1 Building Coherence with Fractions K-5 Fraction Tasks and ModelsDenise Schulz, NCDPI Kitty Rutherford, NCDPI

2 Welcome “Who’s in the Room”Survey participants: first timers, math coaches, classroom teachers, administrators, central office, etc… “Who’s in the Room”

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4 Research

5 We’ve had a tendency in our traditional scope and sequence of math that you teach all this whole-number stuff…and then, all of a sudden, you get to fractions, and it’s a whole new world of what to do—everything they learned in whole numbers has nothing to do with how you do fractions. Linda Gojak

6 High school students’ knowledge of fractions correlates strongly with their overall mathematics achievement in both the UK and the USA. Fifth graders’ fraction knowledge predicts their mastery of algebra and overall mathematics achievement in high school, even after controlling for IQ, reading achievement, working memory, family income and education, and whole number knowledge. Siegler, Fazio, Bailey, and Zhou

7 Fractions are a rich part of mathematics, but we tend to manipulate fractions by rote rather than try to make sense of the concepts and procedures. Researchers have concluded that this complex topic causes more trouble for students than any other area of mathematics. Bezuk and Bieck 1993 Emphasize the bold in the sentence!

8 A representative sample of 1000 US algebra teachers ranked lack of fraction understanding as one of the two largest problems hindering their students’ algebra learning (trailing only ‘word problems’, many of which involve fractions). Siegler, Fazio, Bailey, and Zhou

9 Algebra proficiency is more closely related to conceptual knowledge of fractions than to conceptual knowledge of whole numbers. Siegler, Fazio, Bailey, and Zhou

10 Part-Whole RelationshipPartitioning wholes into equal-size pieces Identifying different units This is the best way to approach learning about fractions in the early grades. It is essential for students to be provided opportunity to reason about the meaning of part-whole relations. Research supports the idea that part-whole relationship, which involve partitioning wholes into equal-size pieces and identifying different units, is the best way to approach learning about fractions in the early grades. students’ experiences with “fair shares” in everyday life is often the starting point for assisting students in understanding some important ideas about size and number of units in a whole and how units can be divided up into smaller and smaller subunits

11 How much of this brownie has been eaten?A teacher drew a picture of a brownie that had been cut with a slice removed and asked a first grader and a third grader to decide how much had been eaten. The first grader studied the picture for a moment, and decided the missing piece was “half of a half”. The third grader said it was an impossible amount, because the pieces were not all the same size and it therefore could not be “1 out of 3”.

12 What does that tell us? Children have some conceptually sound understanding of fractions, even before instruction. Children can learn to ignore conceptual understanding in favor of models introduced in school that portray fractions in narrow ways. If models do not draw on children’s formative experiences of sharing and partitioning, then they are likely to prevent teachers from cultivating the natural insights about quantities that young children have.

13 Write a fraction to show how much of the large square is shaded.Most students know and can explain ¼ Beyond Pieces and Pies

14 Write a fraction to show how much of the large square is shaded.Traditional line of questioning starts with “How many pieces is the whole brownie cut into?” This line of questioning can lead to misconceptions. If your questions focus on the size of the part relative to the whole, rather than the number of parts into which the whole is cut, children learn to look a the relationship between a part and the whole to name the part. Asking “How many of these parts fit into the whole brownie?” prompts children to focus on the relationship between the size of a fractional part and the whole to determine the value of the fraction. Shocked by responses, most students (9% of upper elementary students) responded 1/3. “First I counted the number of shared parts (one) and used that for the numerator” “Just like I did the other one, only this time it’s one out of three instead of one out of four.” Sometimes our instruction makes students lose that natural instinct students have to partition and understand the fair share piece of fractions because of the procedural way we teach fractions – with no meaning! Must provide opportunities for students to work with unequally partitioned areas and number lines. That’s why the partition piece so important in the 1-2 grades. This corresponds to the partitive meaning of 1 divided by 4

15 The use of fraction manipulatives to teach equivalence and order is popular. However, students can use manipulatives such as fraction bars or divided circles to solve fractions problems without understanding the mathematical basis for the relationship. Extending Children’s Mathematics, pg. 116

16 When asked “What is ¼. ” One child said that ¼ was a “little pie shapeExtending Children’s Mathematics, pg. 4

17 The typical American approach to teaching fractions can overemphasize procedures at the expense of an understanding of the relationships among numbers, which is needed for higher math. Lynn Fuchs

18 Partitioning

19 First Grade 1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. Second Grade 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Third Grade 3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. Refer to standards

