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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 l.jpg

Where’s the Math?

Dr. Janet H. Caldwell

Rowan University

[email protected]



Where s the math3 l.jpg
Where’s the Math?

  • Fraction Models

  • Models for Multiplication

  • Models for Division

  • Decimals, Ratio & Percent


Sketch what you first see l.jpg
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 l.jpg
Sketch what you first see

  • One-half

  • Two-thirds

  • Three-fourths


Sketch what you first see6 l.jpg
Sketch what you first see

  • Three-fifths

  • One-sixth


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Sketch what you first see

  • Two and a quarter

  • Two and two-thirds



Part of a set 3 of 13 pieces l.jpg
Part of a Set 3 of 13 pieces


Part of an area 6 of 36 triangles l.jpg
Part of an Area 6 of 36 triangles


Part of an area 1 of 6 hexagons l.jpg
Part of an Area 1 of 6 hexagons


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Part of an area Blue is 1/3 of largest piece


Fraction models part of a whole l.jpg

Set

Area or region

Circles

Clocks

Rectangles

Pattern blocks

Strips

Length

Number line

Ruler

Fraction ModelsPart of a Whole


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Wins

Losses

Other Meanings for Fractions

  • Part-whole

    • Values - eg, money

  • Division

  • Ratio

  • Rate


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Web Resources

  • Fraction model applet

  • Equivalent fractions

  • Fraction game



Make a triangle that is l.jpg
Make a triangle that is:

  • ¼ green and ¾ red

  • 1/3 red and 2/3 green


Slide18 l.jpg

“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 l.jpg

I thought seven and being able to use it flexibly.”

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


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Using Base Ten Blocks to Multiply and being able to use it flexibly.”

24

x 3

12

60

72


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Make an Array and being able to use it flexibly.”

24

x 3

12

60

72


A harder problem l.jpg
A Harder Problem and being able to use it flexibly.”

24

x 13

12

60

40

200

312


Decimals l.jpg
Decimals and being able to use it flexibly.”

0.12 + 0.60 = 0.72

3 x 0.24

0.3 x 0.6


Draw a picture that shows l.jpg
Draw a picture that shows and being able to use it flexibly.”


Array l.jpg

2 of 3 rows and being able to use it flexibly.”

3 of 4 in each row

Array


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Mixed Numbers, too! and being able to use it flexibly.”

8 x 3 ¾

8 x 3 = 24

24 + 6 = 30


1 2 3 x 2 l.jpg
1 2/3 x 2 ¼ = ? and being able to use it flexibly.”


Algebra l.jpg
Algebra and being able to use it flexibly.”

(x + 1) (x + 2)

= x2 + 2x + x + 2

= x2 + 3x + 2

x + 2

x+

1


Sidetrip to geometry area l.jpg
Sidetrip to Geometry - Area and being able to use it flexibly.”

  • Counting squares on a grid

  • What’s the area?


Break it up l.jpg
Break it up and being able to use it flexibly.”

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 l.jpg
Make a Rectangle and being able to use it flexibly.”

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.


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So? and being able to use it flexibly.”

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

Area = (10 x 5) / 2

= 25 sq. units


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Fraction Division and being able to use it flexibly.”

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 l.jpg
How many 1/2s are there in 1¾? and being able to use it flexibly.”

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 l.jpg

A = 1 ¾ and being able to use it flexibly.”

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


Decimals36 l.jpg
Decimals and being able to use it flexibly.”


Percents l.jpg

$60 and being able to use it flexibly.”

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 l.jpg
Percent Bar and being able to use it flexibly.”

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 l.jpg
Another Approach and being able to use it flexibly.”

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 l.jpg

$40 and being able to use it flexibly.”

$24

? %

100%

More on Percent

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


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Using a Table and being able to use it flexibly.”

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


Still more percent l.jpg

? and being able to use it flexibly.”

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 l.jpg
Still more percent and being able to use it flexibly.”

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


Slide44 l.jpg

Pictures and being able to use it flexibly.”

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?


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Where’s the Math? and being able to use it flexibly.”

  • 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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