proportional reasoning lesson study toolkit (abridged edition...

Area of Polygons Lesson Study Toolkit, abridged edition. Mills College Lesson Study Group 2009. May not be reproduced without permission.

Proportional Reasoning

Lesson Study Toolkit (Abridged edition 2009)

Mills College Lesson Study Group School of Education

Mills College 5000 MacArthur Boulevard

Oakland, CA 94613

www.lessonresearch.net 510 430 3350

The Mills College Lesson Study Group (MCLSG) is funded by the National Science Foundation (Grant No. REC-0633945) to develop and test two lesson study toolkits in collaboration with lesson study groups across the United States. The toolkits are designed to make research-based resources on proportional reasoning or on area of polygons available to lesson study groups. The goal of the project is to investigate lesson study’s potential to build a shared professional knowledge base for teaching.

Any opinions, findings, and conclusions or recommendations expressed in the toolkits developed by the MCLSG are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Proportional Reasoning Lesson Study Toolkit, abridged edition. Mills College Lesson Study Group 2009. May not be reproduced without permission.

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INTRODUCTION AND TOOLKIT PURPOSE

This toolkit is designed for mathematics lesson study teams working to improve student learning about proportional reasoning. A series of self-directed activities explore mathematical tasks, instructional materials, and research articles. Our hope is that the toolkit will put these useful materials at your fingertips and support your exploration of them to build your own understanding of proportional reasoning and how to teach it. The toolkit is especially designed to enhance the first part of the lesson study cycle, known as kyouzai kenkyuu or “curriculum study.” For a more complete guide to lesson study, see Lesson Study: A Handbook (Lewis, 2002) or other lesson study accounts (Fernandez & Yoshida, 2004; Wang-Iverson & Yoshida, 2006). You may want to think of lesson study supported by this toolkit in terms of the following rough timeline:

• Curriculum study and planning the research lesson: 6-8 meetings (or more if time is available);

• Teaching the research lesson: 1 class period; • Discussion of the research lesson: 1 meeting; • Revision of the research lesson: 1-2 meetings; • Reteaching the research lesson in a different classroom: 1 class period; • Reflection/reporting on the lesson study cycle: 1-2 meetings.

Most of the principles that guided the design of the toolkit will be familiar to lesson study groups. For example, one basic principle is that examining student learning is central to teaching.1 Many toolkit activities involve predicting and examining student solution methods for mathematical tasks, considering what students need to understand and how they come to understand it, and studying research and video that illuminate student thinking. Another familiar principle of the toolkit is reflection on knowledge; a concept map, notesheets on student tasks and research, and reflection forms provide opportunities for you to record your evolving ideas. However, two principles underlying the toolkit design may be unexpected. One is the emphasis on asking team members to work individually (on mathematics problems, reading research articles, etc.) before team discussion. Recent research supports the importance of wrestling with problems individually – to become aware of our own initial thinking – prior to group discussion.2 Second, we have drawn heavily on materials from Japan in order to provide a relatively compact example of a coherent, long-term trajectory for learning about proportional reasoning. We hope the Japanese materials will provide a useful perspective for analysis of your own or other U.S. mathematics curriculum. Action steps are denoted with a “⇒” throughout this document to help you identify the suggested activities of the toolkit. 1 Darling-Hammond, L. & Bransford, J. (2005) Preparing teachers for a changing world. San Francisco: John Wiley 2 Swan, M. (2006) Collaborative learning in mathematics: A challenge to our beliefs and practices. London: NRDC and Leicester: NIACE.


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⇒ Think about w hat would make this lesson study group a supportive and productive site for your mathematical learning? Consider the following norm-setting protocol. Revisit these ideas in subsequent meeting.

SETTING NORMS IN YOUR LESSON STUDY GROUP

• Jot down a list of characteristics important to you. (It may help to think about characteristics of groups that have functioned well – or poorly – to support your mathematical learning in the past.) You may want to consider some general norms (such as listening and taking responsibility) and some that have been identified as especially important to mathematics, such as

o Exploring and “unpacking” mathematical connections, being curious o Explaining and justifying solutions, agreeing on what constitutes an adequate

justification o Evaluating solution strategies for correctness, efficiency, and insight o Expressing agreement or disagreement

• Share and discuss as a group the ideas generated by team members, taking particular care

to identify and discuss any possible contradictions. For example, if one group member asks for “safe” and another for “challenging my thinking,” talk about how both can be honored.

• Synthesize members’ ideas to a group list of about 5 key norms you all support.

• Record the norms for future reference.

• At the beginning of each meeting, choose one norm to monitor that day. At the end of

your meeting, discuss whether you upheld it and what can be improved.

