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Unit 1, Lesson 10

The Effect of Extremes

Algebra 1

10.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Aspects of Mathematical Modeling

Materials

None

Activity Narrative

This Warm-up prompts students to think about what variables they may use to analyze a situation. Then, students describe data displays they may use to compare two sets of data. Choosing variables and planning a process for comparing data sets engage students in aspects of mathematical modeling (MP4).

Listen for groups that choose a variable other than the number of wins to determine the top players in the game and for groups that select different data displays or ways of comparing data sets, and ask them to share with the whole group.

Launch

Arrange students in groups of 2.

Ask students, “Have you ever played a game where you did really well, but still lost?” Consider looking up or sharing your own stories of great individual performances that still resulted in a loss. Tell students, “There are often many ways to measure performance and compare results.”

Tell students to think quietly about their answers to the questions for about 1 minute before discussing with their partner and then sharing with the whole group.

Activity

None

Several online video games match players with other players at random to compete in a team game.

What information could you use to determine the top players in a team game like this? Explain your reasoning.
There are two games of a similar type. One person claims that the best players play game A. Another person claims that game B has better players. How could you display data to help inform their discussion? Explain your reasoning.

Student Response

to access Student Response.

Activity Synthesis

Select previously identified students to share their solutions. If it does not come up in discussion, ask students how they might interpret a situation in which a small group of players has a significantly greater number of wins or points than the rest of the group. It might mean that most of the players are not very good and there are a few who are, or it might mean that there are a few dominant players who are much better than average players.

10.2

Activity

Instructional Routines

Analyze It MLR5: Co-Craft Questions

Materials

To Gather

Internet-enabled device Statistical technology

Activity Narrative

The mathematical purpose of this activity is to help students understand how measures of center for distributions with different shapes are impacted by changes in the data. Students will create a dot plot, then describe the shape of the distribution and find measures of center. They will investigate how the measures of center change when the data set changes. Students create and investigate a data set from a given set of parameters including the shape of the distribution. Monitor for students using the correct terminology to describe the shape of the distribution.

This activity works best when each student has access to technology that computes measures of center and displays dot plots or histograms easily because students will benefit from seeing the relationship in a dynamic way. If students don't have individual access, projecting the distributions would be helpful during the Launch.

In the digital version of the activity, students use an applet to do much of the visualization and calculation of statistics. The applet allows students to focus on the impact of the values in the data set rather than creating visualizations or doing calculations themselves. Use the digital version if it is available so that students can dynamically see the impact of different values on various statistics.

This is the first time Math Language Routine 5: Co-Craft Questions is suggested in this course. In this routine, students are given a context or situation, often in the form of a problem stem (for example, a story, image, video, or graph) with or without numerical values. Students develop mathematical questions that can be asked about the situation. A typical prompt is, “What mathematical questions could you ask about this situation?" The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers, and to develop students’ awareness of the language used in mathematics problems.

This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.

10.3

Activity

Instructional Routines

Analyze It MLR8: Discussion Supports

Materials

None

Activity Narrative

In this activity students recognize the relationship between measures of center and the shape of the distribution by creating and describing distributions with given measures of center. Listen for students using the terms "symmetric," "uniform," and "skewed." By using technology to create a dot plot with a given mean and median, students are using their understanding of the structure of the distribution (MP7) to adjust individual data values to change the measures of center. Making a spreadsheet available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Keep students in their groups.

Action and Expression: Internalize Executive Functions. To support organization, provide students with grid or graph paper to organize their work with the 3 dot plots.
Supports accessibility for: Language, Organization

Lesson Synthesis

Here are some questions to draw out the relationship between measures of center and the shape of the distribution.

“Why is the median preferred to the mean for a skewed data distribution?” (The values way to the right (or left) in skewed data have a greater effect on the mean, so the median is preferred to better reflect the typical values.)
“When an extreme value is present, why is the median preferred to the mean?” (Extreme values have a greater effect on the mean than on the median, so the median is preferred.)
“When the data distribution is symmetric or approximately symmetric, why is the mean preferred to the median?” (The mean takes into account every data value, so it is the preferred measure when it is representative of what is typical for the data.)

to access Cool-down.

Student Lesson Summary

Is it better to use the mean or median to describe the center of a data set?

The mean gives equal importance to each value when finding the center. The mean usually represents the typical values well when the data have a symmetric distribution. On the other hand, the mean can be greatly affected by changes to even a single value.

The median tells you the middle value in the data set, so changes to a single value usually do not affect the median much. So, the median is more appropriate for data that are not very symmetrically distributed.

We can look at the distribution of a data set and draw conclusions about the mean and the median.

Here is a dot plot showing the amount of time a dart takes to hit a target in seconds. The data produce a symmetric distribution.

