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Mental Math

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Title: Mental Math


1
Mental Math
  • Strand B
  • Grade Five

2
Quick Addition no regrouping
  • Begin at the front end of the numbers and add.
  • Example 56 23
  • Think Add 50 and 20 for 70, then add 6 and 3 for
    9 answer 79
  • Example 2341 3400
  • Think Add 2000 and 3000 for 5000, then add 300
    and 400 for 700, and then finally add 41. The
    answer is 5741.
  • Example 0.34 0.25
  • Think Add .30 and .20 for .50 and then add .04
    and .05 for .09 the answer is 0.59.

3
Quick Addition no regrouping
  • 71 12
  • 44 53
  • 291 703
  • 507 201
  • 5200 3700
  • 4423 1200
  • 0.3 0.6
  • 0.7 0.1
  • 2.45 3.33
  • 0.5 0.1

4
Quick Addition no regrouping
  • 37 51
  • 66 23
  • 234 52
  • 534 435
  • 4067 4900
  • 6621 2100
  • 6200 1700
  • 6334 2200
  • 0.2 0.5
  • 0.45 0.33

5
Front End Addition
  • Add the highest place value first and the add the
    sums of the next place value.
  • Example 450 380
  • Think 400 300 is 700, and 50 and 80 is 130 and
    700 plus 130 is 830.

6
Front End Addition
  • 340 220
  • 470 360
  • 3500 2300
  • 2900 6000
  • 8800 1100
  • 5400 3400
  • 4.9 3.2
  • 3.6 2.9
  • 0.62 0.23
  • 5.4 3.7

7
Front End Addition
  • 607 304
  • 3700 3200
  • 2700 7200
  • 6800 2100
  • 7500 2400
  • 6300 4400
  • 6.6 2.5
  • 0.75 0.05
  • 1.4 2.5
  • o.36 0.43

8
Finding Compatibles
  • Look for pairs of numbers that add to powers of
    10 (10, 100, and 1000).
  • Example 400 720 600
  • Think 400 and 600 is 1000,
  • so the sum is 1720.

9
Finding Compatibles
  • 800 740 200
  • 4400 1600 3000
  • 3250 3000 1750
  • 3000 300 700 2000
  • 290 510
  • 0.6 0.9 0.4 0.1
  • 0.7 0.1 0.9 0.3
  • 0.4 0.4 0.6 0.2 0.5
  • 0.80 0.26
  • 0.2 0.4 0.8 0.6

10
Finding Compatibles
  • 300 437 700
  • 900 100 485
  • 9000 3300 1000
  • 2200 2800 600
  • 3400 5600
  • 02. 0.4 0.3 0.8 0.6
  • 0.25 0.50 0.75
  • .45 0.63
  • 475 25
  • 125 25

11
Break Up and Bridge
  • Begin with the first number and add the values in
    the place values starting with the largest of the
    second numbers.
  • Example 5300 2400
  • Think 5300 and 2000 (from the 2400) is 7300 and
    7300 plus 400 (from the rest of 2400) is 7700.

12
Break Up and Bridge
  • 7700 1200
  • 7300 1400
  • 5090 2600
  • 4100 3600
  • 2800 6100
  • 4.2 3.5
  • 6.1 2.8
  • 4.15 3.22
  • 15.46 1.23
  • 6.3 1.6

13
Break Up and Bridge
  • 17 400 1300
  • 5700 2200
  • 3300 3400
  • 15 500 1200
  • 2200 3200
  • 0.32 0.56
  • 5.43 2.26
  • 43.30 8.49
  • 4.2 3.7
  • 2.08 3.2

14
Compensation
  • Change one number to a ten or hundred, carry out
    the addition, and then adjust the answer to
    compensate for the original change.
  • Example 4500 1900
  • Think 4500 2000 is 6500 but I added 100 too
    many so, I subtract 100 from 6500 to get 6400.

15
Compensation
  • 1300 800
  • 3450 4800
  • 4621 3800
  • 5400 2900
  • 2330 5900
  • 0.71 0.09
  • 0.44 0.29
  • 4.52 0.98
  • 0.56 0.08
  • 0.17 0.59

16
Compensation
  • 2111 4900
  • 6421 1900
  • 15 200 2900
  • 2050 6800
  • 3344 5500
  • 1.17 0.39
  • 0.32 0.19
  • 2.31 0.99
  • 25. 34 0.58
  • 44.23 0.23

17
Quick Subtraction
  • Use this strategy if no regrouping is needed.
    Begin at the front end and subtract.
  • Example 3700 2400
  • Think 3-2 1, 7-4 3, and add two zeros. The
    answer is 1300.

