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© Boardworks Ltd 2008 1 of 51 N10 Written and calculator methods Maths Age 11-14

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© Boardworks Ltd 2008 2 of 51 A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 N10.1 Estimation and approximation Contents N10 Written and calculator methods N10.2 Addition and subtraction N10.3 Multiplication N10.4 Division N10.5 Using a calculator N10.6 Checking results

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© Boardworks Ltd 2008 3 of 51 Estimation four-in-a-line

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© Boardworks Ltd 2008 4 of 51 Martin uses his calculator to work out 39 × 72. The display shows an answer of 1053. How do you know this answer must be wrong? “is approximately equal to” 39 × 72 40 × 70 =2800 The product of 39 and 72 must therefore end in an 8. 9 × 2 = 18. Estimation Also, if we multiply together the last digits of 39 and 72 we have

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© Boardworks Ltd 2008 5 of 51 3.5 × 17.5 can be approximated to: 4 × 20 =80 3 × 18 =54 4 × 17 =68 or between 3 × 17 =51and 4 × 18 =72 How could we estimate the answer to 3.5 × 17.5? Estimation

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© Boardworks Ltd 2008 6 of 51 Using points on a scale to estimate answers Jessica is trying to estimate which number multiplied by itself will give the answer 32. She knows that 5 × 5 = 25 and that 6 × 6 = 36. The number must therefore be between 5 and 6. She draws the following scales to help her find an approximate answer. 252627282930313233343536 56 5.64

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© Boardworks Ltd 2008 7 of 51 Use Jessica’s method to estimate which number multiplied by itself will give an answer of 40. We know that 6 × 6 = 36 and that 7 × 7 = 49. Draw a scale from 36 to 49. Underneath, draw a scale from 6 to 7. 363738394041424344454647484967 6.31 Using points on a scale to estimate answers

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© Boardworks Ltd 2008 8 of 51 Contents N10 Written and calculator methods A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 N10.2 Addition and subtraction N10.3 Multiplication N10.4 Division N10.5 Using a calculator N10.6 Checking results N10.1 Estimation and approximation

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© Boardworks Ltd 2008 9 of 51 Adding and subtracting decimals

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© Boardworks Ltd 2008 10 of 51 Jack is doing some DIY. He buys a 3m length of wood. Jack needs to cut off two pieces of wood – one of length 0.7m and one of length 1.92m. a)What is the total length of wood which Jack needs to cut off? b)What is the length of the piece of wood which is left over? 0.7 1.92 0 + a) 1 262. b) –2.62 3.00 2 1 9 1 83.0 Jack needs to cut off 2.62m altogether. The left-over wood will measure 0.38m (or 38cm). Adding and subtracting decimals

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© Boardworks Ltd 2008 11 of 51 Sums and differences

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© Boardworks Ltd 2008 12 of 51 Contents N10 Written and calculator methods N10.3 Multiplication N10.4 Division N10.5 Using a calculator N10.6 Checking results N10.2 Addition and subtraction N10.1 Estimation and approximation

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© Boardworks Ltd 2008 13 of 51 The grid method for multiplying decimals

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© Boardworks Ltd 2008 14 of 51 Using the standard column method Start by finding an approximate answer: 2.28 × 7 2 × 7 =14 2.28 × 7 is equivalent to 228 × 7 ÷ 100 228 × 7 6 5 9 1 15 Answer 2.28 × 7 = 1596 ÷ 100 =15.96 What is 2.28 × 7?

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© Boardworks Ltd 2008 15 of 51 Using the standard column method Again, start by finding an approximate answer: 392.7 × 0.8 400 × 1 =400 392.7 × 0.8 is equivalent to 3927 × 8 ÷ 100 3927 × 8 6 5 1 2 4 7 31 Answer 392.7 × 0.8 = 31416 ÷ 10 ÷ 10 = 314.16 What is 392.7 × 0.8?

