# Hyde Community College

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1 Hyde Community College Numeracy Booklet 1

3 Table of Contents Topic Page Number 1. Mental Methods (+ - x) 4 2. Written Methods (+ - x ) 7 3. Number Properties Place Value Fractions Percentages Fraction, Decimal & Percentage Equivalence Ratio & Proportion Negative Numbers Coordinates Inequalities Names of Two Dimensional Shapes D Shapes Perimeter Area of 2D Shapes Volume Units of Measurement Time Bearings Displaying Data Averages & Spread 46 Mathematical Dictionary 47 Answers 49 Ideas for help at home 52 3

4 1. Mental methods (+ - x ) Addition Example Method 1 Add the tens, then the units, then add together = = = 81 Method 2 Split the number to be added into tens and units and add separately = = 81 Method 3 Round up to the next 10, then subtract = 84 but 30 is 3 too many therefore subtract = 81 Subtraction Example Method 1 Count on Count on from 56 until you reach 93. This can be done in several ways e.g Answer =

5 Method 2 Break up the number being subtracted e.g. subtract 50 then subtract = = Start Multiplication It is essential that pupils know all of the times tables from 1x1 up to 10x10. These are shown below: X x 9 = 63 5

6 Example 39 x 6 Method 1 Multiply by the tens then by the units 30 x 6 = x 6 = = 234 Method 2 Multiply 40 x 6 then subtract 1 x 6 40 x 6 = is 1 too many so = 234 Subtract 1 x 6 = For you to try 1) ) ) ) ) ) ) 23 x 6 8) 59 x 8 9) 7 x 68 6

7 2. Written methods (+ - x ) Addition Example Place the digits in the correct place value columns with the numbers under each other. Th H T U Begin adding in the units column Subtraction Show any carrying in the next column Example: Place the digits in the correct place value columns with the numbers under each other. Begin subtracting in the units column. 6 7 You can t subtract 9 from 6 so move 1 ten from the 8 tens to the 6 units to make 16 units. Note that the same has happened with the hundreds. Th H T U For you to try 1) ) ) ) ) ) ) ) ) ) ) )

8 Addition of decimals Example Place the digits in the correct place value columns with the numbers under each other. Make sure the decimal points are lined up vertically. Begin adding in the furthest column on the right. T U. 1/10 1/ Subtraction of decimals Example: Fill in any gaps with zeros. Place the digits in the correct place value columns with the numbers under each other. Make sure the decimal points are lined up vertically. Begin subtracting in the furthest column on the right. - T U. 1/10 1/ For you to try 1) ) ) ) ) ) ) ) ) ) ) )

9 Multiplication Method 1 Grid Method Example 56 x 34 Separate the 56 and 34 into tens and units. x 50 6 Multiply the columns with the rows and place the answers in the grey boxes Add the numbers: = 1904 Method 2 Napier s Bones Example 847 x 6 Write 847 across the top and 6 down the side. Multiply each of the digits 8,4 & 7 by the 6, putting the answers in the orange boxes. The answer is obtained by adding up from right to left =5 8+2=10 4+4=8 Write 0 Carry 1 For you to try 1) 36 x 62 2) 82 x 47 3) 156 x 5 4) 263 x 7 5) 556 x 62 6) 452 x 81 9

10 Division Example: Concise method There are 2 fours in 9 with remainder 1 so the answer starts with 2 and the remainder 1 is placed next to the There are 4 fours in 18 with remainder 2. There are 5 fours in 20 with no remainder. The answer is 245 Chunking method We use multiples of 100, 10, 5, 2 and 1 as these are easy to work out. X 4 Total x 4 = 400 which is a great deal less than Another 100 x 4 will make a total of Another 100 x 4 will give a total of 1200 which is more than 980 so we use 10 x 4 = 40 giving a total of x 4 = 40 another 3 times gives a total of x 4 = 20 giving a total of 980 which is what we need. 245 By adding the x column we can see how many 4s there are in 980. For you to try 1) ) ) ) ) )

