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KS3 Mathematics

KS3 Mathematics. N10 Written and calculator methods. N10 Written and calculator methods. Contents. N10.2 Addition and subtraction. N10.3 Multiplication. N10.1 Estimation and approximation. N10.4 Division. N10.5 Using a calculator. N10.5 Checking results. Estimation four-in-a-line.

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KS3 Mathematics

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  1. KS3 Mathematics N10 Written and calculator methods

  2. N10 Written and calculator methods Contents N10.2 Addition and subtraction N10.3 Multiplication N10.1 Estimation and approximation N10.4 Division N10.5 Using a calculator N10.5 Checking results

  3. Estimation four-in-a-line

  4. Estimation “is approximately equal to” 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? 39 × 72  40 × 70 = 2800 Also, if we multiply together the last digits of 39 and 72 we have 9 × 2 = 18 9 × 2 = 18 The product of 39 and 72 must therefore end in an 8.

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

  6. Estimation How could we estimate the answer to 4948 ÷ 58? 4948 ÷ 58 can be approximated to: 5000 ÷ 60 = ? (60 does not divide into 5000) 5000 ÷ 50 = 100 4950 ÷ 50 = 99 or 4800 ÷ 60 = 80

  7. Estimating points on a scale

  8. Using points on a scale to estimate answers 25 26 27 28 29 30 31 32 33 34 35 36 5 6 Jessica is trying to estimate which number multiplied by itself will give the answer 32. She knows that 6 × 6 = 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. 5.64

  9. Using points on a scale to estimate answers 36 37 38 39 40 41 42 43 44 45 46 47 48 49 6 7 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. 6.31

  10. N10 Written and calculator methods Contents N10.1 Estimation and approximation N10.3 Multiplication N10.2 Addition and subtraction N10.4 Division N10.5 Using a calculator N10.5 Checking results

  11. Adding and subtracting decimals

  12. Adding and subtracting decimals • 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 • What is the total length of wood which Jack needs to cut off? • What is the length of the piece of wood which is left over? 2 9 b) a) Jack needs to cut off 2.62m altogether. The left-over wood will measure 0.38m (or 38cm). 1 1 0.7 0 3 .00 + 1.92 – 2.62 0 . 3 8 2 . 6 2 1

  13. Sums and differences

  14. N10 Written and calculator methods Contents N10.1 Estimation and approximation N10.2 Addition and subtraction N10.3 Multiplication N10.4 Division N10.5 Using a calculator N10.5 Checking results

  15. The grid method for multiplying whole numbers

  16. The grid method for multiplying decimals

  17. Using the standard column method What is 2.28 × 7? Start by finding an approximate answer: 2.28 × 7  2 × 7 = 14 2.28 × 7 is equivalent to 228 × 7 ÷ 100. 228 Answer × 7 15 9 6 2.28 × 7 = 1596 ÷ 100 = 15.96 1 5

  18. Using the standard column method What is 392.7 × 0.8? 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 Answer × 8 392.7 × 0.8 = 31416 ÷ 10 ÷ 10 31 4 1 6 = 314.16 7 2 5

  19. Drag and drop multiplication problem

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

  21. 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 8 × 232 = 1856 10 × 232 2320 4176 Answer: 4176 ÷ 100 = 41.76

  22. Multiplying two-digit numbers Calculate 394 × 0.47 Estimate: 400 × 0.5 = 200 Equivalent calculation: 394 × 0.47 × 100 ÷ 100 = 232 × 47 ÷ 100 394 × 47 7 × 394 = 2758 40 × 394 15760 18518 Answer: 18518 ÷ 100 = 185.18

  23. N10 Written and calculator methods Contents N10.1 Estimation and approximation N10.2 Addition and subtraction N10.3 Multiplication N10.4 Division N10.5 Using a calculator N10.5 Checking results

  24. Dividing decimals – Example 1 Dividend Divisor What is 259.2 ÷ 6?

  25. Using repeated subtraction 6 259.2 - 240.0 - 18.0 - 1.2 Start by finding an approximate answer: 259.2 ÷ 6 240 ÷ 6 = 40 6 ×40 19.2 6 ×3 1.2 6 ×0.2 0 Answer: 43.2

