area of polygons lesson study toolkit

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

Area of Polygons 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 lesson study toolkits in collaboration with lesson study groups across the United States. 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


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INTRODUCTION AND PURPOSE OF TOOLKIT

This toolkit is designed for lesson study teams working to improve student learning about area of polygons. A series of self-directed activities explore mathematical tasks, instructional materials, and research articles. Our hope is that the toolkit will put some useful materials at your fingertips and support your exploration of them. 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, meeting reflection forms and final 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 area of polygons. 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.

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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⇒ We suggest that you begin your work together by setting norms.

SETTING NORMS IN YOUR LESSON STUDY GROUP

What would make this lesson study group a supportive and productive site for your mathematical learning?

• 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 the area of polygons. To help you see how your own ideas are changing, this section begins with a blank “concept map” where you can record your initial thoughts about student learning of the area of polygons. Three student tasks from the National Assessment of Educational Progress (NAEP) follow, for you to solve and analyze. These tasks, labeled Carpet, Polygon on a Grid, and Pentagon, together begin to illuminate the wide range of knowledge students need in order to solve problems related to area of polygons. 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 area of polygons, 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

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

⇒ 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. All tasks are from the National Center for Education Statistics, National Assessment of Education Progress (NAEP) http://nces.ed.gov/nationalreportcard/itmrls/startsearch.asp

CARPET TASK - GRADE 4 A rectangle carpet is 9 feet long and 6 feet wide. What is the area of the carpet in square feet?

A) 15 B) 27 C) 30 D) 54

Notesheet 1 on Carpet Task Instructions: Solve the problem individually and keep track of your solution strategies and ideas below, so you can refer to them later in group discussion. My Work on this Problem

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

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Likely Students Approaches to Solving the Carpet Task Update your concept map: Update your concept map to reflect any new thoughts about (1) what understandings students need to develop about area of polygons and (2) what tasks and experiences enable them to develop these understandings. Question for later discussion with team members: What solution methods are particularly important for students to understand, and why?

POLYGON ON A GRID - GRADE 8 What is the area of the shaded figure?

A) 9 square centimeters B) 11 square centimeters C) 13 square centimeters D) 14 square centimeters


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Notesheet 2 on Polygon on a Grid Task Instructions: Solve the problem individually and keep track of your solution strategies and ideas below, so you can refer to them later in group discussion. My Work on this Problem Likely Students Approaches to Solving the Polygon on a Grid Task Update your concept map: Update your concept map to reflect any new thoughts about (1) what understandings students need to develop about area of polygons and (2) what tasks and experiences enable them to develop these understandings. Question for later discussion with team members: What solution methods are particularly important for students to understand, and why?


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PENTAGON – GRADE 8

a) What is the area in square units, enclosed by the pentagon above?

b) On the figure below, draw a different pentagon that has the same area as the one show. (Be sure the pentagon that you draw does not look like the one shown when it is turned in a different direction.)


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Notesheet 3 on the Pentagon Task Instructions: Solve the problem individually and keep track of your solution strategies and ideas below, so you can refer to them later in group discussion. My Work on this Problem Likely Students Approaches to Solving the Task Update your concept map: Update your concept map to reflect any new thoughts about (1) what understandings students need to develop about area of polygons and (2) what tasks and experiences enable them to develop these understandings. Question for later discussion with team members: What solution methods are particularly important for students to understand, and why?


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⇒ As a group, discuss the tasks, with a focus on identifying solution methods you

consider particularly important for students to understand. National student performance data are provided below for each NAEP task. What do the performance data and student work suggest about the aspects of the mathematics that are particularly difficult for students?

CARPET TASK - 2005 NATIONAL PERFORMANCE RESULTS Solution: D) 54

Score Percentage of students Correct 19 % Incorrect 81 %

Omitted 0 %

0 100 Note: These results are for public and nonpublic school students. Percentages may not add

to 100 due to rounding. POLGYON ON A GRID - 2005 NATIONAL PERFORMANCE RESULT Solution: B) 11 squares centimeters

Score Percentage of students Correct 77 % Incorrect 22 %

Omitted | 1 %

0 100

Note: These results are for public and nonpublic school students. Percentages may not add to

100 due to rounding.


