Slides for Bochum 2007. Evolution of Abundances D Be mass fraction 0.
Fraction Mass HOPE April 2013
338
Fractions MassHOPE-TEACH Worcester, MA Saturday, April 27, 2013 2:45pm - 3:45pm Joan A. Cotter, Ph.D. [email protected]
-
Upload
rightstartmath -
Category
Education
-
view
15 -
download
2
description
Transcript of Fraction Mass HOPE April 2013
Fractions
MassHOPE-TEACHWorcester, MA
Saturday, April 27, 20132:45pm - 3:45pm
Joan A. Cotter, [email protected]
Why Learn Fractions
• Sharing pizza
Why Learn Fractions
• Sharing pizza
• Cooking and baking
Why Learn Fractions
• Sharing pizza
• Cooking and baking
• Reading rulers
Why Learn Fractions
• Sharing pizza
• Cooking and baking
• Reading rulers
• Easing into decimals
• Learning algebra
Fractions in the Comics
Fractions in the Comics
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
Fraction Stairs
1
Fraction Stairs
1
12
Fraction Stairs
1
12
13
Fraction Stairs
1
12
13
14
Fraction Stairs
1
12
13
14
15
Fraction Stairs
1
12
13
14
15
16
Fraction Stairs
1
12
13
14
15
1716
Fraction Stairs
1
12
13
14
15
17
18
16
Fraction Stairs
1
12
13
14
15
17
18
16
19
Fraction Stairs
1
12
13
14
15
17
18
110
16
19
Fraction Stairs
Fraction Stairs
Fraction Stairs
A hyperbola.
Fraction Names
• In English, except for half, we use ordinal numbers to name fractions.
Sometimes called quarter.
Fourths
Sometimes called quarter.
• A quarter of a hour (15 min.)
Fourths
Sometimes called quarter.
• A quarter of a hour (15 min.)
• A quarter of a dollar (25¢)
Fourths
Sometimes called quarter.
• A quarter of a hour (15 min.)
• A quarter of a dollar (25¢)
• A quarter of a gallon (quart)
Fourths
Sometimes called quarter.
• A quarter of a hour (15 min.)
• A quarter of a dollar (25¢)
• A quarter of a gallon (quart)
• A Quarter Pounder (4 oz.)
Fourths
Writing Fractions
1 one
Writing Fractions
1 onedivided by
Writing Fractions
13
onedivided bythree
Writing Fractions
13
onedivided bythree
113
13
13
Writing Fractions
13
onedivided bythree
Avoid saying “over” as in 1 over 3.
113
13
13
Unit Fraction War
Objective: To help the children realize a unit fraction decreases as the denominator increases.
Unit Fraction War
Object of the game: To collect all, or most, of the cards with the greater unit fraction.
Objective: To help the children realize a unit fraction decreases as the denominator increases.
Unit Fraction War
15
14
Unit Fraction War
15
14
Unit Fraction War
Unit Fraction War
Unit Fraction War
118
Unit Fraction War
118
Unit Fraction War
Unit Fraction War
16
16
Unit Fraction War
16
16
Unit Fraction War
16
16
13
14
Unit Fraction War
Fraction Chart
How many fourths in a whole?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths in a whole?
12
13
15
16
17
18
19
110
13
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
14
14
12
1
13
12
13
15
16
17
18
19
110
13
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths in a whole?
14
14
12
1
13
Fraction Chart
How many fourths in a whole?
13
15
16
17
18
19
110
13
15
16
17
18
19
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
14
14
12
1
13
14
12
Fraction Chart
How many fourths in a whole?
13
15
16
17
18
19
110
15
16
17
18
19
15
16
17
18
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
14
14
12
1
13
12
13
14
14
Fraction Chart
How many sixths in a whole?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many eighths in a whole?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Concentrating on One Game
Objective: To help the children realize that 5 fifths, 8 eighths, and so forth, make a whole.
Concentrating on One Game
Object of the game: To find the pairs that make a whole.
Objective: To help the children realize that 5 fifths, 8 eighths, and so forth, make a whole.