PART 1: EXPLORING STUDENT TASKS Overview We expect that your study of toolkit resources will result in some changing ideas about the teaching and learning of proportional reasoning. To help you see how your own ideas are changing, this section begins with a blank “concept map” for proportional reasoning where you can record your initial thoughts about student learning of this topic. Four student tasks follow for you to solve and analyze. These tasks, labeled Caterpillars, Containers, Poster, and Mixing Juice illuminate the wide range of knowledge students need about proportional reasoning, and may reveal additional ideas for your concept map. To make best use of individual ideas and group discussion, we suggest that you:


⇒ Initially work individually, spending 5-10 minutes jotting down on the concept map your initial ideas about proportional reasoning, with a focus on what students need to understand and how they learn it. (We suggest you update the concept map as you have new thoughts, or you may prefer to write your ideas on small “stickies” so that they can easily be reorganized on the concept map.)

CONCEPT MAP

Elementary School High School

Sequence of Understandings that Students Must Develop Over Time

Tasks & Experiences that Help Students Develop These Understandings

Students Understand and Solve Complex Problems Related to

Area of Polygons

Proportional Reasoning Lesson Study Toolkit, abridged edition ©Mills College Lesson Study Group, 2009. May not be reproduced without written permission.

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⇒ Continuing to work individually, solve all three tasks and record your solutions and ideas about student solutions on the notesheets following each problem before discussing the tasks as a whole group. If the tasks spark any new thoughts about student understanding and how it develops, modify your concept map as needed.

CATERPILLARS AND LEAVES

A fourth-grade class needs 5 leaves each day to feed its 2 caterpillars. How many leaves would they need each day for 12 caterpillars? Answer: _________________________ Use drawings, words, or numbers to show how you got your answer.

Caterpillars and Leaves task from the National Center for Education Statistics, National Assessment of Education Progress (NAEP)http://nces.ed.gov/nationalreportcard/itmrls/startsearch.asp


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Notesheet 1: Solution Sheet for Caterpillar Task

1. How did you solve the problem? Please describe the method you used. 2. What are some of the ways students might try and solve the problem? Please describe the methods they might use. 3. What understandings (e.g., skills and knowledge) do students need to solve the problem? 4. What tasks or experiences will enable students to develop the skills/knowledge to solve the problem?


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CONTAINERS

If water is poured into these test tubes, looking at these containers do you think the depth of water will be proportional to the amount of water? For container (1) ________________ For container (2) ________________ What do you notice about the numbers in the tables below?

Task reproduced from Book 6A Tokyo Shoseki's Mathematics for Elementary School (p.72). Copyright 2004 Global Education Resources ([email protected]). Do not copy, reproduce or distribute without written permission.

Amount of Water (dl)

1 2 3 4 5 6

Depth of Water (cm)

4 8 12 16 20 24

Amount of Water (dl)

1 2 3 4 5 6

Depth of Water (cm)

4 7 10

13

16

19

(1)

(2)


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Notesheet 2: Solution Sheet for Containers Task

1. How did you solve the problem? Please describe the method you used. 2. What are some of the ways students might try and solve the problem? Please describe the methods they might use. 3. What understandings (e.g., skills and knowledge) do students need to solve the problem? 4. What tasks or activities will enable students to develop the skills/knowledge to solve the problem?


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POSTER

The Poster This problem gives you the chance to:

• calculate sizes in an enlargement

1. A photograph is enlarged to make a poster. The photograph is 10 cm wide and 16 cm high. The poster is 25 cm wide. How high is the poster? _______________ Explain your reasoning. 2. On the poster, the building is 30 cm tall. How tall is it on the photograph? Explain your work. _______________

Poster task provided by Silicon Valley Mathematics Institute’s Mathematics Assessment Collaborative. Reprinted by permission.


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Notesheet 3: Solution Sheet for Poster Task 1. How did you solve the problem? Please describe the method you used. 2. What are some of the ways students might try and solve the problem? Please describe the methods they might use. 3. What understandings (e.g., skills and knowledge) do students need to solve the problem? 4. What tasks or activities will enable students to develop the skills/knowledge to solve the problem?


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Notesheet 4: Solution Sheet for Mixing Juice Task 1. How did you solve the problem? Please describe the method you used. 2. What are some of the ways students might try and solve the problem? Please describe the methods they might use. 3. What understandings (e.g., skills and knowledge) do students need to solve the problem? 4. What tasks or activities will enable students to develop the skills/knowledge to solve the problem?


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⇒ Discuss the tasks and your solutions within your group. To get the most from your discussion, you may want to look at problems one by one and: • Have group members post or otherwise share their solution methods. • Identify both correct and incorrect approaches that students might use. • Explore the mathematics deeply by probing the specifics of how each group

member solved the problem, including the operations used to produce correct and incorrect answers.

