Lesson 9

The purpose of this number talk is to gather strategies and understandings students have for dividing by powers of 10. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to find the mean for various samples.

While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.

Reveal one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all previous 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 ___’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 _____’s strategy?”

• “Do you agree or disagree? Why?”

In this activity, students begin to see numerical evidence that different samples can produce different results and thus different estimates for population characteristics (MP2). Students look at a small population and some different collections of samples from this population. Although the data for this population is small enough that it is not necessary to use a sample, it is helpful to get an idea of how data from a sample compares to the population data.

Arrange students in groups of 2. In each group, one student should be assigned to work with mean as their measure of center and the other should work with median as their measure of center.

Tell students that, often in this unit, the data sets are small enough that sampling is not necessary, but it will be easier to work with small data sets so that we may compare information from the sample to the same information from the population.

The purpose of this discussion is to show that different samples can result in different estimates for a population characteristic as well as a reminder of reasons we might choose one measure of center over another.

Some questions for discussion:

• “What is the population for this situation?” (All of the paintings sold.) 
• “What are the samples used in the calculations?” (The first two paintings sold, those sold at a gallery show, and the oil paintings.)
• “Why did the different samples have different means?” (Because they used different paintings.)
• “Why were the means for the first two paintings sold and those sold at the gallery show so far off from the mean of all the paintings?” (Because they contained the cheapest one and most expensive one, respectively, with only a few other numbers to balance it out.)
• “Based on the numbers in the population, does it make more sense to use median or mean?” (Median since the \$1,200 painting is much greater than the rest of the values, so the measure of center is affected much more by the one painting when using mean.)

In this activity, students begin to see that some samples represent the population better than others. Students compare the dot plot of a population of data with the dot plots of several samples and discuss some aspects that would make some samples better than others (MP7). In the discussion, the phrase representative sample is defined. 

Ask several groups to share things they noticed and wondered about the dot plots. Record responses for all to see. If possible, display the dot plots to refer to while students share.

Consider asking these discussion questions:

• “What are some aspects that make for a good sample? Bad?” (A sample is “good” if it has a similar distribution to the population data. A sample is “bad” if the data does not have a similar distribution to the population data. For example, Sample 2 is bad because it is not centered in the same place.)
• “If you were to find a measure of center to represent a typical value for the population, would you use mean or median?” (Median since the data is not approximately symmetric.)
• “The population in this example has a mean of $2.06 and a median of $1.95. Sample 1 has a mean of $2.09 and median of $2. Sample 2 has a mean of $1.79 and a median of $1.80. Sample 3 has a mean of $2.36 and a median of $2.45. Based on this information, which seems to represent the population the best?” (Sample 1.)

Define representative sample. A representative sample is a sample that has a distribution that closely resembles the population distribution in center, shape, and spread.

Explain that a sample with the same mean as the population is not necessarily representative, since it may miss other important aspects of the population. 

• Example 1: If the population for a question is all of the humans in the world and you use one person from each country as your sample, it may not actually be representative of the population. Larger countries, such as China are under-represented since there are actually many Chinese people, but only 1 is included in our sample. Similarly, a smaller country like Cuba might be over-represented since it has fewer people living there, but is represented in the sample exactly the same as all of the other larger countries.
• Example 2: The average height of men in the world is approximately 70 inches. You might find two men, one who is 95 inches (7 feet 11 inches) tall and one who is 45 inches (3 feet 9 inches) tall. Their mean height may be the same as the world’s, but these two certainly do not represent the heights of most men.

Explain that a representative sample is the ideal type of sample we would like to collect, but if we do not know the data for the population, it will be hard to know if a sample we collect is representative or not. If we do know the population data, then a sample is probably unnecessary. In future lessons, we will explore methods of collecting samples that are more likely to produce representative samples (although they are still not guaranteed).

This activity is additional practice for students to understand the relationship between a sample and population. It may take additional time, and so is included as an optional activity.

In this activity, students attempt to recreate the data from the population data using three given samples (MP2). It is important for students to recognize that this is difficult to do and that some samples are more representative than others. Without knowing the population data, though, it can be difficult to know which samples will be representative. Methods for selecting samples in an unbiased way are explored in future lessons.

Keep students in groups of 2.

Remind students of the activity from a previous lesson where students selected papers (labeled A through O) from the bag and guessed at the sample space. That was an example of trying to interpret information about the population given a sample of information.

Read the first sentence of the task statement: “An online company tracks the number of pieces of furniture they sell each month for a year.” And then ask the students, “How many dots should be represented in the population data for one year?” (12, one for each month of the year.)

Allow students 5 minutes of partner work time followed by a whole-class discussion.

Students may consider that each of the auditors’ samples should be added together to create one larger sample, rather than considering that the auditors may have chosen the same data point in their separate samples.

Therefore, each auditor having a data point at 41,000 may mean that there is only one data point there, and each auditor included it in the sample, or it may mean that there are actually three data points there and each auditor included a different point from the population. 

The purpose of the discussion is for students to understand that getting an understanding of the population data from a sample can be very difficult, especially when it is not known whether samples are representative of the population or not.

Display the population dot plots for all to see.

For furniture sales, the samples came from data represented in this dot plot.

For electronics sales, the samples came from data represented in this dot plot.

Ask students

• “How close was your estimate to the actual dot plot? Consider the shape, center, and spread of the data in your answer.”
• “Were any samples better at mimicking the population than others?”
• “What could the auditors have done to make their samples more representative of the population data without knowing what the population would be?” (They could include more information in their samples. They should also think about how the samples were selected. For example, if the auditors only came on months when there were large sales happening, they may be missing important data.)

We will explore how to be careful about selecting appropriate samples in future lessons.