Japanese Lesson Study Summary

Monday, July 7, 2003

Today we split into two groups again to work on creating lesson plans for our stair- step fractal. Here are the results so far (definitely a work in progress!)

Stair-step Fractal Lesson on scaling (Joyce, Celeste, Tony, and Judy ( New Zealand))

Pre- lesson:
Overarching goal: Making connections between dimensionality and scale factor
Mathematics objective: Students will be able to articulate how changing the length will affect the area and volume.

Hook: You can make some amazing things with paper. Mathematicians often make models to describe or understand mathematical concepts.
Have students make the model.
Key question: What do you see? What observations can you make? What do you wonder?
Collect these responses. Discuss those especially that relate to shapes, similarity, area, volume.
Anticipated student responses:
There are small, medium, and large squares
The little steps are half the size of the medium steps, medium steps are half of larger steps
There are different sized steps.
It's symmetric.
All the shapes are similar and the same kind of shape
It looks like a fractal.
The middle step is centered on the large step in the same way that the small step is centered on the middle step.
Key questions: Calculate the width of each step using the squares as units, the area of the side of each square, and the volume of each step. How do these measurements compare for each of the sizes (small, medium, and large)? How did the area of the side of the step and the volume of the step change as you doubled the length of the step?

Preparation for lesson: Do we want to use graph paper and if so what size? Do we use a paper cutter to ensure that the calculations will be easy to work with? We thought yes, and started working on this.

Teacher question:
If you made another set of cuts, what would stay the same and what would change?
Do you see any relationships between the squares, steps, side lengths, etc.

What different fractals do you see? Linear model like Cantor's Comb, Sierpinski's triangle or gasket, 3-D Sierpinski's gasket, stair step fractal, etc. This is also a solution to the Tower of Hanoi problem (used to demonstrate recursion in beginning programming classes)

Here is the lesson developed by Jennifer, Jill, and Jerry

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 directions on how to construct the 1st iteration of the stair step fractal.
    2. After students finish construction, then the teacher will address the class and ask "Is this interesting?: Well, why not? Well, what happens if we repeat the process? --Walk the students through the process repetition (Note mention # of folds that are inverted) "Pop it up again? More interesting? Now try it again - Try it again on your own. What interesting things do you notice about 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 lloks like stairs - what might these staris be used to model?
    3. What is the relationship between the sizes of the small stair, medium stair and large stair (Note: We have left this question ambiguous on purpose? [Suppose the figure you see are the stairs that lead up to the library. How man cubic inches of concrete would you need to fill the smallest square?]
  6. Student strategies - how might students so the problem
    1. Students might use inspection as their strategy.
      1. The teacher can then push them what do you mean small, medium? Large? Can you quantify that?
    2. Students might measure side length or area or volume
      1. If students do not mention all three, after discussing their observations, ask what measurement is missing
    3. Units - the small one is how many of the medium etc.
  7. Teacher responses - anticipate how you will respond to student questions
    See above
    1. 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.
  8. Summing up - use the kids words not a pre-written script. Otherwise they have no reason to do the activity
    1. What about the case of scaled by 3 (use a model??/)?
  9. 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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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.