1. Computation
Algorithms
Everyday Mathematics

2. 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!

3. Before selecting an algorithm, consider how you would solve the following problem.
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 and then
= = = 847
What was your thinking?

4. 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.

5. 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!

6. Click to proceed at your own speed!
Partial Sums
735
+ 246
900
Add the hundreds ( ) 70
Add the tens ( )
+11
Add the ones (5 + 6)
981
Add the partial sums
( )

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

8. 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.

9. 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

10. 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

11. 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
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12. 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.

13. 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.

14. – 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)

15. 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))
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16. Students begin by choosing partial quotients that they recognize!
I know 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

17. 8 5 R6
another one!
Try
3 2
5 0
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
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18. 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

19. 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
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