Concentrating on One
Concentrating on One
53
Concentrating on One
53
Concentrating on One
53
Concentrating on One
53
25
Concentrating on One
Concentrating on One
38
Concentrating on One
38
Concentrating on One
38
Concentrating on One
38
78
Concentrating on One
Concentrating on One58
Concentrating on One58
Concentrating on One58
Concentrating on One
38
58
Concentrating on One
Concentrating on One
Fraction Chart
Which is more, 3/4 or 4/5?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
Which is more, 3/4 or 4/5?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
Which is more, 3/4 or 4/5?
Fraction Chart
Which is more, 7/8 or 8/9?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
Which is more, 7/8 or 8/9?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
An interesting pattern.
Partial Chart
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Partial Chart
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Partial Chart
Partial Chart
Fraction War
Fraction War
Objective: To practice comparing ones, halves, fourths, and eighths in preparation for reading a ruler.
Fraction War
Object of the game: To capture all the cards.
Objective: To practice comparing ones, halves, fourths, and eighths in preparation for reading a ruler.
Fraction War1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
14
18
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
14
18
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
14
18
Fraction War1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
34
58
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
34
58
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
34
58
Fraction War1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
34
34
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
34
34
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War
34
34
38
14
1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
Fraction War1
14
14
14
14
12
12
18
18
18
18
18
18
18
18
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half?
112
12
14
15
16
17
18
19
110
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half? Eighths?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half? Eighths?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half? Eighths? Sevenths?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
How many fourths equal a half? Eighths? Sevenths?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/4 plus 1/8?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/4 plus 1/8?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/4 plus 1/8?
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/4 plus 1/8? [3/8]
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
If the chance of rain is 3/4, the chance it won’t rain is 1/4.
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
If the chance of rain is 3/4, the chance it won’t rain is 1/4.
Faulty Fractions
Faulty Fractions“Fish Tank” model
Faulty Fractions
2
“Fish Tank” model
Faulty Fractions
25
“Fish Tank” model
Faulty Fractions
= part
whole
CRA model
Faulty Fractions
= part
whole
“Goal: To develop the spatial organization, visually and kinesthetically, to read and write fractions correctly.
CRA model
Faulty Fractions
= part
whole
“Goal: To develop the spatial organization, visually and kinesthetically, to read and write fractions correctly.
“Materials: Red squares and larger black squares are displayed to help with sequencing and number placement.”
CRA model
Faulty Fractions
This is fourths.
“Words” model
Faulty Fractions
This is fourths. This is thirds.
“Words” model
Faulty Fractions
13
13
13
14
14
14
14
“Rounded corners”
Faulty Fractions
The middle fractions are greater than the fractions at the ends!
13
13
13
14
14
14
14
“Rounded corners”
Faulty Fractions
12
112
13
13
13
6 6 6 6 6 617
17
17
17
17
17
17
19
1 10
1 10
1 10
1 10
1 10
1 10
1 10
1 10
1 10
1 10
18
18
18
18
18
18
18
18
14
14
14
19
19
19
19
19
19
19
19
15
15
15
15
15
1 1 1 1 1 1
14
“Color” model
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
13
13
13
15
15
15
15
15
16
16
16
16
16
16
Faulty Fractions
112
112
112
112
112
112
112
112
112
112
112
112
Missing 7ths & 9ths
112
Faulty FractionsMissing 7ths & 9ths
1
12
13
14
15
18
110
16
Faulty Fractions
Are we comparing angles, arcs, or area?
Circles
Faulty Fractions
61
61
61
61
61
61
51
41
21 3
1
51
51
51
51
41
41
41
31
31
21
Try to compare 4/5 and 5/6 with this model.
Circles
Faulty FractionsExperts 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
Circles
Faulty FractionsExperts 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
Circles
Definition of a Fraction
What is the definition of a fraction?
Definition of a Fraction
What is the definition of a fraction?
A part of a set or part of a whole, a small part.
Definition of a Fraction
What is the definition of a fraction?
A part of a set or part of a whole, a small part.
This is the everyday meaning of fraction.
Definition of a Fraction
32What about ?
What is the definition of a fraction?
A part of a set or part of a whole, a small part.
This is the everyday meaning of fraction.
Definition of a Fraction
An expression that indicates thequotient of two quantities.
American Heritage Dictionary:
Definition of a Fraction
An expression that indicates thequotient of two quantities.
This is the mathematical meaning of fraction.
American Heritage Dictionary:
Definition of a Fraction
This is the mathematical meaning of fraction.
