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Numeracy Across the Curriculum Toolkit Addition Subtraction Addition 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 Methods Fractions Proportion Ratio Negative numbers Reading Scales Metric Units Length Metric Units Capacity and Mass Perimeter Area Volume Measure angles Averages What makes a good graph Bar Chart Pictogram Grouped Frequency Diagram Frequency Polygon Pie Chart Scatter Diagram metric and imperial units Llanishen High School - Numeracy toolkit Addition. +70 +80 -1 +9 84 84 154 163 +70 150 154 +4 80 80 + 70 = 150 4 + 9 = 13 85 + 79 = 163 84 + 79 84 + 80 – 1 =163 80 163 164 163 + 9 150 154 163 84 + 79 163 1 1 Adding Words •Sum •Total •Plus •Altogether •More than +4 +24 - 50 -2 56 60 -4 84 Adding on to subtract 24 + 4 = 28 28 30 80 84 Subtraction 84 - 56 84 -50 – 6 =28 80 -50 -6 28 56 +4 +20 + 4 = 84 Answer = 28 150 154 34 163 7 1 84 Answer = 28 84 - 56 28 • • • • • Subtracting Words Difference Take away Take off Minus Less than Worded questions. The hardest thing about worded questions is deciding which operation to use. Here are some ‘operation’ words to help you . Subtracting Adding Words Words Division Words • Sum • Difference • Share • Total • Take away • Each • Plus • Take off • Divide • Altogether • Minus • Quotient • More than • Less • Goes into Multiplication words • Times • Product • Multiply • Of • Lots of Reading numbers. It really helps to leave gaps between the hundred’s, thousand’s and millions. Numbers up to 999 345 – three hundred and forty two 902 – nine hundred and two When reading decimals remember that; 3.45 is three point four five NOT three point forty five. 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. Number Relationships and Divisibility tests. 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 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 4x5 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. 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 324 6 3 6 324 5 3 2 6 324 54 6 3 322 4 6 into 3 doesn’t go so carry the 3 22117 1 x 17 2 x 17 4 x 17 8 x 17 = 17 = 34 = 68 = 136 13 x 17 = 221 6 into 32 goes 5 times remainder 2 6 into 24 goes 4 times no remainder Check: 54 x 6 = 324 So, 221 17 13 Double to divide! Check: 13 x 17 = 221 612 18 6 18 6 1 2 18 into 6 doesn’t go so carry the 6 3 6 7 6 7 18 6 1 2 3 4 18 6 1 2 18 into 72 goes 4 times no remainder Check: 18 x 34 = 612 18 into 61 goes 3 times remainder 7 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! Multiplication and division patterns 7 ´ 5 = 35 16 ´ 2 = 32 67 ´1 = 67 7 ´ 0.5 = 3.5 16 ´ 0.2 = 3.2 67 ´ 0.1 = 6.7 ¸10 ¸10 ¸10 ¸10 ¸10 ¸10 You are multiplying by a number that is 10 times smaller so the answer is ten times smaller. 65 ¸ 5 = 13 16 ¸ 2 = 8 18 ¸1 = 18 65 ¸ 0.5 = 130 16 ¸ 0.2 = 80 18 ¸ 0.1 = 180 ¸10 ´10 ¸10 ´10 ¸10 ´10 You are dividing by a number that is 10 times smaller so the answer is ten times bigger. Bidmas is an acronym explaining the order in which operations should be carried out. BIDMAS Consider: 8 –3x2 8 – 6 =2 (8 – 3) x 2 5 x 2 = 10 5 +2 x 3 B I D M A S brackets ndices division 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 5 x 2 = 10 multiply (8 + 4) ÷ (10 - 7) 12 ÷ 3 =4 addition subtraction. (8 – 3) x ( 4 +3) 5 x 7 = 35 Equivalent Fractions/Decimals/Percentages 1 = 0.5 = 50% 2 1 = 0.25 = 25% 4 1 = 0.1 = 10% 10 1 = 0.01 = 1% 100 1 0.2 20% 5 1 = 0.3 = 33.3% 3 So 3 = 0.75 = 75% 4 9 0.9 90% So 10 So 17 = 0.17 = 17% 100 So 3 0.6 60% 5 2 = 0.6 = 66.6% So 3 25% Half it and half again 10% Divide by 10 30% 10% x 3 Finding a percentage with a calculator. 