• Consider the similarities and differences among the tasks. • Update your individual concept maps to reflect any ideas you want to

remember. ⇒ As you discuss the Poster and Caterpillar tasks, examine the associated student work that follows and consider two issues: • How would you categorize the various solution strategies, and what does each

reveal about student understanding of proportional reasoning?

• How do the tasks and solutions connect to team members’ understanding of the following terms: o “In proportion,” “proportional” o “Multiplicative” o “Additive” o “Fraction” o “Ratio” o “Part-to-part ratio,” “Part-to-whole ratio” Our experience is that mathematics educators often use these terms in different ways, so it may be useful for your team to develop shared terminology and meaning.

⇒ After you have discussed all the problems, consider what students would need to understand in order to solve all of these problems. Update your concept map as needed.


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Student Work


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The Poster


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PART 2: EXPLORING CURRICULUM Overview Through solving and discussing the student tasks you and your team probably began to develop useful ideas about:

• What students need to understand about proportional reasoning; • The tasks and experiences that help develop understanding; and • The sequence of these tasks and experiences.

Curriculum study is a good way to further develop and refine your ideas about all three issues. Resources suggested in this section enable you to explore the sequence of experiences by which students learn about proportional reasoning in Japan. The Japanese example was chosen for the toolkit because it provides a relatively concise, coherent trajectory that has been developed by classroom teachers and mathematicians working in collaboration and extensively tested in lesson study. If time permits, comparing the Japanese materials to your own or other U.S. curricula may enable you to further consolidate and refine your thinking.

PLEASE NOTE: The following sections A and B require three resources: • Japanese Mathematics for Elementary School textbooks: Arithmetic grades 1-

6, available from www.globaledresources.com. Cost: $128.65 plus shipping. • Japanese Grade 7, 8 and 9 Mathematics textbooks, available from

http://ucsmp.uchicago.edu/ (click on Resource Component, Translation Series). Cost: $57.00 plus shipping.

• Units from the Japanese textbook Teachers’ Manual related to proportional reasoning. Free, downloadable from www.lessonresearch.net/nsf_toolkit.html

A. Overview of Japanese Teaching Materials Most of the Japanese materials referred to in this section are drawn from grades one through six: however, the foundational ideas related to proportional reasoning addressed by these materials continue to be a major source of difficulty for secondary students. Secondary teachers may therefore find the materials useful for identifying gaps in student understanding or for highlighting ways to address the gaps. Excerpts from the Japanese middle school textbooks are included to show how knowledge of proportional reasoning builds toward middle and high school math topics, including linear functions, graphing, slope, linear equations, similar figures, rate of change, and the relation between area and volume of similar figures.


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Table 1 (p.24) provides references to textbook page numbers and brief summaries of the units relevant to proportional reasoning in the textbook series Mathematics for Elementary School and in the middle school series Japanese Grades 7, 8, and 9 Mathematics. Since the textbooks have many illustrations and sparse text, the amount of material described in Table 1 is not as extensive as it may appear. Taken together, the textbook segments will enable you to get a fairly complete picture of one trajectory by which students might develop proportional reasoning. Translated excerpts from the Teachers’ Manual for several of the key elementary level units are included in the toolkit. Looking at these along with the elementary textbooks will help you “read between the lines” of the textbook to see what teachers are hoping students will gain from the lessons. The Teachers’ Manual describes the teaching plan, objectives, and “learning activities” for each lesson; cumulatively, these lessons are intended to lead to the goals of the unit. The Teachers’ Manual also discusses the mathematics related to the learning activities. Looking at the translations of the Teachers’ Manual should provide a sense of how the textbook is used – i.e., how questions are posed, and student ideas elicited and discussed, and how much time is devoted to each problem in the textbook. The Teachers’ Manual also provide background discussion on mathematical and pedagogical issues, for example, whether it is appropriate to write an equation showing that a ratio and the value of the ratio are equal (a:b = a/b) or how to help students see proportions as functions and to notice the variables within the function, as well as the constant. Partial or full Teachers’ Manual translations are provided for the following units: Grade 2: Multiplication (1) (17 periods, pp. 14-18, 24 of textbook 2B) Grade 4: Investigating Changes in Quantities (5 periods, pp. 54-59 of textbook 4B) Grade 5: Per Unit Quantity (17 periods, pp. 18-37 of textbook 5B) Grade 6: Ratio and its Value (9 periods, pp 48-57 of textbook 6A) Grade 6: Proportional Relationships (11 periods, pp. 70-82 of textbook 6A) Please note that Teachers’ Manual translations are not provided for all units, and that the translated excerpts are drafts and may contain translation errors; feedback is welcome.

