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Computation Algorithms Everyday Mathematics

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Computation Algorithms in Everyday Mathematics Instead of learning a prescribed (and limited) set of algorithms, Everyday Mathematics encourages students to be flexible in their thinking about numbers and arithmetic. Students begin to realize that problems can be solved in more than one way. They also improve their understanding of place value and sharpen their estimation and mental- computation skills. The following slides are offered as an extension to the parent communication from your child’s teacher. We encourage you to value the thinking that is evident when children use such algorithms—there really is more than one way to solve a problem!

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Before selecting an algorithm, consider how you would solve the following problem. 48 + 799 We are trying to develop flexible thinkers who recognize that this problem can be readily computed in their heads! One way to approach it is to notice that 48 can be renamed as 1 + 47 and then What was your thinking? 48 + 799 = 47 + 1 + 799 = 47 + 800 = 847

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An algorithm consists of a precisely specified sequence of steps that will lead to a complete solution for a certain class of problems. Important Qualities of Algorithms Accuracy –Does it always lead to a right answer if you do it right? Generality –For what kinds of numbers does this work? (The larger the set of numbers the better.) Efficiency –How quick is it? Do students persist? Ease of correct use –Does it minimize errors? Transparency (versus opacity) –Can you SEE the mathematical ideas behind the algorithm? Hyman Bass. “Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician’s Perspective.” Teaching Children Mathematics. February, 2003.

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Table of Contents Partial Sums Partial Products Partial Differences Partial Quotients Lattice Multiplication Click on the algorithm you’d like to see! Trade First

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735 + 246 900 Add the hundreds (700 + 200) Add the tens (30 + 40) 70 Add the ones (5 + 6) Add the partial sums (900 + 70 + 11) +11 981 Click to proceed at your own speed!

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356 + 247 500 Add the hundreds (300 + 200) 90 Add the tens (50 + 40) Add the ones (6 + 7) Add the partial sums (500 + 90 + 13) +13 603

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429 + 989 1300 100 + 18 1418 Click here to go back to the menu.

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56 × 82 4,000 100 480 12 + 4,592 50 X 80 50 X 2 6 X 80 6 X 2 Add the partial products Click to proceed at your own speed!

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52 ×76 3,500 140 300 12 + 70 X 50 70 X 2 6 X 50 6 X 2 3,952 Add the partial products How flexible is your thinking? Did you notice that we chose to multiply in a different order this time?

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502 40 6 200080 12 300 52 × 46 2,000 300 80 12 2,392 Click here to go back to the menu. A Geometrical Representation of Partial Products (Area Model)

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12 7 2 37 2 3 4 5 94 5 9 6 11 2 13 6 4 Students complete all regrouping before doing the subtraction. This can be done from left to right. In this case, we need to regroup a 100 into 10 tens. The 7 hundreds is now 6 hundreds and the 2 tens is now 12 tens. Next, we need to regroup a 10 into 10 ones. The 12 tens is now 11 tens and the 3 ones is now 13 ones. Now, we complete the subtraction. We have 6 hundreds minus 4 hundreds, 11 tens minus 5 tens, and 13 ones minus 9 ones.

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Click here to go back to the menu. 10 8 0 28 0 2 2 7 42 7 4 7 9 5 12 2 8 14 9 4 69 4 6 5 6 85 6 8 8 13 3 16 7 8

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Subtract the hundreds (700 – 200) Subtract the tens (30 – 40) Subtract the ones (6 – 5) Add the partial differences (500 + (-10) + 1) 5 0 05 0 0 – 2 4 5– 2 4 5 1 4 9 14 9 1 1 0 7 3 6

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Subtract the hundreds (400 – 300) Subtract the tens (10 – 30) Subtract the ones (2 – 5) Add the partial differences (100 + (-20) + (-3)) 1 0 01 0 0 – 3 3 5– 3 3 5 7 7 2 0 4 1 2 3 Click here to go back to the menu.

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4 1 1 1 1 0 5 1 9 R3 1 2 01 2 0 6 0 2 3 12 3 1 1 2 Click to proceed at your own speed! 5 1 4 8 3 1 9 Students begin by choosing partial quotients that they recognize! Add the partial quotients, and record the quotient along with the remainder. I know 10 x 12 will work…

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Click here to go back to the menu. 1 01 0 1 1 2 61 1 2 6 5 05 0 2 52 5 8 5 R6 8 0 08 0 0 2 7 2 62 7 2 6 3 2 3 2 63 2 6 3 2 03 2 0 68 58 5 Compare the partial quotients used here to the ones that you chose! 1 6 0 0 1 6 0 0

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53 7 2 2 1 1 0 0 6 8 1 6 53 × 72 3500 100 210 6 3816 + 3 5 3 Compare to partial products! 3 × 7 3 × 2 5 × 7 5 × 2 Add the numbers on the diagonals. Click to proceed at your own speed!

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16 2 3 1 2 0 3 1 8 3 6 8 16 × 23 200 30 120 18 368 + 0 2 Click here to go back to the menu.

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