50% Half it Percentage Non Calculator 5% 10% ÷ 2 Turn the percentage into decimal and multiply! 42% of 540 1% Divide by 100 0.42 x 540 = 235.2 6% of 310 75% 50% + 25% 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? 42 100 70% 60 I do 42÷60 x 100 on the calculator. Fractions Lowest Terms Equivalent fractions 30 55 ´3 5 15 = 7 ´3 21 Is there a number that goes into 30 and 55 exactly? ¸5 Yes 5. So ¸5 25 5 = 40 ¸5 8 This is its lowest terms because only 1 goes into 6 and 11 exactly There are 6 equal parts. 3 pieces are shaded. There are 8 equal parts. 3 pieces are shaded. So, 3 8 is shaded 30 6 = 55 ¸5 11 So, Or 3 6 is shaded 3 1 = 6 2 In its lowest terms 1 ´ 42 = 6 7 42 ¸ 7 3 ´ 60 = 36 5 60 ¸ 5´ 3 Proportion. To make 12 pancakes 100g of plain flour To make 6 pancakes I would HALF all the quantities 2 eggs 200ml of milk 60g of butter 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 24 pancakes I would DOUBLE all of the quantities To make 30 pancakes I would add together the quantities for 24 and 6 pancakes. 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 Negative Numbers. + 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 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 + 9 - - 3 = 12 + -14 - - 3 = -11 6 - + 9 = -3 Multiplying negative numbers Neg No. x Neg No. = Pos. Answer Neg No. x Pos. No. = Pos. Answer Pos. No. x Pos. No. = Pos. Answer Pos. No. x Neg. No. = Pos. Answer The same rules apply when dividing. 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 Metric Units: Length 10mm = 1cm 100cm = 1m 6 cm Is the same as 60 mm 6.2 cm Is the same as 62 mm 5m Is the same as 500cm 0.2cm Is the same as 2 mm Multiply it by 10, so every digit becomes 10 times BIGGER. 0.3m Multiply it by 100, so Is the same as every digit becomes 30cm 100 times BIGGER. EVERY digit moves one place to the LEFT. 5.4m Is the same as 540cm EVERY digit moves two places to the LEFT. 2 mm Is the same as 0.2 cm 53 mm Is the same as 5.3 cm 200 cm Is the same as 2m 50 mm Is the same as 5 cm Divide it by 10, so every digit becomes 10 times SMALLER. 5 cm Divide it by 100, so Is the same as every digit becomes 0.05 m 100 times SMALLER. EVERY digit moves one place to the RIGHT. 62 cm Is the same as 0.62 m EVERY digit moves two places to the RIGHT. 1000m = 1km 7 km Is the same as 7000km 0.75km Is the same as 750m 6.9km Is the same as 6900km Multiply it by 1000, so every digit becomes 1000 times BIGGER. EVERY digit moves three places to the LEFT. 3000m Is the same as 3km 8800m Is the same as 8.8km 51 m Is the same as 0.051km Divide it by 1000, so every digit becomes 1000times SMALLER. EVERY digit moves three places to the RIGHT. Metric Units: Capacity and Mass 1000ml = 1l 4l Is the same as 4000ml 7.4l Is the same as 7400ml 0.89l Is the same as 890ml Multiply it by 1000, so every digit becomes 1000 times BIGGER. EVERY digit moves three places to the LEFT. 6700ml Is the same as 6.7l 35ml Is the same as 0.035l 580ml Is the same as 0.580l Divide it by 1000, so every digit becomes 1000 times SMALLER. EVERY digit moves three places to the RIGHT. 1000g = 1Kg 6 kg Is the same as 6000g 0.54Kg Is the same as 540g 6.9Kg Is the same as 6900g Multiply it by 1000, so every digit becomes 1000 times BIGGER. EVERY digit moves three places to the LEFT. 3700g Is the same as 3.7Kg 62g Is the same as 0.062Kg 530g Is the same as 0.530Kg Divide it by 1000, so every digit becomes 1000 times SMALLER. EVERY digit moves three places to the RIGHT. Perimeter is the distance around the outside edge of a 2D shape Perimeter. 