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PENTAGON – 2005 PERFORMANCE RESULTS Solutions:

a. 16 square units b. Below are some of the possible correct responses

Incorrect – both parts a. and b. incorrect Partial – part a. or part b. correct Correct – parts a. and b. correct

Score Percentage of students Incorrect 48 % Partial 48 % Correct 2 % Omitted 2 % Off task 0 %

0 100

Note: These results are for public and nonpublic school students. Percentages may not add to

100 due to rounding.

⇒ Individually, take a few minutes to update your concept map, to capture any additional ideas you want to remember from the group discussion of the tasks.


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 area of polygons; • 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 in this section enable you to explore the sequence of experiences by which students learn about area of polygons 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. A. Try Out a Problem From the Japanese Curriculum ⇒ As a way to begin exploration of the Japanese curriculum, we suggest that you read pages

68-71 of the Japanese text books Mathematics for Elementary School 5A, in which students build understanding of the area of a parallelogram, through a hands-on task in which they transform a parallelogram into shapes (such as rectangles) for which they know how to calculate the area. The key information from these pages is reproduced here.

⇒ First, students try to transform the parallelogram on the previous page into a rectangle with

the same area as the original parallelogram. If this task is new to you, you might want to try it, using scissors and graph paper. How many different ways can you use your knowledge of rectangle area to find the area of this parallelogram? The next page shows 3 solutions.


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Next, students apply these strategies to a more difficult problem, in which the height of the parallelogram falls outside the base. ⇒ Can you find the area of this parallelogram on Notesheet 4 by changing the shape?

Record your solutions and thinking on Notesheet 4, found on the next page. (When students grapple with this problem, on p.72 of Mathematics for Elementary School 5A, they have already learned formulas for finding the area of rectangles, squares, and parallelogram with height inside, so you can assume knowledge of these).

Subtract areas of two triangles. If you put together the two triangles you can make a rectangle. You can subtract the rectangle from the large rectangle you created originally.


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Notesheet 4: Area of Parallelogram Task Instructions: Solve the problem individually and keep track of your solution strategies and ideas below, so you can refer to them later during group discussion.

(1) Find the area of this parallelogram by changing the shape. Record your work here: (2) Can you explain why the formula for area of parallelogram (developed on page 71 of Japanese textbook Mathematics for Elementary School 5A) Area of parallelogram = Base x Height can be used for this parallelogram?


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Student Work: Area of Parallelogram (Problem from Mathematics for Elementary School 5A, p.72)3

⇒ As you’re waiting for teammates to finish working on the problem, look over the

solution examples produced by 6th graders at Paterson School 2. 3 We are grateful to Makoto Yoshida of Global Education Resources (http://www.globaledresources.com/) for provision of this problem and student work, and to the students of Paterson School Number 2, NJ.


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⇒ Discuss with the team your solutions and ideas from Notesheet 4. Look over the

solution examples produced by 6th graders. ⇒ Once again, update your concept map to reflect any new thinking sparked by the

group discussion.


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PLEASE NOTE: Section B and C of this toolkit require the following resources: • Japanese Mathematics for Elementary School textbooks: Arithmetic grades 1-

6, available from www.globaledresources.com. Cost: $128.65 plus shipping. • Japanese Elementary Course of Study: Arithmetic grades 1-6 available

from www.globaledresources.com. Cost: $12.99 plus shipping. • Units from the Japanese textbook Teachers’ Manual related to

the area of polygons. Free, downloadable from www.lessonresearch.net/nsf_toolkit.html

