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Computation Algorithms Everyday Mathematics 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 mentalcomputation 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! 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 48 + 799 = 47 + 1 + 799 = 47 + 800 = 847 What was your thinking? 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. Table of Contents Partial Sums Partial Products Partial Differences Trade First Partial Quotients Lattice Multiplication Click on the algorithm you’d like to see! Click to proceed at your own speed! Add the hundreds (700 + 200) Add the tens (30 + 40) Add the ones (5 + 6) Add the partial sums (900 + 70 + 11) 735 + 246 900 70 +11 981 Add the hundreds (300 + 200) Add the tens (50 + 40) Add the ones (6 + 7) Add the partial sums (500 + 90 + 13) 356 + 247 500 90 +13 603 Click here to go back to the menu. 429 + 989 1300 100 + 18 1418 Click to proceed at your own speed! 50 X 80 50 X 2 6 X 80 6X2 Add the partial products 56 × 82 4,000 100 480 12 + 4,592 How flexible is your thinking? Did you notice that we chose to multiply in a different order this time? 70 X 50 70 X 2 6 X 50 6X2 Add the partial products 52 × 76 3,500 140 300 12 + 3,952 A Geometrical Representation of Partial Products (Area Model) 50 2 40 2000 80 6 300 12 Click here to go back to the menu. 52 × 46 2,000 300 80 12 2,392 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. 11 13 6 12 723 459 2 64 Now, we complete the subtraction. We have 6 hundreds minus 4 hundreds, 11 tens minus 5 tens, and 13 ones minus 9 ones. 9 12 7 10 802 274 5 28 Click here to go back to the menu. 13 16 8 14 946 568 3 78 Subtract the hundreds (700 – 200) Subtract the tens (30 – 40) Subtract the ones (6 – 5) Add the partial differences (500 + (-10) + 1) 736 – 245 500 10 1 491 Subtract the hundreds (400 – 300) Subtract the tens (10 – 30) Subtract the ones (2 – 5) Add the partial differences (100 + (-20) + (-3)) Click here to go back to the menu. 412 – 335 100 20 3 77 19 R3 12 2 31 1 20 10 1 11 Add the partial Students begin 60 5 quotients, and by choosing record the partial quotients 51 quotient along that they with the 48 4 recognize! remainder. 3 19 Click to proceed at your own speed! I know 10 x 12 will work… 85 R6 32 2726 1 60 0 50 1126 Compare the partial quotients 800 25 used here to the ones that you 326 chose! 3 20 10 6 85 Click here to go back to the menu. Click to proceed at your own speed! 53 5 3 × 72 35 × 7 23 × 7 7 3500 3 5 1 Compare 100 to partial 15 × 2 03 × 2 2 210 products! 8 0 6 + 6 Add 1the numbers 6 3816 on the diagonals. 16 1 6 × 23 1 2 0 200 2 2 30 0 1 3 120 3 3 8 + 18 8 6 368 Click here to go back to the menu.