3 Chapter Whole Numbers and Their Operations Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

3-5 Mental Mathematics and Estimation for Whole-Number Operations Mental Mathematics: Addition Mental Mathematics: Subtraction Mental Mathematics: Multiplication Mental Mathematics: Division Computational Estimation Estimation: Multiplication and Division Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Mental Mathematics and Estimation for Whole-Number Operations The process of producing an answer to a computation without using computational aids. Computational estimation The process of forming an approximate answer to a numerical problem. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Mental Mathematics: Addition 1. Adding from the left 76 70 + 20 = 90 (Add the tens.) + 25 6 + 5 = 11 (Add the ones.) 90 + 11 = 101 (Add the two sums.) 2. Breaking up and bridging 76 76 + 20 = 96 + 25 96 + 5 = 101 Add the first number to the tens in the second number. Add the sum to the units in the second number. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Mental Mathematics: Addition 3. Trading off 76 → 76 + 4 = 80 + 25 → 25 – 4 = 21 80 + 21 = 101 Add 4 to make a multiple of 10. Subtract 4 to compensate. Add the two numbers. 4. Using compatible numbers Compatible numbers are numbers whose sums are easy to calculate mentally. 130 + 50 + 70 + 20 + 50 = 200 + 100 + 20 = 320 200 100 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Mental Mathematics: Addition 5. Making compatible numbers 76 → 75 + 25 = 100 + 25 → 100 + 1 = 101 75 + 25 adds to 100. Add 1 more unit. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Mental Mathematics: Subtraction 1. Breaking up and bridging 2. Trading off 3. Drop the zeros 74 → 74 – 20 = 54 – 26 → 54 – 6 = 48 74 → 74 + 4 = 78 – 26 → 26 + 4 = 30 78 – 30 = 48 7400 → 74 – 6 = 68 – 600 → 7400 – 600 = 6800 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 3-5 Adding up or cashier’s algorithm Noah owed $11 for his groceries. He used a $50 check to pay the bill. While handing Noah the change, the cashier said, “11, 12, 13, 14, 15, 20, 30, 50.” How much change did Noah receive? What the cashier said $11 $12 $13 $14 $15 $20 $30 $50 Amount of money Noah received $1 $5 $10 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 3-5 (continued) The total amount of change that Noah received is $1 + $1 + $1 + $1 + $5 + $10 + $20 = $39 Thus $50 − $11 = $39 because $39 + $11 = $50. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Mental Mathematics: Multiplication 1. Front-end multiplying 2. Using compatible numbers 3. Thinking money 78 → 70  4 = 280  4 → 8  4 = 32 280 + 32 = 312 2  8  5  40  5 = (2  5)  (40  5)  8 = 10  200  8 = 16,000 84 Think of the product  25 as 84 quarters. 21 dollars = 2100 cents Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Mental Mathematics: Division 1. Breaking up the dividend Break up the dividend into parts. Divide both parts by 8. Add the answers. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Mental Mathematics: Division 2. Using compatible numbers Look for numbers that are divisible by 3 and whose sum is 105. Divide both parts by 3, then add the answers. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Mental Mathematics: Division 2. Using compatible numbers (continued) Look for numbers that are divisible by 8 and whose difference is 232. Divide both parts by 8, then subtract the answers. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Computational Estimation 1. Front-end with adjustment Add front-end digits: 4 + 5 + 2 = 11. Place value = 1100. Adjust: 22 + 31 ≈ 50 and 74 ≈ 70, so 50 + 70 = 120. Adjusted estimate is 1100 + 120 = 1220. 474 522 + 231 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Computational Estimation 2. Grouping to nice numbers 23 39 32 64 + 49 About 100 The sum is about 200. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Computational Estimation 3. Clustering Used when a group of numbers cluster around a common value Estimate the “average”: about 5000 4724 5262 5206 4992 + 5331 Multiply the “average” by the number of values: 5  5000 = 25,000 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Computational Estimation 4. Rounding 7262 → 7000 –3806 → – 4000 3000 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Computational Estimation 5. Using the range Problem Low Estimate High Estimate 7262 + 3806 7000 + 3000 10,000 8000 + 4000 12,000 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Estimation: Multiplication and Division 1. Front-end estimation Start multiplying at the front: 400  9 = 3600 Multiply the next digit: 70  9 = 630 Add the two numbers: 3600 + 630 = 4230 2. Compatible numbers 474  9 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.