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!)
Stairstep 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.
Lesson:
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.
Extensions:
What different fractals do you see? Linear model like Cantor's Comb,
Sierpinski's triangle or gasket, 3D 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  StairStep Fractal
 Overarching Goal:
 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
 How does this objective fit into a unit? Students will understand the
relationship between dimensionality and scale factor
 What is the prerequisite knowledge?
How to find the area of a rectangle
How to find the volume of a rectangular solid
 Math problem  hook, problem on which they will work
 When students enter the class, they will be given printed 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 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:
 It looks like stairs  Response: are all the stairs the same?
 I see squares / boxes
 The stairs are getting bigger
 The figure is symmetrical  Response  What type of symmetry? or how?
 It looks like how you make snowflakes
 It's a cool design.  What makes it cool?
 It looks like a building  What type of building
 It lloks like stairs  what might these staris be used to model?
 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?]
 Student strategies  how might students so the problem
 Students might use inspection as their strategy.
 The teacher can then push them what do you mean small, medium? Large? Can
you quantify that?
 Students might measure side length or area or volume
 If students do not mention all three, after discussing their observations,
ask what measurement is missing
 Units  the small one is how many of the medium etc.
 Teacher responses  anticipate how you will respond to student questions
See above
 Discuss w/ students a counterexample (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 sideview. 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.
 Summing up  use the kids words not a prewritten script. Otherwise they
have no reason to do the activity
 What about the case of scaled by 3 (use a model??/)?
 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 DMS0940733 and DMS1441467. 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.
