1. Focusing on Fractions
3 2
5 5
Presented by Kathleen Cotter Lawer
Based on the work of Joan A. Cotter, Ph.D.
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2. Joan A. Cotter, Ph.D.

3. Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin

4. Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin
• Montessori Educator

5. Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin
• Montessori Educator
• Math Games

6. Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin
• Montessori Educator
• Math Games
• Tutor

7. Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin
• Montessori Educator
• Math Games
• Tutor
• AL Abacus

8. Joan A. Cotter, Ph.D.
• Engineer from University of Wisconsin
• Montessori Educator
• Math Games
• Tutor
• AL Abacus
• Middle-School Teacher

9. 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

10. How are fractions
usually taught?
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11. Fraction Model: “Fish Tank”
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12. Fraction Model: “Fish Tank”
2
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13. Fraction Model: “Fish Tank”
2
5
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14. Fraction Model: “Words”
This is fourths.
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15. Fraction Model: “Words”
This is fourths. This is thirds.
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16. Fraction Model: “Words”
This is fourths. This is thirds.
If this were chocolate…
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17. Fraction Model: “Words”
This is fourths. This is thirds.
If this were chocolate…
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18. Fraction Model: “Rounded Corner”
1 1 1
3 3 3
1 1 1 1
4 4 4 4
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19. Fraction Model: “Rounded Corner”
1 1 1
3 3 3
1 1 1 1
4 4 4 4
The middle fractions are
greater than the fractions at
the ends!
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20. Fraction Models: Circles
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21. Fraction Models: Circles
1 1
3 3
1 1
2 2
1
3
1 1 1 1
1 1 5 5 6 6
4 4
1 1
1 1 6 6
1 1 5 5
4 4 1 1
1
6 6
5
Try to compare 4/5 and 5/6 with this model.
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22. Fraction Models: Circles
Are we comparing angles, arcs, or area?
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23. 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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24. 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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25. So how should we
teach fractions?
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26. Fraction Model: Linear Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
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27. Fraction Model: “Color”
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
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28. Fraction Model: “Missing Parts”
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
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29. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
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30. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
Ask the children to put it together like a puzzle.
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31. Definition of a Fraction
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32. Definition of a Fraction
A fraction is part of a set or part of a whole.
Textbook definition
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33. Definition of a Fraction
A fraction is part of a set or part of a whole.
Textbook definition
What about 3 ?
2
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34. Definition of a Fraction
An expression that indicates the
quotient of two quantities.
American Heritage Dictionary
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35. Definition of a Fraction
An expression that indicates the
quotient of two quantities.
American Heritage Dictionary
a
A number in the form bwhere b ≠ 0.
Arizona Standards
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36. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
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37. Fraction Chart
1
1 1
1÷2 2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
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38. Fraction Chart
1
1 1
1÷2 2 2
1 1 1
1÷3 3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
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39. Fraction Chart
1
1 1
1÷2 2 2
1 1 1
1÷3 3 3 3
1 1 1 1
1÷4 4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
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40. Fraction Naming
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41. Fraction Naming
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42. Fraction Naming
One divided into 2 equal parts.
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43. Fraction Naming
One divided into 2 equal parts.
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44. Fraction Naming
One divided into 2 equal parts.
One divided into 3 equal parts.
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45. Fraction Naming
One divided into 2 equal parts.
One divided into 3 equal parts.
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46. Fraction Naming
One divided into 2 equal parts.
One divided into 3 equal parts.
One divided into 4 equal parts.
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47. Fraction Naming
One divided into 3 equal parts.
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48. Fraction Naming
One divided into 3 equal parts.
1 One whole
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49. Fraction Naming
One divided into 3 equal parts.
1 One whole
divided into
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50. Fraction Naming
One divided into 3 equal parts.
1 One whole
divided into
3 three equal parts
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51. Fraction Naming
One divided into 3 equal parts.
1 One
divided by
3 three
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52. Fraction Naming
One divided into 3 equal parts.
1
Read as “one-third.”
3
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53. Fraction Naming
1
1 1 1
3 3 3
One divided into 3 equal parts.
1
Read as “one-third.”
3
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54. Fraction Naming
1
1 1 1
3 3 3
One divided into 3 equal parts.
1
Read as “one-third.”
3
In English, we use ordinal numbers to name
fractions, with the exception of half.
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55. How should we practice
what we learned?
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56. Games
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57. Games
Games
Math
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58. Games
Games
=
Math
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59. Games
Games Books
=
Math Reading
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60. Games
Games Books
=
Math Reading
Games provide interesting repetition
needed for automaticity.
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61. Games
Games Books
=
Math Reading
Games provide interesting repetition
needed for automaticity.
Games also provide an application
for new information.
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62. Unit Fraction War
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63. Unit Fraction War
