Download Updating Montessori Fractions to a Linear Model

Updating Montessori Fractions
to a Linear Model
Joan A. Cotter, Ph.D.
[email protected]"
AMS Annual Conference!
Philadelphia, Pennsylvania!
Saturday, March 14, 2015 (Happy Pi Day)!
8:00 a.m.– 9:30 a.m.!
© Joan A. Cotter, Ph.D.. 2015
Fractions in the Comics
© Joan A. Cotter, Ph.D.. 2015
Fractions in the Comics
© Joan A. Cotter, Ph.D.. 2015
Meanings of a Fraction
•  One or more equal parts of a whole.
•  One or more equal parts of a collection.
•  Division of two whole numbers.
•  Location on a number line.
•  Ratio of two numbers.
© Joan A. Cotter, Ph.D.. 2015
Meanings of a Fraction
Which meaning is used in everyday life?
•  One or more equal parts of a whole.
•  One or more equal parts of a collection.
•  Division of two whole numbers.
•  Location on a number line.
•  Ratio of two numbers.
© Joan A. Cotter, Ph.D.. 2015
Meanings of a Fraction
Which meaning is mathematical?
•  One or more equal parts of a whole.
•  One or more equal parts of a collection.
•  Division of two whole numbers.
•  Location on a number line.
•  Ratio of two numbers.
© Joan A. Cotter, Ph.D.. 2015
Meanings of a Fraction
Which meaning is found in elementary textbooks?
•  One or more equal parts of a whole.
•  One or more equal parts of a collection.
•  Division of two whole numbers.
•  Location on a number line.
•  Ratio of two numbers.
© Joan A. Cotter, Ph.D.. 2015
Definition of a Fraction
An expression that indicates the quotient
of two quantities. American Heritage Dictionary
A number in the form
a
b
where b ≠ 0.
Arizona Standards
© Joan A. Cotter, Ph.D.. 2015
Montessori and Fractions
•  Developed from geometry, not arithmetic.
•  A fraction is always less than one. Psychogeometry p. 165
•  A fraction contains two numbers with “a symbol
between them indicating division.” p. 165
•  We no longer “reduce” fractions; we simplify them.
•  Montessori shows centesimal frame with zero at the
3 o’clock position and increases counterclockwise.
(This is mathematically correct.)
© Joan A. Cotter, Ph.D.. 2015
Problems Teaching Fractions
•  Common meaning is “small amount.”
•  Introduced as part of a whole (not whole story).
•  Fraction names use ordinal words.
•  Are “improper” fractions somehow undesirable?
•  Mixed numbers different in arithmetic, algebra.
•  Our culture implies fractions are difficult.
•  Many adults have only a partial understanding.
© Joan A. Cotter, Ph.D.. 2015
How have fractions
been taught?
© Joan A. Cotter, Ph.D.. 2015
Fraction Model: “Fish Tank”
What fraction of the fish are blue?
© Joan A. Cotter, Ph.D.. 2015
count the blue fish
2
count all the fish
5
Fraction Model: “Words”
This is fourths.
© Joan A. Cotter, Ph.D.. 2015
This is thirds.
Fraction Models: Circles
Are we comparing angles, arcs, or area?
© Joan A. Cotter, Ph.D.. 2015
Fraction Models: Circles
Experts in visual literacy say that comparing
quantities in pie charts is difficult because
most people think linearly. It is easier to
compare along a straight line than compare
pie slices. askoxford.com
Specialists also suggest refraining from using
more than one pie chart for comparison.
www.statcan.ca
© Joan A. Cotter, Ph.D.. 2015
How can we model
fractions mathematically?
