focusing on fractions
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Focusing on Fractions
Presented by Kathleen Cotter LawerBased on the work of Joan A. Cotter, Ph.D.
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Joan A. Cotter, Ph.D.
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Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin
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Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin• Montessori Educator
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Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin• Montessori Educator• Math Games
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Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin• Montessori Educator• Math Games• Tutor
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Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin• Montessori Educator• Math Games• Tutor• AL Abacus
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Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin• Montessori Educator• Math Games• Tutor• AL Abacus• Middle-School Teacher
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Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin• Montessori Educator• Math Games• Tutor• AL Abacus• Middle-School Teacher• Ph.D. in Math Education
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How are fractions usually taught?
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Fraction Model: “Fish Tank”
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Fraction Model: “Fish Tank”
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Fraction Model: “Fish Tank”
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Fraction Model: “Words”
This is fourths.
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Fraction Model: “Words”
This is fourths. This is thirds.
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Fraction Model: “Words”
This is fourths. This is thirds.
If this were chocolate…
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Fraction Model: “Words”
This is fourths. This is thirds.
If this were chocolate…
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Fraction Model: “Rounded Corner”
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Fraction Model: “Rounded Corner”
The middle fractions are greater than the fractions at the ends!
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Fraction Models: Circles
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Fraction Models: Circles
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Try to compare 4/5 and 5/6 with this model.© Activities for Learning, Inc. 2012
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Fraction Models: Circles
Are we comparing angles, arcs, or area?© Activities for Learning, Inc. 2012
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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
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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
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So how should we teach fractions?
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Fraction Model: Linear Chart
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Fraction Model: “Color”
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Fraction Model: “Missing Parts”
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Fraction Chart1
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Fraction Chart
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Ask the children to put it together like a puzzle.
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Definition of a Fraction
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Definition of a Fraction
A fraction is part of a set or part of a whole.Textbook definition
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Definition of a Fraction
32What about ?
A fraction is part of a set or part of a whole.Textbook definition
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Definition of a Fraction
An expression that indicates the quotient of two quantities.
American Heritage Dictionary
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Definition of a Fraction
An expression that indicates the quotient of two quantities.
American Heritage Dictionary
A number in the form where b ≠ 0.Arizona Standards
ab
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Fraction Chart1
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Fraction Chart
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Fraction Chart
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Fraction Chart
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Fraction Naming
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Fraction Naming
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Fraction Naming
One divided into 2 equal parts.
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Fraction Naming
One divided into 2 equal parts.
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Fraction Naming
One divided into 2 equal parts.
One divided into 3 equal parts.
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Fraction Naming
One divided into 2 equal parts.
One divided into 3 equal parts.
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Fraction Naming
One divided into 2 equal parts.
One divided into 3 equal parts.
One divided into 4 equal parts.
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Fraction Naming
One divided into 3 equal parts.
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Fraction Naming
One divided into 3 equal parts.
1 One whole
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Fraction Naming
One divided into 3 equal parts.
1 One whole
divided into
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Fraction Naming
One divided into 3 equal parts.
1 One whole
divided into
3 three equal parts
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Fraction Naming
One divided into 3 equal parts.
1 One
divided by
3 three
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Fraction Naming
One divided into 3 equal parts.
1Read as “one-third.”
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Fraction Naming
One divided into 3 equal parts.
1Read as “one-third.”
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Fraction Naming
One divided into 3 equal parts.
1Read as “one-third.”
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In English, we use ordinal numbers to name fractions, with the exception of half.
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How should we practicewhat we learned?
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Games
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GamesMath
Games
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=GamesMath
Games
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=Books
ReadingGamesMath
Games
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=Books
Reading
Games provide interesting repetition needed for automaticity.
GamesMath
Games
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=Books
Reading
Games provide interesting repetition needed for automaticity.
Games also provide an application for new information.
GamesMath
Games
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Unit Fraction War
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Unit Fraction War
Purpose: Practice for naming and comparing unit fractions.
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Unit Fraction War
Purpose: Practice for naming and comparing unit fractions.
To help the children realize a unit fraction decreases as the denominator increases.
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Unit Fraction War
Purpose: Practice for naming and comparing unit fractions.