18
Quick Subtraction
  • 9800 7200
  • 8520 7200
  • 5600 4100
  • 56 000 23 000
  • 0.38 0.21
  • 0.96 0.85
  • 0.66 0.42
  • 3.86 0.45
  • 0.78 0.50
  • 17.36 0.24

19
Quick Subtraction
  • 4850 2220
  • 78 000 47 000
  • 460 000 130 000
  • 500 000 120 000
  • 0.33 0.23
  • 0.98 0.86
  • 0.66 0.41
  • 3.85 0.43
  • 0.64 0.32
  • 0.76 0.42

20
Back Through 10/100
  • Subtract part of the first number to get to the
    nearest one, ten, hundred, or thousand and then
    subtract the rest of the next number.
  • Use this strategy when the numbers are far apart.
  • Example 530 70
  • Think 530 subtract 30 (one part of the 70) is
    500 and 500 subtract 40 (the other part of the
    70) is 460.

21
Back Through 10/100
  • 420 60
  • 540 70
  • 340 70
  • 760 70
  • 9200 500
  • 7500 700
  • 9500 600
  • 4700 800
  • 800 600
  • 3400 - 700

22
Back Through 10/100
  • 630 60
  • 320 50
  • 6100 300
  • 4200 800
  • 2300 600
  • 9100 600
  • 7600 600
  • 9400 500
  • 4500 600
  • 700 - 500

23
Counting on to Subtract
  • Count the difference between the two numbers by
    starting with the smaller, keeping track of the
    distance to the nearest one, ten, hundred, or
    thousand and add to this amount the rest of the
    distance to the greater number.
  • Note this strategy is most effective when two
    numbers involved are quite close together.
  • Example 2310 1800
  • Think It is 200 from 1800 to 2000 and 310 from
    2000 to 2310 therefore, the difference is 200
    plus 310, or 510.

24
Counting on to Subtract
  • 5170 4800
  • 9130 8950
  • 7050 6750
  • 3210 2900
  • 2400 1800
  • 15.3 14.9
  • 45.6 44.9
  • 34.4 33.9
  • 27.2 26.8
  • 23.5 22.8

25
Counting on to Subtract
  • 1280 900
  • 8220 7800
  • 4195 3900
  • 8330 7700
  • 52.8 51.8
  • 19.1 18.8
  • 50.1 49.8
  • 70.3 69.7
  • 3.25 2.99
  • 24.12 23.99

26
Compensation
  • Change one number to a ten, hundred or thousand,
    carry out the subtraction, and then adjust the
    answer to compensate for the original change.
  • Example 5760 997
  • Think 5760 1000 is 4760 but I subtracted 3
    too many so, I add 3 to 4760 to compensate to
    get 4763.

27
Compensation
  • 8620 998
  • 9850 498
  • 4222 998
  • 4100 994
  • 3720 996
  • 7310 194
  • 5700 397
  • 2900 595
  • 8425 - 990
  • 75 316 - 9900

28
Compensation
  • 854 399
  • 953 499
  • 647 198
  • 523 198
  • 805 398
  • 642 198
  • 763 98
  • 534 488
  • 512 297
  • 7214 - 197

29
Balancing For a Constant Difference
  • Add or subtract the same amount from both the
    first number and the second number so that each
    number is easier to work with.
  • Example 345 198
  • Think Add 2 to both numbers to get 347 200 so
    the answer is 147.

30
Balancing for a Constant Difference
  • 649 299
  • 912 797
  • 631 -499
  • 971 696
  • 563 397
  • 6.4 3.9
  • 4.3 1.2
  • 6.3 2.2
  • 15. 3 5.7
  • 7.6 1.98

31
Balancing for a Constant Difference
  • 486 201
  • 382 202
  • 564 303
  • 437 103
  • 829 503
  • 8.63 2.99
  • 6.92 4.98
  • 7.45 1.98
  • 27.84 6.99
  • 5.40 3.97

32
Break Up and Bridge
  • Begin with the first number and subtract the
    values in the place values, beginning with the
    highest of the second number.
  • Example 8369 204
  • Think 8369 subtract 200 (from the 204) is 8169
    and 816 minus 4 (the rest of the 204) is 8165.

