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

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.

Addition Algorithms Concrete model 14 + 23 37

Addition Algorithms Expanded algorithm Standard algorithm 14 14 14 14 +23 +23 7 (Add ones) 37 +30 (Add tens) 37

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

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

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

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

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

Addition Algorithms Left-to-Right Algorithm for Addition (Partial Sums) 568 568 + 757 + 757 1200 → 1215 → 1325 110 15 1325 32 (500 + 700) → (60 + 50) → (8 + 7) →

Addition Algorithms Lattice Algorithm for Addition

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

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

Addition Algorithms Scratch Algorithm for Addition (continued) 87 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

Addition Algorithms Scratch Algorithm for Addition (continued) 8 7 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

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.

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.

Subtraction Algorithms Concrete model (continued) 243 − 61 182

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 → → → →

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

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

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

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 →

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.

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.

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

Mental Mathematics: Addition 5. Making compatible numbers 76 → 75 + 25 = 100 + 25 → 100 + 1 = 101 75 + 25 adds to 100. Add 1 more unit.

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

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

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.

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

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

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

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

Computational Estimation 5. Using the range Problem Low Estimate High Estimate 7262 + 3806 7000 + 3000 10,000 8000 + 4000 12,000