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Unit 8, Lesson 14

Questioning Experimenting

Algebra 2

14.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

In this Warm-up, students use their understanding of normal distributions to determine whether there is evidence that the difference in means from an experiment is likely due to the treatment. Students must use the structure of the distribution and a normal distribution model to solve the problem (MP7).

Launch

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Activity

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A scientist divides 30 strawberry plants into two groups at random. One group of 15 plants represents the control group and grows in standard greenhouse conditions. The second group of 15 plants represents the treatment group and grows under the same conditions except the type of soil the plants are grown in. After 6 weeks, the total weight (in grams) of the strawberries are measured for each plant. The scientist then performs a randomized experiment to compare the groups.

The data are summarized by these statistics and histogram.

Mean for the control group: 238.67 grams
Mean for the group with different soil: 347.47 grams
Standard deviation of differences in means from randomized groupings: 29.83 grams

<p>Histogram. differences in means from random grouping. </p>

Is there evidence that the difference in means from the original groupings is due to the different soil? Explain your reasoning.

Student Response

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Building on Student Thinking

Activity Synthesis

The purpose of the discussion is to recall how the mean and standard deviation determine a normal distribution, which can then be used to determine whether there is evidence that it is due to the treatment.

Ask students:

"Why is it useful to know the standard deviation for the differences in means from randomized groupings?" (This number gives a sense of how spread out we expect the differences to be if the treatment had no effect. Along with the mean of 0, the standard deviation describes a normal distribution that can be used to determine how likely it is that the observed difference in means or something more extreme would be if the treatment had no effect.)
"Why can we assume that the mean of the difference in means in the randomization distribution is 0?” (For each redistribution of data into Groups A and B, there is another redistribution into Groups B and A. Therefore, each difference in means will also have its opposite in the randomization distribution when all of the possible redistributions are taken into account.)

14.2

Activity

Instructional Routines

Aspects of Mathematical Modeling MLR4: Information Gap Cards

Materials

To Copy (from Blackline Masters)

Is There a Difference Cards

To Gather

Math Community Chart

Activity Narrative

This activity gives students an opportunity to determine and request the information needed to assess whether the results from an experiment are either significant or due to chance groupings.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6). Students must model the sampling distributions using a normal distribution based on the mean and standard deviation (MP4).

This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.

14.3

Activity

Optional

Instructional Routines

MLR8: Discussion Supports

Materials

None

Activity Narrative

This activity is helpful for students to better understand normal distributions, how to understand the relative extremity of values on a normal distribution, and connections between areas under a normal curve.

In this activity, students explore the use of tables to find the area under a normal curve bounded by certain values. Students are first introduced to z-scores and then use a table to find the area under the normal curve that is less than the value associated with that z-score.

Launch

Distribute the tables that show the areas under the standard normal curve based on z-scores. Demonstrate how to use the table to find the area under the curve to the left of these z-scores:

-1.96 (area of 0.0250)
-0.50 (area of 0.3085)
1.65 (area of 0.9505)

MLR8 Discussion Supports. Prior to students solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.
Advances: Reading, Representing

Lesson Synthesis

The purpose of the discussion is to have students practice asking questions needed to analyze data from an experiment. Consider asking:

"What questions are the most important to ask about the data when determining whether there is evidence that the treatment for an experiment causes a difference in means for the original groups?" (What is the difference in means for the original group? After redistributing the data into groups at random, what proportion of these trials had differences in means that are more extreme than the original group?)
"Why is it important to redistribute the data into groups at random many times?" (The more times the data is redistributed, the better we can understand what differences are possible and whether the original difference is likely or not.)

to access Cool-down.

Student Lesson Summary

After collecting data from an experiment, it is important to analyze the data to determine whether there is evidence that the difference in means for the groups is due to the treatment or whether the difference might be explained by the random groupings. There are several things that an experimenter needs to know to determine the possible cause of the differences.

First, the difference in the means for the two groups is important to know. Then the difference can be compared to the differences in means collected from regrouping the data into groups at random. The proportion of differences in means that are more extreme than the original difference can help determine whether the results we observed are inconsistent with the assumption that the treatment had no effect.

The proportion can be determined either from counting the actual number of differences that are more extreme or modeling the differences with a normal distribution.

Launch

Math Community

Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms or invite a student to read them out loud. Tell them that during this activity they are going to choose a norm to focus on and practice. This norm should be one that they think will help themselves and their group during the activity. At the end of the activity, students can share what norm they chose and how the norm did or did not support their group.

Tell students they will continue to examine data from experiments to determine if there is a significant difference in means or whether the difference is due to random groupings. Display the Information Gap graphic that illustrates a framework for the routine for all to see.

Remind students of the structure of the Information Gap routine, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, give a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give students the cards for a second problem and instruct them to switch roles.

Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Information Gap graphic visible throughout the activity or provide students with a physical copy.
Supports accessibility for: Memory, Organization

Activity

None

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because .”

Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.

If your teacher gives you the data card:

Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!

These steps may be repeated.
Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.

Student Response

to access Student Response.

Building on Student Thinking

If students use language that indicates definitive conclusions can be drawn from the data, consider asking:

“Do we know that for sure? How can we describe our conclusions while still acknowledging our uncertainty?”
“What was your initial assumption? Does the evidence support that assumption?”

A factory produces baseballs. The weights of the baseballs produced are approximately normally distributed with a mean weight of 145 grams and a standard deviation of 2 grams. Official rules require the balls to weigh between 142 and 149 grams.

Recall that the proportion of items in an interval of an approximately normally distributed situation can be estimated by the area under the normal curve. A table can be used to determine the area under a normal curve bounded by an interval.

First, the relevant values need to be converted to a z-score. A value's z-score represents the number of standard deviations it is above the mean. In the baseball example, the value 147 grams has a z-score of 1 because it is 1 standard deviation above the mean. The value 140 grams has a z‑score of -2.5 because it is 2.5 standard deviations below the mean.

In general, a z-score can be found using

Find the z-score for 142 grams.
Find the z-score for 149 grams.
What value has a z-score of 1.45?
The table gives the area under the normal curve that is less than the given value. Shade the region that is given by the table for the area related to 142 grams.
Normal distribution graph on a coordinate plane with no grid. Horizontal axis scale 140 to 150 by 2’s, labeled “baseball weight (grams)”. Vertical axis scale 0 point 0 2 to 0 point 1 4 by 0 point 0 2’s. The graph is a normal distribution curve with a mean of 145 and a standard deviation of 2.
Use the z-score for 142 grams and the table to find the area under the normal curve that is less than 142 grams.
Shade the region that is given by the table for the area related to 149 grams.

Normal distribution graph on a coordinate plane with no grid. Horizontal axis scale 140 to 150 by 2’s, labeled “baseball weight (grams)”. Vertical axis scale 0 point 0 2 to 0 point 1 4 by 0 point 0 2’s. The graph is a normal distribution curve with a mean of 145 and a standard deviation of 2.
Use the z-score for 149 grams and the table to find the area under the normal curve that is less than 149 grams.
Use the two areas to find the area under the normal curve between 142 and 149 grams. Explain or show your reasoning.
What proportion of the baseballs that the factory makes is estimated to be within the official rules?

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Lesson 14

Questioning Experimenting

Warm-up

Standards Alignment

Building On

Addressing

HSS-IC.B.5

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSS-IC.B.5

Building Toward

Standards Alignment

Building On

Addressing

HSS-ID.A.4

Building Toward