## Implementing Lesson Study Summary

### Monday - Friday, July 9 - 13, 2007

07.09.2006
Participants: Aki, Joyce, Rudy, Judy, Anastasia, Jim, Oscar, Gloria, Rosemary

To begin, Aki has now joined us after a week in Japan. He endorsed that the problem needs to be decided early on so we can think about the mathematics involved. There are many questions about the mathematics that we need to ask, beyond just finding the answer.

We received feedback from one SSTP participant, in which she tried dropping straws 30 times in a square configuration with 4 cups, and she made it in to the hole twice.

Where is our focus going to be? We have a problem that can have mathematics involving area or can emphasize probability. We had a discussion about where we will start, which of our explorations from Friday we would be going with, and what mathematics we are planning on emphasizing.

Revisit the mathematical objective:
Students will use the area of known shapes to find the area of composite shapes.

Our aim today is to focus our ideas on the mathematics and an appropriate problem.

Question Ideas:

• If you were the fair owner, what configuration would you use and why?
• If you were the patron, what configuration would you use and why?
• Now you are packing up your exhibit, what configuration would you use and why?

Alternate Context:

• Cut out circles from a piece of paper. Involve the leftover space in the problem.
• Cookie cutting. Look at the leftover dough involved.
• Tangible negative space. (waste, leftover)
• Circular crops. Percentage of land in production. Are they better off to make one big circle or several smaller spaces. Using a square piece of land, consider one large crop versus four smaller crops.

We considered what kind of calculations we would expect students to perform in their different approaches to solving this problem. It was decided that the size of the paper would figure in to the types of solutions we see from students.

Hook:
Show Google Map of Circular Fields to introduce the idea and context of the problem.

Problem:
Imagine a farmer has a square plot of land to irrigate. The farmer can buy irrigation systems that irrigate in a circular pattern, anchored at the center and extending as long as the radius. The farmer prefers to buy one size of irrigation pipe so he can move it among his crops (use it for more than one crop). His plot of land is one square mile. (1 mile x 1 mile)

1. If the farmer wants four circular fields, how much fallow, or unused land, would remain?
2. What arrangement of congruent circular fields of any size gives the least unusable land?

(Consider that all do 4 first, then look for other methods (1 or 9 or 16...))

Considerations for tomorrow:
Inacessible land "trapped" in the middle. Find that area. Accessibility of the road. The question we leave with is "How do we bring the focus back to the composite space in the center?" What can we change or add to the problem to increase its complexity and bring it back to our objective?

07.10.2007

One opening question was about working from an interesting problem and then deciding on a mathematical objective versus starting with a mathematical objective

15 students are meeting in the library (for remodeling). Their seating is essentially pairs, with a few on the couch. Their equipment is an overhead with a projector screen. There is not much board space for writing. We may need to bring flip charts. We will be taking t-shirts to the students.

Problem:
Imagine a farmer has a square plot of land to irrigate that is one square mile. The farmer can buy irrigation systems that irrigate in a circular pattern, anchored at the center and extending as long as the radius. The farmer prefers to buy one size of irrigation pipe so he can move it among his crops (use it for more than one crop).

If a crop is watered twice, it is deemed unusable. For this reason the farmer does not overlap his watering.

1. If the farmer wants four circular fields, how much fallow, or unused land, would remain?
2. What arrangement of congruent circular fields of any size gives the least unusable land?

Consider:

• 4 miles x 4 miles (16 square miles)
• Part 2 - Son suggests using the unused space. He wants to add a fifth irrigation about the center.

Which irrigation plan would you choose and why? Which plan has more "wasted" farm land?

07.12.2007

Participants: Gail, Joyce, Judy, Gloria, Oscar, Jim, Rudy, Rosemary, Anastasia

Today the group made big decisions about the lesson itself and the implementation of the lesson. The group decided that Jim Town will be teaching the lesson on Friday to volunteer teacher-students.

For the lesson, students will be receiving a schematic of the two field versions. The group decided to use hand drawn drawings of the two irrigation plans. Students will also be receiving a written copy of the problem with an explanation of the scenario. Some reasons for this were to focus on the mathematics rather than the actual drawing of the irrigation system.

Working off of the lesson study template, our group developed a plan for our lesson. We also decided on other materials that would be given to students. Students will get a printed version of the problem with the main question and their task. After the lesson, students will also receive a half-sheet evaluation page for summary comments about what they learned.

The question is now going to be stated as follows:

A farmer has a square plot of land measuring 16 square miles. He wants to irrigate his crops in a circular pattern with one system that will be moved around. Any land that is double-watered is considered unusable for farming. The farmer has been irrigating his land as four circular fields for many years. His son thinks it may be more efficient to add a fifth irrigation site at the center of his land as illustrated in the given diagrams.

Which irrigation plan has the least "wasted" farm land?

Using words, numbers, and pictures, explain your group solution(s) on chart paper.

07.13.2007

Participants: Gail, Joyce, Judy, Oscar, Jim, Rudy, Rosemary, Anastasia

Today we had our first iteration of lesson study using eight volunteer SSTP participants. After watching the lesson, we evaluated our lesson and discussed elements of the student work that was generated during the lesson as it correlated to our written lesson.

One point to include in our revision is the element of time. There are several places in the lesson where a stopping point would be valuable to allow thinking time for all students. For example, we discussed allowing time when the problem is first handed out for reading and processing. Next would be a two to five minute silent thinking time when students begin working on their problem.

Another topic that we discussed was how status played a role in the group discussions and preparation of group solutions. There are parts of the lesson that can be altered to encourage sharing ideas, thinking independently on multiple solutions, and incorporating more than one solution on the poster.

Concerning materials and their availability and placement, we discussed which materials should be readily available and visible. Materials might influence how students approach the problem, such as with cutting if scissors are made available.

For our summarizing discussion, we considered changing the setup. Now student solutions will all be posted around the room or on a wall. The discussion could be done after a round robin where all students can visit each other's solution and read and compare what they produced. Another approach would be to pull the discussion to include comparison of student strategies.

Before we left we decided that Oscar would be teaching the lesson on Tuesday. Discussion and further development will continue on Monday.

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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.