32
An expression that indicates thequotient of two quantities.
American Heritage Dictionary:
Fractions > 11
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fractions > 11
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
18
Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper FractionsEach row of connected rectangles represents 1.
Write each quantity as a mixed numberand as an improper fraction.
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper FractionsEach row of connected rectangles represents 1.
two 4s
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper Fractions
24
two 4s
Each row of connected rectangles represents 1.
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper Fractions
24
Each row of connected rectangles represents 1.
two 4s + 3
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper Fractions
43
Each row of connected rectangles represents 1.
two 4s + 32
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper Fractions
43
Each row of connected rectangles represents 1.
two 4s + 3 = 112
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper Fractions
43 11
Each row of connected rectangles represents 1.
two 4s + 3 = 112
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper Fractions
4113
Each row of connected rectangles represents 1.
two 4s + 3 = 112
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper Fractions
4113
Each row of connected rectangles represents 1.
two 4s + 3 = 11
four 3s + 2 = 14
2
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper FractionsEach row of connected rectangles represents 1.
4113
two 4s + 3 = 11
four 3s + 2 = 14
2
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper FractionsEach row of connected rectangles represents 1.
4113
four 3s + 2 = 14
two 4s + 3 = 112
Write each quantity as a mixed numberand as an improper fraction.
Mixed to Improper FractionsEach row of connected rectangles represents 1.
24113
four 3s + 2 = 14
two 4s + 3 = 11
four 5s + 3 = 23
Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.
Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.
Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.
Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.
Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.
Improper to Mixed FractionsCircle the wholes and write each quantity asan improper fraction and as a mixed number.
Fraction of Geometric Figures
12Shade
Fraction of Geometric Figures
12Shade
Fraction of Geometric Figures
12
23Shade Shade
Fraction of Geometric Figures
12
23Shade Shade
Fraction of Geometric Figures
12
23
14Shade Shade Shade
Fraction of Geometric Figures
12
23
14Shade Shade Shade
Making the WholeDraw the whole.
13
Making the WholeDraw the whole.
13
13
13
13
Making the WholeDraw the whole.
13
13
13
13
23
Making the WholeDraw the whole.
13
13
13
13
23
23
13
112
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/2 of 1/2?
12
112
13
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/2 of 1/2?
12
14
112
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
What is 1/3 of 1/2?
12
Fraction Chart
What is 1/3 of 1/2?
112
13
14
15
16
17
18
19
110
13
13
14
15
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
12
16
Simplifying Fractions
12
Simplifying Fractions
36 = 1
2
Simplifying Fractions
48 = 1
2
Simplifying Fractions
Simplifying Fractions
912
Simplifying Fractions
912 3
Simplifying Fractions
912 3
Simplifying Fractions
912 = 3
43
Simplifying Fractions
Simplifying Fractions
Simplifying Fractions
Simplifying Fractions
Simplifying Fractions
2128
Simplifying Fractions
2128
Simplifying Fractions
2128
Simplifying Fractions
4572
Simplifying Fractions
4572
Simplifying Fractions
4572
Simplifying Fractions
1216
Simplifying Fractions
1216
Simplifying Fractions
1216
Simplifying Fractions
1216
Simplifying Fractions
1216
Simplifying Fractions
1216
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsTwos
2 4 6 8 10
12 14 16 18 20
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsTwos
2 4 6 8 10
12 14 16 18 20
The ones repeat in the second row.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsFours
4 8 12 16 20
24 28 32 36 40
The ones repeat in the second row.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSixes 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., 2013
Multiples PatternsSixes 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., 2013
Multiples PatternsSixes 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., 2013
Multiples PatternsSixes 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., 2013
Multiples PatternsSixes 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., 2013
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
6 x 4
6 x 4 is the fourth number (multiple).
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80 8 x 7
8 x 7 is the seventh number (multiple).
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsNines
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., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
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., 2013
Multiples PatternsThrees
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., 2013
Multiples PatternsThrees
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., 2013
Multiples PatternsThrees
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., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the “opposites.”
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.
© Joan A. Cotter, Ph.D., 2013
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.
