## Japanese Lesson Study Summary## Friday, July 11, 2003We had a busy and productive day. We first talked about release forms so that our lesson can be videotaped. We had found forms on the New Mexico site that Lisa had suggested, but that form didn't really seem to fit our situation. Jennifer had also found a form on the National Board Certification site. Gail looked over both of them and suggested modifying one that had been used this week for Deborah Ball's class of 5th graders. Here is Gail's revision:
Next, we decided on what our roles as observers should be for both the practice lesson and for the model lesson on Thursday. We decided upon the following: Teacher: Celeste Moderator for debriefing: Gail · How can we improve the questions and better anticipate student responses - Jennifer · Pick 3 students record everything they do - Tony · Time each section of the lesson - Jerry · How did kids use the resources · When / how did the “math appear”? When were the key statements made?- Joyce · Did we match the level of difficulty correctly? / How did students apply their previous knowledge - Jill We decided to have Celeste teach the lesson (both she and Tony had volunteered) because Celeste really thought it would help to go observe the class on Wednesday afternoon and she was willing to do that (it's our afternoon off). We worked on the lesson some more and made some great progress. 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 directions (including a sketch) 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? So, 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?” 1. 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? (Individual work) Find a partner and compile your conclusions in an organized way (In pairs). Record your answers on butcher paper. Be prepared to explain your answers and how you got them. VI. Student strategies – How might students solve the problem? (In teacher responses be sure to be clear about which dimensions students are describing) A. Students might use inspection as their strategy. i. The teacher can then push them to explain, “ What do you mean by small, medium, or 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 the teacher should 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. Evaluation – what evidence will we have that kids understood? A. 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?” With the students conclusions, create a table and ask students to fill in the row for scaling by 3. Deal with misconceptions ( the funny nature of 2). In a whole class discussion, fill in the table for scale factors n > 3 and the general case. **Note introduce the vocabulary of scale after students see the need – 5thng, 6thng etc. **You can keep scaling up to the n case and then go to cases where n < 1 IX. Summing up – use the kids words not a pre-written script. (Otherwise they have no reason to do the activity.) **Mention that this fractal can be used to solve the famous Tower of Hanoi problem. Finally, we spent some time refining our homework questions with some great suggestions from Gail. Here they are.
Name ___________________________________ - Describe examples of scaling you find in your world. You may want to consider examples from architecture, literature, toys, film, science, etc.
- There is a rule which states that your height at age 2 is ˝ your adult height.
- How would you expect your weight to change over the same period?
- If you want your weight to double from age 2 to adult, how would you expect your height to change? Does this seem reasonable?
- Suppose the figure you created in class is a set of stairs. How many people can fit on the stairs if 15 people fit on the top stair?
- Suppose that each dimension of a 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?
We plan on using the room near the main theater past the women's bathroom. We will be using butcher paper sheets to record student responses and will have a bag or box for Celeste to have models ready but out of sight. Joyce will prepare some cubes in case the students need help visualizing volume increasing by eight when the side length is doubled and increasing by 27 when the side length is tripled. We will be using 4 squares per inch graph paper which either Jill or Joyce will supply. Joyce will also round up enough scissors. Art has agreed to videotape both days and Jim says he has a video camera that we may use. We still need to find name tags for the students. PCMI@MathForum Home || IAS/PCMI Home
This material is based upon work supported by the National Science Foundation under Grant No. 0314808. |