8cm 10cm P = 2 x10 +2 x 6 = 32cm P = 8 + 9 +7 + 5 5cm = 29cm 9cm The perimeter of a circle is called the CIRCUMFERENCE 7cm 6cm P=4x6 = 24cm d C = pd or C = 2p r 6cm Area. Area is the number of unit squares that fit onto a 2D shape A =l´w a 1 A = ´ (a + b) ´ h 2 1. Check all values have the same unit before you start. w 2. Always write down the correct formula first. 3. Rewrite the formula with values l h b r A = p r2 4. Use Bodmas when doing the calculation h b´h A= 2 b 5. Give correct units. A = b´h b 2 (cm ) h The surface area of a 3D shape is the total area of all its faces h w l Vol = l ´ w ´ h Volume. Volume is the number of unit cubes that fit onto a 3D shape 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. V = A ´l V= b´h ´l 2 (cm3 ) A prism is a shape with a uniform cross section . Vol. of a prism = area of uniform cross section x length V = CSA ´l V = A ´l V = p r2 ´ l Mean Median Mode The ‘average’, or arithmetic mean, of a set of discrete data is the sum of quantities divided by the number of quantities. The Median is the middle number or value when all values in a set of data are arranged in ascending order. The Mode is the most commonly occurring value. Find the mean of the following data: Find the Median of the following data: Find the Mode of the following data: 2, 3, 4, 5, 4, 6 4, 7, 8, 2, 3, 5, 6, 10, 6, 5, 6 Add the numbers and then divide your answer by how many numbers there are? 2 3 4 5 4 6 24 4 6 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 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 What makes a good graph? A Graph Showing the Results of ………………………… title label with units Temperature (oc) arrow 100 90 80 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 × × 70 60 × 50 × 40 × × 30 × Plot each point as a neat cross × 20 10 arrow 0 0 10 20 30 40 50 60 70 80 Time (s) label with units Bar Chart Favourite Number topic 24- Check list: •Title •A label on each axis •A sensible scale 2220- F r e q u e n c y 18161412- A bar chart is for discrete data; there should be gaps between the bars 108 6 4 2 0 - % long division Bodmas Number topic F’s, D’s and P’s All the bars must be the same width and All the gaps between the bars should be equal Pictogram Vehicle Car Frequency When drawing a pictogram: Van All pictures must be the same shape, size and colour Bike All gaps between the pictures must be the same size. There must be a key Bus Key = 40 Vehicles There are 90 vans And 130 bikes Grouped Frequency Diagram Height of year 9 girls. 242220- F r e q u e n c y Check list: •Title •A label on each axis •A sensible scale 181614- A Grouped Frequency Diagram is for Continuous Data; there should be no gaps between the bars 12108 6 4 2 0 - 1.4m 1.5m 1.6m Height (M) 1.7m 1.8m Notice that the each height interval is equal Frequency Polygon Height of yr9 girls. Frequency 1.4 < m £1.6 9 1.6 < m £ 1.8 1.6 < m 1.8 1.6 < m £ 1.8 21 16 5 £ Height of year 9 girls. Check list: •Title •A label on each axis •A sensible scale 242220- F r e q u e n c y 1816- A Frequency polygon is for continuous data. Points are plotted on the mid-point 1412108 6 4 - Notice that the each height interval is equal 2 0 - 1.4m 1.5m 1.6m Height (M) 1.7m 1.8m Pie Chart Sixty people were asked their favourite number topic. First step is to find out how many degrees will represent each person 60 people = 360 o 360 o 1person = = 6o 60 