20 Developing Meaning In PartitionsStudent was shown a picture with half the cookie colored. Was asked, “The part of the cookie that isn’t colored in is a half of a cookie, how could you show ¼ of the cookie?” For this student, “one-fourth” was the process of partitioning into 4 parts rather than 1 part in specific relationship to a whole. Extending Children’s Mathematics, pg. 5

21 Mario wants to cut the pizza into equal pieces and give his sister a fourth of the pizza to eat. Color the piece of pizza that Mario would give his sister. Christopher wants to give half of his candy bar to his brother. Color the piece of the candy bar that Christopher would give his brother. 1st Grade

22 1st Grade

23 Show two different ways to partition these rectangles into halvesShow two different ways to partition these rectangles into halves. How do you know you made halves? 2nd Grade

24 You and three friends will share this cookieYou and three friends will share this cookie. Show how you will partition the cookie into equal pieces. 2nd Grade

25 You have 3 rectangular cakesYou have 3 rectangular cakes. Cut each cake into fourths in three different ways. Explain how you know that each cake has been partitioned into fourths. 2nd Grade

26 2nd Grade

27 Eliot and Kaylee bought the same kind of candy bar at the storeEliot and Kaylee bought the same kind of candy bar at the store. Eliot ate two halves of his candy bar. Kaylee ate three thirds of her candy bar. Kaylee said she ate more of her candy bar than Eliot. Is she correct? 2nd Grade

28 Mrs. Copeland told her class to fold a square sheet of paper into fourths. The pictures below show how students folded their papers. Circle the squares that show fourths. Explain how you know the squares are folded into fourths. 2nd Grade

29 How do these compare? What is the progression?Circle the pictures that show fourths. Circle the squares that show fourths. Second Grade First Grade

30 3. G. 2 Partition shapes into parts with equal areas3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. 1 4 For example, partition a shape into 4 parts with equal area, and describe the area of each part as of the area of the shape. 1 4 1 4 1 4 Third grade first time students write fractions symbolically, big focus on unit fractions – foundational for fractions! 3rd Grade

31 Mr. Rogers started building a deck on the back of his houseMr. Rogers started building a deck on the back of his house. So far, he finished 1 4 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. 3rd Grade

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33 Rudy was asked to partition a rectangle into thirdsRudy was asked to partition a rectangle into thirds. His solution is below: Did Rudy correctly partition the rectangle into thirds? Justify your answer using pictures, numbers, or words. You may cut the square into parts, if needed. 3rd Grade

34 How does the study of partitioning in K-3 lead to an understanding of fractions?Develop understanding of fractions as numbers. 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. 3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. Grade 3 expectations in this domain are limited to fraction with denominators 2, 3, 4, 6, and 8

35 Students recognize that 3 5 means 3 parts of size 1 5 .3.NF.1: Interpreting 𝒂 𝒃 as the quantity formed by 𝑎 parts of size 𝟏 𝒃 using different fraction models. Students recognize that means 3 parts of size 1 5

36 Students recognize that 3 5 means 3 parts of size 1 5Students recognize that 3 5 means 3 parts of size Recognizing number line interpretations of fractions is essential for measuring skills. Develop understanding of fractions as numbers. 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. 3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. Grade 3 expectations in this domain are limited to fraction with denominators 2, 3, 4, 6, and 8

37 What fraction of the scarf will be complete after three days? Martha is making a scarf for her sister. Each day she knits of a scarf. What fraction of the scarf will be complete after three days? What fractions of the scarf will be complete after six days? How can you use a number line to prove that your answers are correct? 3rd Grade

38 Theresa invented a new see-through cereal box that helps people see how much cereal they have. She wants to pub a number line on the box so it’s easy to see the fraction of the box that is full. Theresa already marked that this box is full. Help Theresa mark the following fractions on the cereal box’s number line: Explain how you decided where to write each fraction on the number line. 3rd Grade

39 Gino has 8 4 feet of licorice to share with his friendsGino has feet of licorice to share with his friends. He decides to give each friend foot of licorice. Draw lines on Gino’s licorice to show where he should cut each foot. 3rd Grade

40 Suni was using the following yard stick to measure pieces of yarn for her art project. This ruler number line shows how much yard she cuts for each color. What fraction of a unit does she need of each color? If the unit was divided into fourths, which colors of string could be measured in fourths? How many fourths is each of those colors? Explain your answer using pictures or words. Are any of the colors equal to of the unit? Write a sentence explaining your reasoning. 3rd Grade

41 Partitioning Students need to build an understanding of partitioning and how the pieces relate to the whole. Developing an understanding of partitioning helps students develop an understanding of unit fractions which will help with decomposing fractions There may be multiple ways to partition, and developing this skill helps building fluency with fractions.

