1. Numeracy Across the Curriculum Toolkit AdditionAddition Subtraction Addition and subtraction of decimalsSubtractionAddition and subtraction of decimals time number lines Worded Problems Reading Numbers Number Relationships Multiplication Division Multiplication and division patterns Bidmas Equivalent fractions, decimals and percentages Percentage Checking Methodstime number linesWorded ProblemsReading NumbersNumber RelationshipsMultiplicationivisionMultiplication and division patterns BidmasEquivalent fractions, decimals and percentages PercentageChecking Methods Fractions Proportion Ratio Negative numbers Reading ScalesFractionsProportion RatioNegative numbersReading Scales Metric Units Length Metric Units Capacity and Mass PerimeterMetric Units Length Metric Units Capacity and MassPerimeter Area Volume Measure angles Averages What makes a good graphAreaVolumeMeasure anglesAverages What makes a good graph Bar Chart Pictogram Grouped Frequency Diagram Frequency Polygon Pie Chart Scatter Diagram metric and imperial unitsBar ChartPictogramGrouped Frequency DiagramFrequency PolygonPie Chart Scatter Diagrammetric and imperial units Llanishen High School - Numeracy toolkit

2. Addition. 84 + 79 80 + 70 = 150 4 + 9 = 13 85 + 79 = 163 84 + 79 163 1 84154163 +70 + 9 84 + 80 – 1 =163 80150154163 84 163 164 +80 Adding Words Sum Total Plus Altogether More than 80150154163 + 4 +70 + 9

3. Subtraction 84 - 56 56 +4 +20 + 4 = 84 Answer = 28 84 - 56 28 Adding on to subtract 24 + 4 = 28 56 +24 84 -50 – 6 =28 80150154163 28 - 50 1 Subtracting Words Difference Take away Take off Minus Less than Answer = 28 28 80 84 - 4 -6 84 60 + 4 -50 34 84 30 - 2 7

4. Worded questions. The hardest thing about worded questions is deciding which operation to use. Here are some ‘operation’ words to help you. Subtracting Words Difference Take away Take off Minus Less Adding Words Sum Total Plus Altogether More than Division Words Share Each Divide Quotient Goes into Multiplication words Times Product Multiply Of Lots of

5. Reading numbers. Numbers up to 999 345 – three hundred and forty two 902 – nine hundred and two 714 – seven hundred and fourteen Numbers up to 999 999 6 703 – six thousand seven hundred and three. 15 243 – fifteen thousand two hundred and forty three 209 312 – two hundred and nine thousand, three hundred and twelve 745 697 – seven hundred and forty five thousand, six hundred and ninety seven Numbers up to 999 999 999 6 703 769 – six million seven hundred and three thousand, seven hundred and sixty nine 89 503 102 – eighty nine million, five hundred and three thousand, one hundred and two. 607 132 617 – six hundred and seven million, one hundred and thirty two, six hundred and seventeen. It really helps to leave gaps between the hundred’s, thousand’s and millions. When reading decimals remember that; 3.45 is three point four five NOT three point forty five.

6. The Multiples of a number are all the numbers in that times table. E.G. The multiples of 6 are: 6, 12, 18,24 … The Multiples of a number are all the numbers in that times table. E.G. The multiples of 6 are: 6, 12, 18,24 … The Factors of a number are all the numbers that goes into it exactly E.G. The factors of 12 are: 1, 2, 3, 4, 6, 12 The Factors of a number are all the numbers that goes into it exactly E.G. The factors of 12 are: 1, 2, 3, 4, 6, 12 Divisibility tests. A number is divisible: By 2 if it is even, i.e., it ends in 0, 2, 4, 6 or 8 By 3 if the sum of the digits is a multiple of 3 By 4 if half of the number is even By 5 if it ends in 0 or 5 By 6 if half the number is divisible by 3 By 8 if half of half the number is even By 9 if the sum of the digits is a multiple of 9 By 10 if it ends in 0. Divisibility tests. A number is divisible: By 2 if it is even, i.e., it ends in 0, 2, 4, 6 or 8 By 3 if the sum of the digits is a multiple of 3 By 4 if half of the number is even By 5 if it ends in 0 or 5 By 6 if half the number is divisible by 3 By 8 if half of half the number is even By 9 if the sum of the digits is a multiple of 9 By 10 if it ends in 0. Number Relationships and Divisibility tests. A prime number is a number with exactly two factors, one and itself E.G. 17 is prime because only 1 and 17 goes into it exactly. The primes are 2, 3,5,7,11,13… Remember that 1 is NOT a prime number because it only has one factor A prime number is a number with exactly two factors, one and itself E.G. 17 is prime because only 1 and 17 goes into it exactly. The primes are 2, 3,5,7,11,13… Remember that 1 is NOT a prime number because it only has one factor The factor pairs of a number are pairs of numbers that times to it. E.G. The factor pairs of 20 are 1 x 20 2 x 10 4 x 5 The factor pairs of a number are pairs of numbers that times to it. E.G. The factor pairs of 20 are 1 x 20 2 x 10 4 x 5

