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Arkansas Common Core State Standards Professional Development Project for Mathematics

Fraction Operations: Multiple Grouping (Grades 3-‐6) This course uses as a required text the book Extending Children’s Mathematics: Fractions and Decimals by Susan B. Empson and Linda Levi. The sessions will explore the mathematics and strategies raised when students tackle “multiple grouping” problems, problems “involving a whole number of equal groups of fractional amounts.” The course includes:

• Examining how students think about and solve multiple grouping problems

• Identifying the mathematics embedded in student work • Using student work to increase understanding of fractions and operations, such as-‐

o Composing, decomposing, and recomposing wholes o Understanding 1/b and its relationship to a/b (repeated addition or multiplication)

o Relating multiplication and division • Recording mathematical thinking and attending to properties of operations.

The course Fraction Concepts: Equal Sharing is required before you are eligible to participate in this course. Special thanks to James Brickwedde, of the Project for Elementary Mathematics, Hamline University, Minnesota, for writing adaptations of his work especially for use in Arkansas professional development. Special thanks to NCTM for granting permission to use articles from Teaching Children Mathematics, Mathematics Teaching in the Middle School, and Mathematics Teacher, highly valuable resources for every mathematics educator, and for their ongoing leadership in mathematics education. Visit them at www.nctm.org for membership information.

Multiple Grouping Problem Work the problem using all five number sets.

• How are these number sets related? • What connections would you like to see students make when working these

problems? Angeles uses ___ of a bag of beads to make a necklace. If she makes ___ necklaces, how many bags of beads will she need?

(1/3, 12) (1/3, 24) (1/3, 36) (2/3, 12) (2/3, 36)

Arkansas CCSSM Fraction Operations - Multiple Grouping

Day 2 Handout Page 1 of 21

Student Work for the Lesson Angeles uses ___ of a bag of beads to make a necklace for her store. If she makes ___ necklaces, how many bags of beads will she need? (1/3; 12) (1/3; 24) (1/3; 36) (2/3; 12) (2/3; 36)

Guillermo Blake

Angeles Cristina



Planning Sheet – Mrs. Kasnicka – Multiple Grouping (Multiplication) p.1

1. Sort student work to determine what mathematics students brought to bear on the problem and what mathematics is available for instruction through a discussion. Sort by…strategy, representation used, level (correct/complete, productive failure), missing elements.

2. Determine where most of the class appears to be in terms of the mathematics they understand and the mathematics they are ready to learn.

3. Select/create a learning goal or goals that will address where your class is. In doing so, think about how to support students that might be working below the rest of the class so the discussion helps them as well.

4. Select student papers that can be used as the basis for a discussion/lesson directed at the learning goal(s). Determine in what order to use the student work. Determine if the student will present the work (P) or if you will allow the class to interpret it (I).

5. Develop questions to pose about the work, either to the student it belongs to (O) or to the class (C).

Selected Work Learning Goal(s) Questions to Pose

Guillermo Drawing of 4 rectangles, with 3 sections in each numbered 1-‐12.

(Focus attention on students who are still struggling with making sense of the problem context) Use a direct modeling strategy to make sense of the problem and to find “12 sets of 1/3”

(I) What did Guillermo do? (C) How did he represent a necklace (1/3 bag of beads)? (C) How did he represent a bag of beads? (C) What do the numbers 1-‐12 in his picture show? (C) How can we write an equation to show his thinking?

Blake Drawing of 8 rectangles, shaded in groups of 2/3 (incomplete drawing), showing 3 sets of 2/3.

Interpret Blake’s incomplete picture to try to relate 3 groups of 2/3 = 2 in a multiplicative way to 12 groups of 2/3 = 8.

(I) What did Blake do? (C) What does the picture with the two rectangles, red shading, arrow at the top and numeral 3 represent? (C) How did he represent a bag of beads? (C) Where is a necklace in his picture? (C) How can we write an equation to show his thinking? (C) How can this help us solve the problem?