11 3. Number Properties Even numbers 2, 4, 6, 8, 10, 12,, etc. Even numbers are the same as the numbers in the two times table. A number is even if it ends in a 2, 4, 6, 8 or 0 e.g is even as it ends in an 8. Odd numbers 1, 3, 5, 7, 11, 13,, etc. Odd numbers are all the numbers that aren t in the two times table. A number is odd if it ends in a 1, 3, 5, 7 or 9 e.g. 673 is odd as it ends in a 3. Square numbers 1 2 = 1 x 1 = = 2 x 2 = = 3 x 3 = = 4 x 4 = = 5 x 5 = 25 The first ten square numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 For you to try From the following list which are a) odd b) even c) square numbers? 7, 11, 18, 25, 30, 36, 100, 285,

12 Multiples A multiple of a number is that number multiplied by any whole number. e.g. 14 is a multiple of 7 because 7 x 2 = 14 6 is a factor of 2 because 2 x 3 = 6. The multiples of a number start with that number and can be thought of as the times table of that number. e.g. The multiples of 5 are: 5, 10, 15, 20, 25,, etc. Note: Multiples of a number go on forever! Factors A factor is a number that divides exactly into another number. e.g. 4 is a factor of 12 because 3 lots of 4 make is a factor of 12 because 2 lots of 6 make 12. All the factors of 12 are: 1, 2, 3, 4, 6 and 12 Prime numbers A prime number has exactly two factors, 1 and itself. e.g. The only factors of 17 are 1 and 17. So 17 is a prime number. The prime numbers between 1 and 20 are: 2, 3, 5, 7, 11, 13, 17, 19 Note: 1 is not a prime number because it only has one factor! For you to try From the following list, which are a) multiples of 6 b) factors of 30 c) prime? 3, 5, 9, 12, 15, 19, 24, 30 12

13 4. Place Value Thousands (1000) Hundreds (100) Tens (10) Units (1). Tenths 1 10 Hundredths Thousandths units = 1 ten 10 thousandths = 1 hundredth 10 tens = 1 hundred 10 hundredths = 1 tenth 10 hundreds = 1 thousand 10 tenths = 1 unit The placement of the digits within the number gives us the value of that digit. e.g. The digit 4 has the value of The digit 5 has the value 4 thousand of 5 tenths ( 5 / 10 or 0.5 ) (4000) The digit 8 has the value The digit 7 has the value 8 tens (80) 7 thousandths ( 7 / 1000 or 0.007) For you to try What is the value of the 7 in each of the following numbers? 1) 756 2) ) ) ) )

14 5. Fractions Understanding Fractions The numerator is the number on the top of the fraction 3 4 The denominator is the number on the bottom Example A necklace is made from black and white beads. What fraction of the beads are black? There are 3 black beads out of a total of 7, so 7 3 of the beads are black. Equivalent fractions All the fractions below represent the same proportion. Therefore they are called equivalent fractions. ½ 2 / 4 4 / 8 14

15 Below are three rows of equivalent fractions. What do you think would come next? = = = = = = = = = = = = =.... =.... =.... You can tell if two fractions are equivalent if the numerator and denominator have both been multiplied by the same amount. Example What fraction of the flag is shaded? 6 out of 12 squares are shaded. So 12 6 of the flag is shaded. It could also be said that 2 1 the flag is shaded. x and are equivalent fractions because 1 x 6 = 6 and 2 x 6 = x 6 15

16 Simplifying Fractions To simplify a fraction you divide the numerator and denominator by the same number. Example (a) (b) 4 16 = = This can be done repeatedly until the numerator and denominator are the smallest possible numbers - the fraction is then said to be in its simplest form. Example Simplify = = = 6 7 (simplest form) Fractions of Quantities To find the fraction of a quantity, divide by the denominator. To find 2 1 divide by 2, to find 3 1 divide by 3, to find 7 1 divide by 7 etc. Example 1 Find 5 1 of of 150 = =

17 Example 2 Find 4 3 of 48 (To find 4 3 of a quantity, start by finding 4 1 ) 1 of 48 = 48 4 = 12 4 so 4 3 of 48 = 3 x 12 = 36 For you to try 1) Write each of the following fractions in their simplest form: 10 a) b) 20 c) d) 80 7 e) f) 40 2) Calculate each of the following: a) of 24 b) of 30 c) of d) 4 3 of 20 e) 5 2 of 40 f) 9 7 of 72 17

