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Mental Math

Mental Math. Strand B – Grade 6. Quick Addition and Subtraction. Use this strategy when no regrouping is needed. Begin the calculation from the front end. Example : 2.327 + 1.441 Think: 2 + 1 = 3, 3 + 4 = 7, 2 + 4 = 6, 7 + 1 = 8 which gives the answer 3.768. Quick Addition and Subtraction.

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Mental Math

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  1. Mental Math Strand B – Grade 6

  2. Quick Addition and Subtraction • Use this strategy when no regrouping is needed. • Begin the calculation from the front end. Example: 2.327 + 1.441 Think: 2 + 1 = 3, 3 + 4 = 7, 2 + 4 = 6, 7 + 1 = 8 which gives the answer 3.768.

  3. Quick Addition and Subtraction 7.406 + 2.592 6.234 + 2.604 8.947 – 2.231 0.735 – 0.214 4.234 + 2.755 32.107 + 10.882 7.076 – 3.055 96.982 – 12.281 12.295 + 7.703 100.236 + 300.543

  4. Quick Addition and Subtraction 12.479 – 1.236 125.443 – 25.123 7.596 – 2.381 31.208 + 2.721 5.9235 + 4.0621 17.5 – 2.1 46.256 + 42.613 10.882 – 6.221 2.314 + 2.685 8.932 – 3.711

  5. Quick Multiplication and Division • Use this strategy when no regrouping is needed. • Begin at the front end. Example: 52 x 3 Think: 150 + 6 = 156 Example: 640 ÷ 2 Think: 300 + 20 = 320

  6. Quick Multiplication and Division 423 x 2 43 x 2 142 x 2 12.3 x 3 1220 x 3 72 x 3 803 x 3 143 x 2 42 000 x 4 84 x 2

  7. Quick Multiplication and Division 360 ÷ 3 105 ÷ 5 490 ÷ 7 328 ÷ 4 2107 ÷ 7 3612 ÷ 6 420 ÷ 2 426 ÷ 6 505 ÷ 5 248 000 ÷ 8

  8. Multiplying and Dividing by 10, 100, and 1000 • For this strategy, you need to keep track of how the place values have changed. • Multiplying by 10 increases all the place values of a number by one place • Multiplying by 100 increases all the place values of a number by two places. • Multiplying by 1000 increases all the place values of a number by three places. Example: 1000 x 45 Think: the 4 tens will increases to 40 thousands and the 5 ones will increase to 5 thousands; therefore, the answer is 45 000.

  9. Multiplying and Dividing by 10, 100, and 1000 10 x 53 10 x 20 92 x 10 100 x 7 100 x 74 10 x 3.3 8.3 x 10 100 x 2.2 7.54 x 10 100 x 0.12

  10. Multiplying and Dividing by 10, 100, and 1000 100 x 8.3 8.36 x 10 100 x 0.41 1000 x 2.2 8.02 x 1000 100 x 9.9 10 x 0.3 100 x 0.07 1000 x 43.8 0.04 x 1000

  11. Dividing by tenths (0.1), hundredths (0.01), and thousandths (0.001) • Dividing by tenths increases all the place values of a number by one place • Dividing by hundredths increases all the place values of a number by two places. • Dividing by thousandths increases all the place values of a number by three places. Example: 3 0.4 ÷ 0.01 Think: the 4 tenths will increases to 4 tens, therefore the answer is 40.

  12. Dividing by tenths (0.1), hundredths (0.01), and thousandths (0.001) 5 ÷ 0.1 46 ÷ 0.1 0.5 ÷ 0.1 0.02 ÷ 0.1 14.5 ÷ 0.1 23 ÷ 0.1 2.2 ÷ 0.1 425 ÷ 0.1 0.15 ÷ 0.1 253.1 ÷ 0.1

  13. Dividing by tenths (0.1), hundredths (0.01), and thousandths (0.001) 4 ÷ 0.01 1 ÷ 0.1 0.2 ÷ 0.01 0.8 ÷ 0.01 8.2 ÷ 0.01 7 ÷ 0.01 05 ÷ 0.01 0.1 ÷ 0.01 6.5 ÷ 0.01 17.5 ÷ 0.01

  14. Dividing by Ten, Hundred, and Thousand • Dividing by 10 decreases all the place values of a number by one place. • Dividing by 100 decreases all the place values of a number by two places. • Dividing by 1000 decreases all the place values of a number by three places. Example: 7500 ÷ 100 Think: 7 thousands will decrease to 7 tens and the 5 hundreds will decrease to 5 ones; therefore, the answer is 75.