8. 3 2 46 6 into 3 doesn’t go so carry the 3 6 into 32 goes 5 times remainder 2 6 into 24 goes 4 times no remainder 5 3 2 46 2 5 4 6 2 3 2 4 Check: 54 x 6 = 324 1 x 17 = 17 2 x 17 = 34 4 x 17 = 68 8 x 17 = 136 13 x 17 = 221 So, Double to divide! 6 1 2 18 Use a ready reckoner. 1x18 = 18 2x18 = 36 3x18 = 54 4x18 = 72 5x18 = 90 6x18 = 108 7x18 = 126 8x18 = 144 9x18 = 162 add 20 takeaway 2! 18 into 6 doesn’t go so carry the 6 18 into 61 goes 3 times remainder 7 18 into 72 goes 4 times no remainder 6 1 2 18 6 1 2 18 3 7 3 4 7 Check: 13 x 17 = 221 Check: 18 x 34 = 612 3 3 6 6 6 3

9. Multiplication and division patterns You are multiplying by a number that is 10 times smaller so the answer is ten times smaller. You are dividing by a number that is 10 times smaller so the answer is ten times bigger.

10. BIDMAS Bidmas is an acronym explaining the order in which operations should be carried out. B brackets I ndices D division M multiply A addition S subtraction. Consider: 5 +2 x 3 If you did this in the order it was written the answer is 21. In fact the answer is 11. This is because we must multiply before we add 8 – 3 x 2 8 – 6 = 2 (8 – 3) x 2 5 x 2 = 10 (8 – 3) x 2 5 x 2 = 10 (8 – 3) x ( 4 +3) 5 x 7 = 35 (8 + 4) ÷ (10 - 7) 12 ÷ 3 = 4

11. So Equivalent Fractions/Decimals/Percentages So

12. Percentage Non Calculator 50% Half it 25% Half it and half again 75% 50% + 25% 10% Divide by 10 1% Divide by 100 5% 10% ÷ 2 30% 10% x 3 Finding a percentage with a calculator. Turn the percentage into decimal and multiply! 42% of 540 0.42 x 540 = 235.2 6% of 310 0.06 x 310 = 18.6 Find one number as a percentage of another with a calculator I got 42 out of 60 in my test What percentage is that? I do 42÷60 x 100 on the calculator.

13. Fractions Equivalent fractions Lowest Terms Is there a number that goes into 30 and 55 exactly? Yes 5. So This is its lowest terms because only 1 goes into 6 and 11 exactly There are 8 equal parts. 3 pieces are shaded. So, is shaded There are 8 equal parts. 3 pieces are shaded. So, is shaded There are 6 equal parts. 3 pieces are shaded. So, is shaded There are 6 equal parts. 3 pieces are shaded. So, is shaded Or In its lowest terms Or In its lowest terms

14. To make 12 pancakes 100g of plain flour 2 eggs 200ml of milk 60g of butter To make 24 pancakes I would DOUBLE all of the quantities To make 6 pancakes I would HALF all the quantities To make 30 pancakes I would add together the quantities for 24 and 6 pancakes. To make 9 pancakes I would divide by 4 to find the quantities for 3 pancakes and then multiply by 3 to find to quantities for 9 To make 5 pancakes I would divide by 12 to find the quantities for 1 pancake and then multiply by 5 to find to quantities for 5 Proportion.

