**Flexible Problem Solving by**

Jon Menzin

Jon Menzin

"There are many ways to solve math problems"

**Addition**

"How else could you solve the same problem?"

**Addition**

Multiplication

**4583 x 3792**

**Multiplication**

**Lattice**

"Math is creative and innovative expression"

Subtraction

using manipulatives

"See if you can figure out another way to solve the same problem"

**Subtraction**

26 + 37

20

6

30

7

50

13

Decomposing & recomposing

**Tree model**

**26 + 37**

26

30

+4

30 + 37 = 67

67

Shifting method

-4

346 - 188

346

**-**

188

3

4

6

1

8

8

2

16

13

**=**

8

5

1

= 158

188

200

+12

346 - 200 = 146

146

+12

= 158

**Left-to-right partial differences**

346

- 188

200

-40

-2

=158

+

+

"There is almost always more than one way to solve a mathematics problem"

**Multiplication**

"Students should not just be memorizing past methods; they need to engage, do, act, perform, and problem solve."

**Division**

Math is "a powerful way of expressing relationships and ideas in numerical, graphical, symbolic, verbal, and pictorial forms. "

**Division**

**Fractions**

4 2/5 + 3 1/6

The Generic Rectangle model

4000

500

80

3

3000

700

90

2

12,000,000

2,800,000

360,000

8,000

1,500,000

350,000

45,000

1,000

240,000

9,000

56,000

7,200

160

2,100

270

6

11,376

296,000 + 7,360

1,896,000

14,800,000 + 368,000

15,168,000

2,199,360

17,368,000 - 640

17,379,376 - 640

17,378,000 + 1,376 - 640

17,378,000 + 736

=17,378,736

4

5

8

3

3

7

9

2

**4583 x 3792**

6

2

7

2

1

9

1

6

7

2

5

6

2

4

1

5

3

5

4

5

1

0

8

3

6

2

8

1

2

6

3

7

8

7

3

7

1

**= 17,378,736**

"Math allows you to investigate problems from a variety of perspectives."

4583

x 3792

**Left-to-right**

Partial Products method

Partial Products method

12000000

1500000

240000

9000

2800000

350000

56000

2100

360000

45000

7200

270

8000

1000

160

6

6

3

7

8

7

3

7

1

6256 94

**Generic Rectangles**

**6000**

**256**

**94**

63 R78

2 R68

63 R78 + 2 R68 = 65 R146

65 R146 = 66 R52

94 x 2 = 188

256 - 188 = 68

94 x 60 = 5640

6000 - 5640 = 360

94 x 3 = 282

360 - 282 = 78

**Partial Quotients**

"People do not need to regurgitate hundreds of standard methods. They need to reason and problem solve, flexibly applying methods in new situations."

"Long and complicated problems...encourage persistence, a critical trait for young people to develop that will stand them in good stead in life and work."

6256 94

94

6256

Math facts for 94

94 x 10 = 940

94 x 20 = 1880

94 x 40 = 3760

94 x 50 = 4700

4700

50

1556

940

10

616

94 x 5 = 470

470

5

146

94

1

52

66 R52 = 66 52/94 = 66 26/47

66 R52 = 66 52/94 = 66 26/47

63

= 63

"Did anyone solve the problem a different way?"

346 - 188

**using the Shifting method**

**Subtraction**

2 8/9 - 1/3

3 5/9 x 2/7

5 2/7

7

= 7 17/30

3

3 - 1/3 = 2 2/3

3

5/9

2/7

6/7

10/63

6/7 + 10/63

54/63 + 10/63

= 64/63

= 1 1/63

"Children need to...use, adapt, and apply standard methods, as well as to make connections between methods and to reason mathematically."

Division

6256 94

47

3128

Math facts for 94

94 x 10 = 940

94 x 20 = 1880

94 x 40 = 3760

94 x 50 = 4700

1880

40

1248

940

20

308

94 x 5 = 470

235

5

73

47

1

26

66 R26 = 66 26/47

**Equivalent Problems**

**=**

3128 47

Cut in half

Cut in half

Math facts for 47

47 x 20 = 940

47 x 40 = 1880

47 x 10 = 470

Partial

Quotients

47 x 5 = 235

Math "requires creativity, original thinking, and ingenuity."

Generic Rectangles

Shifting

+1/9

2 2/3

-1/9

2 6/9 - 1/9

= 2 5/9

How many 2/7 are there in 5?

Count the number of 2/7's, shown as pairs of like-colored stars

6 Blue + 6 Yellow + 5 Red = 17

Remainder: 1 Red star = 1/2 pair = 1/7

5 2/7 = 17 R1/7 = 17 1/2

Draw a picture

Decomposing

12/30

5/30

17/30