42 Adding and Subtracting with Fractions

43 Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 5 6 = 1 6 2 6 3 6 4 6 5 6 5 6 = Understand a fraction 𝒂 𝒃 with 𝑎>1 as a sum of fractions 𝟏 𝒃 .Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 1 6 2 6 3 6 4 6 5 6

44 Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 5 6 = 1 − 1 6 2 6 3 6 4 6 5 6 1 5 6 = − Understand a fraction 𝒂 𝒃 with 𝑎>1 as a sum of fractions 𝟏 𝒃 .Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 1 6 2 6 3 6 4 6 5 6 1 7 6 8 6

45 One way to introduce the addition and subtraction of fractions is to begin with representations of fractions, rather than with problems involving combining or separating.

46 A cake recipe calls for you to use cup of milk, cup of oil, and cup of water. How much liquid was needed to make the cake? milk oil water model = =

47 Tom and Hal will both get 1 piece of cake. 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. 4th Grade

48 1 8 + 1 8 + 2 8 + 2 8 + 2 8 = 8 8 Tom Mary Nancy Bob Hal Mary Nancy= 8 8

49 There are 2 gallons of punch left in the punch bowlThere are 2 gallons of punch left in the punch bowl. It gets divided between 8 students, with each getting a different amount. Micah takes of a gallon Roberta takes 3 times as much as Micah. Steve takes twice as much as Roberta. Yanni takes of a gallon less than Steve. Amy takes of a gallon less that Yanni. The remaining punch is divided between Tom, Jackie, and Henry. Tom and Jackie had the same amount of punch. Henry had less punch than both Tom and Jackie. How much punch did each person take? Draw a picture and write an equation to match this context. 4th Grade

50 Micah 1 12 Yanni Yanni 1 12 Roberta 1 12 Yanni 1 12 Roberta 1 12 Amy 1 12 Roberta 1 12 Amy 1 12 Steve 1 12 Amy 1 12 Steve 1 12 Tom 1 12 Steve 1 12 Tom 1 12 Steve 1 12 Tom 1 12 Steve 1 12 Jackie 1 12 Steve 1 12 Jackie 1 12 Yanni 1 12 Jackie 1 12 Yanni 1 12 Henry 1 12

51 In order to train for the Girls on the Run 5K Race, the girls’ running team at Lincoln Elementary School runs the following distances: Draw a number line to show the distances that the girls ran each week. How far did the girls run in all? Write an equation that matches the story. The girls at Jefferson Elementary School ran 10 miles total during the same time. How much farther did they run that the girls at Lincoln Elementary? Use a picture and an equation to find your answer. Week Distance Week 1 miles Week 2 miles Week 3 miles Week 4 miles 4th Grade

52 =

53 Jeff ran 2 5 of a mile on Wednesday and 4 10 of a mile on FridayJeff ran of a mile on Wednesday and of a mile on Friday. How far did he run on those two days? 1 5 1 10 5th Grade

54 Jeff ran 2 5 of a mile on Wednesday and 4 10 of a mile on FridayJeff ran of a mile on Wednesday and of a mile on Friday. How far did he run on those two days? 1 5 1 10 5th Grade

55 Leland has 2 3 of a cake left over from his birthday partyLeland has of a cake left over from his birthday party. He gave of the entire cake to his friend. How much cake did Leland have left? 5th Grade

56 Leland has 2 3 of a cake left over from his birthday partyLeland has of a cake left over from his birthday party. He gave of the entire cake to his friend. How much cake did Leland have left? 5th Grade

57 Joe and Grace are baking cookiesJoe and Grace are baking cookies. They need a total of 2 cups of sugar for the recipe. Joe has cups of sugar and Grace has 3 4 cups of sugar. Without solving the problem, do they have enough sugar? Explain your thinking. Solve the problem using a model to justify your reasoning. 5th Grade

58 2 2 3 + 1 1 2 Use 2 different strategies to solve this problem: Explain your thinking using pictures, words, or models. 5th Grade

59 Multiplication with Fractions

60 Write an addition equation to show this situation.Katie makes 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. Show your answer with a model. Use numbers or words to explain how your model shows addition. Write a multiplication equation to show this situation. How are your addition and multiplication equations alike? Different? Would you use one over the other? Why or why not? 4th Grade

61 1 4 1 4 1 4 1 4 1 4 1 4 1 4 2 4 1 4 3 4 4 4 5 4 6 4 7 4 = 7 4 1 4 × 7= 7 4

62 Kelly was making curtains for her living roomKelly was making curtains for her living room. She bought four pieces of fabric that were each 2 3 yards 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 yards long. 4th Grade

63 1 3 2 3 × 4 = 8 3 1 3 1 3 1 3 2 3 × 4 =2 2 3

64 1 3 1 3 1 3

65 Putting Multiplication in ContextWith a partner, create a problem that illustrates each of the following: Multiplication of a fraction x whole number Multiplication of a whole number x fraction Exchange problems with another partner group and write equations to match the written problems.

