# Models that Make Math Meaningful

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```					                Where’s the
Math?
Models that Make Math Meaningful

Dr. Janet H. Caldwell
Rowan University
caldwell@rowan.edu
Where’s the Math?
• Fraction Models
• Models for Multiplication
• Models for Division
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                • Length
• Area or region        – Number line
– Circles             – Ruler
– Clocks
– Rectangles
– Pattern blocks
– Strips
Other Meanings for
Fractions
• Part-whole
– Values - eg, money   Wins
Losses
• Division
• Ratio
• Rate
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.
Computational Fluency

7 x 28
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 3

3 4
Array

2 of 3
rows

3 of 4 in each row

shaded area 2 rows of 3 6 1
              
total area   3 rows of 4 12 2
Mixed Numbers, too!
8 x 3 ¾
8 x 3 = 24
3 24
8x    6
4 4

24 + 6 = 30
1 2/3 x 2 ¼ = ?
6

4

2

5    10

5 9 45     9    3
x    3    3
3 4 12    12    4
Algebra
x   +    2   (x + 1) (x + 2)

x
+                 = x2 + 2x + x + 2

1
= x2 + 3x + 2
Sidetrip to Geometry -
Area
• Counting squares on a grid
• What’s the area?

6

4

2
Break it up
Yellow (L) = ½ x 4 = 2
Blue = 2 x 3 = 6
6

Yellow (R) = ½ x 2 = 1   4

Orange = ½ x 2 = 1       2

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   6

UL: ½ x 4 = 2
UR: ½ x 2 = 1
4

LL: ½ x 4 = 2       2

LR: ½ x 2 = 1
Area of pentagon
Total = 6 squares
= 18 – 6 = 12 sq.
So?
Find the area of a
triangle with base
10 and height 5.           5

10

Area = (10 x 5) / 2
= 25 sq. units
Fraction Division
What is the whole if
half is 1¾?

1¾            Measurement model
Need two pieces of
1¾ ÷ 2 = 1¾ x 2    size 1¾, so find
=3½          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)
1¾ ÷½=3½
What’s the length?
The area of a field is
1¾ square miles.
Its width is ½ mile.
1/2    A=1¾

Missing Factor Model
½ x ___ = 1 ¾
SO?
•   Pictures              • Which model(s) are
•   Manipulatives           most meaningful
•   Oral language           for my students?
•   Written symbols       • Which models
promote more
•   Tables                  powerful thinking?
•   Graphs                • In what order
•   Relevant situations     should I use
selected models?
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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