Japanese Lesson Study Summary
Monday  Friday, July 1923, 2004
All four days this week were dedicated to writing and revising our lesson plan. The best record of the week is the change shown in the progress of the lesson. Here are the daily electronic versions of the lesson.
Monday:
To facilitate participation and progress, we formed two smaller groups and thereby, came up with two different visions of the lesson. Megan and Remy submitted the following drafts:
Megan Taylor
Subject: Draft #1 of the Big Question
LESSON STUDY WORKING GROUP INFORMATION
The new Gamecube game, "Superhero City," has just come out in stores.
To win the game you need a certain number of points. At the beginning
of the game you must choose a character. Guinevere starts with 5
points in Level 1 and every level she wins 5 more points. Harry Potter
starts with 1 point in Level 1, and wins the square of the level
number he is on. Spiderman also starts with 1 point in Level 1, but
his points double every new level. When Guinevere beats a level, she
adds her new points to her score. However, Harry Potter and Spiderman
do not.
 How many points would each character have at the end of Level 5?
 Choose one character and graph his/her score from Level 1 to
Level 8.
 Which character would you choose?
Remy's group
PCMI Lesson Study Group 1 Plan (draft)
Over arching goals:
 Lesson will set the stage for development of the meaning of
coefficients and variables
 Lesson will focus on conceptual versus procedural understanding
 Lesson will convey some necessity, importance, beauty of math
Essential questions:
How can change be represented graphically?
Launch:
 Ask students about what does it mean for something to change, e. g.
give examples about something that changes in their life
 Provide time for students to write about their thought
(Need to think of situations that are relevant to students)
Activity 1: Blow a balloon for students to observe change
 First balloon just to observe
 Blow a second balloon for students to draw a picture of change from
their observation
 Ask students what is changing as they see the balloon being blown up
 Provide individual thinking time in regards to what is changing
before they write down their observation
 Have students discuss with partner or in small group
As students are writing about "change," facilitator walks around the
room to make sure who will be called on to get a range of answer and
let them share in the order that will lead to what we want.
What if student draws the change of balloon without using a graph,
what questions can we ask them? (How can we represent change
graphically?)
Talk about how would they represent change and how one variable relate
to another.
Ask students about which representation do they like and why?
Activity 2:
 Students in each group pick a scenario of "change" that they had
shared and graph them
 Exchange the graph with another group to see whether they will be
able to match the scenario
Conclusion:
 Justification of how graph match with scenario
 Debrief about using graph representation
 Ask students about what should we talk about next
Tuesday: Megan took over the typing and combining of the two plans
As you can see, there is a great deal to do in reconciling these two versions. Megan thoughtfully (and expertly) took over the task of compiling the two and with the use of a projector, the lesson study group began combining ideas. The result was this version of the lesson.
Subject: Lesson Study Group TOPIC
PCMI Japanese Lesson Study
July 12  July 31, 2004
Overall Theme: Translation between Math and Real Life
Overall Goals:
 Emphasize the meaning of coefficients and variables
 Lesson will convey some necessity, importance, beauty of Math
 Lesson will focus on conceptual versus procedural understanding
Content Theme: Modeling Change
Overall Goals/ Essential Question(s):
#1
 Students can understand the difference between 2x, x2 and 2x.
"Understanding" means being able to compare the contrast the functions
algebraically, being able to create a table and graph for each
function and being able to describe each with a reallife example.
 Students understand the increase in mathematical complexity as you
move from a linear to exponential model.
vs.
#2
 How can some change be represented graphically?
 Students should understand how change can be represented
graphically. Students can create a reallife scenario for a given
graph. Students can use a description of a situation to sketch a
representative graph.
The three groups came back together and with fantastic work by Megan, we patched the three together and started revising.
Thursday Author: Megan Taylor
Subject: Notes from Lesson Study  7/19
Thursday:
Megan continues to type revisions for us and this is the version we had on Thursday.
Overall Theme: Translation between Math and Real Life
Overall Goals:
 Lesson will emphasize the meaning of coefficients and
variables.
 Lesson will convey some necessity, importance, beauty of Math.
 Lesson will focus on conceptual versus procedural understanding.
Content Theme: Modeling Change
The Essential Question(s):
How can change be represented graphically?
Content Goal(s):
Students should understand how change can be represented graphically.
"Understanding" means students can explain how graphs represent change
and can tell "stories" with graphs. Students can create a reallife
scenario for a given graph and conversely can sketch a graph using a
description of a situation ("a story"). Students should be able to
label graphs and choose quadrants effectively and understand that one
graph can describe more than one situation. Finally, students should
see graphs as showing relationships between two or more variables.