To help the children realize a unit fraction decreases as the denominator increases.
Goal: To collect all, or most, of the cards by comparing unit fractions.
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Unit Fraction War
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Unit Fraction War
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Unit Fraction War
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Unit Fraction War
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Unit Fraction War
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Unit Fraction War
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Unit Fraction War
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Unit Fraction War
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Unit Fraction War
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Unit Fraction War
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Unit Fraction War
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Unit Fraction War
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More Fraction Naming
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More Fraction Naming1
13
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More Fraction Naming
One-third taken 2 times.
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More Fraction Naming
One-third taken 2 times.
1Two
3s
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More Fraction Naming
One-third taken 2 times.
1Two is
323
s
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More Fraction Naming
One-third taken 2 times.
Read as “two-thirds.”
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1Two is
323
s
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More Fraction Naming1
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More Fraction Naming1 1
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More Fraction Naming
Two divided into 3 equal parts.
1 1
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More Fraction Naming
Two divided into 3 equal parts.
1 1
2 Two
divided by3 three
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More Fraction Naming
Two divided into 3 equal parts.
1 1
2 Two
divided by3 three
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More Fraction Naming
Two divided into 3 equal parts.
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1
2 Two
divided by3 three
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More Fraction Naming
Two divided into 3 equal parts.
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1
1two
3s=
2 Two
divided by3 three
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Fraction Chart
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Fraction Chart
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How many fourths in a whole?
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Fraction Chart
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How many fourths in a whole?
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Fraction Chart
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How many fourths in a whole?
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Fraction Chart
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How many fourths in a whole?
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Fraction Chart
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How many fourths in a whole?
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Fraction Chart
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How many sixths?How many fourths in a whole?
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Purpose: To help the children realize that 5 fifths, 8 eighths, and so forth, make a whole.
Concentrating on One Game
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Purpose: To help the children realize that 5 fifths, 8 eighths, and so forth, make a whole.
Goal: To find the pairs that make a whole.
Concentrating on One Game
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Concentrating on One
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Let’s begin to comparefractions….
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Fraction Chart
Which is more, 3/4 or 4/5?
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Fraction Chart
Which is more, 3/4 or 4/5?
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Fraction Chart
Which is more, 3/4 or 4/5?
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Fraction Chart
Which is more, 7/8 or 8/9?
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Fraction Chart
Which is more, 7/8 or 8/9?
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Fraction Chart
Which is more, 7/8 or 8/9?
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Fraction Chart
An interesting pattern.
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Fraction Stairs
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Fraction Stairs
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Fraction Chart
How many fourths equal a half?
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Fraction Chart
How many fourths equal a half?
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Fraction Chart
How many fourths equal a half? Eighths?
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Fraction Chart
How many fourths equal a half? Eighths?
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Fraction Chart
How many fourths equal a half? Eighths? Sevenths?
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Fraction Chart
How many fourths equal a half? Eighths? Sevenths?
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Fraction Chart
What is 1/4 plus 1/8?
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Fraction Chart
What is 1/4 plus 1/8?
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Fraction Chart
What is 1/4 plus 1/8?
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Fraction Chart
What is 1/4 plus 1/8?
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Fraction Chart
What is 1/4 plus 1/8? [3/8]
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Fraction Chart
If the chance of rain is 3/4, the chance it won’t rain is .
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Fraction Chart
If the chance of rain is 3/4, the chance it won’t rain is .
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Fraction Chart
If the chance of rain is 3/4, the chance it won’t rain is 1/4 .
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Partial Chart
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Partial Chart
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Partial Chart
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Partial Chart
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Fraction War
Purpose: To practice comparing ones, halves, fourths, and eighths in preparation for reading a ruler.
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Fraction War
Purpose: To practice comparing ones, halves, fourths, and eighths in preparation for reading a ruler.
Goal: To capture all the cards.
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Fraction War1
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Fraction War
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Fraction War
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Fraction War
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Fraction War1
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Fraction War
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Fraction War
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Fraction War
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Fraction War1
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Fraction War
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Fraction War
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Fraction War
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Fraction War1
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Advanced Fraction War
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What about fractions bigger than 1?