  26. Using short division Start by finding an approximate answer: 259.2 ÷ 6 240 ÷ 6 = 40 0 4 3 . 2 6 2 5 9 . 2 2 1 1 2.59 ÷ 6 = 43.2

  27. Dividing decimals – Example 2 Dividend Divisor What is 714.06 ÷ 9?

  28. Using repeated subtraction 9 714.06 - 630.00 - 81.00 - 2.70 - 0.36 Start by finding an approximate answer: 714.06 ÷ 9 720 ÷ 9 = 80 9 ×70 84.06 9 ×9 3.06 9 ×0.3 0.36 9 ×0.04 0 Answer: 79.34

  29. Using short division 9 7 1 4 . 0 6 Start by finding an approximate answer: 714.06 ÷ 9 720 ÷ 9 = 80 0 7 9 . 3 4 7 8 3 3 714.06 ÷ 9 = 79.34

  30. Drag and drop division problem

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

  32. Find the equivalent calculation

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

  34. Dividing by two-digit numbers 46 812 Calculate 8.12 ÷ 0.46 Estimate: 8 ÷ 0.5 = 16 Equivalent calculation: 8.12 ÷ 0.46= 812 ÷ 46 - 460 10× 46 352 - 322 7× 46 30.0 - 27.6 0.6× 46 2.40 - 2.30 0.05× 46 0.10 Answer: 8.12 ÷ 0.43 = 17.65 R 0.1 = 17.6 to 1 d.p.

  35. N10 Written and calculator methods Contents N10.1 Estimation and approximation N10.2 Addition and subtraction N10.3 Multiplication N10.5 Using a calculator N10.4 Division N10.5 Checking results

  36. Solving complex calculations mentally 3.2 + 6.8 What is ? 7.4 – 2.4 3.2 + 6.8 10 = = 7.4 – 2.4 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?

  37. Using bracket keys on the calculator 3.7 + 2.1 What is ? 3.7 – 2.1 3.7 + 2.1 6  = 3.7 – 2.1 2 We start by estimating the answer: 3 Using brackets we key in: (3.7 + 2.1) ÷ (3.7 – 2.1) = 3.625

  38. Interpreting the calculator display

  39. Finding whole number remainders This is 19.6 recurring or 19.6 . . Number of eggs left over = 0.6 × 12 = 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. How many boxes are filled? How many eggs are left over? Using a calculator: 236 ÷ 12 = 19.66666667 Number of boxes filled = 19 8

  40. Finding whole number remainders My calculator display shows the following: Find the remainder if this answer was obtained by: a) Dividing 384 by 60 0.4 × 60 = 24 b) Dividing 160 by 25 0.4 × 25 = 10 c) Dividing by 2464 by 385 0.4 × 385 = 154

  41. Working with units of time – 2 3 4 5 8 ÷ = 7 × 7 = = What is 248 days in weeks and days? Using a calculator we key in: 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: This give us the answer 3. 248 days = 35 weeks and 3 days.

  42. Converting units of time to decimals 15 18 7 4 minutes days 60 24 3 1 7 4 = = minutes days 4 4 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 = = 7.25 minutes 4 days and 18 hours = = 4.75 days

  43. 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?

  44. N10 Written and calculator methods Contents N10.1 Estimation and approximation N10.2 Addition and subtraction N10.3 Multiplication N10.5 Checking results N10.4 Division N10.5 Using a calculator

  45. Making sure answers are sensible 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. 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. 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. For example:

  46. 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. For example, 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.

  47. Using inverse operations We can use a calculator to check answers using inverse operations. For example, 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 be 45.9

  48. Using inverse operations 4 of 224 = 128 7 We can use a calculator to check answers using inverse operations. For example, We can check the solution to by calculating, 128 × 7 ÷ 4 If the calculation is correct then the answer will be 224.

  49. Using inverse operations We can use a calculator to check answers using inverse operations. For example, We can check the solution to 6 ÷ 13 = 0.4615384 … by calculating, 13 × 0.4615384 If the calculation is correct then the answer will be 6.

  50. Using an equivalent calculation Another way to check answers to calculations is to use an equivalent calculation. For example, we can check 698 × 11 = 7678 with (700 – 2) × 11 = 7700 – 22 or 698 × (10 + 1) = 6980 + 698

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