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Where’s the Math?

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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

Set

Area or region

Circles

Clocks

Rectangles

Pattern blocks

Strips

Length

Number line

Ruler

Wins

Losses

- 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

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.

24

x 3

12

60

72

24

x 3

12

60

72

24

x 13

12

60

40

200

312

0.12 + 0.60 = 0.72

3 x 0.24

0.3 x 0.6

2 of 3 rows

3 of 4 in each row

8 x 3 ¾

8 x 3 = 24

24 + 6 = 30

(x + 1) (x + 2)

= x2 + 2x + x + 2

= x2 + 3x + 2

x + 2

x+

1

- Counting squares on a grid
- What’s the area?

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

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.

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

Area = (10 x 5) / 2

= 25 sq. units

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 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

The area of a field is 1 ¾ square miles.

Its width is ½ mile.

1/2

Missing Factor Model

½ x ___ = 1 ¾

$60

Cost of Food

Tax

and Tip

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?

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%

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

$24

? %

100%

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

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

?

12

48%

100%

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

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

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?

- 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