15. Adding and subtracting negative numbers. If you can put a circle around two signs use the following rules “If the signs are the same we like it!” ADD “If the signs are different we don’t like it!” SUBTRACT -7 + - 3 = 10 -14 - - 3 = -11 9 - - 3 = 12 6 - + 9 = -3 - - + - + + - + +--+ Direct number line sums. -7 + 3 = -4 Start here go up three end up here 4 - 6 = -2 Start here go up three end up here Negative Numbers. Multiplying negative numbers Neg No. x Neg No. = Pos. Answer Pos. No. x Pos. No. = Pos. Answer Neg No. x Pos. No. = Pos. Answer Pos. No. x Neg. No. = Pos. Answer The same rules apply when dividing. - + + - + −

16. Reading Scales. There are 5 small divisions between 0 and 250 so each small division is 50 So 750 and 3 small divisions equals 900

17. Metric Units: Length 10mm = 1cm 6.2 cm Is the same as 62 mm 0.2cm Is the same as 2 mm 6 cm Is the same as 60 mm 53 mm Is the same as 5.3 cm 2 mm Is the same as 0.2 cm 50 mm Is the same as 5 cm EVERY digit moves one place to the LEFT. EVERY digit moves one place to the RIGHT. Multiply it by 10, so every digit becomes 10 times BIGGER. Divide it by 10, so every digit becomes 10 times SMALLER. 100cm = 1m 5.4m Is the same as 540cm 0.3m Is the same as 30cm 5 m Is the same as 500cm 62 cm Is the same as 0.62 m 200 cm Is the same as 2 m 5 cm Is the same as 0.05 m EVERY digit moves two places to the LEFT. EVERY digit moves two places to the RIGHT. Multiply it by 100, so every digit becomes 100 times BIGGER. Divide it by 100, so every digit becomes 100 times SMALLER. 1000m = 1km 6.9km Is the same as 6900km 0.75km Is the same as 750m 7 km Is the same as 7000km 8800m Is the same as 8.8km 3000m Is the same as 3km 51 m Is the same as 0.051km EVERY digit moves three places to the LEFT. EVERY digit moves three places to the RIGHT. Multiply it by 1000, so every digit becomes 1000 times BIGGER. Divide it by 1000, so every digit becomes 1000times SMALLER.

18. Metric Units: Capacity and Mass 1000 ml = 1 l 0.89 l Is the same as 890 ml 4 l Is the same as 4000 ml 580 ml Is the same as 0.580 l 6700 ml Is the same as 6.7 l 35 ml Is the same as 0.035 l EVERY digit moves three places to the LEFT. EVERY digit moves three places to the RIGHT. Multiply it by 1000, so every digit becomes 1000 times BIGGER. Divide it by 1000, so every digit becomes 1000 times SMALLER. Divide it by 1000, so every digit becomes 1000 times SMALLER. 7.4 l Is the same as 7400 ml 1000 g = 1 Kg 0.54 Kg Is the same as 540 g 6 kg Is the same as 6000 g 530 g Is the same as 0.530 Kg 3700 g Is the same as 3.7 Kg 62 g Is the same as 0.062 Kg EVERY digit moves three places to the LEFT. EVERY digit moves three places to the RIGHT. Divide it by 1000, so every digit becomes 1000 times SMALLER. 6.9 Kg Is the same as 6900 g Multiply it by 1000, so every digit becomes 1000 times BIGGER.

19. Perimeter. Perimeter is the distance around the outside edge of a 2D shape The perimeter of a circle is called the CIRCUMFERENCE 6cm 9cm 7cm 5cm 8cm 10cm 6cm P = 8 + 9 +7 + 5 = 29cm P = 4 x 6 = 24cm P = 2 x10 +2 x 6 = 32cm or

20. 1.Check all values have the same unit before you start. 2.Always write down the correct formula first. 3.Rewrite the formula with values 4.Use Bodmas when doing the calculation 5.Give correct units. Area. Area is the number of unit squares that fit onto a 2D shape w h h h bb b a The surface area of a 3D shape is the total area of all its faces

21. Volume. 1.Check all values have the same unit before you start. 2.Always write down the correct formula first. 3.Rewrite the formula with values 4.Use Bodmas when doing the calculation 5.Give correct units. Volume is the number of unit cubes that fit onto a 3D shape A prism is a shape with a uniform cross section. Vol. of a prism = area of uniform cross section x length A prism is a shape with a uniform cross section. Vol. of a prism = area of uniform cross section x length