18 6. Percentages The symbol % Means Out of % Means 100% Means 120% Means 63 / 100 (63 out of 100) 100 / 100 (or 1 whole one) 120 / 100 (Percentages can be more than 100%) Percentages of Amounts Non- Calculator Methods Method 1 Using Equivalent Fractions: 50% = % = % = 10 1 Example a) Find 50% of 2000kg 50% of 2000kg = 2 1 of 2000kg = 2000kg 2 = 1000kg b) Find 25% of % of 640 = 4 1 of 640 = =

19 Method 2 Using 1% In this method, first find 1% of the quantity (by dividing by 100), then multiply to give the required value. Example Find 9% of 200g 1 1% of 200g = 100 of 200g = 200g 100 = 2g so 9% of 200g = 9 x 2g = 18g Method 3 Using 10% This method is similar to the one above. First find 10% (by dividing by 10), then multiply to give the required value. Example Find 70% of 35 10% of 35 = 1 10 of 35 = = 3.50 so 70% of 35 = 7 x 3.50 = For you to try (without a calculator) 1) 50% of 200 2) 25% of 80 3) 10% of 40 4) 20% of 60 5) 30% of 500 6) 70% of 90 7) 3% of 600 8) 15% of 360 9) 67% of

20 Calculator Method To find the percentage of a quantity using a calculator, change the percentage to a fraction, then multiply. Example a) Find 23% of % = so 23% of = x = A fraction can be typed into a calculator as Numerator Denominator b) Find 68% of % = so 68% of 400 = x 400 = Note: We do not use the % button on a calculator during maths lessons! For you to try (with a calculator) 1) 24% of 50 2) 79% of 400 3) 18% of ) 17.5% of 40 5) 47% of ) 135% of 20 20

21 7. Fraction, Decimal & Percentage Equivalence Some fractions and percentages are used very frequently. It is useful to be able to express these as either a fraction, decimal or percentage. Fraction Decimal Percentage % % % % % % 10 2 (= 5 1 ) % % 10 For you to try 1) Change into decimals: a) 40% b) 85% c) ) Change into percentages: a) 0.8 b) 10 5 c) 5 4 3) Change into fractions: a) 90% b) 0.6 c)

22 8. Ratio & Proportion Writing a Ratio Ratio is used to make a comparison between two things. Example In this pattern we can see that there are 3 happy faces to every sad face. We use the symbol : to represent to in the above statement, therefore we write the ratio like this: The ratio of happy faces to sad faces is 3 : 1 The ratio of sad faces to happy faces is 1 : 3 Note: The order of the numbers is important. Ratio is used in a number of situations including In a cooking recipe When mixing concrete or paint In the scale on maps or in models e.g. if a scale of 1 : is used on a map, it means that 1 cm on the map represents cm in reality. 22

23 Simplifying Ratios Ratios can be simplified in much the same way as fractions, by dividing each part of the ratio by the same number Example 1 Purple paint can be made by mixing 10 tins of blue paint with 6 tins of red. The ratio of blue to red can be written as 10 : 6 It can also be written as 5 : 3, as it is possible to split up the tins into 2 groups, each containing 5 tins of blue and 3 tins of red. B B B B B R R R B B B B B R R R We have simplified the ratio 10 : 6 by dividing both numbers by two to get 5 : 3 Example 2 Simplify each ratio: (a) 4:6 (b) 24:36 (c) 6:3:12 (a) 4:6 (Divide by 2) (b) 24:36 (Divide by 12) (c) 6:3:12 (Divide by 3) = 2:3 = 2:3 = 2:1:4 Example 3 Concrete is made by mixing 20 kg of sand with 4 kg cement. Write the ratio of sand : cement in its simplest form The ratio of Sand to Cement = 20 : 4 Which can be simplified (by dividing by 4) to 5 : 1 23

24 Proportion Two quantities are said to be in direct proportion if when one doubles the other doubles. We can use proportion to solve problems. Example 1 A car factory produces 1500 cars in 30 days. How many cars would they produce in 90 days? x3 Days Cars x The factory would produce 4500 cars in 90 days. Example 2 5 adult tickets for the cinema cost How much would 8 tickets cost? Tickets Cost ( ) (5.50 x 8) The cost of 8 tickets is 44 For you to try 1) Simplify the following ratios as much as possible: a) 15 : 12 b) 20 :30 c) 36 : 27 d) 28 : 35 : 14 2) If 3 pens cost 75p, how much would 7 identical pens cost? 3) In a class of 30 pupils there are 18 boys. Write as a ratio in its simplest form the number of boys to the number of girls. 24