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Fractions > 1
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Fractions > 1
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Mixed to Improper Fractions
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
© Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s
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2 =
34
114
Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
two 4s
© Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3
© Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3 = 11
© Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3 = 11
© Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3 = 11 134
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3 = 11 134 =
133
© Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3 = 11 134 =
133
four 3s© Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3 = 11 134 =
133
four 3s + 1 © Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3 = 11 134 =
133
four 3s + 1 = 13© Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3 = 11 134 =
133
four 3s + 1 = 13
353
© Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3 = 11 134 =
133
four 3s + 1 = 13
353 =
185
© Activities for Learning, Inc. 2012
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Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed number and as an improper fraction.
342 =
114
two 4s + 3 = 11 134 =
133
four 3s + 1 = 13
353 =
185
three 5s + 3 = 18
© Activities for Learning, Inc. 2012
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Help the student discover the algorithm.
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Improper to Mixed Fractions
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Improper to Mixed Fractions
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Improper to Mixed FractionsWrite each quantity as an improper fraction,
15
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15
15
15
15
15
15
15
15
15
15
13
13
13
13
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Improper to Mixed FractionsWrite each quantity as an improper fraction, then circle the wholes and write again as a mixed number.
15
13
15
15
15
15
15
15
15
15
15
15
13
13
13
13
© Activities for Learning, Inc. 2012
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Improper to Mixed FractionsWrite each quantity as an improper fraction, then circle the wholes and write again as a mixed number.
15
13
15
15
15
15
15
15
15
15
15
15
13
13
13
13
115
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Improper to Mixed Fractions
15
13
15
15
15
15
15
15
15
15
15
15
13
13
13
13
115
© Activities for Learning, Inc. 2012
Write each quantity as an improper fraction, then circle the wholes and write again as a mixed number.
![Page 194: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/194.jpg)
Improper to Mixed Fractions
15
13
15
15
15
15
15
15
15
15
15
15
13
13
13
13
115 = 21
5
© Activities for Learning, Inc. 2012
Write each quantity as an improper fraction, then circle the wholes and write again as a mixed number.
![Page 195: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/195.jpg)
Improper to Mixed Fractions
15
13
15
15
15
15
15
15
15
15
15
15
13
13
13
13
53
115 = 21
5
© Activities for Learning, Inc. 2012
Write each quantity as an improper fraction, then circle the wholes and write again as a mixed number.
![Page 196: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/196.jpg)
Improper to Mixed Fractions
15
13
15
15
15
15
15
15
15
15
15
15
13
13
13
13
53
115 = 21
5
© Activities for Learning, Inc. 2012
Write each quantity as an improper fraction, then circle the wholes and write again as a mixed number.
![Page 197: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/197.jpg)
Improper to Mixed Fractions
15
13
15
15
15
15
15
15
15
15
15
15
13
13
13
13
53 = 12
3
115 = 21
5
© Activities for Learning, Inc. 2012
Write each quantity as an improper fraction, then circle the wholes and write again as a mixed number.
![Page 198: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/198.jpg)
Remember, let the student discover the algorithm.
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Fraction of Geometric Figures
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Fraction of Geometric Figures
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Fraction of Geometric Figures
12Shade
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Fraction of Geometric Figures
12Shade
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Fraction of Geometric Figures
12
23Shade Shade
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Fraction of Geometric Figures
12
23Shade Shade
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Fraction of Geometric Figures
12
23
14Shade Shade Shade
© Activities for Learning, Inc. 2012
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Fraction of Geometric Figures
12
23
14Shade Shade Shade
© Activities for Learning, Inc. 2012
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Making the Whole
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Making the Whole
13
23
© Activities for Learning, Inc. 2012
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Making the WholeDraw the whole.
13
23
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Making the WholeDraw the whole.
13
13
13
13
23
© Activities for Learning, Inc. 2012
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Making the WholeDraw the whole.
13
13
13
13
23
13
23
© Activities for Learning, Inc. 2012
![Page 212: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/212.jpg)
How about a fraction of a number?
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A Fraction of Twelve
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A Fraction of Twelve
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A Fraction of Twelve
What is 1/2 of 12?
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A Fraction of Twelve
What is 1/2 of 12?
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/2 of 12? [6]
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/3 of 12?
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/3 of 12?
