Lesson 10

The purpose of this Math Talk is to elicit strategies and understandings that students have for working with proportions. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to estimate sample proportions and population proportions.

Math Talks build fluency by encouraging students to think about expressions and rely on what they know about properties of operations to mentally solve a problem.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

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.

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.

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

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.)

The mathematical purpose of this activity is for students to calculate a sample proportion, to estimate the margin of error from a dot plot, and to estimate the margin of error from the mean and standard deviation of the sample proportions resulting from a simulation. In addition, students interpret the meaning of the margin of error in context.

After making an estimation for the population proportion, students are presented with a method for simulating additional samples based on the original sample proportion. Students then use the results from a simulation to measure some expected variability which is used to describe a margin of error.

Arrange students in groups of 2. Quiet work time followed by sharing work with a partner.

Students may be confused about why drawing additional samples might have different proportions when they are drawn from a population that is specifically set up to have the same proportion as the original sample. Remind students that different samples from the same population can still have different statistics and ask students where they may have seen that before (such as in the warm-up for this lesson).

The goal of this discussion is to make sure that students can calculate proportions and estimate the margin of error when given the mean and standard deviation of the sample proportions.

Here are some questions for discussion:

• “How do you think Andre knew to use 200 pieces of paper with 45 of them marked as Mutant?” (He knew that 9 out of 40 is an equivalent proportion to 45 out of 200.)
• "What is the shape of the distribution produced by the simulations?" (It is approximately normal.)
• "Since the distribution is approximately normal, we might be able to use a normal curve to model the data. What proportion of data is within 2 standard deviations of the mean for a normal distribution?" (Close to 95%)
• “What is a different way that you could simulate this experiment?” (You could change the number of slips to be something else is that is an equivalent proportion to 9 out of 40. You could also design a spinner that has a sector that is 81 degrees marked Mutant since that is \(\frac{9}{40}\) of a circle.)
• “What does the margin of error tell you in this context?” (The margin of error allows you to find the interval that you are 95% confident contains the population proportion.)
• “How confident would you be if you used the standard deviation instead of the margin of error to estimate the interval that contains the mean? Explain your reasoning.” (I would be much less confident. In a previous lesson we noticed that approximately 68% of the data is within one standard deviation of the mean for normally distributed data.)