  15. Dividing by Ten, Hundred, and Thousand 80 ÷ 10 420 ÷ 10 1200 ÷ 10 700 ÷ 100 2400 ÷ 100 7000 ÷ 1000 80 000 ÷ 1000 2000 ÷ 1000 60 ÷ 10 790 ÷ 10

  16. Dividing by Ten, Hundred, and Thousand 96 000 ÷ 1000 13 000 ÷ 1000 100 ÷ 10 360 ÷ 10 900 ÷ 100 4000 ÷ 100 37 000 ÷ 100 29 000 ÷ 1000 100 000 ÷ 1000 750 000 ÷ 1000

  17. Think Multiplication when Dividing Example: 60 ÷ 12 Think: What times 12 is 60? -- ? X 12 = 60 (5)

  18. Think Multiplication when Dividing 920 ÷ 40 240 ÷ 12 880 ÷ 40 1470 ÷ 70 3600 ÷ 12 1260 ÷ 60 6000 ÷ 12 660 ÷ 30 690 ÷ 30 650 ÷ 50

  19. Multiplication and Division of tenths, hundredths, and thousandths • Multiplying by 0.1 decreases all the place values of a umber by one place. • Multiplying by 0.01 decreases all the place values of a number by two places. • Dividing by 100 decreases all the place values of a number by two places. • Multiplying by 0.001 decreases all the place values of a number by three places. • Dividing by 1000 decreases all the place values of a number by three places. Example: 5 x 0.01 Think: the 5 ones will decrease to 5 hundredths, therefore the answer is 0.05

  20. Multiplication and Division of tenths, hundredths, and thousandths 3 x 0.1 12 x 0.1 406 x 0.1 0.1 x 10 0.1 x 3.2 330 x 0.001 1.2 x 0.01 0.7 x 0.01 10 x 0.001 46 x 0.01

  21. Multiplication and Division of tenths, hundredths, and thousandths 400 ÷ 100 4200 ÷ 100 9700 ÷ 100 900 ÷ 100 7600 ÷ 100 82 000 ÷ 1000 66 000 ÷ 1000 430 000 ÷ 1000 98 000 ÷ 1000 70 000 ÷ 1000

  22. Compensation • Change one of the factors to a ten, hundred, or thousand, carry out the multiplication, and then adjust the answer to compensate for the change that was made. • This strategy could be carried out when one of the factors is near ten, hundred, or thousand. Example: 6 x $4.98 Think: 6 times 5 dollars less 6 x 2 cents, therefore $30 subtract $0.12 which is $29.88.

  23. Compensation 3.99 x 4 4.98 x 2 9.99 x 8 6.99 x 9 5.99 x 7 19.99 x 3 20.98 x 2 6.98 x 3 49.98 x 4 99.98 x 5

  24. Compensation 3.98 x 3 9.97 x 6 4.99 x 5 6.99 x 8 98.99 x 4 7.98 x 4 19.98 x 2 22.99 x 3 59.98 x 5 9.97 x 7

  25. Halving and Doubling • Halve one factor and double the other factor in order to get two new factors that are easier to calculate. • You may need to record some sub-steps. Example: 42 x 50 Think: one-half of 42 is 21 and 50 doubled is 100; therefore, 21 x 100 is 2100.

  26. Halving and Doubling 500 x 88 12 x 2.5 4.5 x 2.2 140 x 35 86 x 50 500 x 46 18 x 2.5 0.5 x 120 180 x 45 50 x 28

  27. Halving and Doubling 52 x 50 2.5 x 22 3.5 x 2.2 160 35 64 x 500 500 x 70 86 x 2.5 1.5 x 6.6 140 x 15 500 x 22

  28. Front End Multiplication or the Distributive principle in 10s, 100s, and 1000s • Find the product of the single-digit factor and the digit in the highest place value of the second number, and add to this product a second sub-product. Example: 62 x 3 Think: 3 times 6 tens is 18 tens, or 180; and 3 times 2 is 6; so 180 plus 6 is 186

  29. Front End Multiplication or the Distributive principle in 10s, 100s, and 1000s 53 x 3 29 x 2 62 x 4 3 x 503 606 x 6 503 x 2 122 x 4 804 x 6 703 x 8 320 x 3

  30. Front End Multiplication or the Distributive principle in 10s, 100s, and 1000s 3 x 4200 5 x 5100 2 x 4300 4 x 2100 2 x 4300 7 x 2100 6 x 3100 6 x 3200 4 x 4200 410 x 5

  31. Finding Compatible Factors • Look for pairs of factors whose product is a power of ten and re-associate the factors to make the overall calculation easier. • Sometimes this strategy involves factoring one of the factors to get a compatible. Example: 25 x 63 x 4 Think: 4 times 25 is 100, and 100 times 63 is 6300. Example: 25 x 28 Think: 28 (7x4) has 4 as a factor, so 4 times 25 is 100, and 100 times 7 is 700.

