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

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

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

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

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

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Addition Algorithms Concrete model 14 + 23 37 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Addition Algorithms Expanded algorithm Standard algorithm 14 14 +23 +23 7 (Add ones) 37 +30 (Add tens) 37 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Addition Algorithms Expanded algorithm with regrouping 37 +28 15 (Add ones) +50 (Add tens) 65 Expanded algorithm with regrouping Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Addition Algorithms Standard algorithm with regrouping 37 +28 65 (Add ones, regroup, and add the tens) Standard algorithm with regrouping 1 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Addition Algorithms Add two three-digit numbers with two regroupings. 186 + 127 Add the ones and regroup. 6 ones + 7 ones = 13 ones 13 ones = 1 ten + 3 ones 186 + 127 3 1 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Addition Algorithms Add two three-digit numbers with two regroupings. (continued) 186 + 127 Add the tens and regroup. 1 ten + 8 tens + 2 tens = 11 tens 11 tens = 1 hundred + 1 ten 11 186 + 127 13 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Addition Algorithms Add two three-digit numbers with two regroupings. (continued) 186 + 127 Add the hundreds. 1 hundred + 1 hundred + 1 hundred = 3 hundreds 186 + 127 313 11 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Addition Algorithms Left-to-Right Algorithm for Addition 568 568 + 757 + 757 1200 → 1215 → 1325 110 15 1325 32 (500 + 700) → (60 + 50) → (8 + 7) → Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Addition Algorithms Lattice Algorithm for Addition Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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

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

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

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Addition Algorithms Scratch Algorithm for Addition (continued) 2 8 7 6 5 + 4 9 2 0 1 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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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

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

Subtraction Algorithms Concrete model (continued) 243 − 61 182 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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 − 163 255 + 7 − 163 + 7 262 − 170 262 + 30 − (170 + 30) 292 − 200 92 → → → → Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Understanding Addition and Subtraction in Bases Other Than Ten 343five + 2five = 400five 222five – 43five = 124five Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Understanding Addition and Subtraction in Bases Other Than Ten Concrete model 12five + 31five 43five Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Understanding Addition and Subtraction in Bases Other Than Ten Expanded algorithm Standard algorithm 12five + 31five 3 + 40 43five 12five + 31five 43five Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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