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/3 of 12? [4]
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/4 of 12?
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/4 of 12?
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/4 of 12? [3]
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/5 of 12?
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/5 of 12?
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/5 of 12? It’s not even; there is a remainder.
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/6 of 12?
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/6 of 12?
© Activities for Learning, Inc. 2012
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A Fraction of Twelve
What is 1/6 of 12? [2]
© Activities for Learning, Inc. 2012
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A Fraction of 100
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A Fraction of 100100
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A Fraction of 100100
50 50
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A Fraction of 100100
50 50331
3 3313 331
3
© Activities for Learning, Inc. 2012
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A Fraction of 100100
50 50
25 2525 25331
3 3313 331
3
© Activities for Learning, Inc. 2012
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10050 50
25 2525 25
A Fraction of 100
20 2020 20 20
3313 331
3 3313
© Activities for Learning, Inc. 2012
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A Fraction of 100100
50 50
25 2525 25331
3
20 2020 20 20
1212 121
2 1212 121
2 1212 121
2 1212 121
2
3313 331
3
© Activities for Learning, Inc. 2012
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A Fraction of 100100
50 50
25 2525 2520 2020 20 20
10 10 10 10 10 10 10 10 10 10
3313 331
3 3313
1212 121
2 1212 121
2 1212 121
2 1212 121
2
© Activities for Learning, Inc. 2012
![Page 238: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/238.jpg)
Percents
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PercentsPercent means a fraction of 100
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PercentsPercent means a fraction of 100 or per 100.
© Activities for Learning, Inc. 2012
![Page 241: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/241.jpg)
PercentsPercent means a fraction of 100 or per 100.
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![Page 242: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/242.jpg)
PercentsPercent means a fraction of 100 or per 100.
© Activities for Learning, Inc. 2012
![Page 243: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/243.jpg)
PercentsPercent means a fraction of 100 or per 100.
2of 1001
© Activities for Learning, Inc. 2012
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PercentsPercent means a fraction of 100 or per 100.
2of 100 =
1001 50
© Activities for Learning, Inc. 2012
![Page 245: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/245.jpg)
PercentsPercent means a fraction of 100 or per 100.
2of 100 = = 50%
1001 50
© Activities for Learning, Inc. 2012
![Page 246: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/246.jpg)
PercentsPercent means a fraction of 100 or per 100.
© Activities for Learning, Inc. 2012
![Page 247: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/247.jpg)
PercentsPercent means a fraction of 100 or per 100.
© Activities for Learning, Inc. 2012
![Page 248: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/248.jpg)
PercentsPercent means a fraction of 100 or per 100.
4of 100 1
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PercentsPercent means a fraction of 100 or per 100.
4of 100 =
1001 25
© Activities for Learning, Inc. 2012
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PercentsPercent means a fraction of 100 or per 100.
4of 100 = = 25%
1001 25
© Activities for Learning, Inc. 2012
![Page 251: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/251.jpg)
PercentsPercent means a fraction of 100 or per 100.
© Activities for Learning, Inc. 2012
![Page 252: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/252.jpg)
PercentsPercent means a fraction of 100 or per 100.
© Activities for Learning, Inc. 2012
![Page 253: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/253.jpg)
PercentsPercent means a fraction of 100 or per 100.
10of 1001
© Activities for Learning, Inc. 2012
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PercentsPercent means a fraction of 100 or per 100.
10of 100 =
1001 10
© Activities for Learning, Inc. 2012
![Page 255: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/255.jpg)
PercentsPercent means a fraction of 100 or per 100.
10of 100 = = 10%
1001 10
© Activities for Learning, Inc. 2012
![Page 256: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/256.jpg)
Now let’s simplify fractions.