  32. Finding Compatible Factors 5 x 19 x 2 500 x 86 x 2 250 x 67 x 4 2 x 43 x 50 250 x 56 x 4 40 x 37 x 25 4 x 38 x 25 40 x 25 x 33 5000 x 9 x 2 2 x 78 x 500

  33. Finding Compatible Factors 25 x 32 24 x 500 250 x 8 50 x 25 250 x 16 16 x 2500 12 x 25 500 x 36 5000 x 6 68 x 500

  34. Partitioning the Dividend • Partition the dividend into two parts. Both parts need to be easily divided by the given divisor. • Look for ten, hundred or thousand that is an easy multiple of the divisor and that is close to, but less than, the given dividend. Example: 372 ÷ 6 Think: (360 + 12) ÷ 6, so 60 + 2 = 62.

  35. Partitioning the Dividend 3150 ÷ 5 248 ÷ 4 432 ÷ 6 8280 ÷ 9 224 ÷ 7 344 ÷ 8 5110 ÷ 7 504 ÷ 8 1720 ÷ 4 3320 ÷ 4

  36. Balancing for a Constant Quotient • Change a given division question to an equivalent question that will have the same quotient by multiplying both the divisor and the dividend by the same amount. Example: 125 ÷ 5 Think: I could multiply both 5 and 125 by 2 to get 250 ÷ 10, which is easy to do. The answer is 25.

  37. Balancing for a Constant Quotient 120 ÷ 2.5 23.5 ÷ 0.5 140 ÷ 5 110 ÷ 2.5 32.3 ÷ 0.5 120 ÷ 25 320 ÷ 5 40 ÷ 2.5 135 ÷ 0.5 1200 ÷ 25

  38. Estimation--Rounding • Estimate your answers before you use pencil/paper to calculate your answers. • “Ball park” or reasonable answers are very helpful when you need to get an answer quickly. • Use words such as: about, just about, between, a little more than, a little less than, close, close to and near. • Just work with the first digits.

  39. Estimation--Rounding 593 x 41 879 x 22 295 x 59 687 x 52 912 x 11 87 x 371 363 x 82 658 x 66 567 x 88 972 x 87

  40. Estimation--Rounding 411 ÷ 49 651 ÷ 79 233 ÷ 29 360 ÷ 71 810.3 ÷ 89 2601 ÷ 50 3494 ÷ 60 2689 ÷ 90 8220 ÷ 90 1717 ÷ 20

  41. Front End Addition, Subtraction, Division, and Multiplication • Estimate to the nearest whole number. • Work with the first and second digits. Example: 0.093 + 4.236 Think: 0.1 + 4.2 = 4.3 (to the nearest tenth)

  42. Front End Addition, Subtraction, Division, and Multiplication 2.104 + 2.706 0.914 + 0.231 0.442 + 0.231 100.004 + 100.123 3.146 + 2.736 15.3 – 10.1 0.321 – 0.095 5.601 – 4.123 4.312 – 0.98 12.001 – 9.807

  43. Front End Addition, Subtraction, Division, and Multiplication 87.956 x 8 6 x 43.333 6 x 12.013 100.123 x 3 202.273 x 8 735 ÷ 9 182 ÷ 2 735 ÷ 8 276.5 ÷ 9 1701 ÷ 2

  44. Adjusted Front End or Front End with Clustering • You may need to use pencil/paper to record part of the answer. • Use the steps shown below. Example: 93 x 41 Think: 90 x 40 is 40 groups of 9 tnes, or 3600; and 3 x 40 is 40 groups of 3, or 120; 3600 plus 120 is 3720

  45. Adjusted Front End or Front End with Clustering 6.1 x 23.4 47 x 22 61 x 79 672 x 58 86 x 39 222 x 21 481 x 19 58 x 49 584 x 78 352 x 61

  46. Adjusted Front End or Front End with Clustering 38.2 x 5.9 43.1 x 4.1 48.3 x 3.2 73.3 x 4.1 57.2 x 6.9 91.2 x 1.9 55.1 x 5.1 63.1 x 2.1 84.3 x 6.1 87.3 x 6.2

  47. Doubling for Division • Round and double both the dividend and the divisor. Example: 2223 ÷ 5 Think: 4448 ÷ 10 or about 445

  48. Doubling for Division 1333.97 ÷ 5 243 ÷ 5 3212.11 ÷ 5 403 ÷ 5 1343.97 ÷ 5 231.95 ÷ 5 2222.89 ÷ 5 250 ÷ 5 3698.55 ÷ 5 2546.23 ÷ 5

  49. Doubling for Division 524 ÷ 5 9635 ÷ 5 4887 ÷ 5 1236 ÷ 5 565 ÷ 5 897 ÷ 5 1237 ÷ 5 931 ÷ 5 4592 ÷ 5 369 ÷ 5

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