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

7.406 + 2.5926.234 + 2.6048.947 – 2.2310.735 – 0.2144.234 + 2.755

32.107 + 10.8827.076 – 3.055

96.982 – 12.28112.295 + 7.703

100.236 + 300.543

Quick Addition and Subtraction

12.479 – 1.236125.443 – 25.123

7.596 – 2.38131.208 + 2.721

5.9235 + 4.062117.5 – 2.1

46.256 + 42.61310.882 – 6.2212.314 + 2.6858.932 – 3.711

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

Quick Multiplication and Division

423 x 243 x 2

142 x 212.3 x 31220 x 3

72 x 3803 x 3143 x 2

42 000 x 484 x 2

Quick Multiplication and Division

360 ÷ 3105 ÷ 5490 ÷ 7328 ÷ 4

2107 ÷ 73612 ÷ 6420 ÷ 2426 ÷ 6505 ÷ 5

248 000 ÷ 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 45Think: the 4 tens will increases to 40 thousands and the 5 ones will increase to 5 thousands; therefore, the answer is 45 000.

Multiplying and Dividing by 10, 100, and 1000

10 x 5310 x 2092 x 10100 x 7

100 x 7410 x 3.38.3 x 10

100 x 2.27.54 x 10

100 x 0.12

Multiplying and Dividing by 10, 100, and 1000100 x 8.38.36 x 10

100 x 0.411000 x 2.2

8.02 x 1000100 x 9.910 x 0.3

100 x 0.071000 x 43.80.04 x 1000

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.

Dividing by tenths (0.1), hundredths (0.01), and thousandths (0.001)

5 ÷ 0.146 ÷ 0.10.5 ÷ 0.1

0.02 ÷ 0.114.5 ÷ 0.123 ÷ 0.12.2 ÷ 0.1425 ÷ 0.10.15 ÷ 0.1

253.1 ÷ 0.1

Dividing by tenths (0.1), hundredths (0.01), and thousandths (0.001)

4 ÷ 0.011 ÷ 0.1

0.2 ÷ 0.010.8 ÷ 0.018.2 ÷ 0.017 ÷ 0.01

05 ÷ 0.010.1 ÷ 0.016.5 ÷ 0.01

17.5 ÷ 0.01

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.

Dividing by Ten, Hundred, and Thousand

80 ÷ 10420 ÷ 10

1200 ÷ 10700 ÷ 100

2400 ÷ 1007000 ÷ 1000

80 000 ÷ 10002000 ÷ 1000

60 ÷ 10790 ÷ 10

Dividing by Ten, Hundred, and Thousand

96 000 ÷ 100013 000 ÷ 1000

100 ÷ 10360 ÷ 10

900 ÷ 1004000 ÷ 100

37 000 ÷ 10029 000 ÷ 1000

100 000 ÷ 1000750 000 ÷ 1000

Think Multiplication when Dividing

Example: 60 ÷ 12Think: What times 12 is 60? -- ? X 12 = 60 (5)

Think Multiplication when Dividing

920 ÷ 40240 ÷ 12880 ÷ 40

1470 ÷ 703600 ÷ 121260 ÷ 606000 ÷ 12660 ÷ 30690 ÷ 30650 ÷ 50

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 0.01Think: the 5 ones will decrease to 5 hundredths, therefore the answer is 0.05

Multiplication and Division of tenths, hundredths, and thousandths

3 x 0.112 x 0.1

406 x 0.10.1 x 100.1 x 3.2

330 x 0.0011.2 x 0.010.7 x 0.0110 x 0.00146 x 0.01

Multiplication and Division of tenths, hundredths, and thousandths

400 ÷ 1004200 ÷ 1009700 ÷ 100900 ÷ 100

7600 ÷ 10082 000 ÷ 100066 000 ÷ 1000

430 000 ÷ 100098 000 ÷ 100070 000 ÷ 1000

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 $4.98Think: 6 times 5 dollars less 6 x 2 cents, therefore $30 subtract $0.12 which is $29.88.

