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* *Reflection: Problem-based Approaches
The Normal Distribution - Section 2: The Normal Distribution and the Heights of Men in the United States

The purpose of these eight Normal Distribution problems is to help students develop their notes about how to use the normal distribution and standard deviation to estimate population percentages. I explain this to students, and I tell them that I'll be circulating to help make sure that their notebooks are full of useful, well-documented examples.

I let students get to work until questions arise naturally, and as work time begins, I start my laps around the room. I have no prescribed method for how a notebook should look in my class, because these are older students and I like to allow to do what works for them. With that in mind, this time will function as a notebook workshop, as I'll give each student a little feedback and if necessary, some suggestions about how to make their notebook as useful as possible.

For the problems themselves, I've included a brief Explaining the Math document that explains my thinking behind each one.

*Problem-based Approaches: Appreciating the math in today's work*

# The Normal Distribution

Lesson 9 of 11

## Objective: SWBAT use the normal distribution to estimate population percentages.

*75 minutes*

Today's class opens with the question, "Was Michael Jordan really one in a million?" (see today's Prezi, slide #1). Of course, I'll have to qualify this question in a few ways, but its wording certainly draws kids in.

I'll have to joke that I'm old, and that I recognize that Lebron James recently passed Kobe Bryant as the NBA's most popular player, and that by now, my students are probably too young to have actually seen Jordan play. "That's ok," I'll say, "you all know who he is, right?" Once they get to talking about it, I will admit to students that someone's odds of being the MVP of the NBA Finals are much tougher than 1 in a million, but for now I'm just talking height. And no matter who we're talking about specifically, in general, we're talking about a bunch of really tall guys.

So how special is it to be that tall?

**Gathering Some Data: Poll Everywhere**

I give the debates (most of which rightly have less to do with Jordan's height than his legacy) a minute or so to stir the class up before asking everyone to consider a more specific version of the question at hand. I move to slide #2, which poses the question, "What is the probability that an adult male in the United States is 6’6” or taller?"

If you have the technology for it, this question makes a great polling question, whether on clickers or on polleverywhere.com. I've included a screenshot of what this poll looks like (see pollev opener). This question is also well-suited for a turn-and-talk or for a lower-tech version of polling, whether by having students stand up or just counting their show of hands.

I use this question to help students make an initial connection between statistics and probability, and as they learn to use the normal distribution to estimate population percentages, they will continue to explore this connection today.

I do not provide an answer to this question now, but tell students that they can expect to be able to answer it by the end of today's class.

**Gathering Some More Data: Sticky Notes Estimate**

As students were answering the previous question, I handed out sticky notes. When students are ready, I move on to slide #4, which poses the question, "How many people in the United States are 6'6" or taller?" I instruct everyone to write their best estimate on their sticky note and add it to the chalkboard.

This is a rich question. By moving from probability to a population estimate, it requires students to have some idea of how many people, and more specifically, adult males, live in the United States. By seeing how they answer, I learn about the number sense my students are bringing to the class. It's hard to deal with numbers as large as the population of the United States, and some students will struggle to even know if an answer sounds reasonable. This question gives them a chance to grapple with that.

To things up, I muse a little about the idea behind the word *normal*. "Is it normal to be 6'6" tall?" I ask. "How many people do you know who are that tall?" I tell students that today, we're going to spend some time with that word.

#### Resources

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**Learning Target Review**

I post SLT 1.4 on slide #5, and ask for a volunteer to read it. I ask for someone to talk about what they've learned so far with regards to this SLT, then I ask for someone else to talk about what they still need to know about it. If no one has noted it already, I point to the word *normal *in the SLT, and I say that the purpose of today's class is to provide an overview of how to use the normal distibution to estimate population percentages.

**Guided Notes**

I distribute the Normal Distribution Male Heights handout, which consists of an image of the normal curve, a few fill-in-the-blank guided notes, and eight problems for students to solve.

For the first note, I ask students to think of the three data representations we've studied so far: dot plots, box plots, and histograms, and to decide which one most resembles the graph they see here. We agree that it's most like a histogram, but with some important differences. We notice that the tops of the bins are curved. We see that the number line goes from -3 to +3, and that there are percentages written inside of each bin. If no one notices it, I point out that there is a line of symmetry at the number 0.

I tell students that the values on the number line represent **the number of standard deviations** above and below the **mean**, to help them fill in the second bulleted note. We spend a few minutes comparing this normal curve to the results of Linear Practice #1, although the specifics of this conversation depend on the results for a particular class. I hope to find that students are getting comfortable with the idea of "standard deviations above and below the mean" from our previous class and from their work on the current Delta Math assignment, but if I have to return to those ideas I'm ready to do so.

Finally, we consider the height of each bin. In the histograms we've studied so far, the height of bin represents frequency. Here the height represents relative frequency as indicated by a percentage. The idea of relative vs. absolute frequency is one that we will revisit during Unit 2. For now, I tell students that this is the distinction I'd like for them to be able to make, and that relative frequencies allow us to apply this curve to different-sized populations.

#### Resources

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#### Linear Practice 2

*15 min*

With about 20-25 minutes left in class, it's time to generate another data set by running another Linear Practice trial. My brief note to students is on slide #10 of today's Prezi, and the structure is the same as it was first time. I've outlined implementation options in my What's Wrong with Mean? lesson.

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#### Problem Set #3

*5 min*

When the timer for Linear Practice #2 expires, I distribute Stats Problem Set 3. This problem set consists of two parts: one is an analysis of the results from Linear Practice #2, and the other consists of few problems about female height and the normal distribution.

Because I use technology to collect results and automate the scoring of Linear Practice #2, I can post the anonymous results immediately, so students can record them on the top of the problem set. Without technology, it might take a day to turn around these scores and use them on the problem set.

The latter half of the problem set is similar to the practice problems that students tried earlier in this class meeting. If students need help here, I refer them to their notes on those previous problems. If those notes are incomplete or unclear, that's where I provide help, rather than giving away solutions on Problem Set #3.

This problem set is one of the resources that students will use during the next class to review and prepare for the upcoming unit exam.

#### Resources

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I distribute index cards and post this question:

**If you want to learn about a data set, would you prefer to know the median and quartiles or the mean and standard deviation of the data set? Why?**

Students have a few minutes to write their response on an index card, which they'll hand to me as they leave. Of course, I'm curious about their answer to question, but I'm much more interested in how they justify their responses and what light that sheds on their understanding of these topics.

To learn more about index card essays, please see my **Index Card Essay** reflection.

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- UNIT 1: Statistics: Data in One Variable
- UNIT 2: Statistics: Bivariate Data
- UNIT 3: Statistics: Modeling With Exponential Functions
- UNIT 4: Statistics: Using Probability to Make Decisions
- UNIT 5: Trigonometry: Triangles
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- LESSON 1: The Game of Greed
- LESSON 2: The Mastermind Project, Day 1: How to Play
- LESSON 3: The Mastermind Project, Day 2: Choosing the Best Representation
- LESSON 4: The Stroop Effect
- LESSON 5: The Mastermind Project, Day 3: Gathering and Organizing Data
- LESSON 6: The Mastermind Project, Day 4: Interpreting Data and Drawing Conclusions
- LESSON 7: What's Wrong With Mean?
- LESSON 8: Measures of Dispersion
- LESSON 9: The Normal Distribution
- LESSON 10: Review and Problem Solving
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