© Joan A. Cotter, Ph.D.. 2015
Fraction Model: Linear Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
© Joan A. Cotter, Ph.D.. 2015
1
7
1
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1
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1
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1
1
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1
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1
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1
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1
10
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Fraction Model: Distracting Color
1
1
2
1
2
1
3
1
3
1
4
1
4
1
5
1
6
1
7
1
8
1
9
1
10
© Joan A. Cotter, Ph.D.. 2015
1
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1
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1
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1
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1
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1
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1
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1
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1
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1
10
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7
1
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1
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1
9
1
10
1
9
1
10
1
3
1
9
1
10
1
4
1
5
1
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Fraction Model: “Missing Parts”
1
4
1
3
1
1
2
1
4
1
3
1
4
1
1
1
1
1
5
5
5
5
5
1
1
1
1
1
1
6
6
6
6
6
6
1
1
1
1
1
1
1
1
8
8
8
8
8
8
8
8
1
1
1
1
1
1
1
1
1
1
10
10
10
10
10
10
10
10
10
10
1
1
1
1
1
1
1
1
1
1
1
1
12
12
12
12
12
12
12
12
12
12
12
12
© Joan A. Cotter, Ph.D.. 2015
1
4
1
3
1
2
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
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1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Ask the children to put it together like a puzzle.
© Joan A. Cotter, Ph.D.. 2015
Fraction Stairs
1
10
1
9
1
8
1
7
1
6
1
5
1
4
1
3
1
2
1
A hyperbola. © Joan A. Cotter, Ph.D.. 2015
Fraction Stairs
1
10
1
9
1
8
1
7
1
6
1
5
1
4
1
3
1
2
1
Two hyperbolas. © Joan A. Cotter, Ph.D.. 2015
Fraction Stairs: “Missing Parts”
1
12
1
10
1
8
1
6
1
5
1
4
1
3
1
2
1
No pattern. © Joan A. Cotter, Ph.D.. 2015
How do we name
fractions?
© Joan A. Cotter, Ph.D.. 2015
Fraction Naming
•  In English, except for “half,” we use
ordinal numbers to name fractions.
•  Sometimes we use “quarter” for a fourth.
•  A quarter of a hour (15 minutes)
•  A quarter of a dollar (25¢)
•  A quarter of a gallon (quart)
•  A Quarter Pounder (¼ of a pound, or 4 oz)
© Joan A. Cotter, Ph.D.. 2015
Fraction Naming
1
© Joan A. Cotter, Ph.D.. 2015
one
Fraction Naming
1
© Joan A. Cotter, Ph.D.. 2015
one
divided by
Fraction Naming
1
3
© Joan A. Cotter, Ph.D.. 2015
one
divided by
three
Fraction Naming
1
1
3
© Joan A. Cotter, Ph.D.. 2015
1
3
1
3
1
3
one
divided by
three
How can we practice
basic fraction concepts?
© Joan A. Cotter, Ph.D.. 2015
Games
Games
Math
=
Books
Reading
Games provide interesting repetition
within a social setting.
© Joan A. Cotter, Ph.D.. 2015
Unit Fraction War
Aim: To help the children realize a unit fraction
decreases as the denominator increases.
Object of the game:
To collect all, or most, of the cards with the
greater unit fraction.
© Joan A. Cotter, Ph.D.. 2015
Unit Fraction War
1
4
© Joan A. Cotter, Ph.D.. 2015
1
5
Unit Fraction War
1
8
© Joan A. Cotter, Ph.D.. 2015
1
2
Unit Fraction War
1
6
1
3
© Joan A. Cotter, Ph.D.. 2015
1
6
1
4
More Fraction Naming
1
1
3
1
3
1
3
Two one-thirds.
1
2
Two
s is
3
3
Read as “two-thirds.”
© Joan A. Cotter, Ph.D.. 2015
More Fraction Naming
1
1
Two divided into 3 equal parts.
2
3
© Joan A. Cotter, Ph.D.. 2015
two
divided by
three
More Fraction Naming
1
1
3
1
3
1
1
3
1
3
1
3
Two divided into 3 equal parts.
2
3
© Joan A. Cotter, Ph.D.. 2015
two
divided by
three
1 s
= two
3
1
3
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
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1
8
1
9
1
10
1
7
1
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1
9
1
10
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1
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1
5
1
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1
10
1
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1
3
1
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1
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1
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1
10
1
9
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
How many fourths in a whole? © Joan A. Cotter, Ph.D.. 2015
1
4
1
3
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
How many sixths in a whole? How many eighths?
