1. 3 Chapter Numeration Systems and Whole Number Operations
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

2. 3-4 Addition and Subtraction Algorithms, Mental Computation, and Estimation
Models to develop algorithms for addition and subtraction.
Addition and subtraction algorithms including the standard algorithms and how to use them to solve problems.
Number bases other than ten to provide insight into base-ten algorithms.
Mental addition and subtraction computational skills and estimation techniques to check reasonableness of answers.

3. Addition Algorithms
Concrete model 14
+ 23 37

4. Addition Algorithms Expanded algorithm Standard algorithm 14 14
7 (Add ones) 37
+30 (Add tens) 37

5. Addition Algorithms Expanded algorithm with regrouping 37 +28
15 (Add ones)
+50 (Add tens) 65
Expanded algorithm with regrouping

6. Addition Algorithms Standard algorithm with regrouping 37 +28
65 (Add ones, regroup, and add the tens)
Standard algorithm with regrouping 1

7. Addition Algorithms Add two three-digit numbers with two regroupings.
Add the ones and regroup.
6 ones + 7 ones = 13 ones
13 ones = 1 ten + 3 ones
186
+ 127 3 1

8. Addition Algorithms
Add two three-digit numbers with two regroupings. (continued)
Add the tens and regroup.
1 ten + 8 tens tens = 11 tens
11 tens = 1 hundred + 1 ten 11
186
+ 127 13

9. Addition Algorithms
Add two three-digit numbers with two regroupings. (continued)
Add the hundreds.
1 hundred + 1 hundred + 1 hundred = 3 hundreds
186
+ 127
313 11

10. Addition Algorithms
Left-to-Right Algorithm for Addition (Partial Sums)
1200 → →
110 15
1325 32
( ) →
( ) →
(8 + 7) →

11. Addition Algorithms
Lattice Algorithm for Addition

12. Addition Algorithms Scratch Algorithm for Addition 8765
+ 49
Add the numbers in the units place starting at the top. When the sum is 10 or more, record this sum by scratching a line through the last digit added and writing the number of units next to the scratched digit. 2

13. Addition Algorithms Scratch Algorithm for Addition (continued) 8765
+ 49
Continue adding the units, including any new digits written down. When the addition again results in a sum of 10 or more, repeat the process. 2 1

14. Addition Algorithms Scratch Algorithm for Addition (continued) 872 87 65
+ 49 1
When the first column of additions is completed, write the number of units, 1, below the addition line in the proper place value position. Count the number of scratches, 2, and add this number to the second column. 2 1

15. Addition Algorithms Scratch Algorithm for Addition (continued) 8 72
8 7
6 5
+ 4 9
Repeat the procedure for each successive column until the last column with non-zero values. At this stage, sum the scratches and place the number to the left of the current value. 2 1

16. Subtraction Algorithms
Concrete model
243
− 61
Represent 243 with 2 flats, 4 longs, and 3 units, as shown. To subtract 61 from 243, we try to remove 6 longs and 1 unit from the blocks shown in the figure. We can remove 1 unit, but to remove 6 longs, we have to trade 1 flat for 10 longs.

17. Subtraction Algorithms
Concrete model (continued)
Now we can remove, or “take away,” 6 longs and 1 unit, leaving 1 flat, 8 longs, and 2 units, or 182.

18. Subtraction Algorithms
Concrete model (continued)
243
− 61
182

19. Equal-Addends Algorithm
The equal-additions algorithm for subtraction is based on the fact that the difference between two numbers does not change if we add the same amount to both numbers.
255
− 163 −
262
− 170
− ( )
292
− 200 92 → → → →

20. Understanding Addition and Subtraction in Bases Other Than Ten
343five + 2five = 400five 222five – 43five = 124five

21. Understanding Addition and Subtraction in Bases Other Than Ten
Concrete model
12five
+ 31five
43five

22. Understanding Addition and Subtraction in Bases Other Than Ten
Expanded algorithm
Standard algorithm
12five
+ 31five 3
+ 40
43five
12five
+ 31five
43five

23. Understanding Addition and Subtraction in Bases Other Than Ten
32five
− 14five
13five 21
Fives
Ones 3 2
− 1 4
Fives
Ones 2 12
− 1 4 1 3 →

24. 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.

25. 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.

26. 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

27. Mental Mathematics: Addition
5. Making compatible numbers
76 → = 100
→ = 101
adds to 100.
Add 1 more unit.

28. 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

29. Example
Noah owed $11 for his groceries. He used a $50 bill to pay. 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

30. 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.

31. 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

32. Computational Estimation
2. Grouping to nice numbers 23 39 32 64
+ 49
About 100
The sum is about 200.

33. 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

34. Computational Estimation
4. Rounding
7262 → 7000
–3806 → – 4000
3000

35. Computational Estimation
5. Using the range
Problem
Low Estimate
High Estimate
7262
+ 3806
7000
+ 3000
10,000
8000
,000