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Simplifying Fractions
12
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Simplifying Fractions
36
= 12
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Simplifying Fractions
48
= 12
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Simplifying Fractions
© Activities for Learning, Inc. 2012
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Simplifying Fractions
912
© Activities for Learning, Inc. 2012
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Simplifying Fractions
912 3
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Simplifying Fractions
912 3
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Simplifying Fractions
912
= 343
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Simplifying Fractions
© Activities for Learning, Inc. 2012
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Simplifying Fractions
© Activities for Learning, Inc. 2012
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Simplifying Fractions
© Activities for Learning, Inc. 2012
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Simplifying Fractions
© Activities for Learning, Inc. 2012
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Simplifying Fractions
2128
© Activities for Learning, Inc. 2012
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Simplifying Fractions
2128
© Activities for Learning, Inc. 2012
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Simplifying Fractions
2128
© Activities for Learning, Inc. 2012
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Simplifying Fractions
4572
© Activities for Learning, Inc. 2012
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Simplifying Fractions
4572
© Activities for Learning, Inc. 2012
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Simplifying Fractions
4572
© Activities for Learning, Inc. 2012
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Simplifying Fractions
12 16
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Simplifying Fractions
12 16
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Simplifying Fractions
12 16
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Simplifying Fractions
12 16
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Simplifying Fractions
12 16
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Simplifying Fractions
12 16
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So how does multiplying fractions work?
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Fraction Chart
What is 1/2 of 1/2?
1
14
14
14
14
12
12
110
110
110
110
110
110
110
110
110
110
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
19
19
13
13
13
15
15
15
15
15
16
16
16
16
16
16
17
17
17
17
17
17
17
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Fraction Chart
What is 1/2 of 1/2?
1
14
14
14
14
12
12
110
110
110
110
110
110
110
110
110
110
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
19
19
13
13
13
15
15
15
15
15
16
16
16
16
16
16
17
17
17
17
17
17
17
© Activities for Learning, Inc. 2012
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Fraction Chart
What is 1/2 of 1/2?
1
14
14
14
14
12
12
110
110
110
110
110
110
110
110
110
110
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
19
19
13
13
13
15
15
15
15
15
16
16
16
16
16
16
17
17
17
17
17
17
17
© Activities for Learning, Inc. 2012
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Fraction Chart
What is 1/3 of 1/2?
1
14
14
14
14
12
12
110
110
110
110
110
110
110
110
110
110
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
19
19
13
13
13
15
15
15
15
15
16
16
16
16
16
16
17
17
17
17
17
17
17
© Activities for Learning, Inc. 2012
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Fraction Chart
What is 1/3 of 1/2?
1
14
14
14
14
12
12
110
110
110
110
110
110
110
110
110
110
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
19
19
13
13
13
15
15
15
15
15
16
16
16
16
16
16
17
17
17
17
17
17
17
© Activities for Learning, Inc. 2012
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Fraction Chart
What is 1/3 of 1/2?
1
14
14
14
14
12
12
110
110
110
110
110
110
110
110
110
110
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
19
19
13
13
13
15
15
15
15
15
16
16
16
16
16
16
17
17
17
17
17
17
17
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Fraction Chart
What is 1/3 of 1/2? That’s multiplying fractions!
1
14
14
14
14
12
12
110
110
110
110
110
110
110
110
110
110
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
19
19
13
13
13
15
15
15
15
15
16
16
16
16
16
16
17
17
17
17
17
17
17
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Multiplying Fractions
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Multiplying Fractions
Multiplying is not exclusively repeated addition.
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Multiplying Fractions
Multiplying is not exclusively repeated addition.
Area is another facet of multiplication.
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Multiplying Fractions
12
x =12
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Multiplying Fractions
One half of one half
12
x =12
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Multiplying Fractions
One half of one half
12
x =12
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Multiplying Fractions
12
x =12
One half of one half
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Multiplying Fractions
12
x =12
One half of one half
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Multiplying Fractions
12
x =12
One half of one half
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Multiplying Fractions
12
x =12
One half of one half
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Multiplying Fractions
One half of one half
12
x =12
14
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Multiplying Fractions
One half of one half
12
x =12
14
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Multiplying Fractions
23
x = 34
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Multiplying Fractions
23
x = 34
Three fourths of two thirds
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Multiplying Fractions
23
x = 34
Three fourths of two thirds
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Multiplying Fractions
23
x = 34
Three fourths of two thirds
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Multiplying Fractions
23
x = 34
Three fourths of two thirds
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Multiplying Fractions
23
x = 34
Three fourths of two thirds
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Multiplying Fractions
23
34
Three fourths of two thirds
x =
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Multiplying Fractions
Three fourths of two thirds
23
x = 34
612
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Multiplying Fractions
Three fourths of two thirds
23
x = 34
612
© Activities for Learning, Inc. 2012
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Multiplying Fractions
Three fourths of two thirds
23
x = 34
612
© Activities for Learning, Inc. 2012
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Multiplying Fractions
Three fourths of two thirds
23
x = =34
612
12
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Multiplying Fractions
23
x = 34
612
The total number of rectangles is
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Multiplying Fractions
x =
The total number of rectangles is 3 x 4.
