1. 3 Chapter Whole Numbers and Their Operations
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

2. 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.

3. 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.

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

5. Mental Mathematics: Addition
1. Adding from the left
= 90 (Add the tens.)
= 11 (Add the ones.)
= 101 (Add the two sums.)
2. Breaking up and bridging
= 96
= 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.

6. Mental Mathematics: Addition
3. Trading off
76 → = 80
+ 25 → 25 – 4 = 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.
= = 320
200
100
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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

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

9. 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.

10. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Example (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.

11. Mental Mathematics: Multiplication
1. Front-end multiplying
2. Using compatible
numbers
3. Thinking money
78 → 70  4 = 280
 4 → 8  4 = 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.

12. 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.

13. 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.

14. 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.

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

16. 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.

17. 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.

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

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

20. 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: = 4230
2. Compatible numbers
474
 9
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.