(5 min.)
Launch: Make a list of things in your life that change.
Describe how one of the things in your list changes.
EX: Money
Growing/ Height
Skin color (being tan)
Gaining/Losing Weight
Friends
Teacher
Grade
Classes
Weather
Discussion Questions:
Which of these things will continue to change/are always changing?
What causes the change?
Can you measure change?
(20 min.)
Activity 1: Balloon Observation
 Students observe the teacher blowing up of the balloon.
Teacher blows up a second balloon for students to decide what is
changing and how from their observation.
 What is changing about the balloon?
EX: size
shape
air in the balloon
surface area/ volume
color (hue)
 Students discuss lists with partners or in small group.
 As students are writing about the change(s), the teacher walks
around the room to make sure who will be called on to share to get a
range of answers.
 Students are given a number of graphs relating time and ______ of a
balloon.
 Students (in pairs or groups) describe each graph with a story.
What does this graph show? What is happening?
(25 min.)
Conclusion: Revisitation of the idea of change
 Students revisit the lists made previously about change in their
lives.
 Describe how you would keep track of this change.
 Describe one of the changes from your list using a graph.
 Students create graphs that describe some of the changes from their
lists. Students will not be told explicitly whether to label or not.
 Student pairs/groups switch graphs and must describe what is
happening in words. Graphs without labels may be described very
differently than intended, which will be a
(3 min.)
WrapUp: What is change?
Lesson Planning:
Group A: Instruction to Lesson; Transitions; Wrapup (Bill, Luis,
Megan)
Group B: Balloon Graphs and Activity Overview (Claudia, Cheryl,
Susana)
Group C: Revisitation of Change Lists (Remy, JoAnn, Don)
(20 min.)
Activity 1: Balloon Observation
 Students observe the teacher blowing up of the balloon.
I want you to write about changes in the balloon as I blow it up.
Teacher blows up a second balloon, slowly, then quickly, releasing
some air, etc... and students are asked to describe what is changing from
the observation.
 As students are writing about the change(s), the teacher walks
around the room to make sure who will be called on to share to get a
range of answers.
 What was changing about the balloon? Teacher calls on volunteers.
EX: size
shape
air in the balloon
surface area/ volume
color (hue)
 Students are given the two situations below and are asked to sketch
a graph for each with their partners.
 A balloon is being blown up quickly (with a helium tank, for
example)
 A balloon is blown up, tied off, then "runs into" a sharp object
 Students (in pairs or groups) describe each graph with a story.
What does this graph show? What is happening?
Descriptions ‡ Graphs
Graphs ‡ Descriptions
Friday
On Thursday Claudia agreed to teach the lesson to a group of volunteers from the HSTP next Tuesday and Megan agreed to teach the lesson in Provo to David Wright's group of eighth graders.
download: pcmi.basis.doc
Hi to the Hardcore Lesson Study Planners!
I feel like we got a lot done today (in the end) and am glad we cut out the large chunk we did. I have compiled what I gleaned from our discussions this morning into a semicoherent document. Here are the things, specifically, I think you should be ready to comment on by tomorrow. Yes, you have homework!
 The "Questions," "Student Response" and "Materials" columns are vastly incomplete. I didn't even try to fill in most of what we need here. Please think about these spaces, especially the "Questions" column, as this will be the most important part of the whole lesson!
 Where there were holes in the plan, I made stuff up. Yes, not exactly the Lesson Study model, but at least then we'll have less to write from scratch tomorrow. Nothing major of course, for example I made the time divisions sum to 60 and I added a "Wrapup" at the end of the lesson. Do we want one? I didn't know! So I put it in and we can talk about it.
 As we probably won't have time for the third activity, where students read situations and have to graph them, I created a *sample* worksheet for it just in case and if we have extra time tomorrow (hahahaha) then we can edit that as well.
 Any other comments...please please please let me know! You do not have to print this out as Kathy said she'll get copies made.
Good job everyone!
I decided I should be excited and honored to be able to teach our lesson!
Megan
Back to Journal Index
PCMI@MathForum Home  IAS/PCMI Home

© 2001  2013 Park City Mathematics Institute
IAS/Park City Mathematics Institute is an outreach program of the School of Mathematics at the Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540
Send questions or comments to: Suzanne Alejandre and Jim King

With program support provided by Math for America
This material is based upon work supported by the National Science Foundation under Grant No. 0314808.
Any opinions, findings, and conclusions or recommendations expressed
in this material are those of the author(s) and do not necessarily reflect
the views of the National Science Foundation.
