1. Transitioning to the Common Core State Standards – Mathematics
Pam Hutchison

2. Please fill in the lines:
First Name ________Last Name__________
Primary ______Alternate _______ .
School____________District______________

3. AGENDA Fractions/Fractions on a Number Line Assessing Fractions
Naming and Locating
Fractions, Whole Numbers, and Mixed Numbers
Adding and Subtracting Fractions
Multiplying Fractions
Comparing Fractions
Equivalent Fractions
Assessing Fractions
Stoplighting the Standards

4. Spending Spree
David spent of his money on a game. Then he spent of his remaining money on a book. If he has $20 left, how much money did he have at first?

5. Fractions

6. Fractions
3.NF.1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

7. Fraction Concepts

8. So what is the definition of a fraction?

9. Definition of Fraction:
Start with a unit, 1, and split it into ___ equal pieces.
Each piece represent 1/___ of the unit.
When we name the fraction__/__, we are talking about ___ of those 1/___ size pieces .

10. Fraction Concepts

11. 3rd Fractions Task - NC
Mr. Rogers started building a deck on the back of his house. So far, he finished ¼ of the deck. The fraction of the completed deck is below.
Draw 2 pictures of what the completed deck might look like. Use numbers and words to explain how you created your picture.

12. Fraction Task - NC
Martha is making a scarf for her sister. Each day she knits 1/6 of a scarf.
What fraction of the scarf will be complete after three days?
What fraction of the scarf will be complete after six days?
How can you use a number line to prove that your answers are correct?

13. Fractions
on a Number Line

14. 1 ● How many pieces are in the unit? Are all the pieces equal?
So the denominator is
And each piece represents . ● 1 7

15. How far is the point from 0?
So the numerator is
And the name of the point is …… ● 1

16. 1 ● How many pieces are in the unit? Are all the pieces equal?
So each piece represents ● 1

17. 1 ● How far is the point from 0? So the name of the point is ….
How many pieces from 0?
So the name of the point is …. ● 1

18. 4 Definition of Fraction:
When we name the point , we’re talking about a distance from 0 of ___ of those ___ pieces. 4

19. The denominator is
so each piece represents ● 1 5

20. 1 ● How far is the point from 0? So the numerator is
and the fraction represented is ● 1 3

21. Academic Vocabulary What is the meaning of denominator?
What about numerator?
Definitions should be more than a location – the denominator is the bottom number
They should be what the denominator is – the number of equal parts in one unit

22. Student Talk Strategy: Rally Coach
Partner A: name the point and explain
Partner B: verify and “coach” if needed
Tip, Tip, Teach
Switch roles
Partner B: name the point and explain
Partner A: verify and “coach” if needed

23. Explains – Key Phrases Here is the unit. (SHOW)
The unit is split in ___ equal pieces
Each piece represents
The distance from 0 to the point is ___ of those pieces
The name of the point is

24. Partner Activity 1

25. 1 2 7 Definition of Fraction: ● Start with a unit, 1,1
Start with a unit, 1,
Split it into __ equal pieces.
Each piece represents of the unit
The point is __ of those pieces from 0
So this point represents 7 2

26. 1 6 8 Definition of Fraction: ● Start with a unit, 1,1
Start with a unit, 1,
Split it into __ equal pieces.
Each piece represents of the unit
The pointa is __ of those pieces from 0
So this point represents 8 6

27. Partner Activity 1, cont.
Partner B 5B. 6B.
Partner A
5A.
6A.

28. 1 1 2 3 3 | | | | | | | | | The denominator is …….
| | | | | | | | |
The denominator is …….
The numerator is ………
Another way to name this point? 1 2 3 3 1

29. 2 1 2 6 3 | | | | | | | | | The denominator is ……..
| | | | | | | | |
The denominator is ……..
The numerator is ………
Another way to name this point? 1 2 6 3 2

30. 1 1 2 5 3 2 3 | | | | | | | | | The denominator is ……
| | | | | | | | |
The denominator is ……
The numerator is ………
Another way to name this point? 1 2 5 3 1 2 3

31. 2 1 2 7 3 1 3 | | | | | | | | | The denominator is …..
| | | | | | | | |
The denominator is …..
The numerator is ………
Another way to name this point? 1 2 7 3 2 1 3

32. 1 2 15 3 | | | | | | | | | Suppose the line was shaded to 5.
| | | | | | | | |
Suppose the line was shaded to 5.
How many parts would be shaded?
So the numerator would be ……… 1 2 15 3

33. 1 2 30 3 | | | | | | | | | Suppose the line was shaded to 10.
| | | | | | | | |
Suppose the line was shaded to 10.
How many parts would be shaded?
So the numerator would be ……… 1 2 30 3

34. Rally Coach Partner A goes first Partner B: coach SWITCH
Name the point as a fraction and as a mixed number. Explain your thinking
Partner B: coach
SWITCH
Partner B goes
Partner A: coach

35. Rally Coach Part 2 Partner B goes first Partner A: coach SWITCH ROLES
Locate the point on the number line
Rename the point in a 2nd way (fraction or mixed number)
Explain your thinking
Partner A: coach
SWITCH ROLES

36. Rally Coach
Partner B
Partner A 6. 7. 8.

37. Connect to traditional
Change to a fraction.
How could you have students develop a procedure for doing this without telling them “multiply the whole number by the denominator, then add the numerator”?

