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ComputationAlgorithms
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
What was your thinking?
48 + 799 = 47 + 1 + 799 = 47 + 800 = 847
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 ContentsPartial SumsPartial ProductsPartial Differences
Partial QuotientsLattice MultiplicationClick on the algorithm you’d like to
see!
Trade First
735+ 246
900Add the hundreds (700 + 200)
Add the tens (30 + 40) 70Add the ones (5 + 6)
Add the partial sums(900 + 70 + 11)
+11981
Click to proceed at your own speed!
356+ 247
500Add the hundreds (300 + 200)
90Add the tens (50 + 40)
Add the ones (6 + 7)
Add the partial sums(500 + 90 + 13)
+13603
429+ 9891300
100 + 18 141
8Click here to go
back to the menu.
56×82
4,00010048012+
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!
52×76
3,500140300
12+
70 X 50
70 X 2
6 X 50
6 X 2
3,952Add the partial products
How flexible is your thinking? Did you notice that we chose to multiply in a different order this time?
50 2
40
6
2000
80
12
300
52× 46
2,00030080
12
2,392Click here to go back to the
menu.
A Geometrical Representation of Partial
Products (Area Model)
127 2 3
4 5 9
6 11
2
13
64
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.
Click here to go back to the
menu.
108 0 2
2 7 4
7 9
5
12
28
149 4 6
5 6 8
8 13
3
16
78
Subtract the hundreds (700 – 200)Subtract the tens (30 – 40)Subtract the ones
(6 – 5)
Add the partial differences (500 + (-10) + 1)
5 0 0– 2 4 5
14 9
1
1 0
7 3 6
Subtract the hundreds (400 – 300)Subtract the tens (10 – 30)Subtract the ones
(2 – 5)
Add the partial differences (100 + (-20) + (-3))
1 0 0– 3 3 5
7
7
2 0
4 1 2
3
Click here to go back to the
menu.
4
1 1 1 1 0
5
1 9 R3
1 2 0
6 0
2 3 1 1 2Click 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…
Click here to go back to the
menu.
1 0
1 1 2 65 0
2 5
8 5 R6
8 0 0
2 7 2 6 3 2
3 2 63 2 0 6 8 5
Compare the partial
quotients used here to the
ones that you chose!
1 6 0 0
5 3
7
2
2 1
1 0
0 6
8 1
6
53×72
3500100210
63816+
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!
1 6
2
3
1 2
0 3
1 8
3 6
8
16×23
200 30120 18
368+
0 2
Click here to go back to the
menu.