# Performance Assessment Task Picking Fractions Grade 4. Common Core State Standards Math - Content Standards

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2 Picking Fractions This problem gives you the chance to: work with equivalent fractions This is a fraction tree. Under the tree are baskets Equivalent fractions picked from the tree must be placed in the same basket. Put each fraction on the tree into the correct basket. Copyright 2007 by Mathematics Assessment 81 Resource Service. All Rights Reserved.

3 2. Find one new equivalent fraction for each basket and write it on the line that is in front of the basket. 3. Fill in the missing numerator and denominator to make this pair of fractions equivalent. 2 = 10 Explain how you figured it out. 8 Copyright 2007 by Mathematics Assessment 82 Resource Service. All Rights Reserved.

4 Task 5: Picking Fractions Rubric The core elements of performance required by this task are: work with equivalent fractions Based on these, credit for specific aspects of performance should be assigned as follows 1. Puts the fractions into the correct baskets 1/2 = 4/8, 3/6, and 2/4 1/4 = 2/8 and 3/12 3/4 = 6/8 and 9/12 1/3 = 3/9, 2/6 and 4/12 2/3 = 6/9 and 8/12 All correct 5 points Partial credit 9, 10, 11 fractions correct 4 points: 8, 7 fractions correct 3 points: 6, 5 fractions correct 2 points: 4, 3 fractions correct 1 point 2. Puts one more correct equivalent fraction onto each plate All 5 correct Partial credit correct points 5 (4) (3) (2) (1) 2 (1) section points Fills in the missing values such as: denominator 5 and numerator 4 or denominator 4 and numerator 5 or denominator 2 and numerator 10 or denominator 1 and numerator 2- or denominator 20 and numerator 1 and Gives correct explanation such as: They are equivalent fractions. 1 Total Points 8 1 Copyright 2007 by Mathematics Assessment 83 Resource Service. All Rights Reserved.

6 number talks throughout the year? How might these conversations facilitate learning the more formal procedures later in the year? Looking at Student Work on Picking Fractions Student A is one of the few students showing work on how to check for equivalent fractions. While the student got full marks for part 3, Student A is giving a procedure. Do you think this method would work for other cases? Why or why not? Try it for 3/?=?/21. Student A Copyright 2007 by Noyce Foundation 85

7 Some students attempted to use diagrams. It is difficult to draw free-hand accurately enough to solve for all the fractions. Student B has some of the fractions correctly identified, but shows little understanding when choosing new fractions. Only some of the denominators are even divisible by the denominator of the original fraction. In part 3, the student may or may not have a bit of procedural knowledge, but doesn t even recognize the need for a denominator under the two. What questions might you ask this student to further probe their understanding of fractions? Student B Copyright 2007 by Noyce Foundation 86

8 Student C also attempts to draw the fractions. Notice that 2/6 is chosen to represent both 1/3 and 2/3. 6/8 is chosen to represent 1/4 and 3/4. What do you think the student is thinking or understanding about fractions? Where does that thinking break down? What would be your next steps with this student? Student C Copyright 2007 by Noyce Foundation 87

9 Student D seems to be able to think about some denominators that could yield equivalent fractions, but struggles with how to choose a numerator. Notice the student doesn t recognize that 3/6= 1/2. Student D Copyright 2007 by Noyce Foundation 88

10 Student E also has trouble with just a basic recognition of 1/2. In part 3 the student shows confidence, but really does not seem to have a grasp of equivalency. It might be interesting to interview this student to see what the student understands about equivalency with number sentences. For example, how might the student fill in the missing number in this equation: = + 5? Student E Copyright 2007 by Noyce Foundation 89

11 Student F has been shown some tools to help make sense of equivalent fractions but can t apply them to this situation. What do you think this student understands? Is confused about? What might be some good next steps for this student? Student F Copyright 2007 by Noyce Foundation 90

12 Student G uses many improper fractions. This may be a partially learned procedure like invert and multiply for division of fractions. Notice that the student uses 3/6 for 2/3, not recognizing that it is one half. What are some of the common fractions that we want students to just know? Also look at the explanation in part 3. This seems to be more evidence of applying a partially learned procedure with no meaning attached. Student G Copyright 2007 by Noyce Foundation 91

13 Student H does not think in fractional form. The student is just writing whole numbers for the equivalents. The student is probably multiplying the numerator times the denominator. Student H Copyright 2007 by Noyce Foundation 92