33
Break Up and Bridge
  • 736 301
  • 848 207
  • 927 605
  • 622 208
  • 928 210
  • 9275 8100
  • 10 270 8100
  • 3477 1060
  • 6350 4200
  • 15 100 - 3003

34
Break Up and Bridge
  • 647 102
  • 741 306
  • 847 412
  • 3586 302
  • 758 205
  • 38 500 10 400
  • 8461 4050
  • 4129 2005
  • 137 400 6100
  • 9371 - 8100

35
Multiplication and Division
  • When you need to divide, think of the question as
    a multiplication question.
  • Example 12 2
  • Think 2 x ____ 12 -- the answer is 6.
  • 40 5
  • 45 9
  • 56 7
  • 54 6
  • 36 4

36
Division as Multiplication
  • 240 12
  • 880 40
  • 1470 70
  • 3600 12
  • 1260 60
  • 6000 12
  • 660 30
  • 690 30
  • 650 50
  • 920 40

37
Division as Multiplication
  • 480 12
  • 880 11
  • 880 20
  • 490 70
  • 4800 12
  • 2400 60
  • 6000 50
  • 660 11
  • 5400 6
  • 1200 30

38
Using Mulitplication Facts for Tens, Hundreds and
Thousands
  • Multiply the 1-digit number by the one non-zero
    digit in the number.
  • Example 4 x 6000
  • Think 4 x 6 and then add the three zeros for
    an answer of 24 000.
  • If you have two non-zero digits in the question,
    you could mulitply them and then add the
    appropriate number of zeros.
  • Example 30 x 80
  • Think 3 x 8 24 and then add two zeros for
    an answer of 2400.

39
Using Multiplication Facts for Tens, Hundreds and
Thousands
  • 30 x 4
  • 20 x 300
  • 6 x 50
  • 6 x 200
  • 90 x 60
  • 10 x 400
  • 8 x 40
  • 70 x 7
  • 8 x 600
  • 4 x 5000

40
Using Multiplication Facts for Tens, Hundreds,
and Thousands
  • 6 x 900
  • 3 x 70
  • 9 x 30
  • 90 x 40
  • 300 x 4
  • 800 x 7
  • 9 x 800
  • 5 x 900
  • 3 x 2000
  • 6 x 6000

41
Multiplying by 10, 100, and 1000
  • Multiplying by 10 increases all the place values
    of a number by one place.
  • Example 10 x 67
  • Think the 6 tens will increase to 6 hundreds
    and the 7 ones will increase to 7 tens
    therefore, the answer is 670.
  • Multiplying by 100 increases all the place values
    of anumber by two places, and multiplying by 1000
    increases all the place values of a number by
    three places.

42
Multiplying by 10, 100, and 1000
  • 10 x 53
  • 100 x 7
  • 100 x 74
  • 73 x 1000
  • 10 x 3.3
  • 100 x 2.2
  • 100 x 0.12
  • 1000 x 5.66
  • 1000 x 14
  • 100 x 8.3

43
Multiplying by 10, 100, and 1000
  • 8.36 x 10
  • 100 x 0.41
  • 1000 x 2.2
  • 8.02 x 1000
  • 100 x 15
  • 16 x 1000
  • 0.7 x 10
  • 100 x 9.9
  • 100 x 0.07
  • 1000 x 43.8

44
Dividing by 0.1, 0.01, and 0.001
  • Dividing by 0.1, 0.01, and 0.001 is like
    multiplying by 10, 100, and 1000. Dividing by
    tenths increases all the lace values of a number
    by one place, by hundredths by two places, and by
    thousandths by three places.
  • Example 0.4 0.1
  • Think the 4 tenths will increase to 4 ones,
    therefore the answer is 4.
  • Example 3 0.001
  • Think The 3 ones will increase to 3 thousands,
    therefore the answer is 3000.