Favourite Number topic. Frequency Angle Percentage 14 14 x 6o = 84 o Long Division 24 24 x 6o = 144 o Bodmas 17 17 x 6o = 102 o Factors, multiples and primes 5 5 x 6o = 30 o 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 Percentage Bodmas Long Division Check the angles add to 360o 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. Science 10 6 3 9 1 9 4 7 5 6 9 7 2 8 0 7 3 7 5 4 Science Maths 10 9 8 7 6 5 4 3 2 1 0 Negative Correlation Is there a link (correlation) between the maths and science score for each child? × × × Plot the points You can draw a scatter diagram to decide (6,7) (9,8) × × × × × As one value increases the other value decreases × × × × × The data values move in opposite directions (10,9) (3,2) × × × × (1,0) (9,7) × Positive Correlation As one value increases the other value increases The data values move in the same direction × × × × × × × (4,3) × (7,7) (5,5) × × (6,4) 0 1 2 3 4 5 6 7 8 9 10 Maths There is a positive correlation – As the maths mark increases so does the science mark.. × × × × × × ×× × × No Correlation The statistics are not linked at all. The one value has no effect on the other value Metric Units for length: Millimetres Centimetres Metre Kilometres Metric units for capacity: Mililitres Litres Metric Units for mass: Grams Kilograms Metric and Imperial Units 1 inch »2.5 centimetres 1 foot = 12 inches 1 mile 1 pint » 1.6 Kilometre » 0.568 litre Imperial Units for length: Inch Foot Yard Mile Imperial units for capacity: Pint Gallon Imperial Units 1 gallon = 8 pints 1 pound » 0.45 kilograms 1 stone = 14 pounds for mass: Ounce Pound Stone Addition and Subtraction of Decimals. 18.46 + 9.71 18.46 + 9.71 28.17 1 1 14.6 - 9.17 Line up the decimal points! 5 1 14.60 - 9.17 5.43 Fill in the gap with a zero! Line up the decimal points! I buy an ice cream for £1.74. How much change do I get from £5 ? 4 9 1 1 5. 0 0 - 1 .7 4 3.26 £3.00 6p 74p 80p 20p £2.00 £5.00 Adding to subtract: £3.00 + 20p +6p = £3.26 How do I find out how much time has passed between 1947 and 2015 3+50+15 = 68 years 4+30+900+15 = 949 years 2015 + 900 2000 + 30 1100 +4 1070 1066 How do I find out how much time has passed between 1066 and 2013 + 13 + 15 2015 2000 +3 + 50 1950 1947 The easiest way is to use a number line. Checking methods Check by substitution Check by estimating Find a, when b= 6 and t=18 68 X 71= 4828 2b + 3a = t 2x6 + 3xa = 18 So a=2 Check: 70 x 70 = 4900 So 4928 is a sensible answer. Check by substitution; 2x6 + 3x2 = 18 Check using inverse operations 71 X 6 + 5 = 431 4.56 x 5.7 = 25.992 So using inverse operations If you did 431 – 5 and then divided that answer by 6 you should get 71. Three numbers after the decimal point in the question, three numbers after the decimal point in the answer. We are measuring in a clockwise direction so use the outside scale. Measuring angles 30o 50o We are measuring in an anti-clockwise direction so use the inside scale. 60o We are measuring in a clockwise direction so use the outside scale. Ratio Mixing blue and yellow paint in the ratio of 2:3 produces green paint. 2parts blue + 3 parts yellow = makes 5 parts green If I need 300ml of green paint, how much blue and yellow paint would I need? If I need 1.5 litres of green paint, how much blue and yellow paint would I need? The 5 parts green = 300ml So 1 part = 300 ÷ 5 = 60 ml The 5 parts green = 1.5 litres So, 1 part = 1.5 ÷ 5 = 0.3 litres I need 2 parts blue = 120ml And I need 3 parts green = 180ml 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