# Japanese Lesson Study Summary

## Wednesday, July 2, 2003

Today was a big day for us because we our homework assignment was to come ready to defend one of our top three topics in preparation for choosing the topic from which to plan our lesson.

THE BIG THREE are:

FRACTALS

SCALING MEASUREMENT/DIMENSIONALITY

PYTHAGOREAN THEOREM

Early on, the idea was suggested that we could integrate two of the three topics and do a Pythagorean Tree Fractal. Celeste grabbed the quilt poster from our morning classroom to demonstrate this. It was also suggested that we could easily integrate scaling and fractals. Several types of fractals were suggested, but we were concerned that most were too much of a hook while being distracting from the concepts that we really wanted to teach. About this time, Debbie whipped out a piece of paper and cut out a stair step fractal which also graphically demonstrates the solution of the famous Tower of Hanoi problem. This makes a pop-out fractal which creates a very nice example of 3-D prisms which shrink by a factor of two for each successive stage. We would be able to discuss changes in side length, area of the faces, and volume of the steps. Once this idea was suggested, we all enthusiastically approved.

The following comments are Gail's notes from the discussion that led to the choice of this topic.

Fractals with Pythagorean Theorem: Extend Pythagorean Theorem
Similarity, concept of fractal and what this is, enrichment, ties area of Pythagorean theorem
Build a fractal?
Building a snowflake Koch curve, triangle - learn iteration, hands on activity
Tie in triangles and squares in their that are similar and can look at area and dimension?
Combine scaling and fractals?
What is the objective?
Choose a topic that can teach. Use fractals as a way to help students understand scaling. In terms of a larger theme they need to carry issues how change in length, changes area, etc.
Three dimensional Pythagorean theorem: If the major goal is drive the concept of scaling, why would fractals be a good choice as opposed to blocks? They are a hook? Never seen them before, curious. Combines interesting mathematics with the things they can do easily and can see. Catch on to the initial things.
Working with area, volume - squares and cubes are the most important.
How much concrete you bought or something that relates to reality? One idea is to make a function, rotate it around and make hot air balloons. Divides up into 8 sections? As you scale up then it flies? 2 times it went 15 feet into the air; scaled it up 3 times they really went up? Even though they see how big it is the numbers that they conjecture are still off!! The scaling things tried did not work.
Filling a pool with water? Have a scale model of pool and can fill it in certain length of time, how long to fill the pool? Is this too indirect? The volume relation might get over looked?

Once we settled on our stair step fractal, we then discussed how to approach the planning of this lesson. Lots of good discussion was generated and the notes from this follow.

Back to fractals. Tower of Hanoi - What happens to the area, and volume of similar figures as the length of a side changes? Focusing on looking at dimensionality? Do it general? a, half a.

What questions do they ask? What do you see in this figure that is interesting? Fractal as a hook, iteration, repeat the process, what do you notice, what do you see? Here are some examples. Not even tell them what doing, keep fractal to the minimum? Start with the construction, and keep the idea of fractal that comes later? Even iteration can be delayed.

Challenge - want to discover, in the lesson planning, have to carefully think about this. Hoping what will come from this - look alike, 3 sizes, dimensions, area, volume? As comparing may notice the relationship?

What is our objective? We think it is around the effect of changing one dimension and its effect on the area and volume. Are we talking about doubling or halving? Start with dimensions and scale up to volume or go backwards from volume to the dimension? Extension - want half the area and third volume

For tomorrow, our homework is to bring our statement of the mathematical objective for our lesson. We'll share these and come to some kind of compromise. Then we will divide into triples and start working on writing up a lesson to share and critique with others on Monday.

Gail ended by saying that she had never seen a lesson study group so quickly converge on a problem topic before! I think we were all amazed and proud of ourselves.

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