A Note about Notation Students sometimes use the equals sign incorrectly to show proportional relationships, for example in solving #4 on p. 28 of the Grade 5B Japanese curriculum, they might write “1.5 dl = 1 m2” to represent the relationship between 1.5 dl and 1 m2. In fact, 1.5 dl is not equal to 1 m2 . There are several ways to represent this relationship more accurately. For example, Primary Mathematics from Singapore foreshadows the use of function notation by writing “1.5 dl 1 m2. Other ways include tables (see page 72 of the 6A Japanese textbook) and the “double number line” representation (see examples on the bottom of p. 73 of the 4B Japanese textbook).


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⇒ Briefly examine the Flow of Japanese Lesson below to familiarize yourself with a typical structure of a mathematics lesson in Japan.

Flow of Japanese Lesson

Lesson Phase Purposes Introduction (very brief)

Students become interested in the topic, connect the lesson to prior learning and/or daily life experiences

Problem Posing (very brief)

Students understand the problem, become interested in it

Individual Work on Problem (10-30 min)

Students bring their own knowledge to bear, exert effort, understand through grappling with a challenging problem

Presentation of Students’ Solutions, Class Discussion (10-30 min)

Several students present solutions or approaches on the blackboard and explain them. Solutions (sometimes including incorrect approaches) are selected and sequenced by the teacher to illustrate different ways of thinking about a problem. Presentation by teacher often begins with a most widely accessible solution. Class members respond to solutions (supported by teacher questions such as “How many solved it this way” and “Do you agree with this method?). Students contrast solutions, supported by teacher questions such as “What is different about Kyoko’s and Mariko’s solutions?” “What are the good points and difficulties of each of solution method?”

[Application Problem]

Students may apply what they have learned to a new problem; the cycle of individual work and presentation/discussion may be repeated.

Summary/ Consolidation of Knowledge (brief)

Teacher and/or students summarize what has been learned; blackboard, class discussion, and math journals may be used, often ending with a journal writing prompt like “What I learned today.”

B. Activities for this Section ⇒ Individually read textbook segments and the related Teachers’ Manual excerpts listed

in Table 1, taking notes in Table 1 about what students come to understand from each unit, how they build their understanding, and the implications for your team. (If your team has limited time, we suggest you read the ‘Mathematical Understandings Highlighted in the Unit’ listed in column 2 of Table 1 for all grades, then select one or several units to focus on. Starred units (**) are recommended as most critical to proportional reasoning).

⇒ Stop about halfway through your reading (perhaps after grade 5) for group discussion. Key questions might be: o What is especially useful, surprising or puzzling about the Japanese materials? o What are the implications, if any, for your concept maps and your instruction?

⇒ Continue individual reading. Continue to update your concept map as your reading sparks new ideas.


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Table 1: Japanese Learning Trajectory for Proportional Reasoning [** indicates key units] Notes Grade Level, Unit, No. of 45-Minute Periods, Textbook Pages

Mathematical Understandings Highlighted in the Unit

Questions for Reflection and Discussion

How Mathematical Understandings of the Unit Relate to Proportional Reasoning; Implications for Our Teaching and Curriculum (e.g., How does our curriculum build this understanding?)

**2B, Unit 10 - Multiplication (1), 17 periods total, pp. 14-18 (4 periods), p. 25 (1 period)

Three ways of thinking about multiplication -repeated addition of discrete quantities (p.14-17) -tape diagram, using a continuous quantity as the context for multiplication (p.18) -a series of multiplications by 4; each time one group is added the total goes up by 4 (p.25) Understanding that all three ways can be expressed with “times as much/ many.”

Do the three ways of thinking about multiplication differ in how they might lay the groundwork for proportional reasoning?


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Grade Level, Unit, No. of 45-Minute Periods, Textbook Pages




3B, Unit 12 - Division Algorithm: how many times as big?, pp. 22 (1 period)

“Bai,” (times as much) - a way of thinking about ratio where one quantity is a base quantity and the other quantity (“comparison” quantity) is considered in terms of the base quantity (e.g., We have 2 meters of red ribbon and 6 meters of blue ribbon. If we use the red ribbon as a base quantity, how many times longer is the blue one?). Relationship of multiplication and division to proportional reasoning. Tape diagram illustrates mathematical relationships.

**4B, Unit 12 – Investigating Changes in Quantities, 4 periods, pp. 54-58

Relationships between two quantities that change together [proportional and non-proportional situations]. How to represent the relationship in a table and mathematical expression. Focus on what changes and how many times bigger one quantity is than another.

What different types of change do students investigate, and how do they record data on each? What changes in each situation? What remains the same? What would you want students to notice about proportional and non-proportional change?


