standard_description: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
problem_activity: Activity
The mathematical purpose of this activity is to collect samples from a population and to use the mean and standard deviation of the sample proportions to estimate the population proportion and its associated margin of error. Students collect samples by drawing slips of paper from a bag and computing the proportion that fit some criteria. The class aggregates their data to create a dot plot showing all the proportions collected from the class. Using the dot plot, students then estimate the population proportion from which the samples are drawn.
Launch
Arrange students in groups of two. Create a bag for each pair of students. Each bag has 20 slips of paper with 14 of them marked “Pass” and 6 marked “Fail.” Collect data from students about the proportion of slips drawn that are marked Pass.
Tell students that a manufacturer is worried that their product may not be consistently good enough to pass quality control inspections. They are going to take some random samples of their product and have a quality control expert examine the items to determine if they pass or fail. To move forward with production, they need to have 80% of the items pass the inspection.
Provide access to devices that can run GeoGebra or other statistical technology. While they create their dot plots of the data, compute the mean and standard deviation of their sample proportions.
Engagement: Develop Effort and Persistence.
Provide a graphic organizer that focuses on increasing the length of on-task orientation in the face of distractions. For example, include a checklist of the tasks students should complete for each trial (draw 10 slips, count Pass slips, find the proportions) and a table for students to record the proportions for each trial.
Supports accessibility for: Attention; Social-emotional skills
Student Facing
Your teacher will give you a bag with paper slips inside that are marked as either Pass or Fail. Do not empty the bag to look at all of the contents at once.
One partner should hold the bag so that the other partner cannot see inside while they draw a slip of paper. The other partner should draw 10 slips of paper from the bag, one at a time. After the 10 slips are drawn, record the number of slips marked Pass.
From the results of the first trial, estimate the proportion of the slips in the bag that are marked Pass.
Switch roles with your partner and repeat the process until you have run 5 trials. For each trial, compute the proportion of slips you drew that are marked Pass.
|              trial              | 1 | 2 | 3 | 4 | 5 |
|---------------------------------|---|---|---|---|---|
|      number of Pass slips       |   |   |   |   |   |
| proportion of slips marked Pass |   |   |   |   |   |
Create a dot plot based on the trials from the class that shows the proportion of slips drawn that are marked Pass.
From the class dot plot, estimate the proportion of slips marked Pass in the bag. Explain your reasoning.
Student Response
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Activity Synthesis
The purpose of this discussion is for students to share their estimates of the population proportion and to make an estimate for the margin of error. Ask students to share their estimates of the population proportion. Ask students “What could you compute from the data to help you better describe the distribution of the sample proportions? How would these values help estimate the population proportion?” (You could compute the mean and the standard deviation. The mean would be a good estimate and the standard deviation could give an idea of how good of an estimate that is.)
Tell students the mean of all the sample proportions. Ask students, “How confident are you that your estimate is close to the actual population proportion?” Ask students to give a range of values that would be reasonable for the proportion of slips marked Pass. Ask whether they think the manufacturer is likely to have achieved the 80% pass rate based on the samples.
Display the dot plot
[IMG]
Tell students that another group had a dot plot that looked like this.
Here are some questions for discussion:
“The other group estimated a proportion of around 0.7 of the slips marked pass. Should the other group be more or less confident that their estimate is close to the actual population proportion than we should be about our estimate? Explain your reasoning.” (I think that they should be less confident than us because the data looks more spread out than ours. This means that there should be a bigger interval in which the actual value of the population proportion could fall.)
“How would the value for the standard deviation of the sample proportions affect your confidence in the answer?” (A greater standard deviation would tell us that the estimate has a wider range of possibilities, so I would be less confident if there was a large standard deviation.)
Tell students the standard deviation for the class data and, “The standard deviation for the other group in the dot plot is approximately 0.202.” Then ask, “If we want to be 95% confident of the population proportion using the sample proportions, what margin of error should I use for our class data? For the other group’s data?” (The margin of error we would use for each would be two times the standard deviation.)
“What interval could you be approximately 95% confident contains the population proportion for the class data?” (The interval would be from the mean minus the margin of error to the mean plus the margin of error.)
“How is this interval related to the margin of error and the standard deviation?” (It is related to both because the margin of error is two times the standard deviation and the interval is centered around the mean.)
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: “I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Support sense-making


aligns: no