14 4 th Grade Task 5 Picking Fractions Student Task Core Idea 1 Number Properties Work with equivalent fractions. Develop an understanding of fractions as a part of unit whole, as part of a collection, and as a location on a number line. Recognize and generate equivalent forms of commonly used fractions. Based on teacher observation, this is what fourth graders know and are able to do: Find equivalent fractions for 1/2 Make equivalent fractions given just the numerator of one fraction and the denominator of the other. Areas of difficulties for fourth graders: Understanding procedures for equivalencies Choosing appropriate denominators Understanding the role of the numerator in non-unit fractions. Having some partially remembered procedures, but not understanding the meaning or entire procedure Copyright 2007 by Noyce Foundation 93

15 The maximum score available for this task is 8 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Many students, 75%, could find 3 or 4 equivalent fractions in part 1, usually for 1/2 and 1/4. More than half the students, 63%, could find 3 or 4 equivalent fractions and solve part 3 of the task. A little less than half the students, 48%, could find 7 or 8 equivalent fractions and solve part 3. About 21% of the students could find most of the equivalent fractions, write their own equivalent fractions, and solve part 3 of the task. 25% of the students scored no points on the task. 80% of the students with this score attempted the task. Copyright 2007 by Noyce Foundation 94

16 Picking Fractions Points Understandings Misunderstandings 0 80% of the students attempted the Students had difficulty finding equivalent task. fractions. (See data in the beginning of this 1 Students could find 3 or 4 equivalent fractions, usually those equal to 1/2 or unit fractions. 2 Students could find 3 or 4 equivalent fractions and solve part 3 of the task. 4 Students could find 7 or 8 equivalent fractions and solve part 3 of the task. toolkit). (See data in the beginning of this toolkit). (See data in the beginning of this toolkit). (See data in the beginning of this toolkit). 7 Students missed either one or two equivalent fractions or missed part 3 of the task. 8 Students could recognize equivalent fractions, make new fractions that were equivalent to a given fraction, and solve part 3 of the task finding tow equivalent fractions given a numerator of one and a denominator of the other. Copyright 2007 by Noyce Foundation 95

17 Implications for Instruction Students need to understand the concept of equality in whole numbers and fractions. At this grade level students should be most comfortable with unit fractions like 1/2, 1/3, and 1/4. Students should have experiences with a variety of tools such as number lines, fraction strips, fraction circles, and bar models to help them think about equivalency or the relative size of fractions. Students should also be introduced to procedures like reducing fractions to find equivalent forms of the same fraction. Students should have frequent opportunities to think about the concept fractional parts and the quantity represented to develop and deepen these ideas over time. Routines like number talks is one way of facilitating this discussion. Consider making a large number line going from 0 to 1 on the board. 0 1 Write different fraction on index cards (e.g. 1/2, 1/8, 1/4, 2/3 as well as their equivalent fractions names. Choose only 2 or 3 fractions to work with each time you do this routine. Make multiple copies of the same numbers. Give a card to a pair of students (so they will need to discuss ideas / promote more discourse). Give them time to discuss where their number would make sense on the number line. Have one pair of students place their card where they think it belongs on the number line. Students must give a mathematically convincing argument as to why they are placing the number at this location. Have the partners discuss whether they agree or disagree with the placement of the card and why. The class may ask clarifying questions of the pair at the front of the room. Students can then share other strategies. Leave numbers on the number line so students can compare placement with other fractions over time. (This routine for number talks comes from the website for San Diego School District. Ideas for Action Research - Examining Student Justifications Examining student work can help clarify some of the big mathematical ideas needed to understand a concept. In this task, students are trying to make sense of equivalent fractions and justify why this relationship is true. Look at the following examples: What are some of the different strategies students use to correctly find equivalent fractions? What did the student have to understand about fractions to use this procedure? Some students found equivalent fractions using procedures. Will these procedures work for all fractions? Why do these procedures work mathematically? Do they make sense? Students at this grade level have some partial understandings about procedures or fractions. Copyright 2007 by Noyce Foundation 96

18 Where does the thinking break down for each student? What might be changed to make the explanation clearer? What are next steps for different types of errors? What kinds of discussion help students let go of misconceptions? A powerful tool for promoting discourse in the class room is to use snippets of student work and pose a question to get the whole class re-engaged in the mathematics of the task. Pick 2 or 3 pieces of student work and plan your own class discussion. Ariel Copyright 2007 by Noyce Foundation 97

19 Brianna Clayton Copyright 2007 by Noyce Foundation 98

20 Douglas Evita Finn Copyright 2007 by Noyce Foundation 99

21 Georgia Harley Ionia Copyright 2007 by Noyce Foundation 100

22 Joshua Kevin Laurel Copyright 2007 by Noyce Foundation 101

23 Marcus Nadine Reflecting on the Results for Fourth Grade as a Whole Think about student work through the collection of tasks and the implications for instruction. What are some of the big misconceptions or difficulties that really hit home for you? Copyright 2007 by Noyce Foundation 102

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