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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”?. “connection between conceptual understanding and computational proficiency” (NCTM 2000, p. 35). Place value

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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
Place value

Operational properties

Number relationships

Accurate, efficient, flexible use of computation for multiple purposes

Conceptual Computational Understanding Proficiency
slide4

Computation

Algorithms:

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

48 + 799 = 47 + 1 + 799 = 47 + 800 = 847

What was your thinking?

slide7
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
Table of Contents

Partial Sums

Partial Products

Partial Differences

Trade First

Partial Quotients

Lattice Multiplication

Click on the algorithm you’d like to see!

slide9

Add the hundreds(700 + 200)

Add the partial sums

(900 + 70 + 11)

+11

Add the ones (5 + 6)

Click to proceed at your own speed!

Partial Sums

735

+ 246

900

70

Add the tens (30 + 40)

981

slide10

Add the hundreds(300 + 200)

Add the tens (50 + 40)

Add the ones (6 + 7)

Add the partial sums

(500 + 90 + 13)

+13

356

Try another one!

+ 247

500

90

603

slide11

+ 18

Try one on your own!

429

+ 989

Nice

work!

1300

100

1418

Click here to go back to the menu.

slide12

80 X 50

80 X 6

2 X 50

+

2 X 6

Add the partial products

Click to proceed at your own speed!

Partial Products

5

6

×

8

2

4,000

480

100

12

4,592

slide13

5

2

×

7

6

70 X 50

70 X 2

6 X 50

+

6 X 2

Add the partial products

Try another one!

3,500

140

300

12

3,952

slide14

5

2

×

4

6

40

6

A Geometrical Representation of Partial Products (Area Model)

50

2

2,000

80

2000

80

300

12

300

12

2,392

Click here to go back to the menu.

slide15

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

723

459

6

2

4

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.

slide16

Try a couple more!

13

9

16

12

14

10

8

7

946

802

568

274

7

2

3

8

5

8

Click here to go back to the menu.

slide17

10

Partial Differences

736

–245

500

Subtract the hundreds

(700 – 200)

Subtract the tens

(30 – 40)

1

  • Subtract the ones
    • (6 – 5)

491

Add the partial differences

(500 + (-10) + 1)

slide18

20

3

Try another one!

412

–335

100

Subtract the hundreds

(400 – 300)

Subtract the tens

(10 – 30)

  • Subtract the ones
    • (2 – 5)

77

Add the partial differences

(100 + (-20) + (-3))

Click here to go back to the menu.

slide19

19R3

12

231

120

I know 10 x 12 will work…

Partial

Quotients

Click to proceed at your own speed!

10

111

Add the partial quotients, and record the quotient along with the remainder.

60

5

Students begin by choosing partial quotients that they recognize!

51

48

4

19

3

slide20

85R6

Try

another one!

32

2726

50

1600

1126

Compare the partial quotients used here to the ones that you chose!

800

25

326

10

320

6

85

Click here to go back to the menu.

slide21

5

3

2

3

3500

7

1

5

210

1

0

100

2

0

6

6

Click to proceed at your own speed!

Lattice Multiplication

5

3

7

2

×

5× 7

3× 7

3

Compare to partial products!

3× 2

5× 2

8

+

Add the numbers on the diagonals.

6

1

3816

slide22

1

6

0

1

200

2

2

2

120

0

1

30

3

3

8

18

Try Another One!

1

6

2

3

×

3

+

8

6

368

Click here to go back to the menu.

slide23

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.”

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.

benefits
Benefits
  • Should be central part of mathematics curriculum
  • Engaging opportunities for practice
  • Encourages strategic mathematical thinking
  • Encourages efficiency in computation
  • Develops 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 reflection

Games Require Reflection

Games need to be seen as a learning experience

where s the math29
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 thinking
  • Encourage cooperation, not competition
  • Share the game and mathematical goals with parents
extensions
Extensions
  • 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 websites
Games websites
  • www.mathwire.com
  • http://childparenting.about.com/od/makeathomemathgames/
  • http://www.netrover.com/~kingskid/Math/math.htm
  • http://www.multiplication.com/classroom_games.htm
  • http://www.awesomelibrary.org/Classroom/Mathematics/Mathematics.html
  • http://www.primarygames.co.uk/
  • http://www.pbs.org/teachers/math/