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4B, Unit 13-2 Multiplication and Division of Decimal Numbers (16 periods): times as much, pp. 72-73 (2 periods)

“How many times as long” using tape lengths and division. Similar to 3B-12:, if 8 cm is considered 1, 12 cm is what? Here, introduces decimal quotient. Tape diagram illustrates mathematical relationships.

How does the diagram help to illustrate the proportional relationship between the tape lengths?

5A, Unit 3-2 - Multiplication of Decimal Numbers: Times as Much and Multiplication, (11 periods), pp. 35-36 (2 periods)

Further exploration of “times as much” using tape lengths, decimal numbers, and double number line representation. Meaning of multiplication expands from equal groups to “given the amount per unit, how much for so many units.”

How might the double number line help students deal with the decimal number multiplier?

5A, Unit 4-2 - Division of Decimal Numbers: Times as Many/Much and Division, (11 periods), pp. 47-48 (2 periods)

Further exploration of “times as much.” Meaning of division expands from fair-sharing to finding the per one unit amount.

What sequence of tasks prepares students to deal with decimal numbers in later proportion problems? How might the double number line representation support student thinking?


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**5B, Unit 9-2 – Per-Unit Quantity, pp. 23-34 (12 periods)

Rates that include two quantities of different kinds (e.g., price per unit, population density, speed) to create a derived measurement. Per-unit quantity can be found in 2 ways: e.g., price per unit weight or weight per unit price.

What kinds of rates are students asked to find and how are the different rates represented (in pictures, tables, equations, etc.)? Do any of these representations seem especially useful, interesting, or puzzling? [See Note about Notation, p. 21 of toolkit]

5B, Unit 12 – Percentage and Graphs, 7 periods, pp. 60-67

Percent as a specific form of relative value using two quantities of the same kind (as opposed to relative value of two unlike quantities discussed in 5B-12). Percent is a commonly used ratio expression and a helpful equivalent ratio (e.g., if 4 out of 25 students in a class are boys, multiplying both quantities by 4 results in an easily computed percentage of boys (16%).


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6A, Unit 3-2 – Division of Fractions: Times as Much and Division, pp. 22-23 (2 periods)

Further exploration of “times as much” using division and the case of fractions. Similar issues as in previous units on “times as much.”

What difficulties might students face with using fractional numbers?

**6A, Unit 5 – Ratio and Its Value, 8 periods, pp. 48-56

Ratio and the value of the ratio are two ways of expressing the relationship between two quantities. Two methods for finding equivalent ratios are presented: multiplying or dividing both numbers in a ratio by the same number, and finding equivalence of the value of the ratio. Multiplication and division underlying equivalent ratios is revealed. Two distinct student strategies presented for using ratios.

What is a ratio? What is value of a ratio? What are students expected to learn about each of these, and about equivalent ratios? What representations are used to support student learning about ratio?


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**6A, Unit 7 – Proportional Relationships, 11 periods, pp. 70-82

Examples of proportional situations and how they can be represented using table, equation, and graph. Ways in which one quantity may change or stay constant in relation to a second quantity. Properties of “proportion” (e.g., that if y is multiplied by a factor, x is multiplied by the same factor; that the graph of a proportion is a straight line that goes through the origin; that the quotient of any y divided by the corresponding value of x is always the same number).

In each situation presented on pp. 70-71, what changes and what stays constant? How is the definition of “proportional” similar to or different from a definition you have seen in other places (e.g., your own curriculum or other resources)? How might you tell whether or not two quantities are proportional from their equations, tables, or graphs? E.g., which of the following are proportional and how do you know: y = 3x; y = x + 3; y = 2x + 3? How is the table used in this text, and what relationships within it are emphasized? Why might the text use the same numbers to introduce the three representations?


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Gr. 7, Chapter 5 (2) – Functions and Proportions, pp. 103-104

Expressing proportional and inversely proportional situations using mathematical (symbolic) expressions. Constant of proportionality.

What is the constant of proportionality? What is the relationship between the constant of proportionality and the value of the ratio? How does information in this unit relate to material presented in 4B, Unit 12 and 6A, Unit 7?

Gr. 8 Chapter 4 – Linear Functions, pp. 70-98

Characteristics of linear functions and representing phenomena by linear functions. Rate of change in the values of linear functions and characteristics of their graphs. Finding corresponding values using linear functions. Simultaneous equations.

How are linear functions expressed in symbolic notation; what are the component parts of the expression? How do linear functions build on the idea of rate treated in earlier grades? How are graphs of linear functions similar to graphs of proportional situations? What is the relationship between the constant of proportionality and the slope of a linear function?


