3.1Procedures for Mental Computation
Count on and Count Back
Use this technique if one of the numbers to be added or subtracted is 1,2 or 3; 10, 20, or 30; or 100 200 or 300; and so on.
How to use these techniques
Choose Compatible Numbers
Use this technique if one or more pairs of numbers can be easily added; subtract, multiplied or divided.
Or
Use this technique if numbers can be combined to produce multiples of 10, 100 or other numbers that make calculations easy.
Example: (25 x 9) x (11 x 4) =.
Break apart numbers
Use this technique if simple calculations involving basic number facts result when the numbers are broken apart according to the place value of the digits.
How to use this technique
Example:345 + 130
Compensation
Use this technique when a calculation can be chosen that is close to the original one and that is easy to do mentally.
How to use this technique
Example:9 x 12 =
Using the Equal Additions Technique
Use this technique when one of the numbers in a subtraction calculation (usually the number being subtracted) can be changed so that it results in a computation that is easy to do mentally.
How to use this technique
Example:145 – 77 =
3.2Procedures for Estimation
Three main types of Estimation
- Estimating a Quantity – finding how many
- Estimating a Measure – finding how much length, area, volume, etc
- Estimating an Answer – finding a sum, difference, product or quotient.
Rounding
Use this technique when rounded numbers produce a calc that can be done mentally.
How to Use this Technique
Example:
Substitute Compatible Numbers
Use this technique if numbers close to the original number make the estimate easy to do mentally.
How to Use this Technique
Example:358 – 144
Front-End Estimation
Use this technique when an estimate is needed quickly and a rough estimate is acceptable.
How to use this Technique
Example:563 + 325
Clustering Technique
Use this technique to estimate sums when the addends in the calculation cluster around the same number.
How to use this technique
.
Example:Suppose you have the following list
Jan146
Feb154
Mar148
.
.
3.3Algorithms for Addition and Subtraction
Expanded Algorithm in which the values of each place are added first and later combined..
Example:369
+244
Standard Algorithm
369
+244
Scratch Addition
Add each column draw a line when you hit 10, write the value over to the right and start adding from that point. Count the number of “slashes” and that becomes the value to carry.
Example:6 7
5 4
3 5
+ 7 9
Subtraction
Expanded Algorithm for Subtraction
Start with the greatest number and repeatedly take away as much as is possible to do mentally.
Example: 245
- 18
Standard Algorithm for subtraction
362
-244
3.4Algorithms for Multiplication and Division
Using the Area Model
Expanded Algorithm for Multiplication
Break apart the numbers according to the place value of each digit and multiply each digit according to its place value.
Example
24
x13
Standard Algorithm for Multiplication
Example:
24
x 13
Using Base-Ten Blocks to Model Dividing
Example: 105 15
Expanded Algorithm for Division
Subtract the divisor as many times as possible:
Standard Algorithm
1
Student Notes – Math 104