23
34
612
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Multiplying Fractions
23
x = 34
612
The total number of rectangles is 3 x 4.
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Multiplying Fractions
23
x = 34
612
The total number of rectangles is 3 x 4.
The number of double crosshatched rectangles is
© Activities for Learning, Inc. 2012
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Multiplying Fractions
23
x = 34
612
The total number of rectangles is 3 x 4.
The number of double crosshatched rectangles is 2 x 3.
© Activities for Learning, Inc. 2012
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Multiplying Fractions
23
x = 34
612
The total number of rectangles is 3 x 4.
The number of double crosshatched rectangles is 2 x 3.
© Activities for Learning, Inc. 2012
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Multiplying Fractions
23
x = 34
612
The total number of rectangles is 3 x 4.
The number of double crosshatched rectangles is 2 x 3.
© Activities for Learning, Inc. 2012
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What about dividing fractions?
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What is Division?
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Process of separating into equal groups
What is Division?
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Process of separating into equal groups
either number of groups, or
What is Division?
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Process of separating into equal groups
either number of groups, or
size of the groups
What is Division?
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6 ÷ 2 = __
What is Division?
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How many 2s in 6?
6 ÷ 2 = __
What is Division?
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How many 2s in 6?
6 ÷ 2 = __
What is Division?
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How many 2s in 6?
6 ÷ 2 = __
Number of groups of 2s
What is Division?
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How many 2s in 6?
6 ÷ 2 = __
Number of groups of 2s
What is Division?
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How many 2s in 6?
6 ÷ 2 = __
Number of groups of 2s; 3 groups
What is Division?
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Dividing Fractions
112
12
14
14
14
14
13
13
13
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Dividing Fractions÷ =
121
112
12
14
14
14
14
13
13
13
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Dividing Fractions÷ =
121
112
12
14
14
14
14
13
13
13
How many 1/2s in 1?
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Dividing Fractions÷ =
121
112
12
14
14
14
14
13
13
13
How many 1/2s in 1?
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121
112
12
14
14
14
14
13
13
13
How many 1/2s in 1?
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121
112
12
14
14
14
14
13
13
13
How many 1/2s in 1?
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
How many 1/2s in 1?
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
131 ÷ =
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
How many 1/3s in 1?
131 ÷ =
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
How many 1/3s in 1?
131 ÷ =
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
How many 1/3s in 1?
131 ÷ =
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
How many 1/3s in 1?
131 ÷ =
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
How many 1/3s in 1?
131 ÷ =
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
How many 1/3s in 1?
131 ÷ = 3
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
131 ÷ = 3
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Dividing Fractions÷ =
121 2
112
12
14
14
14
14
13
13
13
131 ÷ = 3
231 ÷ =
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
How many 2/3s in 1?
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
How many 2/3s in 1?
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
How many 2/3s in 1?
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
![Page 350: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/350.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
How many 2/3s in 1?
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
![Page 351: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/351.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
How many 2/3s in 1?
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
![Page 352: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/352.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231
How many 2/3s in 1?
112
12
14
14
14
14
13
13
13
12÷ = 1
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
131 ÷ = 3
231
How many 2/3s in 1?
112
12
14
14
14
14
13
13
13
12÷ = 1 =
32
© Activities for Learning, Inc. 2012
![Page 354: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/354.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231
112
12
14
14
14
14
13
13
13
÷ = 32
© Activities for Learning, Inc. 2012
![Page 355: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/355.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2 12
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
![Page 357: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/357.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2 12
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
![Page 358: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/358.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
![Page 359: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/359.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
![Page 360: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/360.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
![Page 361: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/361.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
© Activities for Learning, Inc. 2012
![Page 362: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/362.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
How many 4/3s in 1?
