Addition
Break Up the Numbers Strategy
This strategy is used when regrouping is required. One of the addends is broken up into its expanded form and added in parts to the other addend. For example 57 + 38 might be calculated in this way: 57 + 30 is 87 and 8 more is 95.
Front-End (left to right) Strategy
This commonly used strategy involves adding the front-end digits and proceeding to the right, keeping a running total in your head. For example, 124 + 235 might be calculated in the following way: Three hundred (200 + 100), fifty (20+30) nine (4 + 5).
Rounding for Estimation
Rounding involves substituting one or more numbers with “friendlier” numbers with which to work. For example, 784 + 326 might be rounded as 800 + 300 or 1100.
Front-End Estimation
This strategy involves adding from the left and then grouping the numbers in order to adjust the estimate. For example 5239 + 2667 might be calculated in the following way: Seven thousand (5000 + 2000), eight hundred (600 +200) – no, make that 900 (39 and 67 is about another hundred). That’s about 7900
Compatible Number Strategy
Compatible numbers are number pairs that go together to make “friendly” numbers. That is, numbers that are easy to work with. To add 78 + 25 for example you might add 75 + 25 to make 100 and then add 3 to make 103.
Near Compatible Estimation
Knowledge of the compatible numbers that are used for mental calculations is used for estimation. For example, in estimating 76 + 45 + 19 +26 +52, one might do the following mental calculation: 76 + 26 and 52 + 45 sum to about 100. Add the 19. The answer is about 219.
Balancing Strategy
A variation of the compatible number strategy, this strategy involves taking one or more from one addend and adding it to the other. For example, 68 + 57 becomes 70 + 55 (add 2 to 68 and take 2 from 57)
Clustering in Estimation
Clustering involves grouping addends and determining the average. For example, when estimating 53 + 47 + 48 + 58 +52, notice that the addends cluster around 50. The estimate would be 250 (5 x 50)
Special Tens Strategy
In the early grades, students learn the number of pairs that total ten – 1 and 9, 2 and 8, 3 and 7, and so on. These can be extended to such combinations as 10 and 90, 300 and 700, etc.
Compensation Strategy
In this stage, you substitute a compatible number for one of the numbers so that you can more easily compute mentally. For example, in doing the calculation 47 + 29 one might think (47 + 30) – 1.
Consecutive Number Strategy
When adding three consecutive numbers, the sum is three times the middle number.
Subtraction
Compatible Number Estimation
Knowledge of compatible numbers can be used to find an estimate when subtracting. Look for the near compatible pairs. For example when subtracting 1014 – 766, one might think of the 750 – 250 pairing.
Front-End Strategy
When there is no need to carry, simply subtract from left to right. To subtract 368 – 125 think 300 – 100 = 200, 60 – 20 = 40, 8 – 5 = 3. The answer is 234.
Front-End Estimation
For questions with no carrying in the highest two place values, simply subtract those place values for a quick estimation. For example, the answer to $465.98 - $345.77 is about $120.00
Compatible Numbers Strategy
This works well for powers of 10. Think what number will make the power of 10. For example, to subtract 100 – 54, think what goes with 54 to make 100. The answer is 46.
Equal Additions Strategy for Subtraction
This strategy avoids regrouping. You add the same number to both the subtrahend and minuend to provide a “friendly” number for subtracting, then subtract. For example, to subtract 84 – 58, add too to both numbers to give 86 – 60. This can be done mentally. The answer is 26.
Compensation Strategy for Subtraction
As with addition, subtract the “friendly” number and add the difference. For example, $3.27 - $0.98 – ($3.27 - $1.00) + $0.02 = $2.29
“Counting On” Strategy for Subtraction
Visualize the numbers on a number line. For example, 110 – 44. You need 6 to make 50 from 44, then 50 to make 100, then another 10. The answer is 56.
“Counting On” Estimation
“Counting On” can also be used for estimation. For example, to estimate 894 – 652, think that 652 + 200 gives about 850. Then another 50 gives about 900. The difference is about 250.
