Japanese Lesson Study Lesson Topic: Scaling, Measurement and Dimensionality - Stair-Step Fractal Materials: scissors, quad graph paper (cut to 32 squares by 40 squares), a large fractal model for the 1:2 scale factor, large sticks that can be used to show the height of each stair on the large model, several small models of the 1:3 fractal, a few spare fractals partially created in case students make a mistake, instruction sheets, butcher paper for student presentations if desired)
1. Overarching Goal:
Note - This would be determined by the goals that the school has for its students. We could not determine this for the context of our lesson study. If schools do not have yearly goals, affective goals such as team work, student collaboration, following directions, etc. may work well with this lesson.
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?
Students should be able to find the area of a rectangle and the volume of a rectangular solid
5. Math problem to be presented.
1. Hook: When students enter the class, they will be given printed directions with graphics 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. What can you say about what you've created so far?" (It would be helpful to have a large model of the fractal made from poster board)? "What happens if we repeat the process?" --Walk the students through the 2nd and 3rd iterations - Pop it up again. "What interesting things do you see and what do you wonder when you look at this figure?" This gets students started and helps the teacher determine the extent of students prior knowledge.
Brainstorm individually. Write your observations and thoughts on the back of the instruction sheet (give a minute or two for students to write before taking student answers).

Potential responses: This is an excellent way to anticipate students thinking. If and when students bring up concepts or words such as "length," "area," "volume," the teacher may write these on the board.

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 is it symmetric?
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 - where?
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.
15. The # of each size box is a power of 2
2. Key Question: What relationships exist between the steps in this model (Have the questions written on a flip chart for students)?
Individually, take a few minutes to explore and write down any relationships that you see. This should be monitored by the teacher. If students seem to have difficulty with this question, you may want to ask some of the following sub-questions: What information could you collect about these shapes? -or - How could you explore what relationships exist between the shapes in the model? -or- What kind of measures can you make? - or - Have students brainstorm as a group -or- You may want to collect data with them for the small prism and have them do the large and medium prisms in their groups.
With the people at your table, compile your conclusions in an organized way (Note - if students have difficulty with this instruction ask, "How could you organize this information?" You may want to prepare a blank chart ahead of time on the board or on a sheet to hand out to help students come up with an acceptable result.) - Make conjectures about the relationships you see. Record everything on the paper I give you (note students should be given time to work in groups before butcher paper is handed out). Be prepared to explain your answers and how you got them.
6. Potential student strategies - how might students solve the problem? (In teacher responses, be sure that everyone is the room is clear about which dimensions students are describing)
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? Are they all rectangles? Are there any squares?
2. 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
2. Students might measure side length and/or area and/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. Since the objective is 'scaling,' gently nudge the discussion towards that concept incorporating appropriate mathematical terms whenever these concepts have been expressed or hinted at by students. One note - If students approach the figure in both ways (big box is 1 versus big box is 1600), discuss scale as both a whole # and a fraction - perhaps in the summing up section.
8. Evaluation - what evidence will we have that kids understood?
1. Present students with a model that is scaled by a factor of 3. Have the students work independently to figure out how area and volume change in this case.
Q: "The scale factor in this figure is different. How do you think the scale factor in this model effects the area and volume? Using the students conclusions, create a table and ask students to fill in the scaled by 3 row. Deal with misconceptions (the funny nature of 2 - whether you double it or square it you get the same number! This is not true with other numbers such as 3). In a whole class discussion, fill in the table for scale factors n > 3 and the general case.
**Note introduce the vocabulary of "scaling" if students have still been using the terms - doubling, tripling, quadrupling, 5thng, 6thng etc.
9. Summing up - use the kids words not a pre-written script. (Otherwise they had no reason to do the activity). However, it is also important to encourage them to use appropriate mathematical terms when explaining the concepts.

**You may wish to research connections between this fractal (see reference to the book Fractal Cuts) and the solution to the famous puzzle "The Tower of Hanoi."

This lesson could provide a starting point for a group of teachers at other schools to experience the process of lesson study. Many ideas contained here may not be entirely clear to teachers outside of our lesson study group since the process involves common experiences, collaboration, and over ten hours spent working together towards a common lesson. Though we are excited to try out the lesson with our own classrooms this school year, we are aware that the next phase of the process will be revising our lesson collectively as a result of our experiences. In addition, we hope to apply this process to creating common lessons with teachers in our own buildings.

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This material is based upon work supported by the National Science Foundation under Grant No. 0314808.
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