22. MedianMode Mean The ‘average’, or arithmetic mean, of a set of discrete data is the sum of quantities divided by the number of quantities. Find the mean of the following data: 2, 3, 4, 5, 4, 6 Add the numbers and then divide your answer by how many numbers there are? The Median is the middle number or value when all values in a set of data are arranged in ascending order. Find the Median of the following data: 4, 7, 8, 2, 3, 5, 6, 10, 6, 5, 6 You need to order the data and then find the number in the middle. 2, 3, 4, 5, 5, 6, 6, 6, 7, 8, 10 The Mode is the most commonly occurring value. Find the Mode of the following data: 5, 7, 8, 2, 3, 6, 6, 10, 6, 5, 6 Find the number that appears the most. 2, 3, 5, 5, 6, 6, 6, 6, 7, 8, 10 6 is the mode. The Range is the difference between the greatest value and the least value in a set of numerical data. Remember it is not an average

23. What makes a good graph? 10 50 40 30 20 100 90 80 70 60 0 10 20 30 40 50 60 70 80 0 Time (s) Temperature ( o c) A Graph Showing the Results of ………………………… × × × × × × × title arrow label with units Each graph must have A title A label on both axes An arrow at the end of each axes A sensible scale for both axes Plot each point as a neat cross ×

24. % long Bodmas F’s, D’s division and P’s 24- 22- 20- 18- 16- 14- 12- 10- 8 - 6 - 4 - 2 - 0 - Number topic FrequencyFrequency Favourite Number topic All the bars must be the same width and All the gaps between the bars should be equal Check list: Title A label on each axis A sensible scale Bar Chart A bar chart is for discrete data; there should be gaps between the bars

25. VehicleFrequency Car Van Bike Bus Pictogram Key = 40 Vehicles There are 90 vans And 130 bikes There are 90 vans And 130 bikes When drawing a pictogram: All pictures must be the same shape, size and colour All gaps between the pictures must be the same size. There must be a key When drawing a pictogram: All pictures must be the same shape, size and colour All gaps between the pictures must be the same size. There must be a key

26. 1.4m 1.5m 1.6m 1.7m 1.8m 24- 22- 20- 18- 16- 14- 12- 10- 8 - 6 - 4 - 2 - 0 - Height (M) FrequencyFrequency Height of year 9 girls. A Grouped Frequency Diagram is for Continuous Data; there should be no gaps between the bars Check list: Title A label on each axis A sensible scale Grouped Frequency Diagram Notice that the each height interval is equal

27. 1.4m 1.5m 1.6m 1.7m 1.8m 24- 22- 20- 18- 16- 14- 12- 10- 8 - 6 - 4 - 2 - 0 - Height (M) FrequencyFrequency Height of year 9 girls. Check list: Title A label on each axis A sensible scale Notice that the each height interval is equal Height of yr9 girls.Frequency 1.4 < m 1.69 1.6 < m 1.821 1.6 < m 1.816 1.6 < m 1.85 Frequency Polygon A Frequency polygon is for continuous data. Points are plotted on the mid-point

28. Pie Chart Favourite Number topic. FrequencyAngle Percentage14 14 x 6 o = 84 o Long Division2424 x 6 o = 144 o Bodmas1717 x 6 o = 102 o Factors, multiples and primes 55 x 6 o = 30 o Sixty people were asked their favourite number topic. First step is to find out how many degrees will represent each person Check the angles add to 360 o Percentage Long Division Bodmas F’s, M’s and P’s Remember that pie charts show proportions not the actual numbers so can be misleading. If you look at this chart you can see that Long division in is the most popular topic, but you can not tell how many people said long division

29. A Scatter Diagram helps us to determine whether there is a link between two sets of data. 10 learners take a quiz in two subjects Maths and Science. The results are recorded below. Is there a link (correlation) between the maths and science score for each child? You can draw a scatter diagram to decide MathsScience 10 6 3 9 1 9 4 7 5 6 97280737549728073754 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Maths Science × × × × × × × × × × Plot the points (10,9) (6,7) (3,2) (9,8) (1,0) (9,7) (4,3) (7,7) (5,5) (6,4) There is a positive correlation – As the maths mark increases so does the science mark.. × × × × × ×× × × × × × × × × × × × × × × × × × ××× × Negative Correlation As one value increases the other value decreases The data values move in opposite directions Positive Correlation As one value increases the other value increases The data values move in the same direction No Correlation The statistics are not linked at all. The one value has no effect on the other value

