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Five Seconds
Lesson 6 of 14
Objective: SWBAT construct and analyze histograms that are normally distributed.
Opening
In the next few tasks students will be asked to construct histograms that will trend toward normal distribution. Students will be introduced to normal distribution in the next lesson, so I don't necessarily bring it to their attention now. Instead, I focus on how students decide to group their results in the histograms and what the graph can show them about their data. This work builds on the 6th grade standard Standard 6.SP.A13 where students are asked to work with dot plots and histograms. My purpose for doing this task though is for students to collect and analyze data and them move toward an understanding of standard deviation, which is high school standard HSS.ID.A.4. Additionally, as I work with students who have had significant academic interruptions, some reteaching of the sixth grade standard in service of the high school standard is often necessary. I find this task a nice way to set up that work.
I begin class by telling students that today they will try to measure 5 seconds as accurately as they can. This task can be found in written form in the IMP Year 1 textbook (2009) on page 306. I pair off students and let them know that one partner will watch a clock or watch and the other will work the timer. The partner watching the clock will say start and then stop after five seconds. The partner with the timer will try to follow the partner's commands as accurately as possible. Pairs will record their results to the nearest tenth of a second. Each timer will collect 10 results and then partners will switch roles.
Investigation
Next, I let students get to work on the timing activity. When they are finished, I ask them to figure out how to make a class histogram to show their data. I try to elicit a lot of discussion here about how to group the results into buckets or bins. I might decide to have different groups use different intervals to demonstrate how too many or too few intervals will take away from the "picture" of the graph. We can discuss how this changes the representation of the data. The Timing Five Seconds graphs in the resources section show different bin groupings for a set of classroom data. This can be a good starting point for a discussion about how different bins will lead to graphs that display the data differently. In addition, an extension activity or a way to incorporate technology to this lesson is to use plot.ly to generate the histograms and then adjust the bins from there.
Once we have a class graph, I ask students about what the histogram tells them. I like to begin the discussion by asking students what they notice and what they wonder about. I then follow up with more specific questions like what the histogram tells them about their accuracy. I might ask what percentage of the results fall between certain values  say between 4.9 and 5.1 seconds. If the resulting graph looks like it's somewhat normally distributed, I can ask them about the symmetry of the graph and why it occurs.
Because one of the things that is usually normally distributed is measurement variation, I raise this point here. I might ask students why they have such a range of data when they were all trying to get to the same five seconds. I tell them that this is called "measurement variation" and is something that arises often in math and science. I ask them to come up with another example of an instance when measurement variation might occur.
Resources (3)
Homework Set Up
I tell students that for tonight's homework, they will make a histogram of student stride lengths. I explain that a person's stride is the length of a typical step when they are walking at their normal pace. I demonstrate that they should measure a stride from the front of one foot to the front of the other foot. This activity can be found on page 307 of IMP's Year 1 Textbook (2009).
I have students break into groups and ask them to come up with a method they will use for measuring their own stride. Some students may recommend taking a lot of steps, measuring, and then dividing, while others will try to measure a single step. I have students measure their own stride and then share out the data on a white board so everyone can see it. I ask students to record the stride data so they can make a histogram for tonight's homework.
Closing
Students should be leaving class with a list of the stride length of all of their classmates. For today's reflection activity, I ask them to predict what a histogram of 50 students' stride lengths would look like. They can describe the histogram in words and provide a rough sketch.

This material is adapted from the IMP Teacher’s Guide, © 2010 Interactive Mathematics Program. Some rights reserved.
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