B. Examine the Textbooks Table 1 provides page numbers and brief summaries of the units related to area of polygons in the textbook series Mathematics for Elementary School. Since the textbooks have many illustrations and sparse text, the reading 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 for understanding area of polygons – starting from recognizing and constructing shapes, through transforming shapes while keeping their area the same and to the development of formulas for area of rectangles, parallelograms, triangles, and trapezoids. The Japanese Course of Study treats area of polygons as the product of coordinated understanding of three mathematical domains: geometry, measurement, and quantitative relations (which includes writing mathematical expressions). Translated excerpts from the Teachers’ Manual show how many lessons are devoted to each textbook unit and may enable you to “read between the lines” of the textbook and see what teachers are hoping students will gain from the lessons. The Teachers’ Manual provides a teaching plan, objectives, hatsumon (key questions for students) 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; see, for example, the discussion of what constitutes understanding of “base” and “height” (for parallelograms, triangles, and trapezoids) in connection with the activities intended to build this understanding in Grade 5. ⇒ Briefly examine the Flow of Japanese Lesson on the next page to familiarize yourself

with a typical structure of a mathematics lesson in Japan.


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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.”

⇒ 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 Student Learning Activities listed in column 2 of Table 1 for all grades, then select one or several units to focus on. Grade 4B, Area, pp.22-37 specifically considers the meaning of area.)

⇒ Stop about halfway through your reading (perhaps after grade 3) 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.

⇒ When finished, discuss again with your team, focusing on the questions above. ⇒ Once again, update your concept map to reflect any new thinking sparked by the

group discussion.


Table 1. Learning Trajectory for Area of Polygons in Japanese Curriculum Grade Level, No. of 45-Min. Periods, Textbook Pages

Student Learning Activities

What do students come to understand from this unit? What tasks and experiences build their understanding? Implications for Our Team (e.g., How does our curriculum build these understandings? How do we know if our students have the understandings?)

Gr.1: Playing with Shapes, pp. 53–56 4 periods

Notice, classify, and make 3-D shapes. Trace outlines of block faces to make rectangle, square, triangle, circle. Classify resulting shapes.

Gr.1: Which One is Longer? pp. 57–58 4 periods

Use 3 measurement strategies: direct, indirect and non-standard units. In final task, students figure out starting point for measurement, using grid.

Gr.1: Making Shapes, pp. 101–104 4 periods

Compose, decompose, and manipulate geometric figures, using plastic triangles and squares, sticks, and dot grid.

Gr. 2A Length (1), pp. 25–33 8 periods

Estimate length, measure length with non-standard units and centimeters. Draw straight line, learn what a straight line is. Write expression for measurement of two-segment line.

Gr. 2B Triangles and Quadrilaterals, pp. 2–12 8 periods

Draw, define, and distinguish triangles and quadrilaterals. Make and define right angle. Use set-square. Make, define and distinguish square, rectangle, and right triangle.


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Grade Level, No. of 45-Min. Periods, Textbook Pages



Gr. 2B Shapes of Boxes, pp. 68–73 7 periods

Identify rectangles and lengths of their sides on 3D solids. (Identify number of faces, vertices, and edges on rectangular prisms.)

Gr. 3B Triangles, pp. 2–12 8 periods

Make triangles with sticks, classify and define isosceles and equilateral triangles, and draw both types with compass. Use set-square and folding to compare angles.

Gr. 4A Angles, pp. 16–24 7 periods

Measure angles, using non-standard measure and protractor. Draw angles, write expressions for angle measurements.

Gr. 4A Quadrilaterals, pp. 80–99 14 periods

Identify and draw perpendicular and parallel lines. Understand that the width between parallel lines is always equal and a straight line intersects parallel lines at equal angles. Make trapezoids, parallelograms and rhombuses and investigate their characteristics; draw diagonals of each and investigate their length and intersection angle.

Gr. 4A Tangrams, pp. 100–101 2 periods

Make shapes from paper triangles, square, parallelogram—attending to lengths of sides, measure of angles.


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Grade Level, No. of 45-Min. Periods,

Textbook Pages



Gr. 4B Area, pp. 22–37. See Teachers’ Manual also 11 periods

Consider meaning of area, think about how to measure a rectangle and square, finally using a square centimeter for measurement. See square centimeters that are not square in shape. Build formula for area of rectangle and square, create shapes with a given area, find area of composite figures. Link numerical expressions to areas of decomposed figures. Learn measurement units for large areas.