Planning Sheet – Mrs. Kasnicka – Multiple Grouping (Multiplication) p.2


Angeles Table labeled B (beads) and N (necklaces) with entries beginning with (2, 3) and counting by that “chunk”

Relate Angeles’ table to Blake’s picture and determine how Angeles extended her table to answer the question (multiplicative relationship). Connect table to the number sentences.

(I) Compare Angeles’ work to Blake’s (and Guillermo’s). What connections do you see? (C) What does the first row in Angeles’ table show? What is the 2? What is the 3? (C) How is “2 bags makes 3 necklaces shown in the table? In the picture? In the number sentences? (C) What do the other rows in the table represent? How did Angeles know what to put there?

Cristina Pair of number sentences 3 x 2/3 = 2 and 9 x 2/3 = 6 with a total of 8 circled

Relate Cristina’s number sentences to Angeles’ table and Blake’s drawing. Relate Cristina’ number sentences to each other.

(I) What did Cristina do? (C) How could Cristina’s equations be used to solve the problem? (C) How do Cristina’s equations relate to Angeles’ and Blake’s work? (C) How do Cristina’s number sentences relate to each other?



Planning Sheet

1. Sort student work to determine what mathematics students brought to bear on the problem and what mathematics is available for instruction through a discussion. Sort by…strategy, representation used, level (correct/complete, productive failure), missing elements.

2. Determine where most of the class appears to be in terms of the mathematics they understand and the mathematics they are ready to learn.

3. Select/create a learning goal or goals that will address where your class is. In doing so, think about how to support students that might be working below the rest of the class so the discussion helps them as well.

4. Select student papers that can be used as the basis for a discussion/lesson directed at the learning goal(s). Determine in what order to use the student work. Determine if the student will present the work (P) or if you will allow the class to interpret it (I).

5. Develop questions to pose about the work, either to the student it belongs to (O) or to the class (C).




!

!



Newsletter

Developing Mathematical Notation and Efficiency Using Student Thinking about Fractions

Submitted by Aimee L. Evans Arch Ford Education Service Cooperative

Teachers who employ teaching strategies centered on student thinking have the same goals for their students to be able to use conventional mathematical notation to represent their thinking as traditional mathematics instruction has. The key feature in how they go about developing notation is that their approach is “connecting” conventional notation “to” student thinking versus “substituting” mathematical convention “for” student thinking.

These teachers also have the goal that their students gain efficient methods for performing mathematical calculations. Again, their approach is to help students see predictable, recurring relationships in the problems they solve and the strategies they use in order to help the student become efficient while still holding on to his/her understanding of the problem.

Sometimes the discussion will focus on determining what mathematical equation represents the problem and result. A very different discussion often takes place about how to use conventional notation to represent one’s strategy for thinking through the problem. Teachers must become highly skilled at identifying the underlying thinking in a student’s strategy, determining which mathematical conventions, recurring relationships, and properties are at play, and helping the student link those to his/her thinking. This is a challenging skill set for anyone to develop.

Through the Lens of Student Thinking

The work in this article comes from Arkansas students in intermediate grades working with multiple groups problems. Multiple groups problems are multiplication and division word problems that involve a whole number of groups with a fractional amount in each group (Empson & Levi, 2011).

Problem A: The animal shelter has 4 cans of food. They feed each puppy ½ of a can of food each day. How many puppies can they feed?

The work of two students for Problem A is shown in Figure 1 and Figure 2. The work of both students is very similar in terms of the strategy and the conclusion that the shelter can feed 8 puppies.

(continued on next page)

Figure 1

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However, one student represented the problem as 4 × ½ while the other represented the problem as 4 x 2. This presents the opportunity to bring forward a discussion about the two equations to determine if they can both be correct.

The teacher might prompt the students to think about what each equation means to see which matches the problem, or, in fact, if either one does. The problem does not contain 4 groups of ½, making the equation in Figure 1 incorrect. The solution actually does contain 4 groups of 2, but students need to consider what the 2 means. There are 4 groups of 2 one-‐halves, or 4 × (2 × ½) = 8 × ½.

