Mental Calculation Strategies: Relocation and Rounding

Mental Calculation Strategies: Core Techniques

Relocation Strategy

The Relocation strategy involves mentally changing or grouping numbers that result in units of zero (e.g., multiples of 10 or 100) to simplify addition.

Example: 47 + 86 + 53 + 14

Think: (47 + 53) + (86 + 14) = 100 + 100 = 200

Decomposition Technique

Decomposition is the process of breaking down one or more terms into equivalent, more comfortable parts to facilitate the operation.

Addition Example

77 + 148

Breakdown: 70 + 7 + 130 + 18

Grouping: (70 + 130) + (18 + 7) = 200 + 25 = 225

Subtraction Example

243 - 75

Decompose 243: 100 + (100 - 75) + 43

Calculation: 100 + 25 + 43 = 168

Rounding Strategies

Rounding involves altering the terms of an operation to round at least one of them to a multiple of ten or hundred. Rounding in addition uses compensation, while rounding in subtraction uses conservation.

Rounding in Addition (Compensation)

Compensation requires adding to one term what is simultaneously removed from the other term to maintain the sum.

• Example 1: 57 + 38

  (57 + 3) + (38 - 3) = 60 + 35 = 95

• Example 2: 57 + 38

  (57 - 2) + (38 + 2) = 55 + 40 = 95

Rounding in Subtraction (Conservation)

Conservation maintains the difference by adjusting both the minuend and the subtrahend by the same amount.

Rounding Up (Adding the Same Amount)

Add the same number to both the minuend and the subtrahend to round the subtrahend to a convenient number.

Example: 547 - 189

(547 + 11) - (189 + 11) = 558 - 200 = 358

Rounding Down (Subtracting the Same Amount)

Remove the same amount from both the minuend and the subtrahend to round the subtrahend to a convenient number.

Example: 234 - 63

(234 - 13) - (63 - 13) = 221 - 50 = 171

Counting Strategies (Left-to-Right Calculation)

This technique uses the skill of counting hundreds, tens, and units, proceeding from left to right. It has two main types:

• Ascending (Counting Up): Valid for addition and subtraction.
• Descending (Counting Down): Valid for subtraction (elimination).

Counting Up (Addition)

Count on a sum by adding the hundreds, tens, and units of the second term sequentially.

Example: 283 + 435

1. 283 + 400 = 683
2. 683 + 30 = 713
3. 713 + 5 = 718

Counting Up (Subtraction by Distance)

Measure the distance from the subtrahend to the minuend by counting up in convenient steps.

Example: 326 - 178

• From 178 to 180: 2
• From 180 to 200: 20
• From 200 to 300: 100
• From 300 to 326: 26

Total: 2 + 20 + 100 + 26 = 148

Counting Down (Subtraction by Elimination)

Count down from the minuend by eliminating the hundreds, tens, and units of the subtrahend.

Example: 143 - 84

Steps:

• 143 - 40 = 103
• 103 - 40 = 63
• 63 - 3 = 60
• 60 - 1 = 59

Total difference: 59

Application Examples

1. 324 + 592

   Steps: 324 + 500 = 824; 824 + 90 = 914; 914 + 2 = 916

   Strategy Used: Counting Up

2. 745 - 387

   Transformation using Rounding Up (Conservation): (745 + 13) - (387 + 13) = 758 - 400 = 358

   Strategy Used: Rounding Up (Conservation)

3. 157 + 262

   Decomposition and Relocation: (150 + 250) + (7 + 12) = 400 + 19 = 419

   Strategy Used: Decomposition and Relocation

4. 812 - 543

   Counting Up (Distance): 543 + 7 + 50 + 200 + 12 = 269

   Strategy Used: Counting Up

5. 812 - 543

   Counting Down (Elimination): 812 - 500 = 312; 312 - 40 = 272; 272 - 3 = 269

   Strategy Used: Counting Down (Elimination)