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Where’s the Math?. Dr. Janet H. Caldwell Rowan University [email protected] 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?

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Where’s the Math?

Dr. Janet H. Caldwell

Rowan University

[email protected]


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

  • 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


Set

Area or region

Circles

Clocks

Rectangles

Pattern blocks

Strips

Length

Number line

Ruler

Fraction ModelsPart of a Whole


Wins

Losses

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.


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.

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


2 of 3 rows

3 of 4 in each row

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

  • 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.


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 ÷ 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


A = 1 ¾

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


$60

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

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?


$40

$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

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


?

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

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?

SO?


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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