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Gr. 8, Chapter 7 – Similar Figures, pp. 156-166

Proportional enlargement and reduction of figures to create similar figures with corresponding vertices, sides, and angles. Properties of similar figures, similarity ratio, conditions for similar triangles, using the similarity ratio.

How are ratios helpful in determining whether two figures are similar? How does this treatment of this topic contrast with treatment of the topic in Grade 6?

Gr. 9, Chapter 4 – Functions, pp. 74-77

Rate of change/ velocity for function y=ax2 and comparison with linear functions.

In linear and quadratic functions, what changes and what remains constant? How is change represented graphically in linear and quadratic functions?

Gr. 9, Chapter 6, Figures and Measurement, pp. 118-128

Relation between area and volume of similar figures and the ratio of similarity. Volume and surface area of solid bodies increase in direct proportion when scaling from a smaller to larger model.

What is the relationship between area of similar figures and the ratio of similarity? How does the ratio of similarity help connect the concepts of area, volume, and surface area of polygons and proportional reasoning?


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PART 3. EXPLORING RESEARCH

The resources in this section are designed to help your team flesh out and consolidate its thinking about how students come to understand proportional reasoning. Resources for this work include research articles, video, and lesson plans. A. Read and Discuss Research

Table 2 below and the annotated guide provide a selection of research about proportional reasoning that your group may find interesting. We include a link to one article for you to read in this version of the toolkit: “Making connections: A case for proportionality,” (Cramer & Post, 1993). Internet version available at http://cedh.umn.edu/rationalnumberproject/93_3.html

Table 2. Research Resources and Topics of Interest

Research Resource Topics of Interest Making connections: A case for proportionality, Cramer & Post

- Characteristics of students who reason proportionally - Characteristics of proportional situations - Representations that illuminate multiplicative relationships - Distinguishing proportional and non-proportional situations

Developing concepts of ratio and proportion, Van de Walle

- Definitions (ratio, rate, proportion) and associated mathematical concepts - Contexts (types of problems) - Mathematical notation and terminology - Instructional strategies/ activities - Representations (ratio tables, graphs) - Informal vs. algorithmic problem-solving approaches

Teaching Fractions and Ratio for Understanding, Lamon, Chapters 1 and 9. [Chapters 3, 5, 15-18 may also be quite helpful.]

- Student strategies; samples of proportional reasoning - Mathematical concepts associated with proportional reasoning - Types and examples of student tasks/ proportional reasoning activities - Characteristics of proportional thinkers

An annotated guide to additional mathematical resources on proportional reasoning is available at http://www.lessonresearch.net/nsf_toolkit.html ⇒ Have members report on each article read, and as a group discuss whether and how

these articles add to your thinking about two issues: o What do students need to understand about proportional reasoning? o What tasks and experiences build student understanding?

⇒ Update your individual concept maps to reflect any new ideas from the discussion.


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B. Analyze Lesson Plans and Video The video footage of Caterpillars shows a 6th grade class working on an elementary/introductory proportional reasoning task (the one you did yourself and studied in the first part of this toolkit), is excerpted in two short video clips:

• Excerpt 1 (1:48). This first clip shows one student discussing the thinking behind her solution to the caterpillar task with other students in her group.

• Excerpt 2 (4:33). The second clip shows the class discussion about the caterpillar problem. What ideas does the teacher emphasize?

You can view/download these video clips and the lesson plan at the following website link:

www.lessonresearch.net/nsf_toolkit.html ⇒ Before watching the video, review the lesson plan, rationale and examples of student

work for “Caterpillars,” along with your own and colleagues’ solutions. Consider the following questions:

o How do the ideas discussed in the lesson rational compare and contrast to other

ideas about proportional reasoning you have gleaned in your research? o What aspects of the lesson task might make it easy or challenging for students?

(e.g., how might you adapt the task for students in middle school?) o How might you categorize the student solutions? o How might you sequence the solutions in terms of their level of sophistication? o Can you identify the multiplication and division operations in students’

solutions? ⇒ Watch the video segments from “Caterpillars.” Think about what student responses you saw in the video and what they tell you about what students know and need to know about proportional reasoning. ⇒ Discuss as a group whether and how the lesson plan, video and related materials add

to your thinking about two questions: o What do students need to understand about proportional reasoning? o What tasks and experiences build student understanding?

⇒ Update your individual concept maps to capture new thoughts sparked by the lesson

plan, video, and your group’s discussion.


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C. Develop A Group Theory Based on your work to date solving problems and reviewing instructional materials and research materials: ⇒ Begin to create a shared “group theory” of the major understandings that students

need related to proportional reasoning, how they develop, and in what sequence. A fresh copy of the Concept Map may be a good place to record this group theory. As you discuss your ideas in order to create the group Concept Map, we suggest you highlight or record alongside the Concept Map:

o Aspects of student understanding that are extremely important, in the view of all

team members; o Areas of particular interest to your group – for example, ideas you would like to

learn more about in your lesson study work or apply in the context of your own curriculum;

o Areas of question or disagreement that you want to note for future discussion.