30. Metric and Imperial Units Metric Units for length: Millimetres Centimetres Metre Kilometres Metric units for capacity: Mililitres Litres Metric Units for mass: Grams Kilograms Metric Units for length: Millimetres Centimetres Metre Kilometres Metric units for capacity: Mililitres Litres Metric Units for mass: Grams Kilograms Imperial Units for length: Inch Foot Yard Mile Imperial units for capacity: Pint Gallon Imperial Units for mass: Ounce Pound Stone Imperial Units for length: Inch Foot Yard Mile Imperial units for capacity: Pint Gallon Imperial Units for mass: Ounce Pound Stone 1 inch 2.5 centimetres 1 foot = 12 inches 1 mile 1.6 Kilometre 1 pint 0.568 litre 1 gallon = 8 pints 1 pound 0.45 kilograms 1 stone = 14 pounds

31. Addition and Subtraction of Decimals. 18.46 + 9.71 18.46 + 9.71 28.17 11 Line up the decimal points! 14.6 - 9.17 14.60 - 9.17 5.43 51 Line up the decimal points! Fill in the gap with a zero! I buy an ice cream for £1.74. How much change do I get from £5 ? 5. 0 0 - 1.7 4 3.26 6p20p £3.00 74p 80p £2.00 £5.00 Adding to subtract: £3.00 + 20p +6p = £3.26 4 11 9

32. How do I find out how much time has passed between 1947 and 2015 The easiest way is to use a number line. How do I find out how much time has passed between 1066 and 2013 19471950 2000 2015 + 3 + 50 + 15 3+50+15 = 68 years 106610701100 2015 + 4 + 30 4+30+900+15 = 949 years + 13 2000 + 900

33. Checking methods Check by estimating 68 X 71= 4828 Check: 70 x 70 = 4900 So 4928 is a sensible answer. Check by estimating 68 X 71= 4828 Check: 70 x 70 = 4900 So 4928 is a sensible answer. Check using inverse operations 71 X 6 + 5 = 431 So using inverse operations If you did 431 – 5 and then divided that answer by 6 you should get 71. Check using inverse operations 71 X 6 + 5 = 431 So using inverse operations If you did 431 – 5 and then divided that answer by 6 you should get 71. Check by substitution Find a, when b= 6 and t=18 2b + 3a = t 2x6 + 3xa = 18 So a=2 Check by substitution; 2x6 + 3x2 = 18 Check by substitution Find a, when b= 6 and t=18 2b + 3a = t 2x6 + 3xa = 18 So a=2 Check by substitution; 2x6 + 3x2 = 18 4.56 x 5.7 = 25.992 Three numbers after the decimal point in the question, three numbers after the decimal point in the answer. 4.56 x 5.7 = 25.992 Three numbers after the decimal point in the question, three numbers after the decimal point in the answer.

34. Measuring angles We are measuring in a clockwise direction so use the outside scale. 30 o We are measuring in a clockwise direction so use the outside scale. We are measuring in an anti-clockwise direction so use the inside scale. 60 o 50 o

35. Ratio Mixing blue and yellow paint in the ratio of 2:3 produces green paint. = If I need 300ml of green paint, how much blue and yellow paint would I need? The 5 parts green = 300ml So 1 part = 300 ÷ 5 = 60 ml I need 2 parts blue = 120ml And I need 3 parts green = 180ml 2parts blue3 parts yellow+makes 5 parts green If I need 1.5 litres of green paint, how much blue and yellow paint would I need? The 5 parts green = 1.5 litres So, 1 part = 1.5 ÷ 5 = 0.3 litres I need 2 parts blue = 0.6 litres And I need 3 parts green = 0.9 litres If I had 50ml of blue paint how much yellow paint would I need to make green paint? The two parts blue = 50 ml So 1 part = 50÷2 = 25 ml I need 3 parts yellow 75 ml