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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.