3:2. If there are 180 boys, how many girls are there in the school? 3 : 2 x 60 x : 120 Sharing out: There are three stages to remember: 1. Add the ratio numbers together to find the total number of parts 2. 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 80 was shared between James and Emma in the ratio 3:5. How much money did they each receive? = = James: 3 x 10 = 30 Emma: 5 x 10 = 50

20 Measures, Shape and Space Properties of 2D shapes You need to learn the following 2D (two dimensional flat) shapes: TRIANGLES (Three sided shapes): EQUILATERAL (3 sides equal) ISOSCELES (2 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)

21 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 2

22 Length and Measurement Measures, Shape and Space Length is measured in millimetres (mm), centimetres (cm), metres (m) and kilometres (km). These are all metric units. There are also imperial units (so-called because they were used in the British Empire) such as inches, feet, yards, chains and miles but you do not need learn about these this year. Remember: o 1 km = 1000 m o 1 m = km o 1 m = 100 cm o 1 cm = 0.01 m o 1 cm = 10 mm o 1 mm = 0.1 cm When measuring with your ruler, make sure that you start measuring from zero remember that most rulers have a gap between the end and the zero line. Mind the gap!

23 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 = = 32cm 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 = 20cm² Area of a triangle = base x height 2 5cm 7cm Area = 5 x 7 2 = 35 2 = 17.5cm²

24 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 7cm A B 6cm 4cm = 12 cm 2 Area of B = 6 x 3 = 18cm² Total area = = 30 cm 2 To find the area of a compound shape, add up all of the side lengths. Write in any missing lengths onto the shape. 7cm 3cm 4cm 3cm Work out the missing lengths, labelled here with the arrows. Then add up all of the sides: = 26cm 6cm 3cm

25 Data Handling Probability Probability is the likelihood of something happening. All probabilities can be expressed on a scale from 0 to 1: If the probability of something happening is 0, we say it is impossible the probability of rolling an 8 with a normal die is 0. If the probability of something happening is 1, we say it is certain the probability of rolling a number from 1 to 6 with a normal die is 1. The probability of a coin landing heads up when we toss it is 1. This is because there are 2 two possible outcomes (heads or tails) and they are both equally likely. The probability of throwing a 5 when we roll a die is 1. This is because there are six possible 6 outcomes (1, 2, 3, 4, 5 or 6) and they are all equally likely. Some events are very hard to calculate the probability of, such as the probability of it raining tomorrow or of me scoring 70% on my maths test. This is because there are lots of possible outcomes and they are not all equally likely. Cards: In a standard deck of cards there are 52 cards. It includes thirteen ranks of each of the four suits: clubs ( ), diamonds ( ), hearts ( ) and spades ( ). Each suit includes an ace, depicting a single symbol of its suit; a king, queen and jack, each depicted with a symbol of its suit; and ranks two through ten, with each card depicting that many symbols of its suit.

26 Number Addition and Subtraction of Decimals SPRING TERM It is very easy to add and subtract decimals there are only two things to remember that make these sums different from adding or subtracting whole numbers. 1. We need to line up the columns properly. One easy way to do this is to put the decimal points underneath each other like buttons on a shirt. 2. We need to fill in any missing zeros so that each column has a digit in it. This particularly helps to avoid problems with subtraction

27 Negative Numbers Number It is often useful to think of a number line when working with negative numbers What is ? +6 So = -2 Subtracting negative numbers is the same as adding positive numbers = = 17 EXT: Multiplying and Dividing Negative Numbers: Remind yourself of the rules: 2 x 2 = = 2 2 x 2 = = 2 2 x 2 = = 2 2 X 2 = = 2

28 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 With larger numbers we can use the method below: 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 21. Pull the final zero down from the sum to give you goes into times = 0. You have finished! or consider dividing by factors 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

29 Number Multiplying or Dividing Decimals by a Whole Number Again, 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 x x

30 Measures, Shape and Space Plotting coordinates in all four quadrants (5,4) (-6,-2) 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.

31 Drawing and measuring angles Measures, Shape and Space MEASURING 1. Estimate whether the angle is bigger or smaller than Line up the corner of the angle with the middle point of the protractor 3. Turn the protractor until one of the baselines is line up with one of the lines of the angle 4. Find the zero (0) which is on the line. Follow it round to work out your angle 5. In this example the angle is 75 DRAWING 1. Draw a horizontal line 2. Line up the centre of the protractor with the end of the line 3. Follow the scale round from the zero on the line until you reach the angle you are drawing (here it is 120 ) 4. Make a mark with your pencil by the angle, then remove the protractor 5. Use a ruler to join the mark to the end of the line. Remember to label your angle

32 Calculating Angles Measures, Shape and Space You do not always need to use your protractor to find out how big an angle is. Sometimes, we can use angle rules to calculate (work out) how big the angles are. You need to know these four angle facts: Angles on a straight line add up to 180 : Think of your protractor. In a half turn, it measures from 0 to 180. This should help you remember that angles on a straight line add up to 180. Angle a must be 115 because = a Angles in a triangle add up to 180 : Similarly, the angles in any triangle always add up to 180. If you rip the three angles off a paper triangle, you will find that they fit together to make a straight line. 85 Angle b must be 55 because = 125, and = b

33 Angles about a point add up to 360 : Angles that meet at a point add up to 360 because they make up one whole turn = c = 105 Therefore, angle c must be 105. You should use COLUMN SUMS to make sure your addition and subtraction are correct! Angles in a quadrilateral add up to 360 : Angles in any four sided shape will always add up to 360 (think of a square to help you remember: 4 x 90 = 360 ) = = d Therefore, angle d must be 80. You should use COLUMN SUMS to make sure your addition and subtraction are correct!