The equation 4 × 2 matches the strategy or the thinking, but an equation the students have not considered actually matches the problem: 4÷ ½ = Z or perhaps Z × ½ = 4. In this problem the total number of cans of food is known, as is the amount per puppy (amount in each group). The unknown is the number of puppies (groups). Considering what elements of a situation are given versus what is missing helps students think about what equations fit the problem.

Problem B: Angeles uses 𝟏 𝟑 of a bag of beads for each necklace. How many bags of beads will she need for ___ (12, 24, 36) necklaces?

In Figure 3, the student is solving Problem B for 36 necklaces. She has used the relationship she securely recognizes that 3 groups of 𝟏 𝟑 make 1. In previous problems she had also demonstrated ready knowledge that 2 × ½ = 1 and 4 × ¼ = 1, so she has generalized the grouping of unit fractions to make 1 whole. This generalization could become an element of her using efficient strategies in future problems.

She also strategically distributed the 36 groups of 𝟏 𝟑 so she could count one bag of beads at a time. At this time her teacher might ask her some questions to determine at what point she knew she was going to repeat the expression 3 × 𝟏 𝟑 twelve times. If, for example, she counted by threes and knew

(continued on the next page)

Figure 3

Figure 2

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she would repeat twelve times before she began writing, or if she knew that there are twelve 3s in 36, then it may be feasible to help her think multiplicatively versus additively by asking her to consider this notation and how or if it relates to her thinking:

36 ×𝟏 𝟑 = (her expanded expression from above) = 12 × (3 × 𝟏 𝟑) = 12 × 1 = 12

Once she is able to make sense of the expression 12 × (3 × 𝟏 𝟑) and how it relates to her thinking, she will no longer need to expanded expression that is taking so much time to write.

Juxtapose this approach to how we might “impose” traditional approaches onto this student. If we asked her to think about 36 × 𝟏 𝟑 as 𝟑𝟔 𝟏 × 𝟏 𝟑 = 𝟑𝟔 𝟑 (thought of as 36 ÷ 3), she may not readily see her thinking in the notation.

The student in Figure 4, clearly represents his thinking without using conventional notation for the first number set. When he begins to solve Problem B again for 24 necklaces he uses what he had figured out for 12 necklaces and writes the conventional equation 12 × 𝟏 𝟑= 4 to show his result from the previous work. This equation does not appear to completely reflect his thinking, simply summarizes it. He then duplicates the equation using the reasoning that 12 + 12 is 24 so he will use the result from 12 necklaces twice for 24 necklaces. The last line of his paper is a result of his discussion with his teacher where she helped him write a single

equation that would represent his thinking about the problem. This conventional notation had a direct connection to how he had already thought through the problem.

The student whose work is shown in Figure 5 appears to readily recognize the problem as multiplication. It also initially appears that she is able to multiply a whole number by a unit fraction without interim steps, but when she solves for 36 necklaces she does not use the expression 36 × 𝟏 𝟑. She seems to recognize that doubling one of the factors doubles the product, and thus uses the result of 12 × 𝟏 𝟑 to solve 24 × 𝟏 𝟑. She also recognizes that halving one of the factors halves the product and uses the information to create partial products of 24 × 𝟏 𝟑, 6 × 𝟏 𝟑, and 6 × 𝟏 𝟑 in lieu of 36 × 𝟏 𝟑. In doing so, she is taking advantage of the distributive property to decompose 36 in a way that is useful or easy for her to deal with her.

(continued on page 30)

Figure 4

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(3-‐5 Corner continued from page 8)

The teacher could support this student along a few paths. First, she, like the student from Figure 2, needs support with showing how she combined interim products:

36 × 𝟏 𝟑 = (24 × 𝟏 𝟑) + (6 × 𝟏 𝟑) + (6 × 𝟏 𝟑) = 8 + 2 + 2 = 12

Second, the teacher could support her in looking for relationships between the two factors and the product to see if she might realize that multiplying by 𝟏 𝟑 yields the same results as dividing by 3, or that multiplying by a unit fraction is like dividing by the denominator.

