Math Games to Build Skills and Thinking

Math Games to Build Skills and Thinking

Claran Einfeldt, claran@cmath2.com

Cathy Carter, cathy@cmath2.com

http://www.cmath2.com

What is “Computational Fluency”?

What is “Computational Fluency”?

“connection between conceptual understanding and computational

proficiency”(NCTM 2000, p. 35)

Conceptual Computational Understanding Proficiency Conceptual Computational Understanding Proficiency

Place value

Operational properties

Number relationships

Accurate, efficient, flexible use of computation for multiple purposes

ComputationAlgorithms:

Seeing the Math

Computation Algorithms in Computation Algorithms in

Instead of learning a prescribed (and limited) set of algorithms, we should encourage 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.

Before selecting an algorithm, consider how you would solve the following 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

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 ContentsTable of Contents

Partial 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

4,00048010012+

4,592

80 X 50

80 X 6

2 X 50

2 X 6

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

50 2

40

6

2000

80

12

300

52× 46

2,000

300

80

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

3500

100210

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.

Algorithms“If children understand the mathematics behind the problem, they may very well be able to come up with a unique working algorithm that proves they “get it.” Helping children become comfortable with algorithmic and procedural thinking is essential to their growth and development in mathematics and as everyday problem solvers . . .

Extensive research shows the main problem with teaching standard algorithms too early is that children then use the algorithms as substitutes for thinking and common sense.”

Importance of GamesImportance of Games

Provides . . .Provides . . .

. . .regular experience with meaningful procedures so students develop and draw on mathematical understanding even as they cultivate computational proficiency.

Balance and connection of understanding and proficiency are essential, particularly for computation to be useful in “comprehending” problem-solving situations.

BenefitsBenefits

Should be central part of mathematics curriculumEngaging opportunities for practiceEncourages strategic mathematical thinkingEncourages efficiency in computationDevelops familiarity with number system and compatible numbers (landmark)Provides home school connection

Where’s the Math?Where’s the Math?

What mathematical ideas or understanding does this game promote?

What mathematics is involved in effective strategies for playing this game?

What numerical understanding is involved in scoring this game?

How much of the game is luck or mathematical skill?

Games Require ReflectionGames Require Reflection

Games need to be seen as a learning experience

Where’s the Math?Where’s the Math?

What is the goal of the game? Post this for students.Ask mathematical questions and have students write responses.Model the game first, along with mathematical thinkingEncourage cooperation, not competitionShare the game and mathematical goals with parents

ExtensionsExtensions

Have students create rules or different versions of the games

Require students to test out the games, explain and justify revisions based on fairness, mathematical reasoning

Games websitesGames websites

www.mathwire.comhttp://childparenting.about.com/od/makeathomemathgames/http://www.netrover.com/~kingskid/Math/math.htmhttp://www.multiplication.com/classroom_games.htmhttp://www.awesomelibrary.org/Classroom/Mathematics/Mathematics.htmlhttp://www.primarygames.co.uk/http://www.pbs.org/teachers/math/