38. Connect to traditional
Change to a mixed number.
Again, how could you do this without just telling students to divide?

39. Student Thinking Video Clips 1 – David (5th Grade) Two clips
First clip – 3 weeks after a conceptual lesson on mixed numbers and improper fractions
Second clip – 3.5 weeks after a procedural lesson on mixed numbers and improper fractions

40. Student Thinking Video Clips 2 – Background
Exemplary teacher because of the way she normally engages her students in reasoning mathematically
Asked to teach a lesson from a state-adopted textbook in which the focus is entirely procedural.
Lesson was videotaped; then several students were interviewed and videotaped solving problems.

41. Student Thinking Video Clips 2 – Background, cont.
Five weeks later, the teacher taught the content again, only this time approaching it her way, and again we assessed and videotaped children.

42. Student Thinking Video Clips 2 – Rachel
First clip – After the procedural lesson on mixed numbers and improper fractions
Second clip – 5 weeks later after a conceptual lesson on mixed numbers and improper fractions

43. Classroom Connections
Looking back at the 2 students we saw interviewed, what are the implications for instruction?

44. Research
Students who learn rules before they learn concepts tend to score significantly lower than do students who learn concepts first
Initial rote learning of a concept can create interference to later meaningful learning

45. Discuss at Your Tables
How is this different from the way your book currently teaches fractions?
How does it support all students in deepening their understanding of fractions?

46. Adding Fractions

47. Fraction Computation
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

48. 3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = /8 = 8/8 + 8/8 + 1/8.

49. 3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

50. | | | | | | | | |
Add 3 + 4
Move a distance of 3
From that point, move a distance of 4
You end at 7
So = 7

51. Adding Fractions Remember our initial understanding of fractions
means we have 5 pieces and
each piece is in size

52. Adding Fractions
What are some other ways to decompose the fraction ?

53. Add 1

54. Add 1

55. Add 1

56. Adding Fractions

57. NC Assessment Task
At a party you are giving out 8 pieces of cake. People will get different amounts of cake. Tom and Hal will both get 1 piece of cake. Mary will get 2 pieces of cake. Nancy and Bob share equally the remaining pieces of cake. What fraction of the cake will each person eat? Write an equation to match the situation. Write a sentence explaining the strategy used to solve the problem.

58. NC Assessment Task
There are 12 pieces of candy in the bucket. Maria and Sam each get 2 pieces of candy. Tom gets 5 pieces of candy. Vinny gets the rest of the candy. What fraction does each student get? Write an equation to match this story. Write a sentence to explain the strategy used to solve the problem.

59. Subtracting Fractions

60. 7 - 4

61. 7 - 4

63. / / / / / / / /6

64. / / / / / / / /6

65. | | | | | | | | | | | | | | | | | | | | | | | | |
Subtract:
| | | | | | | | | | | | | | | | | | | | | | | | | 1 2 3 4 5 6

66. Subtract:
| | | | | |
| | | 1 2 3 4 5 6

67. Subtract:
| | | | | |
| | |
| | | 1 2 3 4 5 6

68. | | | | | | | | | | | | | | | | | | | | | | | | |
Subtract:
| | | | | | | | | | | | | | | | | | | | | | | | | 1 2 3 4 5 6

69. | | | | | | | | | | | | | | | | | | | | | | | | |
Subtract:
| | | | | | | | | | | | | | | | | | | | | | | | | 1 2 3 4 5 6

70. Subtracting Fractions

71. Multiplying Fractions

72. Multiplying Fractions
4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

73. 3 x 2 Three groups of two

74. Multiplying Fractions
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

75. Multiplying Fractions
Remember our initial understanding of fractions of fractions
Another way to write this is

76. Multiplying Fractions
What do we normally tell students to be when they multiply a fraction by a whole number?

77. 3 x ½ Three groups of one-half
/ / / /2
1(1/2) (1/2) (1/2)

78. Multiplying Fractions
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 ×(1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

79. | | | | | | | | | 1 2

80. | | | | | | | | | 1 2

81. Multiplying

82. NC Assessment Task
Katie makes 1/4 pound of pasta for each person at her dinner party. If seven people attend the party, how many pounds of pasta will be needed for her guests?
Write an addition equation to show this situation.
Write a multiplication equation to show this situation.
How are your addition and multiplication equations alike? Different?

83. NC Assessment Task
Part 1: Kelly was making curtains for her living room. She bought four pieces of fabric that were each 2/3 yard long. How many yards of fabric did Kelly buy in all? Draw a picture and write an equation to show the total amount of fabric if each piece is 2/3 yard long.

