Japanese Lesson Study Summary

Tuesday, July 8, 2003

Yesterday, Judy, a teacher of teachers from New Zealand, had joined us and was a great resource and inspiration. Today, we were pleased to welcome Lisa Snow, from New Mexico, as an observer and resource. Lisa talked a bit about how lesson study had been used in the schools that she oversees in her district and what a huge impact it has made. From a start of 24 teachers in the program, it has grown to now include 100 teachers. In their district they have groups of three to six teachers, with teachers either working on lessons from scratch or reshaping lessons from their curriculum. She suggested a website for us to use (mathstar.nmsu.edu) and downloaded some forms for us to use as templates.

Jerry made another trip over to Park City High School to talk with Lars and our Geometry students to make sure that they have the schedule for next Thursday. We will be hosting them for lunch from 12:30 to 1:00, teaching the lesson from 1:00 to 2:00, and having them entertained by Jim King and then Leslie Ward for the remaining hour. Their class ends at 3:00 each day. Jerry informed us that they are further ahead than expected in their curriculum and that they will be starting Chapter 11 on Volume right after this lesson.

We discussed whether we wanted our lesson to be an introductory lesson or one that goes all the way to scaling and then dimensionality. Lisa also asked us if we intend to videotape the lesson. If so, we need to get release forms from the students. Gail has said that we would like to videotape the lesson, so we will make arrangements to have the students sign forms. We also need to request volunteers for our practice lesson on Monday. Joyce has agreed to e-mail Lars as an additional contact. We also need to write up observation questions for the observers.

Jennifer started merging our two documents from yesterday so that we can continue working on writing our lesson. Both Celeste and Tony have volunteered to teach the lesson. Here are the notes.

Japanese Lesson Study Lesson

Topic: Scaling, Measurement and Dimensionality - Stair-Step Fractal

  1. Overarching Goal:
  2. Mathematical Objective- When considering the measurements length, area, and volume, students will be able to clearly articulate the effect of scaling one measurement on the two remaining measurements
  3. How does this objective fit into a unit? Students will understand the relationship between dimensionality and scale factor
  4. What is the pre-requisite knowledge?
    How to find the area of a rectangle
    How to find the volume of a rectangular solid
  5. Math problem - hook, problem on which they will work
    1. When students enter the class, they will be given printed (and picture) directions on how to construct the 1st iteration of the stair step fractal. After students finish construction, then the teacher will address the class and say: You can make some amazing things with paper. Mathematicians often make models to describe or understand mathematical concepts. "Is this amazing?" Well, why not? Well, what happens if we repeat the process? -- Walk the students through the 2nd and 3rd iterations "Pop it up again? amazing? What interesting things do you see and what do you wonder when you look at this figure?"
      Potential responses:
      1. It looks like stairs – Response: are all the stairs the same?
      2. I see squares / boxes
      3. The stairs are getting bigger
      4. The figure is symmetrical – Response” What type of symmetry? or how?
      5. It looks like how you make snowflakes
      6. It's a cool design. - What makes it cool?
      7. It looks like a building - What type of building
      8. It looks like stairs - what might these stairs be used to model?
      9. The little steps are half the size of the medium steps, medium steps are half of larger steps
      10. There are small, medium, and large squares
      11. All the shapes are similar and the same kind of shape
      12. There are different sized steps.
      13. It looks like a fractal.
      14. The middle step is centered on the large step in the same way that the small step is centered on the middle step.
    2. Key Question: What relationships exist between the stairs in this model?

_____________This is where we ended on Tuesday______________

VI.  Student strategies – how might students so the problem
  A. Students might use inspection as their strategy.  
     i.	The teacher can then push them what do you mean small, medium? Large?  Can 
you quantify that?
  B. Students might measure side length or area or volume
     i.	If students do not mention all three, after discussing their observations, 
ask what measurement is missing
  C. Units - the small one is how many of the medium etc.


VII. Teacher responses – anticipate how you will respond to student questions
          See above
  A. Discuss w/ students a counter-example (We need to bring a model for this).  
Area the same (side view) but front view shows that there is more than small medium 
large.  Also show an example from front view but not side-view.  What makes our 
figure unique?  Why do we only have S, M and L (1 to 2 from the side and 1 to 4 from 
the front) - similar figures.     

VIII. Summing up - use the kids words not a pre-written script.  Otherwise they 
have no reason to do the activity
  A. What about the case of scaled by 3 (use a model??/)?

IX. Evaluation - what evidence will we have that kids understood?

Potential HW Questions:
Suppose that each dimension of the figure is scaled with a different scale factor, 
without actually doing extensive computation, can you predict the relationship 
between the volume of the original figure and the scaled figure?


Suppose the figure we see is a set of stairs. How many people can fit on the stairs 
if ___ fit on the top stair?

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IAS/Park City Mathematics Institute is an outreach program of the School of Mathematics
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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.
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