25 9. Negative Numbers The negative sign ( - ) tells us the number is below zero e.g. -4. The number line is useful when working with negative numbers. Below is a part of the number line. Negative direction Positive direction The numbers on the right are greater than the numbers on the left e.g. 5 is greater than 2 and 2 is greater than -3. Note that -3 is greater than -8. Adding and subtracting with directed numbers Example: Start Start at -3. Move 7 in the positive direction Answer : 4 Example 2: Start Start at -4. Move 2 in the negative direction Move 7 in the positive direction Answer : 1 25

26 Multiplying and dividing negative numbers We multiply and divide negative numbers in the usual way whilst remembering these very important rules: Two signs the same, a positive answer. Two different signs, a negative answer Note: If there is no sign before the number, it is positive. Examples: 5 x -7 = -35 (different signs give a negative answer) -4 x -8 = 32 (two signs the same give a positive answer) 48-6 = -8 (different signs give a negative answer) = 12 (two signs the same give a positive answer) For you to try 1) ) ) ) ) ) -5 x 6 7) -4 x -8 8) )

27 10. Coordinates We use coordinates to describe location. We write a coordinate as two numbers in a bracket separated by a comma. The first number is the x-coordinate (across) and the second number is the y-coordinate (up or down). y B 3 2 A C D x -3 Example The coordinates of the points are: A=(1,2) (1 across, 2 up) B=(-2,3) C=(-2,-2) D=(3,-2) Note: There is a special name for the point (0,0). It is called the origin. For you to try Plot each of the following points on the coordinate grid above: 1) E = (3,3) 2) F = (1,-2) 3) G = (-2,1) 4) H = (-3,0) 5) I = (2,3) 6) J = (-1,-3) 27

28 11. Inequalities We us the = sign to show that two sums are equal. If one sum is greater than or less than the other we use inequalities: < less than > greater than < less than or equal to > greater than or equal to Examples : 5 < 8 43 > 6-3 > -10 For you to try Put the correct symbol, either < or > in between each of the following pairs of numbers: 1) 3 5 2) ) ) 8-4 5) )

29 12. Names of two dimensional shapes A polygon is a closed shape made up of straight lines. A regular polygon has all of its sides equal in length and all of its angles equal in size. Equilateral triangle Right angled triangle Isosceles triangle Scalene triangle Square Rectangle Parallelogram Rhombus Trapezium Opposite sides Opposite sides One pair of opposite parallel and equal. parallel, all sides equal. sides parallel. Kite Pentagon Hexagon Heptagon Octagon Circle Note: All 2D shapes with 4 sides are known as quadrilaterals 29

30 13. 3D shapes 3D means three dimensions 3D shapes have length, width and height. Shape Name Faces Edges Vertices (corners) Cube Cuboid Square based pyramid Triangular prism

31 14. Perimeter Perimeter is the distance around the outside of a shape. We measure the perimeter in millimetres (mm), centimetres (cm), metres (m), etc. This shape has been drawn on a 1cm grid. Starting on the orange circle and moving in a clockwise direction, the distance travelled is = 14cm Perimeter = 14cm If you know the length of the sides of a shape then to find the perimeter you simply add the lengths together. Example In the rectangle on the right the perimeter = 12cm = 34cm 5cm Note: we added the 12cm twice as the bottom edge is equal in size to the top and similarly we added the 5cm twice as the left and right edges are equal. Example 2 8cm 8cm In the triangle on the left the perimeter = = 20cm 4cm 31

32 15. Area of 2D Shapes The area of a shape is how much surface it covers. We measure area in square units e.g. centimetres squared (cm 2 ) or metres squared (m 2 ). Areas of irregular shapes Given an irregular shape, we estimate its area through drawing a grid and counting the squares that cover the shape Whole square count as one. Half a square or more count as one. Less than half a square - ignore. Area = 11cm². Note: This answer is approximate and not the exact answer. 32

33 Area formulae Rectangle Triangle Multiply the length with the width. Multiply the base with the height and divide by two. b x h Area = l x w Area = 2 Trapezium Parallelogram Add the parallel sides, multiply with the height and divide by two. Multiply the base with the height. Area = (a + b) h 2 Area = b x h Circle Multiply the radius with itself, then multiply with π. Area = r x r x π = πr² 33

