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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
- 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

Set

Area or region

Circles

Clocks

Rectangles

Pattern blocks

Strips

Length

Number line

Ruler

Fraction ModelsPart of a WholeWeb Resources

- Fraction model applet
- Equivalent fractions
- Fraction game

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.

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 FluencySidetrip to Geometry - Area

- Counting squares on a grid
- What’s 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.

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

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

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?

x

$60

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?

$24

? %

100%

More on PercentJosie 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?

12

48%

100%

Still more percentJamal 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?

Where’s 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

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