##
* *Reflection: Grappling with Complexity
Fast Hands - Section 1: Introduction

As educators we frequently have an opportunity to spend our time defining the problems we want students to solve. I put a lot of effort into this part of my practice. From my perspective, this ("defining the problem") is often the best part of mathematics. For me, the creation of a problem is a fun and awesome experience. In today's lesson, my goal is to let my students experience the joy that I feel when I define a great problem. I want my class to experience the powerful feeling of defining an interesting/clever/fun/unique problem to solve.

*The Joy of Problem Definition*

*Grappling with Complexity: The Joy of Problem Definition*

# Fast Hands

Lesson 6 of 8

## Objective: SWBAT use their understanding of slope, proportion, and linear functions to analyze and predict the result of a clapping experiment.

## Big Idea: When we can't predict an exact result, we use patterns to predict a reasonable range in which our result lives.

*60 minutes*

#### Introduction

*15 min*

When students enter the room, they have about a minute or so to consider the question, "how many claps can you complete in one minute?" As students consider the question (and begin to clap and talk about the problem at hand), we talk about the complexity of the experiment. I ask them to consider how they would count and record the number of claps. Typically students talk about how they might use video and listen for the claps in the recording. Once this is established, I will shift the conversation towards the focus of the lesson: **how to predict the result of an "almost linear" function**. In today's lesson, I will ask students to *predict* the number of claps if they only have data for a portion of a minute.

To further motivate the lesson, I will show them a small portion of this video on You Tube:

**Source**: http://youtu.be/Z571ByeNbPQ (accessed April 18, 2014)

The video shows the **world's fastest clapper** breaking the current record. Of course, students don't know this. At this point in the lesson, students are unaware of the problem (we are going to find out how many claps he will reach in one minute) and only somewhat aware of the context of the problem (the almost linear rate of clapping). I believe it is important to let students define the problem as a class.

As a class, we watch the video above. My goal is to set my students up to define the problem themselves (see Grappling with Complexity: The Joy of Problem Solving reflection). I stop the video after about 20 seconds and ask them to write down any questions they might have. These questions might range from "what is he doing?" to "is his clapping rate consistent." Check out the Questions about Clapping to get a sense of the questions your students might generate.

After a minute or so I ask them to turn-and-talk to share their questions. Then, we share the questions as a class. They share their questions and I type them out as they share. I make sure to quote them as well (again, see Questions about Clapping). After all the questions are shared, I start to address them with the class.

I usually start with questions that are the least relevant to our investigation. Questions like, "why is he wearing headphones?" I try and answer these if I can, but often I put it to the class. I ask, "why do you think he is wearing headphones? Does this impact what he is doing?" I find that even these types of questions help us define the problem. For example, I might say "He is wearing headphones to signal the start and end of the experiment." This leads us to the question, "What is this experiment about?" and the critical question of "How long is this experiment?" As I address these questions we eventually define the problem we are solving, which is: **How many claps will he reach in 60 seconds?**

#### Resources

*expand content*

#### Solving the Problem

*20 min*

In the introduction to today's lesson, we defined the problem we are solving as a class: "how many claps will he reach in 60 seconds?" Now, we transition to strategies. I ask the class, "what do you need to solve this problem?" I plan to use the students' questions as guidance. They brainstormed questions like "is he always clapping at the same rate" and "how long is the experiment?" As a class we agree that the answers to these questions are critical and we reflect on what we need to know if order to solve our problem. It is important for students to be consciously aware of the fact that these key questions are about rate and time. Once they make this realization, I expect that they will be able to determine that we need to know his clapping rate and how long and how consistent this rate goes on for.

I don't give them this information I want them to figure it out for themselves. For me, the key is that they *ask *for the information. I don't want to give students all the parameters and pieces needed to solve a problem. I find that my students are so much more invested in a problem when they define it and identify the parameters, models, and tools they need to solve it.

When I feel that the class is ready, I distribute this handout Act 2 Fast Hands and I ask the students, "What is a reasonable amount of claps he will reach after 60 seconds?" I avoid the word "estimate" since it may lead to guessing or foster confusion. Sometimes my students hear the word "estimate" and think, "what am I supposed to do, what is the correct way to estimate?"

**Teaching Note**: In my classroom I want estimations to be based on the context of a problem, not on a list of arbitrary rules. I want students to adjust their estimation based on the context of the problem. I want them to chose a way of estimating and then deal with the consequences of their choices. For example, if students round up or down as they work, they need to consider the "what if" questions attached to their thinking:

"What if he speeds up at the end?"

"What if he stops before 60 seconds?"

#### Resources

*expand content*

#### Summary and Review

*25 min*

I plan to start off today's summary by talking about the work of solving our problem. We start by sharing their estimates for the reasonable ranges they found.

- We talk about "really safe" lows like 791 claps (since we see him reach this number at 59 seconds)
- We talk about "really safe" highs like 900 (since he only reached 791 claps at 59 seconds and averages around 14 claps a second, there is no way he can reach this high number)

Then, I ask the class to narrow down this safe range. We share different high and low ranges and I ask students to justify their reasoning. For example, many students pick 810 as a high since 791 + 14 (a reasonable average of his clapping rate) = 805. In my experience, some students tend to round up in case he has a "burst of energy" at the end. Other students debate the point and leave 805 alone, reasoning that "he has to get tired."

I typically list several different possible ranges on the board. Then, when we have all had a chance to share, I show them the actual count by playing the full clapping video. Students watch in anticipation as we approach the 60 second mark and then often scream and cheer (and sometimes swear) when they compare their predictions to the result in the video.

We then review the mathematics of the lesson. In particular, we focus on the following (see my PowerPoint presentation):

- When you pick a particular point in time and divide his claps by the seconds, why do your predictions change with each point you choose?
- When you find the slope (change in claps:change in seconds) between two time stamps, why does your result change with each pair of points you choose?
- Is there a way to pick all the points at once (linear regression)? What would this mean?
- Could we have consistently predicted the number of claps he would reach? Why did we emphasize a "reasonable range"?

If the lesson has gone as planned, at this point the students have received a solid introduction into the intuition of linear regression. We end it around these big ideas and then follow up with a more detailed analysis in the following lesson.

*expand content*

The idea for this lesson stems from Fawn Nguyen: https://twitter.com/fawnpnguyen/statuses/343018683834175488

A teacher named Mr. Kraft saw her post and turned it into a lesson: https://mrkraft.wikispaces.com/Fast+Clapper

This lesson is based on their ideas, so thank you Mr. Kraft and Fawn Nguyen!

*expand content*

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