Subtracting Fractions
4684–2372
Preliminary understanding
Subtracting Fractions
4684–2372
2000
Preliminary understanding
Subtracting Fractions
4684–2372
2000300
Preliminary understanding
Subtracting Fractions
4684–2372
200030010
Preliminary understanding
Subtracting Fractions
4684–2372
200030010
2
Preliminary understanding
Subtracting Fractions
4684–2372
200030010
22312
Preliminary understanding
Subtracting Fractions
4684–2372
4684–2879
200030010
22312
Preliminary understanding
Subtracting Fractions
4684–2372
4684–2879
200030010
22312
2000
Preliminary understanding
Subtracting Fractions
4684–2372
4684–2879
200030010
22312
2000–200
Preliminary understanding
Subtracting Fractions
4684–2372
4684–2879
200030010
22312
2000–200
10
Preliminary understanding
Subtracting Fractions
4684–2372
4684–2879
200030010
22312
2000–200
10 –5
Preliminary understanding
Subtracting Fractions
4684–2372
4684–2879
200030010
22312
2000–200
10 –51805
Preliminary understanding
Subtracting Fractions
35
4–
Subtracting Fractions
35
4–
325
Subtracting Fractions
35
4–
325
57
5– 2
17
Subtracting Fractions
35
4–
325
3
57
5– 2
17
Subtracting Fractions
35
4–
325
3– 4
7
57
5– 2
17
Subtracting Fractions
35
4–
325
3– 4
7
2 37
57
5– 2
17
Multiplying Fractions
Multiplying Fractions
• Multiplication is more than repeated addition.
Multiplying Fractions
4 x 4 = 4 + 4 + 4 + 4
• Multiplication is more than repeated addition.
Multiplying Fractions
4 x 4 = 4 + 4 + 4 + 4
• Repeated addition doesn’t work well with fractions.
• Multiplication is more than repeated addition.
Multiplying Fractions
• Repeated addition doesn’t work well with fractions.
12 x = + ?1
212
4 x 4 = 4 + 4 + 4 + 4
• Multiplication is more than repeated addition.
Multiplying Fractions
Area is a better model.
4 x 4 =
Multiplying Fractions
12 x =1
2
The square represents 1.
Multiplying Fractions
12 x =1
2
Multiplying Fractions
12 x =1
214
The solution is the double-crosshatched area.
Multiplying Fractions
23 x =3
4
Multiplying Fractions
23 x =3
4
Multiplying Fractions
23 x =3
4
Multiplying Fractions
23 x =3
4 6 12
Multiplying Fractions
x = 34
12= 2
3 6 12
Multiplying Fractions
23 x =3
4
The total number of rectangles is 3 x 4.
Multiplying Fractions
23
34
The number of double-crosshatched rectangles is 2 x 3.The total number of rectangles is 3 x 4.
x =
Multiplying Fractions
23
34
This is why we multiply fractions by multiplying numerators and denominators.
x =
Dividing Fractions
Dividing Fractions÷ =1
21
Dividing Fractions÷ =1
21
1 ÷ 1/2 means how many 1/2s in 1.
Dividing Fractions÷ =1
21
112
12
14
14
14
14
13
13
13
1 ÷ 1/2 means how many 1/2s in 1.
Dividing Fractions÷ =1
21
112
12
14
14
14
14
13
13
13
1 ÷ 1/2 means how many 1/2s in 1.
Dividing Fractions÷ =1
21
112
12
14
14
14
14
13
13
13
1 ÷ 1/2 means how many 1/2s in 1.
Dividing Fractions÷ =1
21 2
112
12
14
14
14
14
13
13
13
1 ÷ 1/2 means how many 1/2s in 1.
Dividing Fractions÷ =1
21 2
÷ =131
112
12
14
14
14
14
13
13
13
1 ÷ 1/3 means how many 1/3s in 1.
Dividing Fractions÷ =1
21 2
÷ =131
112
12
14
14
14
14
13
13
13
1 ÷ 1/3 means how many 1/3s in 1.
Dividing Fractions÷ =1
21 2
÷ =131
112
12
14
14
14
14
13
13
13
1 ÷ 1/3 means how many 1/3s in 1.
Dividing Fractions÷ =1
21 2
÷ =131
112
12
14
14
14
14
13
13
13
1 ÷ 1/3 means how many 1/3s in 1.
Dividing Fractions÷ =1
21 2
÷ =131 3
112
12
14
14
14
14
13
13
13
1 ÷ 1/3 means how many 1/3s in 1.
