Math Games to Build Skills and Thinking

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Math Games to Build Skills and Thinking - PowerPoint PPT Presentation

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

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

Operational properties

Number relationships

Accurate, efficient, flexible use of computation for multiple purposes

Conceptual Computational Understanding Proficiency

Computation

Algorithms:

Seeing the Math

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.

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

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.

Partial Sums

Partial Products

Partial Differences

Partial Quotients

Lattice Multiplication

Click on the algorithm you’d like to see!

(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

Add the tens (50 + 40)

Add the ones (6 + 7)

(500 + 90 + 13)

+13

356

Try another one!

+ 247

500

90

603

+ 18

429

+ 989

Nice

work!

1300

100

1418

80 X 50

80 X 6

2 X 50

+

2 X 6

Click to proceed at your own speed!

Partial Products

5

6

×

8

2

4,000

480

100

12

4,592

5

2

×

7

6

70 X 50

70 X 2

6 X 50

+

6 X 2

Try another one!

3,500

140

300

12

3,952

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

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.

Try a couple more!

13

9

16

12

14

10

8

7

946

802

568

274

7

2

3

8

5

8

10

Partial Differences

736

–245

500

Subtract the hundreds

(700 – 200)

Subtract the tens

(30 – 40)

1

• Subtract the ones
• (6 – 5)

491

(500 + (-10) + 1)

20

3

Try another one!

412

–335

100

Subtract the hundreds

(400 – 300)

Subtract the tens

(10 – 30)

• Subtract the ones
• (2 – 5)

77

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

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

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

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

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

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 Games

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
• 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?
• 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 need to be seen as a learning experience

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
• 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
• www.mathwire.com