34 16. Volume Volume is the amount of space that an object contains or takes up. The object can be a solid, liquid or gas. Volume is measured in cubic units e.g. cubic centimetres (cm 3 ) and cubic metres (m 3 ). This cube has a volume of 1 cm 3 Cuboid Note that a cuboid has six rectangular faces. Volume of a cuboid = length x width x height Prism A prism is a 3-dimensional object that has the same shape throughout its length i.e. it has a uniform cross-section. Volume of a prism = area of cross-section x length 34

35 17. Units of Measurement Metric (new) units of length Millimetre Mm 10 mm = 1 cm Centimetre Cm 100 cm = 1 m Metre M m = 1 km Kilometre Km Imperial (old) units of length Inch in or 12 in = 1 ft Foot ft or 3 ft = 1 yd Yard Yd yd = 1 mile Mile Metric units of mass Milligram Mg mg = 1 g Gram G g = 1 kg Kilogram Kg kg = 1 t Metric tonne T Imperial units of mass Ounce Oz 16 oz = 1 lb Pound Lb 14 lb = 1 st Stone St Metric units of volume Millilitre Ml ml = 1 l Litre L Imperial units of volume Pint Pt 8 pt = 1 gal Gallon Gal 35

36 Converting between imperial and metric units Length 1 inch 2.5 cm 1 foot 30 cm 1 mile 1.6 km 5 miles 8 km Weight/Mass 1 pound ~ 454 g 2.2 pounds ~ 1 kg Volume 1 gallon 4.5 litre 1 pint 0.6 litre(568 ml) 1¾ pints 1 litre For you to try 1) Change into centimetres: a) 40 mm b) 230 mm c) 1.2 m 2) Change into metres: a) 300 cm b) 1.5 km c) 70 cm 3) Change into grams: a) 2 kg b) 5 ½ kg c) 0.3 kg 4) Change into miles: a) 16 km b) 80 km c) 32 km 36

37 18. Time 1000 years = 1 millennium 100 years = 1 century 10 years = 1 decade 60 seconds = 1 minute 60 minutes = 1 hour 24 hours = 1 day 7 days = 1 week 12 months = 1 year 52 weeks 1 year 365 days 1 year 366 days 1 leap year The Yearly Cycle Season Month Days January 31 February 28 March 31 April 30 May 31 June 30 July 31 August 31 September 30 October 31 November 30 December 31 37

38 The 24 hour and 12 hour clock 24 hour 12 hour Midnight 00: a.m. Midnight The 24 hour clock always uses 4 digits to show the time. The 24 hour system does not use a.m. nor p.m. 01:00 1:00 a.m. 02:00 2:00 a.m. 03:00 3:00 a.m. 04: a.m. 05:00 5:00 a.m. 06:00 6:00 a.m. 07:00 7:00 a.m. 08:00 8:00 a.m. 09:00 9:00 a.m. 10:00 10:00 a.m. 11:00 11:00 a.m. The 12 hour clock shows the time with a.m. before midday and p.m. after mid-day. Mid-day 12:00 12:00 p.m. Mid-day 13:00 1:00 p.m. 14:00 2:00 p.m. 15:00 3:00 p.m. 16:00 4:00 p.m. 17:00 5:00 p.m. 18:00 6:00 p.m. 19:00 7:00 p.m. 20:00 8:00 p.m. 21: p.m. 22: p.m. 23:00 11:00 p.m. Time vocabulary 02:10 Ten past two in the morning 2:10 a.m. 07:15 Quarter past seven in the morning 7:15 a.m. 15:20 Twenty past three in the afternoon 3:20 p.m. 21:30 Half past nine in the evening 9:30 p.m. 14:40 Twenty to three in the afternoon 2:40 p.m. 21:45 Quarter to ten at night 9:45 p.m. 38

39 19. Bearings A bearing describes direction. A compass is used to find and follow a bearing. The diagram below shows the main compass points and their bearings NW N NE W E 090 SW S SE N North, S South, E East, W - West The bearing is an angle measured clockwise from the North. Bearings are always written using three figures. e.g. if the angle from the North is 5, we write