Dividing Fractions÷ =1
21 2
÷ =131 3
÷ =231
112
12
14
14
14
14
13
13
13
Dividing Fractions÷ =1
21 2
÷ =131 3
÷ =231
112
12
14
14
14
14
13
13
13
1 ÷ 2/3 means how many 2/3s in 1.
Dividing Fractions÷ =1
21
÷ =131
÷ =231
112
12
14
14
14
14
23
2
3
1 ÷ 2/3 means how many 2/3s in 1.
Dividing Fractions÷ =1
21 2
÷ =131 3
÷ =231
112
12
14
14
14
14
23
23
1 ÷ 2/3 means how many 2/3s in 1.
Dividing Fractions÷ =1
21
÷ =131
÷ =231
112
12
14
14
14
14
23
23
32
2
3
1 ÷ 2/3 means how many 2/3s in 1.
Dividing Fractions÷ =1
21
÷ =131
÷ =231
112
12
14
14
14
14
23
23
32
2
3
1 ÷ 2/3 also must be half of 1 ÷ 1/3.
Dividing Fractions÷ =1
21
÷ =131
÷ =1 3
÷ =231 3
2
112
12
14
14
14
14
13
13
13
2
3
Dividing Fractions÷ =1
21
÷ =131
÷ =1 3
÷ =231 3
2
112
12
14
14
14
14
13
13
13
2
3
1 ÷ 3 is simply the definition of a fraction.
112
12
14
14
14
14
13
13
13
Dividing Fractions÷ =1
21
÷ =131
13÷ =1 3
÷ =231 3
2
2
3
1 ÷ 3 is simply the definition of a fraction.
Dividing Fractions÷ =1
21
÷ =131
÷ =1 3
÷ =1 4
÷ =231
112
12
14
14
14
14
13
13
13
32
2
3
13
Dividing Fractions÷ =1
21
÷ =131 1
4
÷ =1 3
÷ =1 4
÷ =231
112
12
14
14
14
14
13
13
13
32
2
3
13
Dividing Fractions1314
÷ =1 3
÷ =1 4
÷ =1 43
121
131
÷ =231
÷ =
÷ =
112
12
14
14
14
14
13
13
13
32
2
3
Dividing Fractions÷ =1 3
÷ =1 4
÷ =1 43
121
131
÷ =231
÷ =
÷ =
112
12
14
14
14
14
13
13
13
13
1314
32
2
3
Dividing Fractions÷ =1 3
÷ =1 4
÷ =1 43
121
131
÷ =231
÷ =
÷ =
112
12
14
14
14
14
13
13
13
13
1314
32
2
3
Only 3/4 of the 4/3 fits into the 1.
Dividing Fractions÷ =1 3
÷ =1 4
÷ =1 43
121
131
÷ =231
÷ =
÷ =
112
12
14
14
14
14
13
13
13
13
1314
32
2
3
34
Only 3/4 of the 4/3 fits into the 1.
Dividing Fractions÷ =1 3
÷ =1 4
÷ =1 43
121
131
÷ =231
÷ =
÷ =
1314
32
2
3
34
Dividing Fractions÷ =1
21 2
÷ =131 3
1314
÷ =1 3
÷ =1 4
34÷ =1 4
3÷ =231 3
2
Dividing Fractions÷ =1
21 2
÷ =131 3
1314
÷ =1 3
÷ =1 4
34÷ =1 4
3÷ =231 3
2
The colored pairs are reciprocals of each other. A reciprocal may be called a multiplicative inverse.
Dividing Fractions÷ =1
21 2
÷ =131 3
1314
÷ =1 3
÷ =1 4
34÷ =1 4
3÷ =231 3
2
The colored pairs are reciprocals of each other. A reciprocal may be called a multiplicative inverse. When multiplied together, they equal 1.
Dividing Fractions÷ =1
21 2
÷ =131 3
1314
÷ =1 3
÷ =1 4
34÷ =1 4
3÷ =231 3
2
The colored pairs are reciprocals of each other. A reciprocal may be called a multiplicative inverse. When multiplied together, they equal 1. In the equation 6 ÷ 2 = 3, 6 = 2 x 3.
Dividing Fractions÷ =1
21 2
÷ =131 3
1314
÷ =1 31
÷ =1 41
34÷ =1 4
3÷ =231 3
2
Dividing Fractions
Sometimes textbooks put a 1 under a whole number to make it look like a fraction, but it is not necessary.