<p>Dot plot from 0.9 to 1.9 by 0.1’s. Time to hit dartboard in seconds. Beginning at 0.9, number of dots above each increment is 0, 1, 1, 2, 4, 6, 4, 2, 1, 1, 0.</p>

When a distribution is symmetric, the median and mean are both found in the middle of the distribution. Since the median is the middle value (or the mean of the two middle values) of a data set, you can use the symmetry around the center of a symmetric distribution to find it easily. For the mean, you need to know that the sum of the distances away from the mean of the values greater than the mean is equal to the sum of the distances away from the mean of the values less than the mean. Using the symmetry of the symmetric distribution you can see that there are four values 0.1 second above the mean, two values 0.2 seconds above the mean, one value 0.3 seconds above the mean, and one value 0.4 seconds above the mean. Likewise, you can see that there are the same number of values the same distances below the mean.

Here is a dot plot using the same data, but with two of the values changed, resulting in a skewed distribution.

<p>Dot plot from 0.2 to 1.7 by 0.1’s. Time to hit dartboard in seconds. Beginning at 0.2, number of dots above each increment is 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 2, 4, 6, 4, 2, 0.</p>

When you have a skewed distribution, the distribution is not symmetric, so you are not able to use the symmetry to find the median and the mean. The median is still 1.4 seconds since it is still the middle value. The mean, on the other hand, is now about 1.273 seconds. The mean is less than the median because the lower values (0.3 and 0.4) result in a smaller value for the mean.

The median is usually more resistant to extreme values than is the mean. For this reason, the median is the preferred measure of center when a distribution is skewed or if there are extreme values. When using the median, you would also use the IQR as the preferred measure of variability. In a more symmetric distribution, the mean is the preferred measure of center, and the MAD is the preferred measure of variability.

Launch

Tell students to close their other books or devices.

Arrange students in groups of 2. Use Co-Craft Questions to give students an opportunity to familiarize themselves with the context, and to practice producing the language of mathematical questions.

Display only the data, without revealing the question.
Ask students, “What mathematical questions could you ask about this situation?”
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before their comparing questions with a partner.
As pairs discuss, support students in using conversation skills to generate and refine their questions collaboratively by seeking clarity, referring to students’ written notes, and revoicing oral responses, as necessary. Listen for how students use language about measures of center and variability.
Invite several groups to share one question with the class, and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify questions that focus on center and spread.
Reveal the questions, and give students a couple of minutes to compare them to their own question and those of their classmates. Identify similarities and differences.
Consider providing these prompts:
- “Which of your questions is most similar to or different from the ones provided? Why?”
- “Is there a main mathematical concept that is present in both your questions and the ones provided? If so, describe it.”
- “How do your questions relate to one of the lesson goals of measuring corresponding points to determine congruence?”
Invite students to choose one question to answer (from the class or from the curriculum), and then have students move on to the Activity Synthesis.

Provide access to devices that can run GeoGebra or other statistical technology.

After students have completed the problem that asks them to add values to the original data, give each group one of these distribution descriptions:

Uniform distribution with data between 4 and 12
Skewed right with most of the values at 10
Skewed left with most of the values at 10
Symmetric with most of the values at 4 and 16

Remind students of the words used to describe shapes of distributions: "symmetric," "skewed," "bell-shaped," "uniform," and "bimodal."

Activity

None

Activity Synthesis

The purpose of this discussion is for students to understand why the median is the preferred measure of center when a distribution is skewed or if there are extreme values, and the mean is the preferred measure of center when a distribution is symmetric and there are no extreme values.

For each description, select 2–3 groups to share their dot plots.

Here are some questions for discussion.

“For the first and second dot plot, what do the distribution shapes have in common? Why do we choose the mean as the more appropriate measure of center?” (Symmetric. The mean of a set of data gives equal importance to each value to find the center, so it is a preferred measure of center when it accurately represents typical values for data.)
“What do the shapes of the dot plots have in common when the mean is greater than the median?” (Skewed right.)
“What information does the shape of the skewed distributions tell you about the median and mean?” (When distributions are skewed right, they will likely have a mean that is greater than the median because the values to the right disproportionately affect the mean. When distributions are skewed left, they will likely have a mean that is less than the median because the values to the left disproportionately affect the mean.)

Speaking: MLR 8 Discussion Supports. As groups share their dot plots with the whole class, revoice student ideas for determining the appropriate measure of center based on the shape of the distribution. Be sure to amplify mathematical uses of language by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, “Can someone else explain why the median is used for skewed data?” Ask for details in students’ explanations by requesting students to elaborate on an idea or give an example from their data representation. This will help students define sequences with equations and answer questions about the context.
Design Principle(s): Support Sense-Making

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Lesson 10

The Effect of Extremes

Warm-up

Standards Alignment

Building On

Addressing

HSS-ID.A.2

Building Toward

HSS-ID.A.3

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Building on Student Thinking

Launch

Activity

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSS-ID.A.1

HSS-ID.A.3

Building Toward

Standards Alignment

Building On

Addressing

HSS-ID.A.1

HSS-ID.A.3

Building Toward

Student Response

Building on Student Thinking