## Lesson Study Summary

### Monday - Friday, July 4 - 8, 2005

DAY 5 (7/5/05)

Small groups began debriefing at the beginning of the meeting. Joyce, Jeremy and Amy liked keeping the problem the way it was, with an extension of asking students what piece(s) of information could be taken away for the problem to still be solvable.

Megan, Claudia and Allen described a similar problem to the first group. They also asked students what would happen if the 90-degree angle were changed to a 30-degree, 60-degree or 89-degree angle.

Rey, Ellie and Mary explored a problem involving two similar isosceles triangles. They discussed giving descriptions of the triangles or providing a picture. The theme of their problem was a treasure map as well, but students have to get to a treasure in an indirect way (as in there is a moat of crocodiles blocking the direct path to the treasure).

The group decided to combine the ideas to write the actual problem that will be posed to the students. Allen gave reasons for both giving the students a diagram and for giving the students a description of the diagram, forcing them to draw it themselves. Rey drew the previous treasure map problem (groups 1 and 2) and the revised one (group 3). The whole group discussed both problems, to decide which one would be best.

The group decided to flesh out the original treasure map problem. Aki asked "how can students tell where C is?" Students could use similar triangles, 30-60-90 triangle knowledge, trigonometry or measurement extrapolation. To get these responses, Mary suggested asking students how far C is from A or how far C is from B. Ellie and Allen suggested just asking "how many ways can you find the distance from A to C?"

Jeremy and Amy suggested the follow-up question of "what is the minimum amount of information needed to find the distance from A to C?" The group decided that finding different methods might not help students relate them. Aki modified the problem by pulling the small similar triangle out of the bigger one and talking about two people walking to C and A from B at different speeds (see PCMI Lesson Study Graphics). After more discussion, the group decided to split into two smaller groups to (1) flesh out the problem and (2) write out rationale for the problem that will introduce the lesson.

Allen, Ellie, Rey, Claudia and Amy went to work on writing the actual problem. Mary, Jeremy, Joyce, Megan and Aki worked on the rationale for the lesson. The groups came back together after 30 minutes and shared their work. The rationale ended as follows:

• Students will have been introduced to right triangle trigonometry the day before the lesson and it is assumed that students have explored similarity before in some way.
• Understanding of similarity and trigonometry is in the Utah state standards.
• From teaching experience, students seem to struggle with special right triangles, similarity, logical argument and trigonometric concepts.
• Most curricula do not link similarity and trigonometry, despite the power of the connection.
• Teachers would like to help students integrate prior knowledge with new concepts and ideas; students should be able to bridge similarity and trigonometry.
• Students should understand that similarity is a basis for trigonometry in that any ratio in one triangle is the same in any other triangle similar to it; with similar triangles, the focus is on a scale factor; with trigonometry, the focus is on ratios.
• Trigonometry should not be taught as a new concept, but as a special case of similarity.
• Students should be able to compare various methods for solving a problem and defend the most efficient one for a particular case.
• All students can be included and are more likely to be engaged if the posed problem has multiple access points.

The group working on the actual problem shared their work.
Carly and John live on Bla Road. John lives northeast of Carly's house. The distance between their homes is 6 blocks. John walks west to West High School while Carly walks North to North Middle School. Every morning, Carly and john leave their houses at the same time for school. After 5 minutes of travel they meet three blocks North of Carly's house. They wave to each other and continue moving in their same directions at their same previous speeds. How far apart are North Middle School and West High School if Carly still has... 15 minutes to go...20 minutes to go...30 minutes to go...?

As the group began discussing the problem, many issues came up that could not be resolved. Rey suggested a new version of the problem. Finally, the group decided that for homework, everyone would take home with them the rationale, the two versions of the problem and try to write a problem that would works as "the" problem for the lesson. Every member agreed to do this and the next day would start with the sharing of the problems.

DAY 6 (7/6/05)

Aki shared that we need a draft lesson plan ready by the end of the period tomorrow. The teachers who would teach the practice and final lesson needed to be chosen by the end of the meeting as well.

Megan shared the problem she had written (see below) and explained that her focus had moved from a problem where students would make the connection between similarity and trigonometry themselves to a problem that would provide access to a discussion around that topic. She explained that the lesson could move toward the connection between similarity and trigonometry via sharing of solutions and/or a more specific extension problem.

Joyce shared two problems that were de-contextualized. One showed three squares sitting next to each other, the left-most square two by two, the middle square three by three and the question is to find the dimensions of the third and largest square. The other problem showed a figure composed of a five by five square sitting next to (to the left of) a ten by ten square The question asked students to find the distance from the bottom left corner of the figure to the top right hand corner.

The group discussed all three problems and decided to use a modified version of the treasure map problem. The group began suggesting ways to edit it to create the problem that would be given to students. In the middle of the discussion, Aki suggested the group decide who would teach the teacher group and who would teach the student group.

Each member shared his/her opinion about teaching the lesson:

Claudia: would love to teach again but feels selfish doing it; would like to observe
Ellie: would like to observe; feels like a sponge!
Amy: as an elementary school teacher, skeptical of content knowledge
Allen: worries about management, but would like to teach
Rey: wants someone who is experienced to teach it; feels like a sponge too
Mary: willing to teach it; experience is part of it
Jeremy: happy to do either
Megan: will video, can't do it!
Joyce: is a bit shy but would learn a lot and would like to do it

Joyce reminded the group that the lesson study is not about the teacher - the focus is on the lesson and making the lesson better. Claudia and Mary shared some of the experiences changing the lesson between the first time and the second. The group continued discussing who should teach the lessons and eventually decided to vote by paper ballot. Each person voted for two people and the votes were tallied by Aki. The group voted for Jeremy and Mary, where Jeremy would teach the students and Mary would teach the teachers.