40 20. Displaying Data Collecting and recording We can record data in a list e.g. here are the numbers of pets owned by pupils in form 9C: 1, 2, 1, 1, 2, 3, 2, 1, 2, 1, 1, 2, 4, 2, 1, 5, 2, 3, 1, 1, 4, 10, 3, 2, 5, 1 A frequency table (or tally chart) is more structured and helps with processing the information. Displaying In order to communicate information, we use statistical diagrams. Some of the ones we use are: Pictogram Bar Chart Pie Chart Line Graph Conversion Graph Scatter diagram 40

41 Pictogram A pictogram uses symbols to represent frequency. We include a key to show the value of each symbol. Example The diagram below shows the number of pets owned by pupils in 9C. We can see that there are 10 pupils that have 1 pet (5 pictures each worth 2). There are 8 pupils that have 2 pets. There are 3 pupils that have 3 pets (The ½ picture is worth 1 pupil). There are 2 pupils that have 4 pets. There are 2 pupils that have 5 pets. There is 1 pupil that has more than 5 pets. 41

42 Bar chart The height of each bar represents the frequency. All bars must be the same width and there must be gaps between the bars, also of an equal size. The scale of the frequency starts from 0 every time and the numbers go next to the lines, not the spaces. 42

43 Pie chart The complete circle represents the total frequency. The angles for each sector are calculated as follows: Here is the data for the types of pets owned by 9C Type of pet Frequency Angle of the sector Divide 360 by the total Cats X 10 = 130 of the frequency: Dogs X 10 = 110 Birds 5 5 X 10 = = 10 Fish 7 7 X 10 = 70 Therefore 10 Total represents one animal Remember to check that the angles of the sectors add up to

44 Line graph The temperature of water was measured every minute as it was heated and left to cool. A cross shows the temperature of the water at a specific time. Through connecting the crosses with a curve we see the relationship between temperature and time. The line enables us to estimate the temperature of the water at times other than those plotted e.g. at 6½ minutes the temperature was approximately 40 C. Conversion graph We use a conversion graph for two variables which have a linear relationship. We draw it in the same way as the above graph but the points are connected with a straight line. From the graph, we see that 8 km is approximately 5 miles. 44

45 Scatter diagram We plot points on the scatter diagram in the same way as for the line graph. We do not join the points but look for a correlation between the two sets of data. Positive correlation No correlation Negative correlation If there is a correlation, we can draw a line of best fit on the diagram and use it to estimate the value of one variable given the other. The following scatter graph shows a positive correlation between the weights and heights of 12 pupils. The line of best fit estimates the relationship between the two variables. Notice that the line follows the trend of the points. There are approximately the same number of points above and below the line. We estimate that a pupil 155 cm tall has a weight of 60 kg. 45

46 21. Averages & Spread Averages The average is a measure of the middle of a set of data. We use the following types of average: Mean - We add the values in a set of data, and then divide by the number of values in the set. Median - Place the data in order starting with the smallest then find the number in the middle. This is the median. If you have two middle numbers then find the number that s halfway between the two. Mode - This is the value or values that appear most often. Spread The spread is a measure of how close together the items of data are. We use the range to measure spread: Range - The range of a set of data is the difference between the highest and the lowest value. 46

47 Example Find the mean, median, mode and range of the following set of numbers: 4, 3, 2, 0, 1, 3, 1, 1, 4, 5 Mean = 2 4 Median 0, 1, 1, 1, 2, 3, 3, 4, 4, = 2 5 Mode 0, 1, 1, 1, 2, 3, 3, 4, 4, 5 = 1 Range 0, 1, 1, 1, 2, 3, 3, 4, 4, = 5 For you to try Find the mean, median, mode and range of the following set of numbers: 1) 8, 11, 6, 8, 2, 15, 20 2) 6, 7, 8, 10, 3, 12, 15, 8, 6, 5 47

48 Mathematical Dictionary (Key words): Add; Addition (+) a.m. Approximate Calculate Data Denominator Difference (-) Division ( ) To combine 2 or more numbers to get one number (called the sum or the total) Example: = 88 (ante meridiem) Any time in the morning (between midnight and 12 noon). An estimated answer, often obtained by rounding to nearest 10, 100 or decimal place. Find the answer to a problem. It doesn t mean that you must use a calculator! A collection of information (may include facts, numbers or measurements). The bottom number in a fraction (the number of parts into which the whole is split). The amount between two numbers (subtraction). Example: The difference between 50 and 36 is = 14 Sharing a number into equal parts = 4 Double Multiply by 2. Equals (=) Makes or has the same amount as. Equivalent fractions Fractions which have the same value. 6 1 Example and are equivalent fractions 12 2 Estimate To make an approximate or rough answer, often by rounding. Evaluate To work out the answer. Even A number that is divisible by 2. Even numbers end with 0, 2, 4, 6 or 8. Factor Frequency Greater than (>) Least Less than (<) Maximum A number which divides exactly into another number, leaving no remainder. Example: The factors of 15 are 1, 3, 5, 15. How often something happens. In a set of data, the number of times a number or category occurs. Is bigger or more than. Example: 10 is greater than > 6 The lowest number in a group (minimum). Is smaller or lower than. Example: 15 is less than < 21. The largest or highest number in a group. 48