34 Time Measures, Shape and Space There are two different ways of writing the time. 12-hour clock: Times go from 0:01 am (just after midnight) to pm (just before midnight). am (stands for ante meridiem, before midday) and should be used for the times after midnight but before midday. pm (stands for post meridiem, after midday) and should be used for the times after midday but before midnight. To avoid confusion, it is best to write 12 noon and 12 midnight for 12.00! 24-hour clock: Times go from 00:01 (just after midnight) to 23:59 (just before midnight). Midnight itself is 00:00 and midday is 12:00. Always use four figures so 8.45am would be 08:45. Add 12 to the number of hours to work out times in the afternoon quickly eg 5.45pm would be 17:45.

35 Metric units and scales Measures, Shape and Space As well as the unit of length, you should also know the following metric conversions. MASS: Kg = kilograms g = grams mg = milligrams o 1kg = 1000g o 1g = 0 001kg o 1g = 1000mg o 1mg = 0 001g o 1kg = mg o 1mg = kg CAPACITY: l = litre cl = centilitre ml = millilitre o 1l = 1000ml o 1l = 100cl o 1cl = 10ml 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. When reading scales, you need to work out how many divisions there are between two numbers On this number line there are five divisions between 7 and 7 5, so each one is worth 0 1. Here the arrow points to On this number line there are five divisions between 2 and 3, so each one is worth 0 2. Here the arrow points to On this number line there are four divisions between 4 and 5, so each one is worth Here the arrow points to 4 75.

36 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 = 24 Bottom: 8 x 3 = 24 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: Volume = Length Width Height Or V = L W H Example: l = 3 cm Volume = l w h h = 2 cm Volume = w = 1 cm Volume = 6 cm 3

37 Number Addition and Subtraction of Fractions SUMMER TERM It is very easy to add or subtract fractions if their denominators are the same = 5 7 Sometimes you may need to simplify your answer or to convert it to a mixed number eg = 4 8 = 10 7 = 1 2 = If the denominators are not the same, then you should use your knowledge of equivalent fractions to make them the same = Think of the smallest number which is a multiple of both 4 and 3. In this case it is 12. Convert both fractions to twelfths. Here, we multiply both numbers in the first fraction by 3 and in the second fraction by 4. = 5 12 EXT: Add or subtract any whole numbers before you start = = = In this case, the lowest common denominator is 20. 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 ( 23 ) 20 into a mixed number (1 3 ) and add the 1 onto the 7 to make 8. 20

38 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 (b) 4 x 3 2 = = 4 x 9 = = 36 = 6 (c) 1 2 x = 1 2 x = 8 12 = - 4 Data Handling

39 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. 2, 7, 4, 6, 6 13, 9, 11, 9, 9, 15 MEAN = 25 5 = = 66 6 = 11 MEDIAN 2, 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 2 = = 6

40 Average temp in C Frequency Home Bar graphs and line graphs Data Handling Bar graphs are used to display data which is in categories favourite colour, favourite food, hair colour. Make sure you do not leave a space between the bars! Label the categories in the middle of the bars. Remember titles and labels! Bar Graph showing favourite chocolate bars 0 Dairy Milk Crunchie Mars Snickers Chocolate bar Line graphs are used to display continuous data, and usually show changes over time eg temperature, height, frequency. Plot your points with a cross, and join them with a ruler. Label the categories on the lines. Remember titles and labels! 25 Line Graph showing average temperature in London Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month

41 Algebra Introduction to 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 + 2a = 15a 3b²c + 9b²c 5b²c = 4b²c 2d d = 11d 23 6g² + 9g 2g = 6g² + 7g 5ef + 7fe + 2ef = 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 2a + 3b + 3c 3b²c + 9bc² 2d 23 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 6g² + 9g g² and g are not the same and cannot be combined 5ef + 7EF Capital letters are not the same as lower case letters!

42 EXT: 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 42cd 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

43 Solving Equations Algebra 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. One-stage equations x + 4 = 7 (- 4) (- 4) x = 3 Subtract 4 from both sides to leave x on its own. x - 7 = 6 (+ 7) (+ 7) x = 13 Add 7 to both sides to leave x on its own. 6x = 30 ( 6) ( 6) x = 5 Divide both sides by 6 to leave x on its own. x = 9 4 Multiply both sides by 4 to leave x on its own. (x4) (x4) x = 36 Solving equations - the negative x! Negative terms can be confusing - the best thing to do when faced with a negative x is to add it to both sides so that I end up with a positive x. Example: 6 x = 4 (+ x) (+ x) 6 = 4 + x (-4) (- 4) 2 = x x = 2 x + x = 0 Think about if x = 2: = 0 If x = = 0

44 EXT: Two stage equations 2x + 3 = 9 (-3) (-3) 2x = 6 ( 2) ( 2) x = 3 8 x = 11 (+ x) (+ x) 8 = 11 + x (- 11) (- 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 12 = 3 (+ 12) (+ 12) 3x = 15 ( 3) ( 3) x = 5 Start by multiplying out the brackets. Then continue as normal.