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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 l.jpg

Where’s the Math?

Dr. Janet H. Caldwell

Rowan University

[email protected]


Models that make math meaningful l.jpg

Models that Make Math Meaningful


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


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Sketch what you first see

  • Three-fifths

  • One-sixth


Sketch what you first see7 l.jpg

Sketch what you first see

  • Two and a quarter

  • Two and two-thirds


What fraction is blue l.jpg

What fraction is blue?


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


Part of an area blue is 1 3 of largest piece l.jpg

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


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Playing Fraction Tracks


Make a triangle that is l.jpg

Make a triangle that is:

  • ¼ green and ¾ red

  • 1/3 red and 2/3 green


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“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.


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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 l.jpg

Using Base Ten Blocks to Multiply

24

x 3

12

60

72


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Make an Array

24

x 3

12

60

72


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A Harder Problem

24

x 13

12

60

40

200

312


Decimals l.jpg

Decimals

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


Array l.jpg

2 of 3 rows

3 of 4 in each row

Array


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Mixed Numbers, too!

8 x 3 ¾

8 x 3 = 24

24 + 6 = 30


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1 2/3 x 2 ¼ = ?


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Algebra

(x + 1) (x + 2)

= x2 + 2x + x + 2

= x2 + 3x + 2

x + 2

x+

1


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Sidetrip to Geometry - Area

  • Counting squares on a grid

  • What’s the area?


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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 l.jpg

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 l.jpg

So?

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

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


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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


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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


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Decimals


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$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?


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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 l.jpg

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 l.jpg

$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 l.jpg

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 l.jpg

?

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?


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Still more percent

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


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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?


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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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