© Joan A. Cotter, Ph.D.. 2015
Concentrating on One Game
Aim:
To help the children realize that 5 fifths, 8 eighths,
and so forth, make a whole.
Object of the game:
To find the pairs that make a whole.
© Joan A. Cotter, Ph.D.. 2015
Concentrating on One
3
5
2
5
© Joan A. Cotter, Ph.D.. 2015
Concentrating on One
3
8
© Joan A. Cotter, Ph.D.. 2015
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
Which is more, 4/5 or 5/6?
© Joan A. Cotter, Ph.D.. 2015
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Fraction Circles
1
2
1
2
1
3
1
3
1
3
1
4
1
4
1
5
1
4
1
4
1
5
1
6
1
5
1
5
1
5
1
6
1
6
1
6
1
6
1
6
Which is more, 4/5 or 5/6? Which model is easier?
© Joan A. Cotter, Ph.D.. 2015
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
Which is more, 7/8 or 8/9?
© Joan A. Cotter, Ph.D.. 2015
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
An interesting pattern. © Joan A. Cotter, Ph.D.. 2015
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
How many fourths equal a half?
© Joan A. Cotter, Ph.D.. 2015
1
4
1
3
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
How many fourths equal a half? Eighths?
© Joan A. Cotter, Ph.D.. 2015
1
3
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
How many fourths equal a half? Eighths? Sevenths?
© Joan A. Cotter, Ph.D.. 2015
Fraction Model: Distracting Color
1
1
2
1
2
1
3
1
3
1
4
1
4
1
5
1
6
1
7
1
8
1
9
1
10
© Joan A. Cotter, Ph.D.. 2015
1
4
1
5
1
5
1
6
1
6
1
7
1
7
1
8
1
9
1
10
1
7
1
8
1
8
1
9
1
10
1
9
1
10
1
3
1
9
1
10
1
4
1
5
1
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
4
1
6
1
7
1
10
1
9
What is 1/4 plus 1/8?
© Joan A. Cotter, Ph.D.. 2015
1
3
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
4
1
6
1
7
1
10
1
9
What is 1/4 plus 1/8?
© Joan A. Cotter, Ph.D.. 2015
1
3
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
What is 1/4 plus 1/8? [3/8]
© Joan A. Cotter, Ph.D.. 2015
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Partial Chart
1
8
© Joan A. Cotter, Ph.D.. 2015
1
4
1
1
2
1
8
1
8
1
4
1
8
1
8
1
4
1
2
1
8
1
8
1
4
1
8
Partial Chart
1
8
© Joan A. Cotter, Ph.D.. 2015
1
4
1
1
2
1
8
1
8
1
4
1
8
1
8
1
4
1
2
1
8
1
8
1
4
1
8
Partial Chart
© Joan A. Cotter, Ph.D.. 2015
Partial Chart
1
© Joan A. Cotter, Ph.D.. 2015
2
3
4
5
6
Fraction War
Aim:
To practice comparing ones, halves, fourths,
and eighths in preparation for reading a ruler
with inches.
Object of the game:
To capture all the cards.
© Joan A. Cotter, Ph.D.. 2015
Fraction War
1
1
2
1
2
1
4
1
8
1
8
© Joan A. Cotter, Ph.D.. 2015
1
4
1
8
1
8
1
4
1
8
1
8
1
4
1
8
1
8
1
4
1
8
Fraction War
1
1
2
1
2
1
4
1
8
5
8
© Joan A. Cotter, Ph.D.. 2015
1
4
1
8
1
8
1
4
1
8
1
8
1
4
1
8
1
8
3
4
1
8
Advanced Fraction War
© Joan A. Cotter, Ph.D.. 2015
Fractions > 1
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
© Joan A. Cotter, Ph.D.. 2015
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
1
8
Mixed to Improper Fractions
Each row of connected rectangles represents 1.