Purpose: Practice for naming and
comparing unit fractions.
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64. 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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65. 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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66. Unit Fraction War
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67. Unit Fraction War
1 1
4 5
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68. Unit Fraction War
1 1
4 5
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69. Unit Fraction War
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70. Unit Fraction War
1 1
8 2
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71. Unit Fraction War
1 1
8 2
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72. Unit Fraction War
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73. Unit Fraction War
1 1
6 6
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74. Unit Fraction War
1 1
6 6
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75. Unit Fraction War
1 1
6 6
1 1
3 4
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76. Unit Fraction War
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77. Unit Fraction War
1 1
2 3
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78. More Fraction Naming
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79. More Fraction Naming
1
1 1 1
3 3 3
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80. More Fraction Naming
1
1 1 1
3 3 3
One-third taken 2 times.
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81. More Fraction Naming
1
1 1 1
3 3 3
One-third taken 2 times.
Two
1 s
3
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82. More Fraction Naming
1
1 1 1
3 3 3
One-third taken 2 times.
Two
1 s is
2
3 3
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83. More Fraction Naming
1
1 1 1
3 3 3
One-third taken 2 times.
Two
1 s is
2
3 3
Read as “two-thirds.”
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84. More Fraction Naming
1
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85. More Fraction Naming
1 1
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86. More Fraction Naming
1 1
Two divided into 3 equal parts.
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87. More Fraction Naming
1 1
Two divided into 3 equal parts.
2 Two
divided by
3 three
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88. More Fraction Naming
1 1
Two divided into 3 equal parts.
2 Two
divided by
3 three
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89. More Fraction Naming
1 1
1 1 1 1 1 1
3 3 3 3 3 3
Two divided into 3 equal parts.
2 Two
divided by
3 three
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90. More Fraction Naming
1 1
1 1 1 1 1 1
3 3 3 3 3 3
Two divided into 3 equal parts.
2 Two 1
3
divided by = two 3 s
three
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91. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
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92. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
How many fourths in a
whole?
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93. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
How many fourths in a
whole?
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94. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
How many fourths in a
whole?
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95. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
How many fourths in a
whole?
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96. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
How many fourths in a
whole?
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97. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
How many fourths in a How many sixths?
whole?
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98. Concentrating on One Game
Purpose: To help the children realize that
5 fifths, 8 eighths, and so forth, make a
whole.
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99. Concentrating on One Game
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.
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100. Concentrating on One
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101. Concentrating on One
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102. Concentrating on One
3
5
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103. Concentrating on One
3
5
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104. Concentrating on One
3
5
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105. Concentrating on One
3
5
2
5
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106. Concentrating on One
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107. Concentrating on One
3
8
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108. Concentrating on One
3
8
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109. Concentrating on One
3
8
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110. Concentrating on One
7
8
3
8
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111. Concentrating on One
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112. Concentrating on One
5
8
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113. Concentrating on One
5
8
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114. Concentrating on One
5
8
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115. Concentrating on One
5
8
3
8
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116. Concentrating on One
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117. Let’s begin to compare
fractions….
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118. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
Which is more, 3/4 or
4/5?
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119. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
Which is more, 3/4 or
4/5?
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120. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
Which is more, 3/4 or
4/5?
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121. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
Which is more, 7/8 or
8/9?
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122. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
Which is more, 7/8 or
8/9?
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123. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
Which is more, 7/8 or
8/9?
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124. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
An interesting pattern.
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125. Fraction Stairs
1
10
1
9
1
8
1
7
1
6
1
5
1
4
1
3
1
2
1
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126. Fraction Stairs
1
10
1
9
1
8
1
7
1
6
1
5
1
4
1
3
1
2
1
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127. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
How many fourths equal a half?
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128. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
How many fourths equal a half?
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129. Fraction Chart
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 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
7 7 7 7 7 7 7
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
9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
How many fourths equal a half? Eighths?
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