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Mental Math Strand B – Grade 6

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Quick Addition and Subtraction Use this strategy when no regrouping is needed. Begin the calculation from the front end. Example: Think: = 3, = 7, = 6, = 8 which gives the answer

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Quick Addition and Subtraction – – – –

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Quick Addition and Subtraction – – – – – – 3.711

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

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Quick Multiplication and Division 423 x 2 43 x x x x 3 72 x x x x 4 84 x 2

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Quick Multiplication and Division 360 ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 8

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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

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Multiplying and Dividing by 10, 100, and x x x x x x x x x x 0.12

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Multiplying and Dividing by 10, 100, and x x x x x x x x x x 1000

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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: ÷ 0.01 Think: the 4 tenths will increases to 4 tens, therefore the answer is 40.

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Dividing by tenths (0.1), hundredths (0.01), and thousandths (0.001) 5 ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 0.1

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Dividing by tenths (0.1), hundredths (0.01), and thousandths (0.001) 4 ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 0.01

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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.

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Dividing by Ten, Hundred, and Thousand 80 ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 10

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Dividing by Ten, Hundred, and Thousand ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 1000

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

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Think Multiplication when Dividing 920 ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 50

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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 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

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Multiplication and Division of tenths, hundredths, and thousandths 3 x x x x x x x x x x 0.01

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Multiplication and Division of tenths, hundredths, and thousandths 400 ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 1000

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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.

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Compensation 3.99 x x x x x x x x x x 5

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Compensation 3.98 x x x x x x x x x x 7

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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.

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Halving and Doubling 500 x x x x x x x x x x 28

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Halving and Doubling 52 x x x x x x x x x 22

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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

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Front End Multiplication or the Distributive principle in 10s, 100s, and 1000s 53 x 3 29 x 2 62 x 4 3 x x x x x x x 3

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Front End Multiplication or the Distributive principle in 10s, 100s, and 1000s 3 x x x x x x x x x x 5

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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 Example: 25 x 28 Think: 28 (7x4) has 4 as a factor, so 4 times 25 is 100, and 100 times 7 is 700.

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Finding Compatible Factors 5 x 19 x x 86 x x 67 x 4 2 x 43 x x 56 x 4 40 x 37 x 25 4 x 38 x x 25 x x 9 x 2 2 x 78 x 500

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Finding Compatible Factors 25 x x x 8 50 x x x x x x 6 68 x 500

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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: ( ) ÷ 6, so = 62.

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Partitioning the Dividend 3150 ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 4

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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.

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Balancing for a Constant Quotient 120 ÷ ÷ ÷ ÷ ÷ ÷ ÷ 5 40 ÷ ÷ ÷ 25

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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.

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Estimation--Rounding 593 x x x x x x x x x x 87

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Estimation--Rounding 411 ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 20

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

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Front End Addition, Subtraction, Division, and Multiplication – – – – – 9.807

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Front End Addition, Subtraction, Division, and Multiplication x 8 6 x x x x ÷ ÷ ÷ ÷ ÷ 2

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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

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Adjusted Front End or Front End with Clustering 6.1 x x x x x x x x x x 61

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Adjusted Front End or Front End with Clustering 38.2 x x x x x x x x x x 6.2

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

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Doubling for Division ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 5

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Doubling for Division 524 ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ 5

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