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 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!

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?

Important Qualities of Algorithms 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.

Click on the algorithm you’d like to see! 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! Partial Sums 735 + 246 900 Add the hundreds (700 + 200) 70 Add the tens (30 + 40) +11 Add the ones (5 + 6) 981 Add the partial sums (900 + 70 + 11)

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

Click here to go back to the menu. Try one on your own! 429 + 989 Nice work! 1300 100 + 18 1418 Click here to go back to the menu.

Click to proceed at your own speed! Partial Products Click to proceed at your own speed! 5 6 × 8 2 4,000 50 X 80 100 50 X 2 480 6 X 80 12 + 6 X 2 Add the partial products 4,592

How flexible is your thinking How flexible is your thinking? Did you notice that we chose to multiply in a different order this time? 5 2 × 7 6 Try another one! 3,500 70 X 50 140 70 X 2 300 6 X 50 12 + 6 X 2 Add the partial products 3,952

A Geometrical Representation of Partial Products (Area Model) 5 2 × 4 6 50 2 2,000 300 40 6 2000 80 80 12 300 12 2,392 Click here to go back to the menu.

Trade-First 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. 11 13 6 12 7 2 3 4 5 9 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. 2 6 4 Now, we complete the subtraction. We have 6 hundreds minus 4 hundreds, 11 tens minus 5 tens, and 13 ones minus 9 ones.

Click here to go back to the menu. Try a couple more! 9 12 13 16 7 10 8 14 8 0 2 9 4 6 2 7 4 5 6 8 5 2 8 3 7 8 Click here to go back to the menu.

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

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

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

8 5 R6 another one! Try 3 2 2 7 2 6 1 6 0 0 5 0 1 1 2 6 Compare the partial quotients used here to the ones that you chose! 8 0 0 2 5 3 2 6 3 2 0 1 0 6 8 5 Click here to go back to the menu.

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

Click here to go back to the menu. Try Another One! 1 6 1 6 × 2 3 1 2 2 2 3 200 30 120 18 368 + 3 1 8 3 6 8 Click here to go back to the menu.