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

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

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© Boardworks Ltd of 51 Martin uses his calculator to work out 39 × 72. The display shows an answer of 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 of × 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 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

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© Boardworks Ltd 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 Using points on a scale to estimate answers

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© Boardworks Ltd 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 of 51 Adding and subtracting decimals

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© Boardworks Ltd 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? a) b) – 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 of 51 Sums and differences

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© Boardworks Ltd 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 of 51 The grid method for multiplying decimals

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

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© Boardworks Ltd of 51 Using the standard column method Again, start by finding an approximate answer: × 0.8 400 × 1 = × 0.8 is equivalent to 3927 × 8 ÷ × Answer × 0.8 = ÷ 10 ÷ 10 = What is × 0.8?

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© Boardworks Ltd of 51 Drag and drop multiplication problem

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

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© Boardworks Ltd 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 ÷ × Answer: 4176 ÷ 100 = × 232 = 10 × 232 =

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© Boardworks Ltd of 51 Multiplying two-digit numbers Calculate 394 × Estimate: 400 × 0.5 = 200 Equivalent calculation: 394 × 0.47 × 100 ÷ 100 = 394 × 47 ÷ × Answer: ÷ 100 = × 394 = 40 × 394 =

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

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© Boardworks Ltd of 51 Using repeated subtraction Start by finding an approximate answer: ÷ 6 240 ÷ 6 = × 40– × 3– × 0.2– Answer:43.2

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© Boardworks Ltd of 51 Using short division Start by finding an approximate answer: ÷ 6 240 ÷ 6 = ÷ 6 = 43.2

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

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© Boardworks Ltd of 51 Using repeated subtraction Start by finding an approximate answer: ÷ 9 720 ÷ 9 = × 70– × 9– × 0.3– × 0.04– Answer:79.34

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© Boardworks Ltd of 51 Using short division Start by finding an approximate answer: ÷ 9 720 ÷ 9 = ÷ 9 = 79.34

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© Boardworks Ltd of 51 Drag and drop division problem

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© Boardworks Ltd 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 = × × ÷ 0.4 is equivalent to 368 ÷ 4 =92 What is 36.8 ÷ 0.4?

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© Boardworks Ltd of 51 Find the equivalent calculation

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© Boardworks Ltd 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 = R – × – × – × – × 31 = 24.3 to 1 d.p.

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© Boardworks Ltd of 51 Dividing by two-digit numbers Calculate 8.12 ÷ Estimate: 8 ÷ 0.5 = 16 Equivalent calculation: 8.12 ÷ 0.46 = 812 ÷ 46 Answer: 8.12 ÷ 0.43 = R – × – × – × – × 46 = 17.7 to 1 d.p.

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

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

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© Boardworks Ltd of 51 Interpreting the calculator display

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© Boardworks Ltd 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 = This is 19.6 recurring or 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 of 51 Finding whole number remainders Find the remainder if this answer was obtained by: a) Dividing 384 by × 60 = 24 b) Dividing 160 by × 25 = 10 c) Dividing by 2464 by × 385 = 154 My calculator display shows the following:

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© Boardworks Ltd 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 weeks. We have 35 whole weeks. To find the number of days left over we key in: –35=×7= This give us the answer days = 35 weeks and 3 days.

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© Boardworks Ltd 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 = minutes = = 7.25 minutes 4 days and 18 hours = days = = 4.75 days

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

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© Boardworks Ltd 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 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 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 × 0.45 is Could this be correct? × 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 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 = by calculating ÷ 34.2 If the calculation is correct then the answer will be45.9.

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© Boardworks Ltd 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 be of 224 = 128

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

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