1. 3 Chapter Whole Numbers and Their Operations
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2. 3-2 Algorithms for Whole-Number Addition and Subtraction
Addition Algorithms
Subtraction Algorithms
Equal-Addends Algorithm
Understanding Addition and Subtraction in Bases Other Than Ten
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3. Fluency with Multidigit Addition and Subtraction
Children use their understanding of addition to develop quick recall of basic addition facts and related subtraction facts.
They solve arithmetic problems by applying their understanding of models of addition and subtraction (such as combining or separating sets or using number lines), relationships and properties of number (such as place value), and properties of addition (commutativity and associativity).
NCTM Curriculum Focal Points, p. 14
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4. Fluency with Multidigit Addition and Subtraction
(continued)
Children develop, discuss, and use efficient, accurate, and generalizable methods to add and subtract multidigit whole numbers.
They select and apply appropriate methods to estimate sums and differences or calculate them mentally, depending on the context and numbers involved.
NCTM Curriculum Focal Points, p. 14
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5. Fluency with Multidigit Addition and Subtraction
(continued)
They develop fluency with efficient procedures, including standard algorithms, for adding and subtracting whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems.
NCTM Curriculum Focal Points, p. 14
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6. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Addition Algorithms
Concrete model 14
+ 23 37
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7. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Addition Algorithms
Expanded algorithm Standard algorithm
7 (Add ones) 37
+30 (Add tens) 37
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8. Addition Algorithms Expanded algorithm with regrouping 37 +28
15 (Add ones)
+50 (Add tens) 65
Expanded algorithm with regrouping
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9. Addition Algorithms Standard algorithm with regrouping 37 +28
65 (Add ones, regroup, and add the tens)
Standard algorithm with regrouping 1
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10. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
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
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11. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
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
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12. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
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
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13. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Addition Algorithms
Left-to-Right Algorithm for Addition
1200 → →
110 15
1325 32
( ) →
( ) →
(8 + 7) →
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14. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Addition Algorithms
Lattice Algorithm for Addition
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15. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Addition Algorithms
Scratch Algorithm for Addition 87 65
+ 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
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16. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Addition Algorithms
Scratch Algorithm for Addition (continued) 87 65
+ 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
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17. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Addition Algorithms
Scratch Algorithm for Addition (continued) 2 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
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18. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Addition Algorithms
Scratch Algorithm for Addition (continued) 2
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
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19. 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.
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20. 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.
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21. Subtraction Algorithms
Concrete model (continued)
243
− 61
182
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22. 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 → → → →
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23. Understanding Addition and Subtraction in Bases Other Than Ten
343five + 2five = 400five 222five – 43five = 124five
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24. Understanding Addition and Subtraction in Bases Other Than Ten
Concrete model
12five
+ 31five
43five
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25. Understanding Addition and Subtraction in Bases Other Than Ten
Expanded algorithm
Standard algorithm
12five
+ 31five 3
+ 40
43five
12five
+ 31five
43five
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26. 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 →
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