models that make math meaningful
TRANSCRIPT
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Where’s the Math?
Dr. Janet H. Caldwell
Rowan [email protected]
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Models that Make
Math Meaningful
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Where’s the Math?
Fraction Models
Models for Multiplication
Models for Division
Decimals, Ratio & Percent
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Sketch what you first see
One-half
Two-thirds
Three-fourths
Three-fifths
One-sixth
Two and a quarter
Two and two-thirds
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Sketch what you first see
One-half
Two-thirds
Three-fourths
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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
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What fraction is blue?
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Part of a Set
3 of 13 pieces
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Part of an Area
6 of 36 triangles
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Part of an Area
1 of 6 hexagons
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Part of an area
Blue is 1/3 of largest piece
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Fraction Models
Part of a Whole
Set
Area or region
Circles Clocks
Rectangles
Pattern blocks Strips
Length
Number line
Ruler
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Other Meanings for
Fractions
Part-whole
Values - eg, money
Division Ratio
Rate
WinsLosses
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Web Resources
Fraction modelapplet
Equivalent fractions Fraction game
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Playing Fraction Tracks
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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 learnedand being able to use itflexibly.”
- Hiebert, in Lester & Charles,
Teaching Mathematics through
Problem Solving , 2004.
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Computational Fluency
7 x 28
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Using Base Ten Blocks to
Multiply
24
x 312
60
72
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Make an Array
24
x 312
60
72
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A Harder Problem
24x 13
126040
200312
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Decimals
3 x 0.24
0.3 x 0.6
0.12 + 0.60 = 0.72
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Draw a picture that shows
4
3
3
2
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Array
2 of 3
rows
3 of 4 in each row
2
1
12
6
43
32
of rows
of rows
areatotal
areashaded
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Mixed Numbers, too!
8 x 3 ¾
8 x 3 = 24
64
24
4
38 x
24 + 6 = 30
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1 2/3 x 2 ¼ = ?
6
4
5 10
5 9 45 9 3
x 3 33 4 12 12 4
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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 ona 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 = 1Orange = ½ x 2 = 1
Red = ½ x 4 = 2
2 + 6 + 1 + 1 + 2
= 12 square units
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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.
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So?
Find the area of atriangle with base 10and height 5.
5
10
Area = (10 x 5) / 2
= 25 sq. units
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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
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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
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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
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Decimals
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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
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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
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More on Percent
Josie needs $40 for a new sweater. Shehas $24. What percent does she haveof what she needs?
$40
100%
$24
? %
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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%
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Still more percent
Jamal has 48% of his homework done. Hehas done 12 problems. How many
problems did the teacher assign?12 ?
48% 100%
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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
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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?
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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