Multiplication
Multiplying by 10, 100 and 1000 Strategy
Instead of counting zeros and adding them on, students use the concept of annexing zeros. For example, multiplying tens by tens gives hundreds, tens by hundreds gives thousands, hundreds by hundreds results in ten thousands and thousands by thousands results in millions.
Multiplying by 0.1, 0.01, 0.001 Strategy
Students need to realize that these decimals represent 1/10, 1/100 and 1/1000. They should think about groups of 10’s, 100’s and 1000’s.
Compatible Factors Strategy
This strategy involves using the Associative Property and looking for “friendly” combinations to multiply. For example, in multiplying 4 x 76 x 250, one might rearrange the numbers to make the calculation easier. 4 x 250 = 1000 and 1000 multiplied by 76 gives 76 000.
Multiple Compatible Factors Strategy
Students show the numbers as their factors and then regroup to develop numbers that are easier to work with. For example, 16 x 75 can be written as 4 x 4 x 3 25. 4 x 25 = 100 and 4 x 3 = 12. The answer is 1200.
Squaring Numbers Strategy
Students learn that there is a pattern when squaring numbers that end in 5. For example, the answer always ends with 25.
Round to Estimate Multiplication
Use rounding to estimate factors with two digits. For example, when multiplyi8ng 58 x 32, round to 60 x 30. The answer is about 1800.
Percentage/Fraction Connection
To find common percentages, think of the percentage as a fraction and divide by the denominator. For example, 50% of $25 is half of $25. Divide by 2. The answer is $12.50
Estimating Percent Using 1%, 10%, and 100%
As in multiplying 0., students need to consider that they are looking to 1/10 of a number.
Front-End Multiplication Strategy
This is usually used when one factor is a single digit and there is no regrouping. For example, 3 x 2313 = 6000 + 900 + 30 + 9 = 6939
Compensation Strategy for Multiplication
As with addition and subtraction, work with “friendly” numbers. For example, 5 x 29 – 5 x 30 -5 – 145.
Double and Half Strategy
Make numbers easier to multiply by doubling one factor and halving the other to provide a “nice” number. For example, 16 x 35 = 8 x 70 = 560.
Multiplying by 11 Strategy
Have students look for a pattern in the product. They will see that, in answers to questions such as 44 x 11, the first number of the answer is the tens digit of the factor that is not 11, the middle number is the sum of the two numbers of the factor that is not 11, and the final number is the ones digit of the factor that is not 11. The answer is 484.
Further Multiplying by 11 Strategy
When the sum of the middle number above is greater than 9, add the remainder to the tens digit of the factor that is not 11 and proceed as above. So 84 x 11 = 924.
Division
The Percentage/Fraction Connection
Students learn that a knowledge of common fractions is helpful when calculating percentages. For example, 20% is 1/5 and 25% is ¼. So to find 20%, divide by 5, etc.
Break Dividend Into Parts Strategy
For many simple computations, divide the dividend into parts and divide. For example, 1515 / 5 = (1500 / 5) + (15 / 5) = 300 + 3 = 303.
Double and Half Estimation
Double both number of the dividend to get “friendly” numbers and then estimate. For example, 72 / 3.5. 72 doubled is about 140. 3.5 doubles is 7. The answer is about 20.
Double and Half Strategy
This can be used to simplify dividing. For example, 48 / 5 is the same as 96 / 10.
“Think Multiplication” Estimation
For example, to divide 2088 by 7, think what number you multiply 7 by to get 2088. Seven times 300 is 2100.
Dividing by 10, 100 and 1000
Students learn when dividing by powers of 10 occurs, the place value of the last digit of the dividend changes according to the divisor. For example, dividing tens by tens gives units, hundreds by tens gives tens, etc.
Dividing by 0.1, 0.01, 0.001
Students should recognize that when dividing by powers of 10 with negative exponents they can write an equivalent multiplication statement using powers of 10. for example, dividing by 0.1 is the same as multiplying by 10.
Common Zeros
You can factor our powers of ten from the dividend and divisor for an expression that is easier to calculate. For example 3600 / 120 is the same as 360 / 12.
Never Divide by 5 Again!
Have students use the double and half strategy to simplify all division by 5. for example 520 / 5 is the same as 1040 / 10.