84. Comparing Fractions

85. Fractions Extend understanding of fraction equivalence and ordering.
2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

86. Comparing Fractions
A B.

87. Compare Fractions
Using
Sense Making

88. Comparing Fractions
B A.
Common Numerator

89. Comparing Fractions
A B.
Common Numerator

90. Hidden Common Numerator
Comparing Fractions
B A.
Hidden Common Numerator

91. Hidden Common Numerator
Comparing Fractions
A B.
Hidden Common Numerator

92. Benchmark Fractions | | | 0 ½ 1 How can you tell if a fraction is:
| | |
½ 1
How can you tell if a fraction is:
Close to 0?
Close to but less than ½?
Close to but more than ½?
Close to 1?

93. Comparing Fractions
B. A. A. B.

94. Comparing Fractions
A. B.

95. NC Assessment Task
You need 3/4 of a Liter of soda to make punch for a party. Which containers have enough soda in them to make punch? Write a sentence explaining your thinking.
Container A- 2/4 of a Liter
Container B- 2/3 of a Liter
Container C- 5/6 of a Liter
Container D- 11/12 of a Liter
Container E- 7/12 of a Liter

96. Equivalent Fractions

97. Locate on the top number line.● 1
Page 95

98. Copy onto the bottom number line.● 1 ●

99. Are the lengths equal? ● 1 ●

100. Equivalent fractions can be constructed by partitioning equal fractional parts of a whole into the same number of equal parts.
The length of the whole does not change; it has only been partitioned into more equal sized pieces. Since the length being specified has not changed, the fractions that describe that length are equal.

101. CaCCSS
Fractions are equivalent (equal) if they are the same size or they name the same point on the number line. (3.NF3a)

102. ● 1 ●

103. ● 1 ●

104. So ● 1 ●

105. Order Matters! Locate 1st fraction on number line
Duplicate on 2nd number line
“Are they equal?”
Split 2nd number line
Name point on 2nd number line
So Fraction 1 = Fraction 2

106. ● 1 ●

107. Equivalent Fractions
Let’s try a couple more

108. Equivalent Fractions Partner Activity 3 – Rally Coach 3A. 3B.
What language are you listening for?
What are the key parts of the explanation?

109. Equivalent Fractions
Partner Activity 3 – Rally Coach 4A. 4B.

110. Connect to traditional
How can you help students develop a procedure for finding equivalent fractions without telling them?

111. 1 4 8 4 | | | | Connect to Traditional For example, | | | | | | | | |
| | | |
| | |
| | |
| | | 1 4 8 4

112. Fractions Extend understanding of fraction equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

113. Fraction Families

114. NC Assessment Task There is two-thirds of a pizza left.
How many pieces of pizza are left if the original pizza had a total of 3 slices? 6 slices? 12 slices?
Write a sentence to explain your thinking.

115. NC Assessment Task
Sally has a piece of rope that is 3/4 of a foot long. Tomas has a piece of rope that is 1/2 of a foot long. Mitch has a piece of a rope that is 1/3 of a foot long. How many inches is each piece of rope? Write a sentence explaining your thinking.

116. Equivalent Fractions
3A. Find 3 fractions equivalent to 3B. Find 3 fractions equivalent to

117. Simplifying Fractions

118. Factors and GCF
Well before we want to introduce simplifying fractions, student should learn or review factors and greatest common factors.

119. Which is simpler? 20 $1 bills OR 1 $20 bill
3 gallons of milk OR 12 quarts of milk
4 $1 bills OR 16 quarters
15 dimes OR 6 quarters

120. Simplify 1 Locate on the top number line. ●1
Can we regroup the pieces to make larger groups with an EQUAL number of pieces?
What size groups can we make?

121. Simplify 1 Questions for students:
Can we make equal groups of 2 pieces (with both the shaded part and the whole unit)? 1 ●
STOP! We missed 9!

122. Simplify 1 Can we make groups of 3 pieces? ●1
We can group both 9 and 12 evenly into groups with 3 pieces each.

123. Simplify 1 Mark the larger groups on the number line ● ●1 ●
Now, duplicate just the larger parts onto the bottom number line.
Then name the point.

124. Simplify ● 1 ●
So the new name is
Therefore:

125. One More
Simplify

126. Simplify Fractions
Practice with your partner using rows

127. Back to ● 1 ● So

128. Because 2 is not a factor of 9.
Simplify
Why couldn’t we break 9 into groups of 2 pieces?
Because 2 is not a factor of 9.

129. So groups of 3 works because
Back to
We were able to make groups of 3 pieces? Why?
Let’s look at 2 important questions:
Is 3 a factor of 9?
Is 3 a factor of 12?
So groups of 3 works because
3 is a factor of both 9 and 12.

130. A Variation on the Traditional
Simplify
| | | | | | | | | | | | | | | | | 1 4 3 4 4

131. Simplify:
What is the greatest common factor of 16 and 24? 1 2 3 SO

132. Simplify using the alternative method:1. 2.

133. CCSS-M SMARTER Items
Fraction Items