Chapter 3: Estimation and Computation

3.1 Strategies and Procedures for Mental Computation

3.1.1. When to Use Mental Computation

3.1.1.1. Description of Mental Computation: Mental computation is the process of finding an exact answer to a computation mentally, without paper, pencil, calculator, or any other computational aid.

3.1.1.2. There are NO set rules about when to use mental computation

3.1.2. Mental Computation Techniques

3.1.2.1. Count on and count back

3.1.2.1.1. To count on, start by saying the larger addend and then count on to find the sum

3.1.2.1.2. To count back, start by saying the minuend or sum and then count back to find the difference or missing addend

3.1.2.1.3. Procedure for using the count on and count back techniques:

3.1.2.1.3.1. When you might use these techniques: 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.

3.1.2.1.3.2. How to use these techniques:

3.1.2.1.3.2.1. Begin by saying the larger number

3.1.2.1.3.2.2. Count on to add or count back to subtract 1, 2, or 3; 10, 20, or 30; or 100, 200, or 300; and so on.

3.1.2.1.4. See example 3.1 p. 121-122

3.1.2.1.5. Your turn p. 122: Do the practice and the reflect

3.1.2.2. Choose compatible numbers

3.1.2.2.1. Easy numbers to compute mentally are called compatible numbers

3.1.2.2.2. Compatible numbers are user dependent – compatible for me may not be compatible for you and vice versa

3.1.2.2.3. Procedure for using the choose compatible numbers technique:

3.1.2.2.3.1. When might you use this technique: Use this technique if one or more pairs of numbers can be easily added, subtracted, 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

3.1.2.2.3.2. How to use this technique:

3.1.2.2.3.2.1. Look for pairs of numbers that are easy to calculate for the operation required. Do these calculations first

3.1.2.2.3.2.2. Look for the other number combinations that can be calculated easily

3.1.2.2.4. See example 3.2 p. 122-123

3.1.2.2.5. Your turn p. 123: Do the practice and the reflect

3.1.2.2.6. See example 3.3 p. 123

3.1.2.2.7. Your turn p. 123: Do the practice and the reflect

3.1.2.3. Break apart numbers

3.1.2.3.1. breaking apart numbers to permit the use of the commutative, associative, and distributive properties

3.1.2.3.2. See example 3.4 p. 125

3.1.2.3.3. Your turn p. 125: Do the practice and the reflect

3.1.2.3.4. Procedure for break apart numbers technique

3.1.2.3.4.1. When you might use this technique: 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

3.1.2.3.4.2. How to use this technique:

3.1.2.3.4.2.1. Think about each digit in a number according to its place value

3.1.2.3.4.2.2. Do calculations with each ones value, tens value, hundreds value, or combinations of these values

3.1.2.3.4.2.3. Recombine the parts to get the final answer

3.1.2.4. Use compensation

3.1.2.4.1. Uses compatible numbers that are NOT present and then adjusts the answer accordingly

3.1.2.4.2. Procedure for using the compensation technique

3.1.2.4.2.1. When you might use this technique: Use this technique when a calculation can be chosen that is close to the original one and that is easy to do mentally

3.1.2.4.2.2. How to use this technique:

3.1.2.4.2.2.1. Change the original calculation to one that is easy to do mentally. Changing only one number usually makes the adjustment at the end easier

3.1.2.4.2.2.2. Keep track of how you adjusted the original calculation

3.1.2.4.2.2.3. Find the answer to the original calculation by compensating the answer to the adjusted calculation

3.1.2.4.3. See example 3.5 p. 126-127

3.1.2.4.4. Your turn p. 126: Do the practice and the reflect

3.1.2.5. Use equal additions

3.1.2.5.1. Based on the idea that the difference between two numbers does not change if the same number is added to both of the original numbers

3.1.2.5.2. Procedure for using the equal additions technique

3.1.2.5.2.1. When you might use this 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

3.1.2.5.2.2. How to use this technique:

3.1.2.5.2.2.1. Identify a number that can be added to one of the numbers in the original calculation to give a new computation that is easy to do mentally

3.1.2.5.2.2.2. Add this number to both numbers in the original calculation and then compute

3.1.2.5.3. See example 3.6 p. 127-128

3.1.2.5.4. Your turn p. 128: Do the practice and the reflect

3.1.2.6. Procedure for choosing a mental computation technique

3.1.2.6.1. Related to the problem solving strategy: choose an operation

3.1.2.6.2. See example 3.7 p. 128-129

3.1.2.6.3. Your turn p. 130: Do the practice and the reflect