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Mental Math. Strand B Grade Five. 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

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Mental math

Mental Math

Strand B

Grade Five


Quick addition no regrouping

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.


Quick addition no regrouping1

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


Quick addition no regrouping2

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


Front end addition

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.


Front end addition1

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


Front end addition2

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


Finding compatibles

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.


Finding compatibles1

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


Finding compatibles2

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


Break up and bridge

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.


Break up and bridge1

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


Break up and bridge2

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


Compensation

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.


Compensation1

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


Compensation2

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


Quick subtraction

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.


Quick subtraction1

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


Quick subtraction2

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


Back through 10 100

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.


Back through 10 1001

Back Through 10/100

420 – 60

540 – 70

340 – 70

760 – 70

9200 – 500

7500 – 700

9500 – 600

4700 – 800

800 – 600

3400 - 700


Back through 10 1002

Back Through 10/100

630 – 60

320 – 50

6100 – 300

4200 – 800

2300 – 600

9100 – 600

7600 – 600

9400 – 500

4500 – 600

700 - 500


Counting on to subtract

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.


Counting on to subtract1

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


Counting on to subtract2

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


Compensation3

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.


Compensation4

Compensation

8620 – 998

9850 – 498

4222 – 998

4100 – 994

3720 – 996

7310 – 194

5700 – 397

2900 – 595

8425 - 990

75 316 - 9900


Compensation5

Compensation

854 – 399

953 – 499

647 – 198

523 – 198

805 – 398

642 – 198

763 – 98

534 – 488

512 – 297

7214 - 197


Balancing for a constant difference

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.


Balancing for a constant difference1

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


Balancing for a constant difference2

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


Break up and bridge3

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.


Break up and bridge4

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


Break up and bridge5

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


Multiplication and division

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


Division as multiplication

Division as Multiplication

240 ÷ 12

880 ÷ 40

1470 ÷ 70

3600 ÷ 12

1260 ÷ 60

6000 ÷ 12

660 ÷ 30

690 ÷ 30

650 ÷ 50

920 ÷ 40


Division as multiplication1

Division as Multiplication

480 ÷ 12

880 ÷ 11

880 ÷ 20

490 ÷ 70

4800 ÷ 12

2400 ÷ 60

6000 ÷ 50

660 ÷ 11

5400 ÷ 6

1200 ÷ 30


Using mulitplication facts for tens hundreds and thousands

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.


Using multiplication facts for tens hundreds and thousands

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


Using multiplication facts for tens hundreds and thousands1

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


Multiplying by 10 100 and 1000

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.


Multiplying by 10 100 and 10001

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


Multiplying by 10 100 and 10002

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


Dividing by 0 1 0 01 and 0 001

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.


Dividing by 0 1 0 01 and 0 0011

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


Dividing by 0 1 0 01 and 0 0012

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


Multiplying by 0 1 0 01 and 0 001

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.


Multiplying by 0 1 0 01 and 0 0011

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


Multiplying by 0 1 0 01 and 0 0012

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


Dividing by 10 100 and 1000

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.


Dividing by 10 100 and 10001

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


Dividing by 10 100 and 10002

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


Front end multiplication or the distributive principle

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.


Front end multiplication or the distributive principle1

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


Front end multiplication or the distributive principle2

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


Compensation6

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.


Compensation7

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


Compensation8

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


Finding compatible factors

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.


Finding compatible factors1

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


Finding compatible factors2

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


Open frames

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 = 


Open frames1

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


Open frames2

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


Estimation in addition subtraction multiplication and division rounding

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.


Estimation in addition subtraction multiplication and division rounding1

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


Estimation in addition subtraction multiplication and division rounding2

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


Front end estimation

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


Front end estimation1

Front End Estimation

71 + 14

647 + 312

423 + 443

4275 + 2105

1296 + 6388

823 – 240

743 – 519

718 – 338

823 – 240

743 - 519


Front end estimation2

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


Front end estimation3

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


Adjusted front end estimation

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.


Adjusted front end estimation1

Adjusted Front End Estimation

251 + 445

642 + 264

5695 + 2450

5240 + 3790

589 + 210

645 – 290

935 – 494

9145 – 4968

6210 – 2987

6148 - 3920


Adjusted front end estimation2

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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