How Close to Pi Can You Get?

By this point in the unit, my students should have mastered Problems (1), (2) and (3). Problem (4) is slightly more challenging because the information given is different, but this is something that they should fully understand.

As students work on this Warm-Up, I ask them:

• Does this answer make sense to you?
• Can you show your answer in more than one way? Can you do this without a graph? Can you do this with a graph?
• What does this answer actually mean?
• How could you verify your answer?

I will ask students these questions whether or not they are getting the answers “right,” but I specifically like to ask these questions when students get right answers, because I want to show them that they should be thinking deeply about their work. Often students think that if they are able to accurately apply the algorithms, no other thinking is needed. I want to communicate to them that thinking is always a part of class—and asking these questions and expecting high quality answers helps do that. If students seem to brush off these questions, by saying things like, “Yeah, I get it,” or “Sure,” I ask them to explain more clearly or thoroughly.

If students struggle with these problems that should be review content, I ask them:

• What would be a good resource to help you with this problem?
• Where could you look to get support with this?

When students work on Problem (4), they may need more guidance, so I ask them:

• How can you use the given information to find the circle’s equation?
• Would it help you to create a graph of this situation?
• Is it possible to solve this problem without using a graph?

Problem (5) is the big new idea today. The goal is to create formulas for each section of the table from yesterday, though some students may have done that already. Even if they have already done it, I ask them to justify their formulas again, or recreate them without looking at their notes from the previous day. If students struggle to generate these formulas, I ask them to explain their process with numbers. For instance, if n = 6, how do you find each piece of information? Often walking through this helps students understand their process more abstractly. This problem is the perfect example of MP2 abd MP8—turn the repeated calculations from yesterday into abstract generalizations.

Because problem (5) is so important, I like to ask students to discuss this problem specifically with a partner. If I find that their sense of urgency has waned after 30 minutes working on the warm-up, I assign them a new partner (one easy way to do this is to have a set of cards with different shapes on them that have two of each shape) and ask them to spend 5 minutes debriefing problem (5) with this new partner. This can be a good way to increase the sense of urgency and give students a change to reset. It also makes more clear the specific focus of the day’s lesson.

Today’s closing should give students the chance to articulate some of the challenging ideas in the day’s lesson. Because the entire lesson focused on finishing this investigation, the write-up questions for the day’s task can serve as the closing. Students may not have time to finish these questions today, or they may be a little burnt out, so the Exit Ticket questions give them a chance to think about those same big ideas.

Basically, the first two questions reflect essential understandings of this investigation, and the last question is giving them a chance to think about MP5. This question could be a good question at the end of many lessons—“How could technology help us today?” Even if it seems simple, I want students to start articulating things like, “I used the formula I wrote to set up a data table and find the outputs for larger and larger values of n.” This shows deep understanding of how their thinking work relates to the technology they used.