© Activities for Learning, Inc. 2012
![Page 363: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/363.jpg)
13
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
How many 4/3s in 1?
© Activities for Learning, Inc. 2012
![Page 364: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/364.jpg)
13
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
How many 4/3s in 1?
© Activities for Learning, Inc. 2012
![Page 365: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/365.jpg)
13
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
How many 4/3s in 1? [Less than 1]
© Activities for Learning, Inc. 2012
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13
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
How many 4/3s in 1?
© Activities for Learning, Inc. 2012
![Page 367: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/367.jpg)
13
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
12
12
How many 4/3s in 1?
© Activities for Learning, Inc. 2012
![Page 368: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/368.jpg)
13
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
How many 4/3s in 1?
© Activities for Learning, Inc. 2012
![Page 369: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/369.jpg)
13
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
14
14
14
14
How many 4/3s in 1?
© Activities for Learning, Inc. 2012
![Page 370: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/370.jpg)
13
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
14
14
14
How many 4/3s in 1?
© Activities for Learning, Inc. 2012
![Page 371: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/371.jpg)
13
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
14
14
14
How many 4/3s in 1?
34
© Activities for Learning, Inc. 2012
![Page 372: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/372.jpg)
13
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
112
12
14
14
14
14
13
13
13
14
14
14
34
© Activities for Learning, Inc. 2012
![Page 373: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/373.jpg)
Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
34
Each green pair is a reciprocal,
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
34
Each green pair is a reciprocal, sometimes called a multiplicative inverse.
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
34
Each green pair is a reciprocal, sometimes called a multiplicative inverse.
When multiplied together, they equal 1.
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
34
Sometimes textbooks put a 1 under a whole number to make it look like a fraction,
© Activities for Learning, Inc. 2012
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Dividing Fractions÷ =
121 2
131 ÷ = 3
231 ÷ =
32
÷ =1 2
÷ =1 3
12
13
÷ =1 43
34
Sometimes textbooks put a 1 under a whole number to make it look like a fraction, but it is not necessary.
1
1
1
1
© Activities for Learning, Inc. 2012
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Dividing Fractions
÷ =34
112
12
14
14
14
14
13
13
13
23
© Activities for Learning, Inc. 2012
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Dividing Fractions
÷ =34
112
12
14
14
14
14
13
13
13
Is the answer more or less than 1?
23
© Activities for Learning, Inc. 2012
![Page 380: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/380.jpg)
Dividing Fractions
÷ =34
112
12
14
14
14
14
13
13
13
Is the answer more or less than 1?
23
© Activities for Learning, Inc. 2012
![Page 381: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/381.jpg)
Dividing Fractions
÷ =34
112
12
14
14
14
14
13
13
13
Is the answer more or less than 1?
23
© Activities for Learning, Inc. 2012
![Page 382: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/382.jpg)
Dividing Fractions
÷ =34
112
12
14
14
14
14
13
13
13
23
89
© Activities for Learning, Inc. 2012
![Page 383: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/383.jpg)
1
14
14
14
14
Fraction Chart
12
12
110
110
110
110
110
110
110
110
110
110
18
18
18
18
18
18
18
18
19
19
19
19
19
19
19
19
19
13
13
13
15
15
15
15
15
16
16
16
16
16
16
17
17
17
17
17
17
17
© Activities for Learning, Inc. 2012
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Fraction Chart
© Activities for Learning, Inc. 2012
Allow the student to explore the whole picture
![Page 385: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/385.jpg)
Fraction Chart
© Activities for Learning, Inc. 2012
Allow the student to explore the whole picture
and relationships within the whole
![Page 386: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/386.jpg)
Fraction Chart
© Activities for Learning, Inc. 2012
Allow the student to explore the whole picture
and relationships within the whole using the
linear perspective.
![Page 387: Focusing on Fractions](https://reader038.vdocuments.net/reader038/viewer/2022102710/545688f2af7959bc078b52f2/html5/thumbnails/387.jpg)
Focusing on Fractions
Presented by Kathleen Cotter LawerBased on the work of Joan A. Cotter, Ph.D.
35
25
© Activities for Learning, Inc. 2012