Computation Algorithms Everyday Mathematics. Computation Algorithms in Everyday Mathematics Instead of learning a prescribed (and limited) set of algorithms,

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.