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Wheres the Math?

- Dr. Janet H. Caldwell
- Rowan University
- caldwell_at_rowan.edu

Models that Make Math Meaningful

Wheres the Math?

- Fraction Models
- Models for Multiplication
- Models for Division
- Decimals, Ratio Percent

Sketch what you first see

- One-half
- Two-thirds
- Three-fourths
- Three-fifths
- One-sixth
- Two and a quarter
- Two and two-thirds

Sketch what you first see

- One-half
- Two-thirds
- Three-fourths

Sketch what you first see

- Three-fifths
- One-sixth

Sketch what you first see

- Two and a quarter
- Two and two-thirds

What fraction is blue?

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

Fraction ModelsPart of a Whole

- Set
- Area or region
- Circles
- Clocks
- Rectangles
- Pattern blocks
- Strips

- Length
- Number line
- Ruler

Other Meanings for Fractions

- Part-whole
- Values - eg, money
- Division
- Ratio
- Rate

Web Resources

- Fraction model applet
- Equivalent fractions
- Fraction game

Playing Fraction Tracks

Make a triangle that is

- 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.

Computational Fluency

Using Base Ten Blocks to Multiply

- 24
- x 3

12 60 72

Make an Array

- 24
- x 3
- 12
- 60
- 72

A Harder Problem

- 24
- x 13
- 12
- 60
- 40
- 200
- 312

Decimals

0.12 0.60 0.72

- 3 x 0.24

0.3 x 0.6

Draw a picture that shows

Array

Mixed Numbers, too!

- 8 x 3 ¾

8 x 3 24

24 6 30

1 2/3 x 2 ¼ ?

Algebra

- (x 1) (x 2)
- x2 2x x 2
- x2 3x 2

x 2

x

1

Sidetrip to Geometry - Area

- Counting squares on a grid
- Whats the area?

Break it up

- 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

- 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?

- Find the area of a triangle with base 10 and

height 5.

Area (10 x 5) / 2 25 sq. units

Fraction Division

- What is the whole if half is 1¾?

Measurement model Need two pieces of size 1¾,

so find 1¾ x 2 3 ½

1¾

1¾ 2 1¾ x 2 3 ½

How many 1/2s are there in 1¾?

- 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

Whats 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

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

- 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?

60

x

100

125

Another Approach

- 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?

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

- Josie needs 40 for a new sweater. She has 24.

What percent does she have of what she needs?

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

- Jamal has 48 of his homework done. He has done

12 problems. How many problems did the teacher

assign?

SO?

- Pictures
- 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?

Wheres the Math?

- 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