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© Boardworks Ltd 2008 16 of 51 Drag and drop multiplication problem

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© Boardworks Ltd 2008 17 of 51 Multiplying two-digit numbers Calculate 57.4 × 24. Estimate: 60 × 25 = 1500 Equivalent calculation: 57.4 × 10 × 24 ÷ 10 = 574 × 24 ÷ 10 574 × 24 11480 2296 13776 Answer: 13776 ÷ 10 = 1377.6 4 × 574 = 20 × 574 =

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© Boardworks Ltd 2008 18 of 51 Multiplying two-digit numbers Calculate 23.2 × 1.8. Estimate: 23 × 2 = 46 Equivalent calculation: 23.2 × 10 × 1.8 × 10 ÷ 100 = 232 × 18 ÷ 100 232 × 18 2320 1856 4176 Answer: 4176 ÷ 100 = 41.76 8 × 232 = 10 × 232 =

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© Boardworks Ltd 2008 19 of 51 Multiplying two-digit numbers Calculate 394 × 0.47. Estimate: 400 × 0.5 = 200 Equivalent calculation: 394 × 0.47 × 100 ÷ 100 = 394 × 47 ÷ 100 394 × 47 15760 2758 18518 Answer: 18518 ÷ 100 = 185.18 7 × 394 = 40 × 394 =

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© Boardworks Ltd 2008 20 of 51 Contents N10 Written and calculator methods A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 N10.4 Division N10.5 Using a calculator N10.6 Checking results N10.2 Addition and subtraction N10.3 Multiplication N10.1 Estimation and approximation

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© Boardworks Ltd 2008 21 of 51 Dividing decimals – Example 1 What is 259.2 ÷ 6? What is 259.2 ÷ 6? DividendDivisor

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© Boardworks Ltd 2008 22 of 51 Using repeated subtraction Start by finding an approximate answer: 259.2 ÷ 6 240 ÷ 6 =40 259.26 6 × 40– 240.0 19.2 6 × 3– 18.0 1.2 6 × 0.2– 1.2 0 Answer:43.2

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© Boardworks Ltd 2008 23 of 51 Using short division Start by finding an approximate answer: 259.2 ÷ 6 240 ÷ 6 =40 2 5 9. 26 0 2 4 1 3. 1 2 2.59 ÷ 6 = 43.2

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© Boardworks Ltd 2008 24 of 51 Dividing decimals – Example 2 What is 714.06 ÷ 9? What is 714.06 ÷ 9? DividendDivisor

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© Boardworks Ltd 2008 25 of 51 Using repeated subtraction Start by finding an approximate answer: 714.06 ÷ 9 720 ÷ 9 =80 714.069 9 × 70– 630.00 84.06 9 × 9– 81.00 3.06 9 × 0.3– 2.70 0.36 9 × 0.04– 0.36 0 Answer:79.34

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© Boardworks Ltd 2008 26 of 51 Using short division Start by finding an approximate answer: 714.06 ÷ 9 720 ÷ 9 =80 7 1 4. 0 69 0 7 7 8 9. 3 3 3 4 714.06 ÷ 9 = 79.34

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© Boardworks Ltd 2008 27 of 51 Drag and drop division problem

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© Boardworks Ltd 2008 28 of 51 Writing an equivalent calculation This will be easier to solve if we write an equivalent calculation with a whole number divisor. We can write 36.8 ÷ 0.4 as 36.8 0.4 = ×10 368 ×10 4 36.8 ÷ 0.4 is equivalent to 368 ÷ 4 =92 What is 36.8 ÷ 0.4?

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© Boardworks Ltd 2008 29 of 51 Find the equivalent calculation

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© Boardworks Ltd 2008 30 of 51 Dividing by two-digit numbers Calculate 75.4 ÷ 3.1. Estimate: 75 ÷ 3 = 25 Equivalent calculation: 75.4 ÷ 3.1 = 754 ÷ 31 Answer: 75.4 ÷ 3.1 = 24.32 R 0.08 75431 – 620 20 × 31 134 – 124 4 × 31 10.0 – 9.3 0.3 × 31 0.70 – 0.62 0.08 0.02 × 31 = 24.3 to 1 d.p.

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© Boardworks Ltd 2008 31 of 51 Dividing by two-digit numbers Calculate 8.12 ÷ 0.46. Estimate: 8 ÷ 0.5 = 16 Equivalent calculation: 8.12 ÷ 0.46 = 812 ÷ 46 Answer: 8.12 ÷ 0.43 = 17.65 R 0.1 81246 – 460 10 × 46 352 – 322 7 × 46 30.0 – 27.6 0.6 × 46 2.40 – 2.30 0.10 0.05 × 46 = 17.7 to 1 d.p.

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© Boardworks Ltd 2008 32 of 51 Contents N10 Written and calculator methods A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 N10.5 Using a calculator N10.6 Checking results N10.4 Division N10.2 Addition and subtraction N10.3 Multiplication N10.1 Estimation and approximation

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© Boardworks Ltd 2008 33 of 51 Solving complex calculations mentally What is? 3.2 + 6.8 7.4 – 2.4 3.2 + 6.8 7.4 – 2.4 = 10 5 =2 We could also write this calculation as: (3.2 + 6.8) ÷ (7.4 – 2.4). How could we work this out using a calculator?