Blank concepts maps available at http://www.lessonresearch.net/nsf_toolkit.html


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PART 4: CHOOSING AND STUDYING THE RESEARCH LESSON

A. Choose the Focus for the Research Lesson and Write a Lesson Rationale At this point, it may be time to plan a research lesson that builds on your group’s work thus far. For example, your group may want to:

o Study an aspect of student understanding of proportional reasoning that seems especially important, difficult, or currently underemphasized;

o Test a task you found in the research or curriculum materials that you think may be useful for “diagnosing” student thinking or for supporting changes in student thinking;

o Focus on an element in the concept map that was new or surprising to you.

⇒ As a group, think about and discuss what you would like to learn from this cycle of lesson study. Write a brief rationale for your choice of lesson to help clarify what you want to learn from the research lesson. As you discuss and think about the rationale, consider:

o Why have you chosen this focus for the lesson? o What do you expect to learn about students, and about supporting student learning?

A place to write your rationale is provided on the Teaching-Learning Plan for the Research Lesson. A Teaching-Learning plan template, along with other lesson study resources to support planning, observations and debriefing, can be downloaded at the following web link: www.lessonresearch.net/nsf_toolkit.html. If you are new to lesson study we suggest you explore these resources. B. Completing the Lesson Study Cycle ⇒ Develop a teaching-learning plan for the research lesson. ⇒ Conduct the research lesson gathering data for the post lesson discussion. ⇒ If time and circumstances permit, revise the research lesson to incorporate what you

learned in your first teaching, and teach the revised lesson to another class. C. Reflection and Reporting on the Lesson Study Cycle Notesheet 5 provides a place to summarize and consolidate your learning from this lesson study cycle, so that it will be available to you, your team members, and other educators outside of your immediate group. To stimulate your reflection on the lesson study cycle, it will be useful to review materials from the whole cycle, including:

o Your notes about the proportional reasoning tasks, readings, etc.; o Teaching-learning plan; o Student work from the research lesson and notes from post-lesson discussion. ⇒ Reserve at least an hour to review these materials and what you learned from the cycle

and write your reflections on Notesheet 5. ⇒ Set aside additional time for group discussion of your individual learning from the

lesson study cycle.


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Notesheet 5: Reflection on the Lesson Study Cycle Learnings from the Lesson Study Cycle

Implications/Action Steps for my own practice or more broadly

- About Mathematics /Proportional Reasoning

- About Curriculum

- About Students

- About Learning With Colleagues


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PART 5: REFERENCES

American Association for the Advancement of Science (AAAS). (2001). Atlas of science

literacy. Washington, DC: Project 2061 & National Science Teachers Association (NSTA).

Balanced Assessment Mathematics Collaborative (2001). Balanced assessment elementary/ middle grades/ high school/ advanced high school assessment packages 1 & 2, Balanced Assessment Project. White Plains, NY Dale Seymour Publications

Ball, D. L., & Bass, H. (2000a). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics (pp. 83-104). Westport, CT: Ablex.

Ball, D. L., & Bass, H. (2000b). Toward a practice-based theory of mathematical knowledge for teaching. In B. David & E. Simmt (Eds.), Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group, (pp. 3-14). Edmonton, AB: CMESG/ GCEDM.

Ball, D., Bass, H., Hill, H., & Thames, M. (2006). What is special about knowing mathematics for teaching and how can it be developed? Presentation to the American Federation of Teachers, Teachers’ Program and Policy Council, May 2006, Washington, DC. Downloaded October, 2006 from: http://www-personal.umich.edu/~dball/presentations/053106_Shanker.pdf

Bright, G. & Litwiller, B. (2002). Classroom activities for making sense of fractions, ratios, and proportions: 2002 Yearbook. Reston, VA: National Council of Teachers of Mathematics.

Cai, J., & Sun, W. (2002). Developing students’ proportional reasoning: A Chinese perspective. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 NCTM Yearbook (pp. 195-205). Reston, VA: National Council of Teachers of Mathematics.

Cameron, A., Jacob, B., Fosnot, C. T., & Hersch, S. B. (2006). Young mathematicians at work: Working with the ratio table, grades 5–8. (Facilitators Guide) Portsmouth, NH: Heinemann.

Cobb, P., Yackel, E., &Wood, T. (1989). Young children’s emotional acts while doing mathematical problem solving. In D. B. McLeod & V. M. Adam (Eds.), Affect and mathematical problem solving: A new perspective (pp. 117-148). New York: Springer-Verlag.;

Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10(1&2), 113-163.