÷ =121 2
÷ =131 3
1314
÷ =1 31
÷ =1 41
34÷ =1 4
3÷ =231 3
2
Dividing Fractions
÷ = __235To find
Dividing Fractions
÷ = __235To find
First think about finding 1 ÷ 2/3.
Dividing Fractions
÷ = __235
÷ =231First find
To find
Dividing Fractions
÷ = __235
÷ =231 3
2First find
To find
Dividing Fractions
÷ = 235
÷ = __235
÷ =231 3
2First find
To find
Then
Dividing Fractions
÷ = __235
÷ =231 3
2First find
To find
Then ÷ = 235 5 (1 )x ÷
23
Dividing Fractions
= x =
325
÷ = __235
÷ =231 3
2First find
To find
Then ÷ = 235 5 (1 )x ÷
23
Dividing Fractions
= x =
32
1525
÷ = __235
÷ =231 3
2First find
To find
Then ÷ = 235 5 (1 )x ÷
23
Dividing Fractions
= x = =
32
125 7
÷ = __235
÷ =231 3
2First find
To find
Then ÷ = 235 5 (1 )x ÷
23152
Dividing Fractions
= x = =
32
125 7
÷ = __235
÷ =231 3
2First find
To find
Then ÷ = 235 5 (1 )x ÷
23152
Does the answer make sense? About how many 2/3s are in 5?
23
Dividing Fractions
÷ = __Find 34
23
Dividing Fractions
÷ = __Find 34
(Is the answer more or less than 1?)
1
14
14
14
14
12
12
13
13
13
23
Dividing Fractions
÷ = __Find 34
(Is the answer more or less than 1?)
1
14
14
14
14
12
12
13
13
13
23
Dividing Fractions
÷ = __Find 34
(Is the answer more or less than 1?)
1
14
14
14
14
12
12
13
13
13
23
Dividing Fractions
÷ = __Find 34
(The answer must be less than 1.)
Dividing Fractions
To find 23 ÷ = __3
4
÷ =341First find
Dividing Fractions
To find 23 ÷ = __3
4
÷ =341 4
3First find
Dividing Fractions
÷ =341 4
3First find
To find
Then
÷ = __34
23
÷ =
34
23
Dividing Fractions
÷ =341 4
3First find
To find
Then
÷ = __34
23
34
23
34
23÷ = x (1 ÷ )
Dividing Fractions
÷ =341 4
3First find
To find ÷ = __34
23
Then 34
23
34
23÷ = x (1 ÷ )
= x =
43
23
= x =
Dividing Fractions
÷ =341 4
3First find
To find ÷ = __34
23
Then 34
23
34
23÷ = x (1 ÷ )
89
43
23
= x =
Dividing Fractions
÷ =341 4
3First find
To find ÷ = __34
23
Then 34
23
34
23÷ = x (1 ÷ )
89
43
23
The answer should be < 1 and it is.
= x =
Dividing Fractions
÷ =341 4
3First find
To find ÷ = __34
23
Then 34
23
34
23÷ = x (1 ÷ )
89
43
23
The extra step of dividing by 1 can be omitted.
÷ = x (1 ÷ )
= x =
Dividing Fractions
÷ =341 4
3First find
To find ÷ = __34
23
Then 34
23
34
23
89
43
23
The extra step of dividing by 1 can be omitted.
Dividing Fractions
It’s ours to reason why We invert and multiply.
Fraction Meanings
Fraction Meanings• One or more equal parts of a whole.
Fraction Meanings• One or more equal parts of a whole.
• One or more equal parts of a collection.
Fraction Meanings• One or more equal parts of a whole.
• One or more equal parts of a collection.
• Division of two whole numbers.
Fraction Meanings• One or more equal parts of a whole.
• One or more equal parts of a collection.
• Location on a number line.
• Division of two whole numbers.
Fraction Meanings• One or more equal parts of a whole.
• Ratio of two numbers.
• One or more equal parts of a collection.
• Location on a number line.
• Division of two whole numbers.
Fractions
MassHOPE-TEACHWorcester, MA
Saturday, April 27, 20132:45pm - 3:45pm
Joan A. Cotter, [email protected]