45
Dividing by 0.1, 0.01, and 0.001
  • 5 0.1
  • 46 0.1
  • 0.5 0.1
  • 0.02 0.1
  • 14.5 0.1
  • 4 0.01
  • 1 0.01
  • 0.2 0.01
  • 0.8 0.01
  • 8.2 0.01

46
Dividing by 0.1, 0.01, and 0.001
  • 7 0.01
  • 9 0.01
  • 0.3 0.01
  • 5.2 0.01
  • 5 0.001
  • 0.2 0.001
  • 7 0.001
  • 3.4 0.001
  • 1 0.001
  • 0.1 0.001

47
Multiplying by 0.1, 0.01, and 0.001
  • Multiplying by 0.1 decreases all the place values
    of a number by one place.
  • Multiplying by 0.01 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.
  • Example 5 x 0.01
  • Think the 5 ones will decreases to 5 hundredths,
    therefore the answer is 0.05.
  • Example 0.4 x 0.01
  • Think the 4 tenths will decrease to 4
    thousandths, therefore the answer is 0.004.

48
Multiplying by 0.1, 0.01, and 0.001
  • 6 x 0.01
  • 9 x 0.1
  • 72 x 0.1
  • 0.7 x 0.1
  • 1.6 x 0.1
  • 6 x 0.01
  • 0.5 x 0.01
  • 2.3 x 0.01
  • 100 x 0.01
  • 8 x 0.01

49
Multiplying by 0.1, 0.01, and 0.001
  • 3 x 0.001
  • 21 x 0.001
  • 62 x 0.001
  • 7 x 0.001
  • 45 x 0.001
  • 9 x 0.001
  • 0.4 x 0.001
  • 3.9 x 0.001
  • 330 x 0.01
  • 1.2 x 0.01

50
Dividing by 10, 100, and 1000
  • 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 the 7 thousands will decreases to 7 tens
    and the 5 hundreds will decreases to 5 ones
    therefore, the answer is 75.

51
Dividing by 10, 100, and 1000
  • 70 10
  • 200 10
  • 90 10
  • 800 10
  • 40 10
  • 100 10
  • 400 100
  • 4200 100
  • 9700 100
  • 900 100

52
Dividing by 10, 100, and 1000
  • 7600 100
  • 4400 100
  • 6000 100
  • 8500 100
  • 10 000 100
  • 82 000 1000
  • 66 000 1000
  • 430 000 1000
  • 98 000 1000
  • 70 000 1000

53
Front End Multiplication or the Distributive
Principle
  • Find the product of the single-digit factor and
    the digit in the highest place value of the
    second factor, and adding to this product a
    second sub-product.
  • Example 3 x 62
  • Think 3 times 6 is 18 tens or 180, and 3 times
    2 is 6 so, 180 plus 6 is 186.

54
Front End Multiplication or the Distributive
Principle
  • 53 x 3
  • 29 x 2
  • 62 x 4
  • 32 x 4
  • 83 x 3
  • 3 x 503
  • 606 x 6
  • 309 x 7
  • 410 x 5
  • 209 x 9

55
Front End Multiplication or the Distributive
Principle
  • 3 x 4200
  • 5 x 5100
  • 2 x 4300
  • 4 x 2100
  • 2 x 4300
  • 4.6 x 2
  • 8.3 x 5
  • 7.9 x 6
  • 3.7 x 4
  • 8.9 x 5

56
Compensation
  • This strategy can be used when one of the factors
    is near ten, hundred or thousand.
  • 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.
  • Example 7 x 198
  • Think 7 times 200 is 1400, but this is 14 more
    than it should be because there were 2 extra in
    each of the 7 groups therefore, 1400 subtract 14
    is 1368.

57
Compensation
  • 6 x 39
  • 2 x 79
  • 4 x 49
  • 8 x 29
  • 6 x 89
  • 5 x 399
  • 9 x 198
  • 3 x 199
  • 8 x 698
  • 4 x 198

58
Compensation
  • 7 x 598
  • 9 x 69
  • 5 x 49
  • 7 x 59
  • 29 x 50
  • 49 x 90
  • 39 x 40
  • 79 x 30
  • 89 x 20
  • 59 x 60

59
Finding Compatible Factors
  • Look for pairs of factors whose product is a
    power of ten and then re-associate the factors to
    make the overall calculation easier.
  • Example 25 x 63 x 4
  • Think 4 times 25 is 100, and 100 times 63 is
    6300.