Cramer, K. & Post, T. (1993). Making connections: A case for proportionality. Arithmetic Teacher, 60(6), 342–346.

Cramer, K., Post, Thomas, & Currier, S. (1993). Learning and teaching ratio and proportion: research implications. In D. Owens, (Ed.) Research ideas for the classroom (pp. 159-178). NY: Macmillan Publishing Company.

DeAnda, L., Hurd, J., & Fisher, L. (2004). Proportional reasoning lesson for 5/10/04. San Jose, CA: Silicon Valley Mathematics Initiative.


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Dolk, M. & Fosnot, C.T. (2006). Young mathematicians at work: Working with the ratio table, grades 5–8. (CD-ROM) Portsmouth, NH: Heinemann.

Fernandez, C., & Yoshida, M. (2004). Lesson Study: A case of a Japanese approach to improving instruction through school-based teacher development. Mahwah, NJ: Lawrence Erlbaum.

Foster, D., Noyce, P. & Spiegel, S. (2007). When assessment guides instruction: Silicon Valley’s Mathematics Assessment Collaborative. In Alan Schoenfeld (Ed.), Assessing mathematical proficiency (pp. 137-154). New York: Cambridge University Press.

Hironaka, H. & Sugiyama, Y. (2006). Mathematics 1–6. [English translation of Shintei Atarashii Sansu.] Tokyo: Tokyo Shoseki.

Jackson, B, (2004). Equivalent Ratios (lesson plan). Paterson, NJ: Paterson School 2. Kaput, J. & West, M. (1994). Missing-value proportional reasoning problems: factors

affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 237–292). Albany, NY: State University of New York Press.

Kenney, P. A., Lindquist, M. M., & Heffernan, C. L. (2002). Butterflies and caterpillars: Multiplicative and proportional reasoning in the early grades. In B. Litwiller & G. Bright (Eds.), Making sense of fractions, ratios, and proportions: 2002 NCTM Yearbook (pp. 87-99). Reston, VA: National Council of Teachers of Mathematics.

Kodaira, K. (Ed.). (1992). UCSMP textbook translations, Japanese Grades 7, 8, and 9 Mathematics. Chicago: The University of Chicago School Mathematics Project. [Note 3 textbooks; 1 for each grade]

Lamon, S. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Second edition. Mahwah, NJ: Erlbaum.

Lamon, S., (2005) More in-depth discussion of the reasoning activities in “Teaching fractions and ratios for understanding.” Second edition. Mahwah, NJ: Erlbaum.

Langrall, C.W. & Swafford, J. (2000). Three balloons for two dollars: Developing proportional reasoning. Mathematics Teaching in the Middle School 6, 254–261.

Litwiller, B. & Bright, G. (2002). Making sense of fractions, ratios, and proportions: 2002 Yearbook. Reston, VA: National Council of Teachers of Mathematics.

Lo, J., Watanabe, T., & Cai, J. (2004). Developing ratio concepts: An Asian perspective. Mathematics Teaching in the Middle School, 9(7), 362–367.

Lappan, G., Fey, J.T., Fitzgerald, W.M., Friel, S.N., & Phillips, E.D. (2002). Connected mathematics. Upper Saddle River, NJ: Prentice Hall.

Lewis, C. (2002). Lesson study: A handbook of teacher-led instructional change. Philadelphia: Research for Better Schools.

Mills College Lesson Study Group. (2004). Caterpillars [video, 6 minutes]. MCLSG: Oakland, CA.

National Research Council. (2001). Adding it up: Helping children learn mathematics (pp. 241-244). J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.


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Rational Number Project (n.d.). Rational Number Project URL. Available at: http://www.education.umn.edu/rationalnumberproject/

Smith, M.S., Silver, E. A., Stein, M. K., Boston, M., Henningson, M. A., & Hillen, A. F. (2005). Improving instruction in rational numbers and proportionality. Using cases to transform mathematics teaching and learning (Vol. 1). New York: Teachers College Press.

Tournaire, F., & Pulos, S. (1985). Proportional reasoning: a review of the literature. Education Studies in Mathematics, 16, 181–204.

Van de Walle, J. A. (2004) Developing concepts of ratio and proportion (Ch. 18). In Elementary and middle school mathematics: Teaching developmentally. Fifth Edition Boston: Pearson Education, Inc.

Wang-Iverson, P., & Yoshida, M. (2005). Building our understanding of lesson study. Philadelphia: Research for Better Schools.

Watanabe, T. (2003). Teaching multiplication: An analysis of elementary school mathematics teachers’ manuals from Japan and the United States. The Elementary School Journal, 104, 2, 111-125.

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