The group split into two groups from there. Jeremy, Mary, Ellie and Megan worked on working out possible solutions and solution methods while the rest of the group worked on re-writing the rationale to fit the problem. Both groups shared after about thirty minutes of work time. After sharing work, the group finished the day discussing the exact question posed by the problem, in order to focus in on what students will actually take away from the lesson.

DAY 8 (7/7/05)

Joyce shared new information with the group about a discussion with the teacher of the class of students that would be used. She shared that the teacher would disseminate the photo/video permission forms and was happy with the topic decided on for the lesson study. The group talked about logistics of the lesson, such as nametags, seating arrangement and supplies.

The group split up to work on pieces of the lesson plan for the entire meeting time. Jeremy and Gail worked on re-writing the problem. Mary and Allen worked on possible student responses.

Aki shared what the observers should be looking for:

• Was there a variety of solution methods that merged?
• Was this a good way to connect new material to previous knowledge?
• Did students connect similarity and trigonometry?
• Observers will have roles:
• Time-keeper
• Questions for the teacher
• Student responses

PRACTICE LESSON (7/8/05)

9 students participated in the practice lesson, with Mary as the teacher. The lesson lasted one hour and the group began the debrief immediately following the lesson.

Mary began by reading the lesson's goals and commenting on each. A general comment she made was that she was unaware of the error in her numbers, which a few students brought up.

Aki, the facilitator of the lesson, began the group discussion by asking, "Where is a place we need to work hard to improve this lesson?" The group shared the following:

• Posing of the problem
• What should students write down?
• Would the distance along Main street be better to find that the distance along Scorpion Trail
• Confused about whether or not they were allowed to ask questions
• Summing up
• What is the purpose of the summation? What should be addressed?
• Is this a place for new material, information?
• Predicting student hang-ups and dealing with them

Summing It Up - Aki suggested that each member share their idea of what the summary should be and what should be discussed. Rey shared that it could be a time for the teacher to ask questions of the students about what had been done, extending their work into the link between similarity and trigonometry.

Joyce suggested having another big, blank version of the map for the summary, for the teacher to show multiple solutions on the same poster. Teachers were trying to answer the question If this lesson had been ideal, what would the summary have looked like?

Rey said that the goal of the lesson would be met in his eyes if students could identify side ratios on different triangles. Allen suggested grouping the posters by method. Amy and Claudia suggested writing all the angle and distance clues on the map instead of giving them out one by one, but Allen thought that maybe all values should be taken away completely from the problem, forcing students to focus on the method.

Aki said the lesson needs to push students beyond finding an answer and being done. He said this is dependent on the question - a good question is key to students making connections.

Gail went to the map and wrote in the given 60-degree angle. She said "suppose we've already been through the presentations, [we could] take all the information people had and put it on [the big map]...that could be the summary." Gail wrote all the given information from the problem on the big map. "How do we use this to get students to connect similarity and trigonometry?"

Jeremy drew just the two similar triangles on the board to show the relationship and Gail highlighted the same triangles on the map, suggesting the teacher could do this to focus on them. Aki repeated "what question could we ask for students to see these triangles?" Allen said we could write on the board the question what is the connection between similarity and right triangle trigonometry? Megan suggested that the teacher asks the student to focus on the small triangle somehow, still with the idea of trying to find the treasure, forcing students to visualize both the small and big triangles and the connection between them. Taking her idea, Allen suggested giving students the 60-degree angle in the smaller triangle as a clue, instead of the same angle in the bigger triangle.

Gail and Aki tried to keep the group focused on a question that would lead students to making the connection desired and sum up the lesson. "We're thinking 'what should I tell them,' but we need to think 'what should I ask them so that they'll tell me." From that direction, the group decided on the question: What is the connection between this solution that used cosine (trig) and this solution that used side ratios? The teacher would draw out of the students two equations that showed how tangent of an angle ends up being the ratio of the two legs of either triangle.

Posing the Problem - Gail and Allen liked the idea of a "clue grab bag," where students would pull clues relaying angles or side lengths but where there would only be so many of each clue in the bag. Aki suggested keeping the old method of giving the information students ask for, but giving them only two or three minutes to figure out what they need. Mary took Aki's idea and said the teacher's directions could be that the problem will be posed with the caveat that two minutes later information will be given out. Ellie suggested five minutes instead of two, to take different learning styles into account and make sure all students have time to digest the problem.

Gail summarized the suggestions as shortening the problem-solving time, changing the information given or the question asked or wanting students to find the answer in as many ways a possible.

Claudia, Ellie and Allen offered to work on the lesson plan [download the first draft as a MS Word document], fleshing it out over the weekend. Joyce made a list of needs for Tuesday, specifically questions for the class' teacher.

Back to Journal Index

PCMI@MathForum Home || IAS/PCMI Home

 © 2001 - 2016 Park City Mathematics Institute IAS/Park City Mathematics Institute is an outreach program of the Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540 Send questions or comments to: Suzanne Alejandre and Jim King

With program support provided by Math for America

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.