Finally, the teacher could help the student represent the doubling/halving idea through notation that makes use of the associative property, such as:

24 × 𝟏 𝟑 = (2 × 12) × 𝟏 𝟑= 2 × (12 × 𝟏 𝟑) = 2 × 4; and

6 × 𝟏 𝟑= (𝟏 𝟐 × 12) × 𝟏 𝟑= 𝟏 𝟐 × (12 × 𝟏 𝟑) = 𝟏 𝟐 × 4

Problem C: Mrs. E. is tying bows on gifts. Each gift will take 𝟏 𝟓 of a roll of ribbon. How much ribbon will you need for ____(25, 75) gifts?

In Figure 6, the student uses a drawing to work through Problem C. The drawing indicates that she knows that 1 roll of ribbon will make a bow for 5 gifts. The tally marks indicate gifts, the circles represent rolls of ribbon, and she keeps a running total of how many gifts remain to be decorated.

Her equation 25 ÷ 5 = 5 is a record of her solution, not her strategy. However, and she does not draw a picture or do any other work on the next problem. She simply writes the equation 75 ÷ 5 = 15. The teacher will need to ask questions to determine what assumptions or generalizations the student relied on to make this leap.


Figure 6

Figure 5

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Did she, for example, recognize that 75 is three times 25 so the result would be three times the previous result? If so, the student should see and consider this notation to see how or if it connects to her thinking:

75 ÷ 5 = (3 × 25) ÷ 5 = 3 × (25 ÷ 5) = 3 × 5 = 15

Or did she instead recognize that multiplying by a unit fraction can be done by dividing by the denominator? If so, exposing her to the notation in the paragraph above may not be the appropriate next step. Instead the teacher would choose to push on this idea of multiplying by a unit fraction and why it works.

Problem D: Mrs. Moore is painting ornaments for her friends. She uses ¾ of a bottle of paint for each ornament. How much paint will she use if she paints 24 ornaments?

The student whose work is shown in Figure 7 displays some of his thinking about Problem D in his illustration, but there is also some embedded or implied strategy evident. We see that he recognizes that 4 ornaments will use 3 bottles of paint, but he does not tell us how he came to that conclusion. He then uses that assumption to carry out the problem for 24 ornaments.

It would first be important to find out how he concluded that 4 x ¾ = 3. If the thinking behind his work is that 4 × ¾ = 4 × (3 × ¼) = 4 × ¼ × 3 = 1 × 3, it might be possible for this student to generalize from that for other problems, such as 3 × 𝟐 𝟑 or 5 × 𝟒 𝟓. If his thinking is more concrete or based in repeated addition, it might be useful to have him explore how many times he repeats a non-‐unit fraction to come up with the next whole number.

Summary

It is critical to make students’ thinking explicit and strongly bring out the properties and relationships they are using. Often children can explain their thinking verbally using mathematical relationships, but when we say “Can you show me your thinking on paper?” they resort to drawing pictures. There will always be times when students (and adults) need to use pictures, diagrams, or simpler strategies to make sense of new problems or to find a starting place in their thinking, but providing them with notational tools will help them develop ways to communicate their thinking with expressions and equations. Ultimately these tools also aid with efficiency.

There are several teacher skills that are needed to guide students toward notation and efficiency connected to their thinking in this manner. The most relevant is the skill of interpreting the details of


Figure 7

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student thinking, both as it is presented in written and verbal form. It is essential to know when you have enough information to make assumptions about their thinking from their written work and when you need to find out more, either through watching them work or asking them questions. Another vital skill is that of knowing the mathematical properties and notation that fit with the thinking a student is using, and being able to present a logical progression for the student to consider.