© Joan A. Cotter, Ph.D.. 2015
Mixed to Improper Fractions
Each row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
© Joan A. Cotter, Ph.D.. 2015
Mixed to Improper Fractions
Each row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
3
11
24 = 4
two 4s + 3 = 11
© Joan A. Cotter, Ph.D.. 2015
Mixed to Improper Fractions
Each row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
3
11
24 = 4
two 4s + 3 = 11
1
13
43 = 3
four 3s + 1 = 13
© Joan A. Cotter, Ph.D.. 2015
Mixed to Improper Fractions
Each row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
3
24
=
3
35
11
4
two 4s + 3 = 11
1
13
43 = 3
four 3s + 1 = 13
© Joan A. Cotter, Ph.D.. 2015
=
18
5
three 5s + 3 = 18
Improper to Mixed Fractions
Write each quantity as an improper fraction, then circle the wholes and write again as a mixed number.
1
5
1
3
© Joan A. Cotter, Ph.D.. 2015
1
5
1
5
1
3
1
5
1
5
1
3
1
5
1
3
1
5
1
5
1
3
1
5
1
5
1
5
Improper to Mixed Fractions
Write each quantity as an improper fraction, then circle the wholes and write again as a mixed number.
1
5
1
3
© Joan A. Cotter, Ph.D.. 2015
1
5
1
5
1
3
1
5
1
5
1
3
1
5
1
3
1
5
1
5
1
3
1
5
1
5
1
5
11 =
5
2 15
Improper to Mixed Fractions
Write each quantity as an improper fraction, then circle the wholes and write again as a mixed number.
1
5
1
3
© Joan A. Cotter, Ph.D.. 2015
1
5
1
5
1
3
1
5
1
5
1
3
1
5
1
3
1
5
1
5
1
3
1
5
1
5
1
5
5
3
11 =
5
= 1 23
2 15
Fraction of Geometric Figures
1
Shade 2
© Joan A. Cotter, Ph.D.. 2015
Fraction of Geometric Figures
1
Shade 2
© Joan A. Cotter, Ph.D.. 2015
2
Shade 3
Fraction of Geometric Figures
1
Shade 2
© Joan A. Cotter, Ph.D.. 2015
2
Shade 3
1
Shade 4
Fraction of Geometric Figures
1
Shade 2
© Joan A. Cotter, Ph.D.. 2015
2
Shade 3
1
Shade 4
Making the Whole
Draw the whole.
1
3
© Joan A. Cotter, Ph.D.. 2015
Making the Whole
Draw the whole.
1
3
© Joan A. Cotter, Ph.D.. 2015
1
3
1
3
1
3
Making the Whole
Draw the whole.
1
3
1
3
2
3
2
3
1
3
© Joan A. Cotter, Ph.D.. 2015
1
3
1
3
Now let’s simplify fractions
Multiples makes it simpler.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
•  Patterns are integral to mathematics.
•  Patterns make learning easier. •  Knowing the multiples is essential for
simplifying fractions and algebra.
•  The multiples should NOT be used to
count up for multiplication facts.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Twos
2
4
6
8
10
12
14
16
18
20
The ones repeat in the second row.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Fours
4
8
12
16
20
24
28
32
36
40
The ones repeat in the second row.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Sixes and Eights
© Joan A. Cotter, Ph.D.. 2015
6
12
18
24
30
36
42
48
54
60
8
16
24
32
40
48
56
64
72
80
Multiples Patterns
Sixes and Eights
6
12
18
24
30
36
42
48
54
60
8
16
24
32
40
48
56
64
72
80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Sixes and Eights
6
12
18
24
30
36
42
48
54
60
8
16
24
32
40
48
56
64
72
80
6 × 4 is the fourth number (multiple).
© Joan A. Cotter, Ph.D.. 2015
6×4
Multiples Patterns
Sixes and Eights
6
12
18
24
30
36
42
48
54
60
8
16
24
32
40
48
56
64
72
80
8 × 7 is the seventh number (multiple).
© Joan A. Cotter, Ph.D.. 2015
8×7
Multiples Patterns
Nines
9
18
27
36
45
90
81
72
63
54
The second row is written in reverse order.