Gr. 5A Congruent Shapes, pp. 52–67 See Teachers’ Manual also 10 periods

Find, manipulate, investigate and draw congruent triangles and quadrilaterals, with focus on congruent shapes, sides, and angles, to build perception of shape. Investigate which quadrilaterals have congruent triangles when divided by diagonal. Find angle sum of triangle, quadrilateral, pentagon, hexagon.

Gr. 5A Area of Quadrilaterals and Triangles, pp. 68-84. See Teachers’ Manual also 15 periods

Use rectangle area formula and transformation to find formulas for areas of parallelograms, triangles, trapezoids. Use grid, then base and height. Coordinate mathematical expressions, measurement, and geometry.


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C. Examine the Japanese Course of Study The Course of Study is densely packed with information. You may want to become aware of it as a reference work, rather than trying to read and digest it now. Its big ideas provide a roadmap for looking at the textbooks. For example, b. Overview of contents on page 32 summarizes the geometry content of the Japanese elementary curriculum, showing the grade level at which each geometric figure is introduced, and the ideas (e.g., composition-decomposition, equality of angles or sides, noticing parallel and perpendicular) to be emphasized. One big idea from this document is that in grades 4 and 5 students must coordinate understandings from three mathematical domains – geometry, measurement, and quantitative relations – in order to understand and use area formulas. Prior to these grades, the necessary understandings are developed in large part separately, allowing students to focus on separate aspects (e.g., measuring length, perceiving congruent shapes) and preparing students to coordinate them and use them fluently in grades 4 and 5. ⇒ Take a look at the Japanese Course of Study to get an overview of its contents.

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 area of polygons. Resources for this work include research articles, video, and lesson plans. A. Read and Discuss Research Table 2 below provides a selection of articles that your group may find interesting. We include one article for you to read in this version of the toolkit; Research Summary: Measurement of Geometric Objects © 2008 by Education Development Center, Inc. from Fostering Geometric Thinking. Portsmouth, NH: Heinemann. Used with permission. All rights reserved. This article can be downloaded at the following website link: www.lessonresearch.net/nsf_toolkit.html


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Table 2 (part 1). Quick Reference Guide to Suggested Resources Resource, Author(s)

Focal grade level(s)

Elements of student understanding

Recommended Reading 1. Developing measurement

concepts, Van de Walle Elementary, Middle

Focus for discussion: This comprehensive chapter treats measurement of different attributes (length, area, volume, etc.). What understandings are important to measurement of area? What experiences develop these understandings?

2. Research Summary: Measurement of Geometric Objects from the Fostering Geometric Thinking (FGT) project

Elementary, Middle

Focus for discussion: This article reviews challenges for students in understanding linear and area measurement, and points out that students may not be able to mentally structure the space within a rectangle even if they can use a formula to calculate rectangle area. What experiences help students learn to mentally structure the space within a rectangle?

3. The Van Hiele Model of the Development of Geometric Thought, by Mary Crowley.

All levels Focus for discussion: This article outlines levels of student understanding of geometric figures but asserts that “progress through the levels is more dependent on the instruction received than on age or maturity.” What kinds of instruction are recommended? Are there implications for your concept map?

Additional Reading 4. Linear and area measurement

in PreK to grade 2, Stephan & Clements

Pre-K–2 Understanding length measurement; partitioning; unit iteration; conservation of area

5. Young children’s composition of geometric figures…, Clements, Wilson & Sarama

Pre-K–2 Trajectory of student behaviors associated with understanding pattern block geometric shapes; consideration of side length, angles

6.Cultures of mathematics instruction in Japanese and American elementary classrooms, Stigler, Fernandez, & Yoshida

Gr. 5 Area of triangle, square units, grid representation (in U.S. class), types of triangles

7.The van Hiele model of thinking in geometry among adolescents, Fuys, Geddes, & Tischler

Gr. 6, 9 Van Hiele levels, classification of two-dimensional shapes, grouping shapes, geometric language (e.g., congruence), angle measurement and angle relationships in polygons, properties of shape, area of rectangles and other polygons, grids of squares,

