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Where’s the Math?. Dr. Janet H. Caldwell Rowan University caldwell@rowan.edu. Models that Make Math Meaningful. Where’s the Math?. Fraction Models Models for Multiplication Models for Division Decimals, Ratio & Percent. Sketch what you first see. One-half Two-thirds Three-fourths

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### Where’s the Math?

Dr. Janet H. Caldwell

Rowan University

caldwell@rowan.edu

### Models that Make Math Meaningful

• Fraction Models

• Models for Multiplication

• Models for Division

• Decimals, Ratio & Percent

• One-half

• Two-thirds

• Three-fourths

• Three-fifths

• One-sixth

• Two and a quarter

• Two and two-thirds

• One-half

• Two-thirds

• Three-fourths

• Three-fifths

• One-sixth

• Two and a quarter

• Two and two-thirds

Part of a Set 3 of 13 pieces

Part of an Area 6 of 36 triangles

Part of an Area 1 of 6 hexagons

Part of an area Blue is 1/3 of largest piece

Area or region

Circles

Clocks

Rectangles

Pattern blocks

Strips

Length

Number line

Ruler

Fraction ModelsPart of a Whole

Losses

Other Meanings for Fractions

• Part-whole

• Values - eg, money

• Division

• Ratio

• Rate

• Fraction model applet

• Equivalent fractions

• Fraction game

• ¼ green and ¾ red

• 1/3 red and 2/3 green

“Understanding is the key to remembering what is learned and being able to use it flexibly.”

- Hiebert, in Lester & Charles,

Teaching Mathematics through

Problem Solving, 2004.

I thought seven and being able to use it flexibly.”

25’s - that’s 175.

Then I need seven 3’s or 21. So the answer is 175 + 21 = 196

7 x 20 is 140 and 7 x 8 is 56 56 + 140 is 196

7 x 28

I did 7 x 30 first. That’s 210. Then take off seven 2’s or 14. So it’s 196.

Computational Fluency

Using Base Ten Blocks to Multiply and being able to use it flexibly.”

24

x 3

12

60

72

Make an Array and being able to use it flexibly.”

24

x 3

12

60

72

A Harder Problem and being able to use it flexibly.”

24

x 13

12

60

40

200

312

Decimals and being able to use it flexibly.”

0.12 + 0.60 = 0.72

3 x 0.24

0.3 x 0.6

Draw a picture that shows and being able to use it flexibly.”

2 of 3 rows and being able to use it flexibly.”

3 of 4 in each row

Array

Mixed Numbers, too! and being able to use it flexibly.”

8 x 3 ¾

8 x 3 = 24

24 + 6 = 30

1 2/3 x 2 ¼ = ? and being able to use it flexibly.”

Algebra and being able to use it flexibly.”

(x + 1) (x + 2)

= x2 + 2x + x + 2

= x2 + 3x + 2

x + 2

x+

1

Sidetrip to Geometry - Area and being able to use it flexibly.”

• Counting squares on a grid

• What’s the area?

Break it up and being able to use it flexibly.”

Yellow (L) = ½ x 4 = 2

Blue = 2 x 3 = 6

Yellow (R) = ½ x 2 = 1

Orange = ½ x 2 = 1

Red = ½ x 4 = 2

2 + 6 + 1 + 1 + 2

= 12 square units

Make a Rectangle and being able to use it flexibly.”

Area of rectangle

= 3 x 6 = 18 squares

Areas of triangles

UL: ½ x 4 = 2

UR: ½ x 2 = 1

LL: ½ x 4 = 2

LR: ½ x 2 = 1

Total = 6 squares

Area of pentagon

= 18 – 6 = 12 sq.

So? and being able to use it flexibly.”

Find the area of a triangle with base 10 and height 5.

Area = (10 x 5) / 2

= 25 sq. units

Fraction Division and being able to use it flexibly.”

What is the whole if half is 1¾?

Measurement model

Need two pieces of

size 1¾, so find

1¾ x 2 = 3 ½

1¾ ÷ 2 =1¾ x 2

= 3 ½

How many 1/2s are there in 1¾? and being able to use it flexibly.”

How many cakes can you make with 1 ¾ cups of sugar if each cake requires ½ cup?

Partitive Model (Sharing)

1 ¾ ÷ ½ = 3 ½

A = 1 ¾ and being able to use it flexibly.”

What’s the length?

The area of a field is 1 ¾ square miles.

Its width is ½ mile.

1/2

Missing Factor Model

½ x ___ = 1 ¾

Decimals and being able to use it flexibly.”

\$60 and being able to use it flexibly.”

Cost of Food

Tax

and Tip

Percents

A group of students has \$60 to spend on dinner. They know that the total cost, after adding tax and tip, will be 25% more than the food prices shown on the menu. How much can they spend on the food so that the total cost will be \$60?

Percent Bar and being able to use it flexibly.”

A group of students has \$60 to spend on dinner. They know that the total cost, after adding tax and tip, will be 25% more than the food prices shown on the menu. How much can they spend on the food so that the total cost will be \$60?

x

\$60

100%

125%

Another Approach and being able to use it flexibly.”

A group of students has \$60 to spend on dinner. They know that the total cost, after adding tax and tip, will be 25% more than the food prices shown on the menu. How much can they spend on the food so that the total cost will be \$60?

\$40 and being able to use it flexibly.”

\$24

? %

100%

More on Percent

Josie needs \$40 for a new sweater. She has \$24. What percent does she have of what she needs?

Using a Table and being able to use it flexibly.”

Josie needs \$40 for a new sweater. She has \$24. What percent does she have of what she needs?

? and being able to use it flexibly.”

12

48%

100%

Still more percent

Jamal has 48% of his homework done. He has done 12 problems. How many problems did the teacher assign?

Still more percent and being able to use it flexibly.”

Jamal has 48% of his homework done. He has done 12 problems. How many problems did the teacher assign?

Pictures and being able to use it flexibly.”

Manipulatives

Oral language

Written symbols

Tables

Graphs

Relevant situations

Which model(s) are most meaningful for my students?

Which models promote more powerful thinking?

In what order should I use selected models?

SO?

Where’s the Math? and being able to use it flexibly.”

• Models help students explore concepts and build understanding

• Models provide a context for students to solve problems and explain reasoning

• Models provide opportunities for students to generalize conceptual understanding