49 Mean The arithmetic average of a set of numbers (see p46) Median Another type of average - the middle number of an ordered set of data (see p46) Minimum The smallest or lowest number in a group. Minus (-) To subtract. Mode Most Multiple Multiply (x) Negative Number Numerator Another type of average the most frequent number or category (see p46) The largest or highest number in a group (maximum). A number which can be divided by a particular number, leaving no remainder. Example Some of the multiples of 4 are 8, 16, 48, 72 To combine an amount a particular number of times. Example 6 x 4 = 24 A number less than zero. Shown by a minus sign. Example -5 is a negative number. The top number in a fraction. Odd Number A number which is not divisible by 2. Odd numbers end in 1,3,5,7 or 9. Operations The four basic operations are addition, subtraction, multiplication and division. Order of operations Place value p.m. Prime Number Product Remainder Share Sum Total The order in which operations should be done remembered with the acronym BIDMAS. The value of a digit dependent on its place in the number. Example: in the number , the 5 has a value of 500. (post meridiem) Any time in the afternoon or evening (between 12 noon and midnight). A number that has exactly 2 factors (can only be divided by itself and 1). Note that 1 is not a prime number as it only has 1 factor. The answer when two numbers are multiplied together. Example: The product of 5 and 4 is 20. The amount left over when dividing a number. To divide into equal groups. The total of a group of numbers (found by adding). The sum of a group of numbers (found by adding). 49

50 Answers Page 6 1) 77 2) 85 3) 514 4) 21 5) 16 6) 45 7) 138 8) 472 9) 476 Page 11 a) 7, 11, 25, 285 b) 18, 30, 36, 100, 3498 c) 25, 36, 100 Page 7 1) 625 2) 715 3) ) ) ) ) 508 8) 183 9) ) ) ) 4584 Page 8 1) ) ) ) ) ) ) ) ) ) ) ) Page 9 1) ) ) 780 4) ) ) Page 10 1) 186 2) 156 3) 38 4) 259 5) ) 467 Page 12 a) 12, 24, 30 b) 3, 5, 15, 30 c) 3, 5, 19 Page 13 1) 700 2) 70 3) ) 0.7 or 7/10 5) 0.07 or 7/100 6) Page 17 1) a) 5/8 b) 3/4 c) 2/3 d) 1/4 e) 1/3 f) 3/5 2) a) 6 b) 10 c) 9 d) 15 e) 16 f) 56 Page 19 1) 100 2) 20 3) 4 4) 12 5) 150 6) 63 7) 18 8) 54 9)

51 Page 20 Page 27 1) 12 2) 316 3) 360 4) 7 5) ) 27 Page 21 1) a) 0.4 b) 0.85 c) 0.7 2) a) 80% b) 50% c) 80% G H y 0 I E x 3) a) 9/10 b) 6/10 or 3/5 c) 4/10 or 2/5 J F Page 24 1) a) 5:4 b) 2:3 c) 4:3 d) 4:5:2 Page 28 2) ) 3:2 Page 26 1) 4 2) -9 3) -8 4) -6 5) -8 6) -30 7) 32 8) -5 9) 9 1) 3 < 5 2) 65 > 28 3) -5 > -12 4) 8 > -4 5) -7 > -10 6) -4.5 < -3 Page 36 1) a) 4cm b) 23cm c) 120cm 2) a) 3m b) 1500m c) 0.7m 3) a) 2000g b) 5500g c) 300g 4) a) 10miles b) 50miles c) 20miles Page 47 1) Mean = 10, Median = 8, Mode = 8 Range = 18 2) Mean = 8, Median = 7.5, Mode = 6&8 Range = 12 51

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