Also the digits in each number add to 9.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Threes
3
6
9
12
15
18
21
24
27
30
The 3s have several patterns:
Observe the ones.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Threes
3
6
9
12
15
18
21
24
27
30
The 3s have several patterns:
The tens are the same in each row.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Threes
3
6
9
12
15
18
21
24
27
30
The 3s have several patterns:
Add the digits in the columns.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Threes
3
6
9
12
15
18
21
24
27
30
The 3s have several patterns:
Add the digits in the columns.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Threes
3
6
9
12
15
18
21
24
27
30
The 3s have several patterns:
Add the digits in the columns.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Threes
3
6
9
12
15
18
21
24
27
30
The 3s have several patterns:
Add the “opposites.”
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Threes
3
6
9
12
15
18
21
24
27
30
The 3s have several patterns:
Add the “opposites.”
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Sevens
7
14
21
28
35
42
49
56
63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Sevens
7
14
21
28
35
42
49
56
63
70
Look at the tens.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Sevens
7
14
21
28
35
42
49
56
63
70
Look at the tens.
© Joan A. Cotter, Ph.D.. 2015
Multiples Patterns
Sevens
7
14
21
28
35
42
49
56
63
70
Look at the tens.
© Joan A. Cotter, Ph.D.. 2015
Simplifying Fractions
1
2
© Joan A. Cotter, Ph.D.. 2015
Simplifying Fractions
3=1
6 2
© Joan A. Cotter, Ph.D.. 2015
Simplifying Fractions
4=1
8 2
© Joan A. Cotter, Ph.D.. 2015
Simplifying Fractions
9
12
© Joan A. Cotter, Ph.D.. 2015
Simplifying Fractions
9
3
=
12 3
4
© Joan A. Cotter, Ph.D.. 2015
Simplifying Fractions
1
2
3
4
5
6
7
8
9
10
© Joan A. Cotter, Ph.D.. 2015
2
4
6
8
10
12
14
16
18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9 10
18 20
27 30
36 40
45 50
54 60
63 70
72 80
81 90
90 100
Simplifying Fractions
1
2
3
4
5
6
7
8
9
10
© Joan A. Cotter, Ph.D.. 2015
2
4
6
8
10
12
14
16
18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9 10
18 20
27 30
36 40
45 50
54 60
63 70
72 80
81 90
90 100
21
28
Simplifying Fractions
1
2
3
4
5
6
7
8
9
10
© Joan A. Cotter, Ph.D.. 2015
2
4
6
8
10
12
14
16
18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9 10
18 20
27 30
36 40
45 50
54 60
63 70
72 80
81 90
90 100
45
72
Simplifying Fractions
1
2
3
4
5
6
7
8
9
10
© Joan A. Cotter, Ph.D.. 2015
2
4
6
8
10
12
14
16
18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9 10
18 20
27 30
36 40
45 50
54 60
63 70
72 80
81 90
90 100
12
16
Simplifying Fractions
1
2
3
4
5
6
7
8
9
10
© Joan A. Cotter, Ph.D.. 2015
2
4
6
8
10
12
14
16
18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9 10
18 20
27 30
36 40
45 50
54 60
63 70
72 80
81 90
90 100
12
16
Simplifying Fractions
1
2
3
4
5
6
7
8
9
10
© Joan A. Cotter, Ph.D.. 2015
2
4
6
8
10
12
14
16
18
20
3
6
9
12
15
18
21
24
27
30
4
8
12
16
20
24
28
32
36
40
5
10
15
20
25
30
35
40
45
50
6
12
18
24
30
36
42
48
54
60
7
14
21
28
35
42
49
56
63
70
8
16
24
32
40
48
56
64
72
80
9 10
18 20
27 30
36 40
45 50
54 60
63 70
72 80
81 90
90 100
12
16
Subtracting Fractions
Preliminary understanding
4684
–2372
2000
300
10
2
2312
© Joan A. Cotter, Ph.D.. 2015
Subtracting Fractions
Preliminary understanding
4684
–2372
2000
300
10
2
2312
© Joan A. Cotter, Ph.D.. 2015
4684
–2879
2000
–200
10
–5
1805
Subtracting Fractions
4
3
–5
2
35
© Joan A. Cotter, Ph.D.. 2015
Subtracting Fractions
4
3
–5
2
35
© Joan A. Cotter, Ph.D.. 2015
–
–
1
57
5
27
3
4
7
3
27
So how does multiplying
fractions work?
