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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

Where’s the Math?

Dr. Janet H. Caldwell

Rowan University

[email protected]

where s the math3
Where’s the Math?
  • Fraction Models
  • Models for Multiplication
  • Models for Division
  • Decimals, Ratio & Percent
sketch what you first see
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 see5
Sketch what you first see
  • One-half
  • Two-thirds
  • Three-fourths
sketch what you first see6
Sketch what you first see
  • Three-fifths
  • One-sixth
sketch what you first see7
Sketch what you first see
  • Two and a quarter
  • Two and two-thirds
fraction models part of a whole
Set

Area or region

Circles

Clocks

Rectangles

Pattern blocks

Strips

Length

Number line

Ruler

Fraction ModelsPart of a Whole
other meanings for fractions

Wins

Losses

Other Meanings for Fractions
  • Part-whole
    • Values - eg, money
  • Division
  • Ratio
  • Rate
web resources
Web Resources
  • Fraction model applet
  • Equivalent fractions
  • Fraction game
make a triangle that is
Make a triangle that is:
  • ¼ green and ¾ red
  • 1/3 red and 2/3 green
slide18
“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

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
make an array
Make an Array

24

x 3

12

60

72

a harder problem
A Harder Problem

24

x 13

12

60

40

200

312

decimals
Decimals

0.12 + 0.60 = 0.72

3 x 0.24

0.3 x 0.6

array

2 of 3 rows

3 of 4 in each row

Array
mixed numbers too
Mixed Numbers, too!

8 x 3 ¾

8 x 3 = 24

24 + 6 = 30

algebra
Algebra

(x + 1) (x + 2)

= x2 + 2x + x + 2

= x2 + 3x + 2

x + 2

x+

1

sidetrip to geometry area
Sidetrip to Geometry - Area
  • Counting squares on a grid
  • What’s the area?
break it up
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
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.

slide32
So?

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

Area = (10 x 5) / 2

= 25 sq. units

fraction division
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 1
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

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

percents

$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
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
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

$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
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

?

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 percent43
Still more percent

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

slide44
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 math45
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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