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© Boardworks Ltd 2008 34 of 51 Using bracket keys on the calculator What is? 3.7 + 2.1 3.7 – 2.1 We start by estimating the answer: 3.7 + 2.1 3.7 – 2.1 3 6 2 = Using brackets we key in: (3.7 + 2.1) ÷ (3.7 – 2.1) =3.625

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© Boardworks Ltd 2008 35 of 51 Interpreting the calculator display

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© Boardworks Ltd 2008 36 of 51 Finding whole number remainders Sometimes, when we divide, we need the remainder to be expressed as a whole number. For example, 236 eggs are packed into boxes of 12. Using a calculator:236 ÷ 12 = 19.66666667 This is 19.6 recurring or 19.6. Number of boxes filled = 19. Number of eggs left over = 0.6 × 12 = 8 How many boxes are filled? How many eggs are left over?

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© Boardworks Ltd 2008 37 of 51 Finding whole number remainders Find the remainder if this answer was obtained by: a) Dividing 384 by 600.4 × 60 = 24 b) Dividing 160 by 250.4 × 25 = 10 c) Dividing by 2464 by 3850.4 × 385 = 154 My calculator display shows the following:

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© Boardworks Ltd 2008 38 of 51 Working with units of time What is 248 days in weeks and days? Using a calculator we key in: 248÷7= Which gives us an answer of 35.42857143 weeks. We have 35 whole weeks. To find the number of days left over we key in: –35=×7= This give us the answer 3. 248 days = 35 weeks and 3 days.

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© Boardworks Ltd 2008 39 of 51 Converting units of time to decimals When using a calculator to work with with units of time it can be helpful to enter these as decimals. For example: 7 minutes and 15 seconds = 7 15 60 minutes = 7 1 4 = 7.25 minutes 4 days and 18 hours = 4 18 24 days = 4 3 4 = 4.75 days

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© Boardworks Ltd 2008 40 of 51 Find the correct answer Four people used their calculators to work out. 9 + 30 15 – 7 Tracy gets the answer 4. Fiona gets the answer 4.875. Andrew gets the answer –4.4. Sam gets the answer 12.75. Who is correct? What did the others do wrong?

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© Boardworks Ltd 2008 41 of 51 Contents N10 Written and calculator methods A1A1 A1A1 A1A1 A1A1 A1A1 A1A1 N10.6 Checking results N10.5 Using a calculator N10.4 Division N10.2 Addition and subtraction N10.3 Multiplication N10.1 Estimation and approximation

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© Boardworks Ltd 2008 42 of 51 Making sure answers are sensible When we complete a calculation, whether using a calculator, a mental method or a written method we should always check that the answer is sensible. Use checks for divisibility when you multiply by 2, 3, 4, 5, 6, 8 and 9. For example, if you multiply a number by 9 the sum of the digits should be a multiple of 9. Make sure that the sum of two odd numbers is an even number. When you multiply two large numbers together check the last digit. For example, 329 × 842 must end in an 8 because 9 × 2 = 18.

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© Boardworks Ltd 2008 43 of 51 Using rounding and approximation We can check that answers to calculations are of the right order of magnitude by rounding the numbers in the calculation to find an approximate answer. Sam calculates that 387.4 × 0.45 is 174.33. Could this be correct? 387.4 × 0.45 is approximately equal to 390 × 0.5 = 195 This approximate answer is a little larger than the calculated answer but since both numbers were rounded up, there is a good chance that the answer is correct.

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© Boardworks Ltd 2008 44 of 51 Using inverse operations We can use a calculator to check answers using inverse operations. We can check the solution to 34.2 × 45.9 = 1569.78 by calculating 1569.78 ÷ 34.2 If the calculation is correct then the answer will be45.9.

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© Boardworks Ltd 2008 45 of 51 Using inverse operations We can use a calculator to check answers using inverse operations. We can check the solution to by calculating 128 × 7 ÷ 4 If the calculation is correct then the answer will be224. 4 7 of 224 = 128

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© Boardworks Ltd 2008 46 of 51 Using inverse operations We can use a calculator to check answers using inverse operations. We can check the solution to 6 ÷ 13 = 0.4615384 … by calculating 13 × 0.4615384 If the calculation is correct then the answer will be6.

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