© Joan A. Cotter, Ph.D.. 2015
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
9
1
10
1
4
1
5
1
8
1
6
1
7
1
10
What is 1/2 of 1/2?
© Joan A. Cotter, Ph.D.. 2015
1
3
1
9
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
What is 1/3 of 1/2? That’s multiplying fractions! © Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
Multiplying is not exclusively repeated addition.
4×4=4+4+4+4
1 ×1 = 1 +?
2 2
2
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
Multiplying is not exclusively repeated addition.
Area is a better model.
4×4=
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
1 ×1 =
2 2
One half of one half
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
1 ×1 =
2 2
One half of one half
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
1 ×1 =1
2 2 4
One half of one half
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
1 ×1 =1
2 2 4
One half of one half
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
2 ×3 =
3 4
Three fourths
of two thirds
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
2 ×3 =
3 4
Three fourths
of two thirds
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
2 ×3 = 6
3 4 12
Three fourths
of two thirds
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
2 ×3 = 6 = 1
3 4 12 2
Three fourths
of two thirds
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
2 ×3 = 6
3 4 12
The total number of rectangles is
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
2 ×3 = 6
3 4 12
The total number of rectangles is 3 × 4.
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
2 ×3 = 6
3 4 12
The total number of rectangles is 3 × 4.
The number of colored crosshatched rectangles is
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
2 ×3 = 6
3 4 12
The total number of rectangles is 3 × 4.
The number of colored crosshatched rectangles is 2 × 3.
© Joan A. Cotter, Ph.D.. 2015
Multiplying Fractions
2 ×3 = 6
3 4 12
The total number of rectangles is 3 × 4.
The number of colored crosshatched rectangles is 2 × 3.
© Joan A. Cotter, Ph.D.. 2015
What about dividing fractions?
© Joan A. Cotter, Ph.D.. 2015
What is Division?
Process of separating into equal groups
•  either number of groups, or
•  size of the groups
© Joan A. Cotter, Ph.D.. 2015
What is Division?
6 ÷ 2 = __
How many 2s in 6?
Number of groups of 2s
© Joan A. Cotter, Ph.D.. 2015
What is Division?
6 ÷ 2 = __
How many 2s in 6?
Number of groups of 2s; 3 groups
© Joan A. Cotter, Ph.D.. 2015
Dividing Fractions
1
1÷ 2 =
How many 1/2s in 1?
1
1
2
1
2
1
3
1
4
© Joan A. Cotter, Ph.D.. 2015
1
3
1
4
1
3
1
4
1
4
Dividing Fractions
1
1÷ 2 = 2
1
1
2
1
2
1
3
1
4
© Joan A. Cotter, Ph.D.. 2015
1
3
1
4
1
3
1
4
1
4
Dividing Fractions
1
1÷ 2 = 2
1
1÷ 3 =
How many 1/3s in 1?
1
1
2
1
2
1
3
1
4
© Joan A. Cotter, Ph.D.. 2015
1
3
1
4
1
3
1
4
1
4
Dividing Fractions
1
1÷ 2 = 2
1
1÷ 3 = 3
1
1
2
1
2
1
3
1
4
© Joan A. Cotter, Ph.D.. 2015
1
3
1
4
1
3
1
4
1
4
Dividing Fractions
1
1÷ 2 = 2
1
1÷ 3 = 3
1 ÷ 14 = 4
1
1
2
1
2
1
3
1
4
© Joan A. Cotter, Ph.D.. 2015
1
3
1
4
1
3
1
4
1
4
Dividing Fractions
1
1÷ 2 = 2
1
1÷ 3 = 3
1 ÷ 14 = 4
1
1÷ 5 = 5
1
1÷ 6 = 6
© Joan A. Cotter, Ph.D.. 2015
Dividing Fractions
1
1÷ 2 = 2
1
1÷ 3 = 3
1 ÷ 14 = 4
1
1÷ 5 = 5
1
1÷ 6 = 6
© Joan A. Cotter, Ph.D.. 2015
1 ÷ 23 =
How many 2/3s in 1?