60
Finding Compatible Factors
  • 2 x 78 x 500
  • 5 x 450 x 2
  • 5 x 19 x 2
  • 500 x 86 x 2
  • 2 x 43 x 50
  • 250 x 56 x 4
  • 4 x 38 x 25
  • 40 x 25 x 33
  • 2 x 50 x 300
  • 400 x 5 x 40

61
Finding Compatible Factors
  • 2 x 69 x 500
  • 5 x 400 x 2
  • 5 x 25 x 2
  • 500 x 87 x 2
  • 2 x 45 x 50
  • 250 x 65 x 4
  • 4 x 83 x 25
  • 40 x 25 x 44
  • 2 x 50 x 600
  • 400 x 5 x 20

62
Open Frames
  • Open frames in addition think subtraction.
  • Open frames in subtraction think addition.
  • Open frames in multiplication think division.
  • Open frames in division think multiplication.
  • Example 25 ? 85
  • Think 85 25 ?

63
Open Frames
  • 0.4 ? 0.9
  • 29 000 ? 30 000
  • 163 ? 363
  • .032 0. ?6 0.88
  • 5? 000 30 000 87 000
  • 36 - ? 29
  • 487 - ?35 252
  • 3567 - ?222 1345
  • 46 -2? 23
  • ?7 35 22

64
Open Frames
  • 2.24 - ? 2.00
  • 25 x ? 50
  • 30 x ? 60
  • 5? x 168
  • 9 x ? 81
  • 10 ? 5
  • 120 ? 12
  • 6.3 ? 63
  • ? 3 8
  • 3? 5 6

65
Estimation in Addition, Subtraction,
Multiplication, and Division- Rounding
  • Round each number to the highest or the highest
    two places values.
  • Example 348 230
  • Think 348 rounds to 300 and 230 rounds to 200,
    so 300 plus 200 is 500.

66
Estimation in Addition, Subtraction,
Multiplication, and Division- Rounding
  • 28 57
  • 303 49
  • 490 770
  • 8879 4238
  • 6110 3950
  • 427 198
  • 594 301
  • 834 587
  • 4768 3068
  • 4807 - 1203

67
Estimation in Addition, Subtraction,
Multiplication, and Division- Rounding
  • 4 x 59
  • 9 x 43
  • 889 x 3
  • 7 x 821
  • 7 x 22
  • 370 9
  • 458 5
  • 638 7
  • 409 6
  • 732 8

68
Front End Estimation
  • Find a ball-park answer by working with only
    the values in the highest place value.
  • Example 4276 3237
  • Think 4000 plus 3000 is 7000

69
Front End Estimation
  • 71 14
  • 647 312
  • 423 443
  • 4275 2105
  • 1296 6388
  • 823 240
  • 743 519
  • 718 338
  • 823 240
  • 743 - 519

70
Front End Estimation
  • 6.7 1.2
  • 0.2 4.9
  • 5.32 0.97
  • 0.86 0.93
  • 4.8 4.1
  • 6.1 2.2
  • 4.1 0.9
  • 1.9 0.2
  • 5.9 3.1
  • 12.3 10.1

71
Front End Estimation
  • 467 x 4
  • 63 x 8
  • 44 x 7
  • 613 x 6
  • 481 x 9
  • 121 6
  • 141 7
  • 102 5
  • 357 5
  • 75 3

72
Adjusted Front End Estimation
  • Begin by getting a Front End estimate and then
    adjust the estimate to get a closer estimate by
    considering the second highest place values.
  • Example 437 541
  • Think 400 plus 500 is 900, but 37 and 41 would
    account for about another 100 therefore, the
    adjusted estimate is 900 100 or 1000.

73
Adjusted Front End Estimation
  • 251 445
  • 642 264
  • 5695 2450
  • 5240 3790
  • 589 210
  • 645 290
  • 935 494
  • 9145 4968
  • 6210 2987
  • 6148 - 3920

74
Adjusted Front End Estimation
  • 2220 5120
  • 4087 2120
  • 6060 3140
  • 4140 5050
  • 7 x 341.25
  • 3 x 943.19
  • 6 x 280.53
  • 2 x 722.56
  • 8 x 776.43
  • 9 x 371.05
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