8.Understanding area and area formulas, Battista

High school Mathematical properties of area; relationship between units of length and units of area (i.e., square units); area involves covering space

9.Using classroom assessment tasks and student work as a vehicle for teacher professional development, Shannon

High school Area formula for triangles and quadrilaterals; additive nature of area; coordination of symbolic expressions with geometric figures

10.Proofs without words, Nelsen High school Relationships of area to formulas in high school algebra

⇒ 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 area of polygons? 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 Video and Lesson Plan From Can you Find the Area?, a series of three fourth-grade classroom lessons on area of quadrilaterals, we have selected several short video segments. The segments highlight issues in learning about area of polygons, including: measurement of sides; decomposing and reassembling shapes to find their areas; and associating numerical expressions with areas. 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 for “Can You Find the Area?” and

consider the following questions: o In the Day 1 lesson procedure, why are students asked to make a 4 X 5 rectangle

on a geoboard? How might they respond? o In the Day 1 lesson procedure, what strategies might students choose to find the

area of the composite figure? o On Day 3, students are asked to make a four-sided shape with the same area as a

given shape. What might be some reasons to pose this problem? ⇒ Watch the video segments from “Can you Find the Area?” The five video clips

range from 1-4 minutes and are taken from three 1-hour lessons taught on consecutive days to the same class: o Clips 1-3 come from lesson 1. We see how students measure a rectangle side on a

geoboard, how they see and measure area, and how they connect area to numerical expressions.

o Clips 4 and 5 are from the subsequent lessons. They explore how students transform a parallelogram to find its area.

⇒ Discuss as a group whether and how the lesson plan and video add to your thinking

about two questions: o What do students need to understand about area of polygons? 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. 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 area of polygons, 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


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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 D. [Suggested Additional Step for Secondary Teachers: Examine TIMSS Grade 8

Video and Other Secondary Resources] Secondary teachers (and others, if time permits) may find it useful to see a secondary lesson and to read about activities specific to older students. Five items from Table 2 are especially recommended for this purpose: three articles denoted “high school” (#’s 8-10, by Battista, Shannon and Nelsen); the TIMSS video (#14); and the Japanese grade 7- 9 textbook excerpts (#15). You may wish to begin with the TIMSS Japanese grade 8 lesson, in which students use their knowledge of polygons and area to transform a quadrilateral into a triangle of equal area, and a pentagon into a quadrilateral of equal area.

Table 2 (part 2) Quick Reference Guide to Resources

Lesson Plans and Video 11. Can You Find the Area? (Lesson plan and video, MCLSG)

Gr. 4 Measurement of quadrilateral sides, meaning of area, transformation of quadrilateral to find area, expression. (available at www.lessonresearch.net/nsf_toolkit.html)

12. Which is Longer? (video, 14 minutes), Global Education Resources

K–2 Comparison of unit lengths (available at http://www.globaledresources.com/resources.html)

13. Seeing Shapes Within Shapes [Gr 2 lesson], MCLSG (video)

Gr. 2 Composition of shapes from other shapes, direct comparison of shapes, comparison of areas (available at www.lessonreserch.net/nsf_toolkit.html)

14. TIMSS Eighth-Grade Mathematics Lessons: Japan, US Dept. Educ, OERI) (video)

Gr. 8 Using knowledge of area and polygons to transform a quadrilateral (or pentagon) into a triangle (or quadrilateral) with the same area. (available at www.lessonresearch.net/nsf_toolkit.html)

15. Japanese Grade 8,9 textbook, Kodaira

Gr.8.9 Relationships of area to HS algebra (available from http://ucsmp.uchicago.edu/Transl.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 area of polygons 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 area of polygons 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.


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⇒ Set aside additional time for group discussion of your 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 /Area of Polygons

- About Curriculum

- About Students

- About Learning With Colleagues


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PART 5: REFERENCES Battista, M. (1982). Understanding area and area formulas. Mathematics Teacher, 75(5),

362–368. Clements, D., Wilson, D. C., & Sarama, J. (2004). Young children’s composition of

geometry figures: A learning trajectory. Mathematical Thinking and Learning, 6(2), 163–184.

