Models That Make Math Meaningful

8/2/2019 Models That Make Math Meaningful

http://slidepdf.com/reader/full/models-that-make-math-meaningful 1/45

Where’s the Math?

Dr. Janet H. Caldwell

Rowan [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




One-half

Two-thirds

Three-fourths




Three-fifths

One-sixth




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



Fraction Models

Part of a Whole

Set

Area or region

Circles Clocks

Rectangles

Pattern blocks Strips

Length

Number line

Ruler



Other Meanings for

Fractions

Part-whole

Values - eg, money

Division Ratio

Rate

WinsLosses



Web Resources

Fraction modelapplet

Equivalent fractions Fraction game

http://illuminations.nctm.org/ActivityDetail.aspx?ID=45










Playing Fraction Tracks

http://e/Presentations%20Folder/Eexamples/SOURCE/fract1.htm



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 learnedand being able to use itflexibly.”

- Hiebert, in Lester & Charles,

Teaching Mathematics through

Problem Solving , 2004.



Computational Fluency

7 x 28



Using Base Ten Blocks to

Multiply

24

x 312

60

72



Make an Array

24

x 312

60

72



A Harder Problem

24x 13

126040

200312



Decimals

3 x 0.24

0.3 x 0.6

0.12 + 0.60 = 0.72



Draw a picture that shows

4

3

3

2



Array

2 of 3

rows

3 of 4 in each row

2

1

12

6

43

32

of rows

of rows

areatotal

areashaded



Mixed Numbers, too!

8 x 3 ¾

8 x 3 = 24

64

24

4

38 x

24 + 6 = 30



1 2/3 x 2 ¼ = ?

6

4

5 10

5 9 45 9 3

x 3 33 4 12 12 4



Algebra

(x + 1) (x + 2)

= x2 + 2x + x + 2

= x2 + 3x + 2

x + 2

x

+

1



Sidetrip to Geometry -

Area

Counting squares ona grid

What’s the area?



Break it up

Yellow (L) = ½ x 4 = 2

Blue = 2 x 3 = 6

Yellow (R) = ½ x 2 = 1Orange = 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 trianglesUL: ½ x 4 = 2

UR: ½ x 2 = 1

LL: ½ x 4 = 2LR: ½ x 2 = 1

Total = 6 squares

Area of pentagon

= 18 – 6 = 12 sq.



So?

Find the area of atriangle with base 10and height 5.

5

10

Area = (10 x 5) / 2

= 25 sq. units



Fraction Division

What is the whole ifhalf is 1¾?

1¾ Measurement model

Need two pieces of

size 1¾, so find

1¾ x 2 = 3 ½1¾ ÷ 2 = 1¾ x 2

= 3 ½



How many 1/2s are there

in 1¾?

How many cakescan you make with

1 ¾ cups of sugarif each cakerequires ½ cup?

Partitive Model (Sharing)

1 ¾ ÷ ½ = 3 ½



What’s the length?

The area of a field is1 ¾ square miles.

Its width is ½ mile.A = 1 ¾ 1/2

Missing FactorModel

½ x ___ = 1 ¾



Decimals



Percents

A group of students has $60 to spend on dinner. Theyknow 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 totalcost will be $60?

$60

Cost of Food Tax

and Tip





Another Approach

A group of students has $60 to spend on dinner. Theyknow that the total cost, after adding tax and tip, will be

25% more than the food prices shown on the menu. Howmuch can they spend on the food so that the total cost will

be $60?

Dinner $40 $4 $8 $48

Tax & tip $10 $1 $2 $12

Total $50 $5 $10 $60



More on Percent

Josie needs $40 for a new sweater. Shehas $24. What percent does she haveof what she needs?

$40

100%

$24

? %



Using a Table

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

Needs $40 $40 $40

Has $4 $20 $24

Percent 10% 50% 60%



Still more percent

Jamal has 48% of his homework done. Hehas done 12 problems. How many

problems did the teacher assign?12 ?

48% 100%



Still more percent

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

% done 48% 48% 48%

Total 100 50 25

# done 48 24 12



SO?

Pictures

Manipulatives

Oral language Written symbols

Tables

Graphs Relevant situations

Which model(s) aremost meaningful formy students?

Which modelspromote morepowerful thinking?

In what order shouldI use selectedmodels?



Where’s the Math?

Models help students explore conceptsand build understanding

Models provide a context for students tosolve problems and explain reasoning

Models provide opportunities for students

to generalize conceptual understanding

Documents

Models That Make Math Meaningful