Japanese Lesson Study Summary

Thursday, July 10, 2003

Today we welcomed Stacey and Roy from Provo School District to our group. They were gracious enough to try out our lesson with us.

We received an e-mail from Lars saying the plans look great and he wished he wasn't missing the lesson (he'll be at a conference.

Suzanne took our picture and asked us if we would be willing to write an article for AP Central an on-line website. We agreed that we would like to pursue this and asked Suzanne to get us details.

We continued to work on our lesson. The following is the work to date:

Japanese Lesson Study Lesson

Topic: Scaling, Measurement and Dimensionality - Stair-Step Fractal
I. Overarching Goal: 

II. 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

III. How does this objective fit into a unit? Students will understand the 
relationship between dimensionality and scale factor

IV. What is the pre-requisite knowledge?
    How to find the area of a rectangle
    How to find the volume of a rectangular solid
V. Math problem - hook, problem on which they will work
      A. 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?"
          i. Brainstorm individually before taking student answers.
          Potential responses:
          ii.  It looks like stairs – Response: are all the stairs the same? 
          iii. I see squares / boxes
          iv.  The stairs are getting bigger
          v.   The figure is symmetrical – Response” What type of symmetry? 
               or how? 
          vi.  It looks like how you make snowflakes
          vii. It's a cool design. - What makes it cool?
          viii.It looks like a building - What type of building
          ix.  It looks like stairs - what might these stairs be used to model?
          x.   The little steps are half the size of the medium steps, medium 
               steps are  half of larger steps
          xi.	There are small, medium, and large squares
          xii.	All the shapes are similar and the same kind of shape
          xiii.	There are different sized steps.
          xiv.	It looks like a fractal.
          xv.	The middle step is centered on the large step in the same way 
                that the small step is centered on the middle step.
      B. Key Question: What relationships exist between the stairs in this model? 
      Find a partner and compile your conclusions in an organized way (In Pairs)
VI. Student strategies - how might students solve the problem
(In teacher responses be sure to be clear about which dimensions students are 
      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? Are they all rectangles?  Are there any squares?
          ii.   Students may say 1 large, 2 medium, 4 small.  In this case teacher 
                should say affirm it but not lead students to believe that it is 
                the key idea
      B.  Students might measure side length and/or area and/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  

VIII.	Summing up - use the kids words not a pre-written script.  (Otherwise they 
had no reason to do the activity)
Create a t

IX.	Evaluation - what evidence will we have that kids understood?
      A. Create a table and ask students to fill in the scaled by 3 row.  
      Present students with the scaled by a factor of 3.  Deal with misconceptions.

**Note introduce the vocabulary of scale by after students see the need – 5thng, 
6thing etc.

**You can keep scaling up to the n case and then go to cases where n < 1
Potential HW Questions:
    1) 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?
    2) Suppose the figure we see is a set of stairs. How many people can fit on the 
stairs if ___ fit on the top stair?
    3) If 1/4 inch on your model represents 1 foot, how many cubic yards of concrete would it 
take to make this a solid staircase?
    4)	How many cubic yards of concrete would it take to create the staircase if 
each face is created with a wall of concrete 6 inches thick?   
    5)  You are building a deck in your backyard.  You order 1 cubic yard of concrete.  
It turns out that you only use 1/2 of the concrete you ordered.  How long of a 
sidewalk can you pour with the leftover concrete  if the sidewalk is 3 feet wide and 
4 inches deep?

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. 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.