66 Three-fourths of the class is boysThree-fourths of the class is boys. Two-thirds of the boys are wearing tennis shoes. What fraction of the class are boys with tennis shoes? This question is asking what is of ? Or, what is x ? 1 4 1 2 3 4 1 1 4 1 2 3 4 1 1 4 1 2 3 4 1 Another model for students, must be expose to various models

67 1 4 1 2 3 4 1 1 4 1 2 3 4 1 1 4 1 2 3 4 1

68 Three-fifths of the students at Laughlin Middle School participate in sports, one-half of them participate in basketball. Draw a model to find the fractional part of the students that play basketball. 1 5 5th Grade

69 Three-fifths of the students at Laughlin Middle School participate in sports, one-half of them participate in basketball. Draw a model to find the fractional part of the students that play basketball. 1 2 5th Grade

70 Three-fifths of the students at Laughlin Middle School participate in sports, one-half of them participate in basketball. Draw a model to find the fractional part of the students that play basketball. 1 5 1 5 1 5 1 5 1 5 1 2 1 2 5th Grade

71 3 10 1 5 2 5 3 5 4 5 1

72 Brooks used 2 3 of a can of paint to paint the bird feederBrooks used of a can of paint to paint the bird feeder. A full can of paint contains of a gallon. What fraction of the can of paint did Brooks use? 5th Grade

73 2 3 ∙ 7 8 1 3 1 3 Another example 1 3

74 2 3 ∙ 7 8 1 3 1 3 Another example 1 3 1 8

75 2 3 ∙ 7 8 1 3 1 3 Another example 1 3 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8

76 Multiplication with partitioning 2 5 × 3 46 20

77 Division with Fractions

78 What does it mean to divide by a fraction?What are some misconceptions that students may have about division, or ideas they attempt to carry over from whole number division, that make this concept difficult for them?

79 Mrs. Sullivan owns a bakeryMrs. Sullivan owns a bakery. One of her customers cancelled their cake order after the cake was already made. Mrs. Sullivan gave half of the cake to her employees to eat. She brought the other half home for her family to eat. If there are 5 members of the Sullivan family, and they share the cake equally, how much of the original cake will each family member get to eat? 5th Grade

80 1 2 1 2 ÷5

81 David needs a half yard of rope to tie a bundle of papersDavid needs a half yard of rope to tie a bundle of papers. If he has five yards of rope, how many bundles can he make? 5th Grade

82 1 2 3 4 5 6 7 8 9 10 5 ÷ 1 2

83 Putting Division in ContextWith a partner, create a problem that illustrates each of the following: Division of whole number by a fraction Division of a fraction by a whole number Exchange problems with another partner group and write equations to match the written problems.

84 For each of the pictures below write the equation and a story problem to match the picture. The shaded part of each model represents the answer. A. B. C. 5th Grade

85 Why Don’t We Just Flip and Multiply?

86 5 ÷ = 5 ∙ ∙ 3 1 5 ∙ 3 1 ∙ 3 3 5 ∙3

87 Maintain the Value of the QuotientInvert and multiply vs. developing the concept of the procedure Learning the algorithm does not mean that you understand the mathematics, but understanding the mathematics can help you learn and remember the algorithm.

88 In class, Sarah and Tony are talking about the difference between “ 1 3 times 6” compared to “ 1 3 divided by 6.” Their teacher asks them to draw a picture and to write a story problem for each expression. What would Sarah and Tony’s work look like? 5th Grade

89 1 3 x 6 1 3 ÷ 6

90 Turn & Talk How does the concept of fractions develop across K-5?Why is it important for students to have conceptual understanding of fractions before entering middle school?

91 What questions do you have?

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93 DPI Mathematics SectionKitty Rutherford Elementary Mathematics Consultant Denise Schulz Lisa Ashe Secondary Mathematics Consultant Joseph Reaper Dr. Jennifer Curtis K – 12 Mathematics Section Chief Mathematics Program Assistant 93

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