Crowley, M. (1987). The van Hiele model of the development of geometric thought. In M. Lindquist & A. P. Shulte (Eds.), Learning and teaching geometry, K–12 (1987 yearbook, National Council of Teachers of Mathematics, pp. 1–16). Reston, VA: National Council of Teachers of Mathematics.

Driscoll, M. & Mark, J. (2007) Fostering Geometric Thinking (FGT) Toolkit. 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.

Fuys, D., Geddes, D. , & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents (National Council of Teachers of Mathematics Monograph No. 3). Reston, VA: National Council of Teachers of Mathematics.

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

Japanese Ministry of Education, Culture and Sports. (2004). Elementary school teaching guide for the Japanese course of study: Arithmetic (grades 1–6) (A. Takahashi, T. Watanabe, & M. Yoshida, Trans.) Madison, NJ: Global Education Resources. (Original work published 1989.)

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]

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

Lewis, C. (2003). Can You Find the Area? [video: 47 minutes]. Oakland, CA: Mills College Lesson Study Group

Mills College Lesson Study Group (2007). Seeing Shapes Within Shapes. [video: 8 minutes]. Oakland, CA: Mills College Lesson Study Group.

National Center for Education Statistics (2007). NAEP Questions. Available from http://nces.ed.gov/nationsreportcard/about/.

Nelson, R. B. (1993). Proofs without words: Exercises in visual thinking. Washington, DC: Mathematical Association of America.

Shannon, A. (2003). Using classroom assessment tasks and student work as a vehicle for teacher professional development. In Next steps in mathematics teacher professional development, grades 9—12: Proceedings of a workshop. CD available from the National Academy Press.

Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.


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Stephan, M. & Clements, D. (2003). Linear and area measurement in prekindergarten to grade 2. In D. Clements (Ed.), National Council of Teachers of Mathematics yearbook on measurement (pp. 3–16). Reston, VA: National Council of Teachers of Mathematics.

Stigler, J. W., Fernandez, C., & Yoshida, M. (1996). Cultures of instruction in Japanese and American classrooms. In T. Rohlen & G. LeTendre (Eds.), Teaching and learning in Japan (pp. 213–247). New York: Cambridge University Press.

Strutchens, M., Martin, W. G., & Kenney, P. (2003). What students know about measurement: Perspectives from the National Assessment of Educational Progress. In D. Clements (Ed.), National Council of Teachers of Mathematics yearbook on measurement (pp. 195–207). Reston, VA: National Council of Teachers of Mathematics.

Takahashi, A. (2002). Lesson plan (for lessons in video Can You Find the Area?). Grade 4: Finding the area of shapes. Lesson Study Workshop, San Mateo, April 15–18, 2002, http://lessonresearch.net/AreaGeoboard_4thgrade.pdf

Tanaka, H. (2007) Which is longer? [video], retrieved July 18, 2007 from www.globaledresources.com/resources.html# Tokyo Shoseki Co Ltd. Editorial Division (2000). Shintei Atarashii Sansu Kyoushiyou

Shidousho. (Teachers’ Manual, Mathematics for Elementary School) Tokyo: Tokyo Shoseki Co., Ltd. (Draft translation excerpts by Mills College Lesson Study Group)

U.S. Department of Education (1997). Eighth-Grade Mathematics Lesson: Japan. From video Attaining Excellence: Eighth-Grade Mathematics Lessons: United States, Japan, and Germany, ORAD 97-1023, Washington, DC: US Department of Education, OERI. Lesson graph downloaded from www.rbs.org/media/mathsci/timss/resource_guide/lesson_graphs/JP2lessongraph.pdf on July 6, 2007.

Van de Walle, J. A. (2004) Elementary and middle school mathematics: Teaching developmentally. Fifth Edition (Ch. 19, Developing Measurement Concepts). Boston: Pearson Education, Inc.

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

area of polygons lesson study toolkit

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