The work presented above is from multiple groups problems, but the same ideas of developing mathematical notation and efficiency apply to other types of problems, such as multi-‐digit whole number operations or work with decimals. § Reference: Empson, S. B. & Levi, L. (2011). Extending children’s mathematics fractions and decimals: innovations in cognitively guided instruction. Portsmouth, NH: Heinemann.

The Arkansas Mathematics Teacher Professional Development Scholarship provides financial

assistance to Arkansas mathematics teachers to attend a state, regional or national conference whose primary focus is mathematics and to provide financial assistance to Arkansas

mathematics teachers who wish to participate in other Professional Development opportunities

whose primary focus is mathematics.

The Arkansas Council of Teachers of Mathematics Teacher Special Project Grant provides financial assistance to Arkansas mathematics teachers to fund special

projects for their students.

For more information, visit www.actm.net and click on FORMS.

Next Deadline September 1; Award by October 31

Questions? Email Tracy Watson at [email protected].

ACTM Special Project Grant Application

ACTM Professional Development Scholarship

Did you know that ACTM has $$ for you?

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Looking Ahead to Partial Groups Problems Solve this problem: A bag of gumdrops weighs pound. Kellie has bags of candy. How many pounds of candy does she have?

• What type of problem is this? • Write the other two types of problems for this context.

Solve this problem:

• Think about the “traditional algorithm” you were taught for solving a problem like this. Is that how you worked the problem?

• Can you think about it another way? • Examine in your text Figure 8-‐12 on page 196 and Figure 8-‐13 on page 197.

Compare and contrast these two approaches.



Solve each series of problems. For each problem try to create a drawing to represent or make sense of the problem, use a grouping or combining strategy, and write a number sentence. Series 1:

1a) A zookeeper has cups of frog food. His frog gets cups of food each day. How long can he feed the frog before the food runs out? 1b) A zookeeper has cups of frog food. His frog gets cup of food each day. How long can he feed the frog before the food runs out? 1c) A zookeeper has cups of frog food. His frog gets cup of food each day. How long can he feed the frog before the food runs out? 1d) A zookeeper has cups of frog food. His frog gets cup of food each day. How long can he feed the frog before the food runs out?



Series 2: 2a) Angeles uses 2 bags of beads to make a necklace. If she made 12 necklaces, how many bags of beads did she use? 2b) Angeles uses of a bag of beads to make a necklace. If she made 12 necklaces, how many bags of beads did she use? 2c) Angeles uses of a bag of beads to make a necklace. If she made of a necklace, how many bags of beads did she use? 2d) Angeles uses of a bag of beads to make a necklace. If she made necklaces, how many bags of beads did she use?



OPPORTUNITY TO EXTEND – Beyond Day 2:

1. Read from the text pages 93-‐109 to look inside Mrs. Perez’s classroom and see some exchanges around relational thinking.

2. Reflect on the videos from Mrs. Kasnicka’s class and the vignettes from Mrs. Perez’s class in the reading. • What is relational thinking? • Why is it important to make the relational thinking explicit? • How does the teacher go about planning a lesson around student work to

support relational thinking?

3. Pose some open number sentences to your class. Consider their work.

4. Think about the work of your students on the multiple grouping problems and/or open number sentences you have tried. Use the Planning Sheet to make a draft instructional plan. • Develop your learning goal(s) using student work as the focus of a lesson.

Connect these to the CCSSM. • What mathematics would you like to bring out in a discussion of a few pieces

of their work? Describe what pieces of student work you would use in the discussion and why.

• Plan what questions you would ask to get students to grasp the learning goals and think critically about the strategies and the fractions concepts in the lesson.

• Teach the lesson.

5. Reflect on the lesson. Interview some of your students to determine what they learned. • Do you think you selected an appropriate learning goal? Why or why not? • Did the student work and questions you planned lead to the mathematics

you intended? • What do you think your students understand and don’t understand? • What do you think you should do next related to the learning goal?



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