Compensation

3.99 x 44.98 x 29.99 x 86.99 x 95.99 x 7

19.99 x 320.98 x 26.98 x 3

49.98 x 499.98 x 5

Compensation

3.98 x 39.97 x 64.99 x 56.99 x 8

98.99 x 47.98 x 4

19.98 x 222.99 x 359.98 x 59.97 x 7

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.

Halving and Doubling

500 x 8812 x 2.54.5 x 2.2140 x 3586 x 50

500 x 4618 x 2.5

0.5 x 120180 x 4550 x 28

Halving and Doubling

52 x 502.5 x 223.5 x 2.2160 3564 x 500500 x 7086 x 2.51.5 x 6.6140 x 15500 x 22

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

Front End Multiplication or the Distributive principle in 10s, 100s, and

1000s53 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

Front End Multiplication or the Distributive principle in 10s, 100s, and

1000s3 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

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.

Finding Compatible Factors

5 x 19 x 2500 x 86 x 2250 x 67 x 42 x 43 x 50

250 x 56 x 440 x 37 x 254 x 38 x 25

40 x 25 x 335000 x 9 x 22 x 78 x 500

Finding Compatible Factors

25 x 3224 x 500250 x 850 x 25

250 x 1616 x 2500

12 x 25500 x 36 5000 x 668 x 500

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.

Partitioning the Dividend

3150 ÷ 5248 ÷ 4432 ÷ 6

8280 ÷ 9224 ÷ 7344 ÷ 8

5110 ÷ 7504 ÷ 8

1720 ÷ 43320 ÷ 4

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.

Balancing for a Constant Quotient

120 ÷ 2.523.5 ÷ 0.5

140 ÷ 5110 ÷ 2.532.3 ÷ 0.5120 ÷ 25320 ÷ 540 ÷ 2.5

135 ÷ 0.51200 ÷ 25

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.

Estimation--Rounding

593 x 41879 x 22295 x 59687 x 52912 x 1187 x 371363 x 82658 x 66567 x 88972 x 87

Estimation--Rounding

411 ÷ 49651 ÷ 79233 ÷ 29360 ÷ 71

810.3 ÷ 892601 ÷ 503494 ÷ 602689 ÷ 908220 ÷ 901717 ÷ 20

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)

Front End Addition, Subtraction, Division, and Multiplication

2.104 + 2.7060.914 + 0.2310.442 + 0.231

100.004 + 100.1233.146 + 2.736

15.3 – 10.10.321 – 0.0955.601 – 4.1234.312 – 0.98

12.001 – 9.807

Front End Addition, Subtraction, Division, and Multiplication

87.956 x 86 x 43.3336 x 12.013

100.123 x 3202.273 x 8

735 ÷ 9182 ÷ 2735 ÷ 8

276.5 ÷ 91701 ÷ 2

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

Adjusted Front End or Front End with Clustering

6.1 x 23.447 x 2261 x 79

672 x 5886 x 39

222 x 21481 x 1958 x 49

584 x 78352 x 61

Adjusted Front End or Front End with Clustering

38.2 x 5.943.1 x 4.148.3 x 3.273.3 x 4.157.2 x 6.991.2 x 1.955.1 x 5.163.1 x 2.184.3 x 6.187.3 x 6.2

Doubling for Division

Round and double both the dividend and the divisor.

Example: 2223 ÷ 5

Think: 4448 ÷ 10 or about 445

Doubling for Division

1333.97 ÷ 5243 ÷ 5

3212.11 ÷ 5403 ÷ 5

1343.97 ÷ 5231.95 ÷ 5

2222.89 ÷ 5250 ÷ 5

3698.55 ÷ 52546.23 ÷ 5

Doubling for Division

524 ÷ 59635 ÷ 54887 ÷ 51236 ÷ 5565 ÷ 5897 ÷ 5

1237 ÷ 5931 ÷ 5

4592 ÷ 5369 ÷ 5