Dividing Fractions
1
1÷ 2 = 2
1
1÷ 3 = 3
1 ÷ 14 = 4
1
1÷ 5 = 5
1
1÷ 6 = 6
© Joan A. Cotter, Ph.D.. 2015
1 3
2
1 ÷ 3 = 12 = 2
How many 2/3s in 1?
Dividing Fractions
1
1÷ 2 = 2
1
1÷ 3 = 3
1 ÷ 14 = 4
1
1÷ 5 = 5
1
1÷ 6 = 6
© Joan A. Cotter, Ph.D.. 2015
1 ÷ 23 = 32
Dividing Fractions
1
1÷ 2 = 2
1 ÷ 23 = 32
1
1÷ 3 = 3
1 ÷ 14 = 4
1 ÷ 34 =
1
1÷ 5 = 5
1
1÷ 6 = 6
© Joan A. Cotter, Ph.D.. 2015
How many 3/4s in 1?
Dividing Fractions
1
1÷ 2 = 2
1 ÷ 23 = 32
1
1÷ 3 = 3
1 ÷ 14 = 4
1 ÷ 34 = 113 = 43
1
1÷ 5 = 5
1
1÷ 6 = 6
© Joan A. Cotter, Ph.D.. 2015
How many 3/4s in 1?
Dividing Fractions
1
1÷ 2 = 2
1 ÷ 23 = 32
1
1÷ 3 = 3
1 ÷ 14 = 4
1 ÷ 34 = 43
1
1÷ 5 = 5
1
1÷ 6 = 6
© Joan A. Cotter, Ph.D.. 2015
Dividing Fractions
© Joan A. Cotter, Ph.D.. 2015
1
1÷ 2 = 2
1 ÷ 23 = 32
1
1÷ 3 = 3
1 ÷ 14 = 4
1 ÷ 34 = 43
1
1÷ 5 = 5
1
1÷ 6 = 6
5 8
1÷ 8= 5
4 7
1÷ 7= 4
1 ÷ 25 = 52
Dividing Fractions
1
1÷ 2 = 2
1 ÷ 23 = 32
1
1÷ 3 = 3
1 ÷ 14 = 4
1 ÷ 34 = 43
1
1÷ 5 = 5
1
1÷ 6 = 6
5 8
1÷ 8= 5
4 7
1÷ 7= 4
1 ÷ 25 = 52
Guide the child to making the discovery that the
answers are the inverted form of the divisor, the
number by which another number is to be divided.
© Joan A. Cotter, Ph.D.. 2015
Fraction Division War
Aim: To help the child realize that the quotient
resulting from dividing 1 by a fraction is the
inverted form of the divisor.
Object of the game:
To collect the most cards with the greater
quotient.
© Joan A. Cotter, Ph.D.. 2015
Fraction Division War
© Joan A. Cotter, Ph.D.. 2015
1
1
2
1
1
4
Fraction Division War
© Joan A. Cotter, Ph.D.. 2015
1
1
2
=2
1
1
4
=4
Fraction Division War
1
1
© Joan A. Cotter, Ph.D.. 2015
Fraction Division War
© Joan A. Cotter, Ph.D.. 2015
1
3
8
1
2
3
Fraction Division War
© Joan A. Cotter, Ph.D.. 2015
1
3
8
8
=3
1
2
3
3
2
=
More Dividing Fractions
1
1÷4=4
© Joan A. Cotter, Ph.D.. 2015
More Dividing Fractions
1
1÷4=4
1
2÷4=8
© Joan A. Cotter, Ph.D.. 2015
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 )
© Joan A. Cotter, Ph.D.. 2015
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 )
=2×
© Joan A. Cotter, Ph.D.. 2015
4
=8
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 ) = 2 × 4 = 8
© Joan A. Cotter, Ph.D.. 2015
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 ) = 2 × 4 = 8
3 ÷ 14 = 12
© Joan A. Cotter, Ph.D.. 2015
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 ) = 2 × 4 = 8
3 ÷ 14 = 3 × (1 ÷ 14 )
=3×
© Joan A. Cotter, Ph.D.. 2015
4
= 12
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 ) = 2 × 4 = 8
3 ÷ 14 = 3 × (1 ÷ 14 ) = 3 × 4 = 12
© Joan A. Cotter, Ph.D.. 2015
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 ) = 2 × 4 = 8
3 ÷ 14 = 3 × (1 ÷ 14 ) = 3 × 4 = 12
1 1
2÷4 =2
© Joan A. Cotter, Ph.D.. 2015
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 ) = 2 × 4 = 8
3 ÷ 14 = 3 × (1 ÷ 14 ) = 3 × 4 = 12
1 1
1
1
÷
=
×
(1
÷
2 4
2
4)
= 12 ×
© Joan A. Cotter, Ph.D.. 2015
4
=2
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 ) = 2 × 4 = 8
3 ÷ 14 = 3 × (1 ÷ 14 ) = 3 × 4 = 12
1 1
1
1
1
÷
=
×
(1
÷
)
=
2 4
2
4
2×4=2
© Joan A. Cotter, Ph.D.. 2015
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 ) = 2 × 4 = 8
3 ÷ 14 = 3 × (1 ÷ 14 ) = 3 × 4 = 12
1 1
1
1
1
÷
=
×
(1
÷
)
=
2 4
2
4
2×4=2
1 1
1
1
1
4
÷
=
×
(1
÷
)
=
×
4
=
3 4
3
4
3
3
© Joan A. Cotter, Ph.D.. 2015
More Dividing Fractions
1
1÷4=4
1
1
2 ÷ 4 = 2 × (1 ÷ 4 ) = 2 × 4 = 8
3 ÷ 14 = 3 × (1 ÷ 14 ) = 3 × 4 = 12
1 1
1
1
1
÷
=
×
(1
÷
)
=
2 4
2
4
2×4=2
1 1
1
1
1
4
÷
=
×
(1
÷
)
=
×
4
=
3 4
3
4
3
3
3 1
3
1
3
12
4 ÷ 4 = 4 × (1 ÷ 4 ) = 4 × 4 = 4 = 3
© Joan A. Cotter, Ph.D.. 2015
Harder Fraction Division War
© Joan A. Cotter, Ph.D.. 2015
1
6
1
2
2
3
1
4
Summary
Ways to look at fractions:
•  One or more equal parts of a whole.
•  One or more equal parts of a collection.
•  Division of two whole numbers.
•  Location on a number line.
•  Ratio of two numbers.
© Joan A. Cotter, Ph.D.. 2015
Summary: Fraction Chart
1
4
1
2
1
3
1
5
1
6
1
7
1
8
1
9
1
10
© Joan A. Cotter, Ph.D.. 2015
1
7
1
8
1
9
1
10
1
1
6
1
5
1
9
1
10
1
8
1
3
1
4
1
6
1
7
1
10
1
9
1
8
1
10
1
5
1
7
1
9
1
2
1
4
1
3
1
4
1
1
5
5
1
1
1
6
6
6
1
1
1
7
7
7
1
1
1
1
8
8
8
8
1
1
1
1
9
9
9
9
1
1
1
1
1
10
10
10
10
10
Summary
Fraction chart:
•  Allows fractions to be greater than 1.
•  Gives the big picture of fractions.
•  Provides practice in organizing fractions.
•  Shows the inverse relationship pattern.
•  Makes comparing fractions visible.
•  Makes fractions visualizable.
© Joan A. Cotter, Ph.D.. 2015
Updating Montessori Fractions
to a Linear Model
Joan A. Cotter, Ph.D.
[email protected]"
AMS Annual Conference!
Philadelphia, Pennsylvania!
Saturday, March 14, 2015 (Happy Pi Day)!
8:00 a.m.– 9:30 a.m.!
© Joan A. Cotter, Ph.D.. 2015