Lesson 12

The mathematical purpose of this activity is for students to anticipate distributions of values from rolling standard number cubes so that they may compare the values from an actual experiment they will do in the next activity. Ideally, students should anticipate a uniform distribution of values from the number cube, but some variability should not be unexpected.

Show students a standard number cube which is a cube with the faces numbered 1 through 6.

The purpose of this activity is for students to start thinking about variability and an expected mean after repeated trials. If students do not mention that the distribution should be approximately uniform, bring this up to the class.

Here are some questions for discussion:

• “What do you think the distribution will look like?” (I think it will be relatively uniform but it cannot be perfectly uniform because there is an odd number of rolls and there will be some variability based on the actual rolls.)
• “When you roll the dice 35 times in the next activity, do you think it will look exactly like the dot plot you created?” (I do not think so. There is no way to predict exactly how many of each number you will get on the number cube, but on average it should be uniform.)
• “Do you think the distribution would be more uniform if the die is rolled a million times? Explain your reasoning.” (I think it would be. There might still be some irregularity. However, if you look at it relative to the number of values in each column the difference should be pretty small.)
• “What does this activity make you think about the concept of variation?” (It really makes me wonder about what the standard deviation would be for a million rolls versus 35 rolls. I wonder if I we could pool the class data so that we could look at the standard deviation for several hundred rolls.)

The mathematical purpose of this activity is for students to create and analyze a distribution and estimate the mean for rolls of a number cube. Students should recognize the distribution of sample means to be approximately normal which will be important in understanding the margin of error attached to estimates.

It can be easy for students to get confused about find the standard deviation and mean of mean values. Encourage students to slow down in their thinking and examine one level at a time. The dot plot consists of mean values and the distribution can be described by its center and spread by using the mean and standard deviation.

The goal of this discussion is for students to think about how sample means usually produce an approximately normal distribution with some variability. Collect all of the means from the class and create a dot plot. If it does not come up, ask students to describe the distribution shape. It should be approximately normal (bell-shaped and symmetric around 3.5).

Here are some questions for discussion:

• “Why do you think the mean is approximately 3.5?” (The center of the distribution is near 3.5.)
• “Which interval contains most of the means for your group’s data?” (Most of our means were between 2.8 and 4.2.)
• “What percentage of the means are in that interval?” (21 out 28 or approximately 78%)
• “Estimate the percentage of the class data that is in this interval?” (It looks like close to 90% of the means are in this interval.)
• “How does the class distribution compare to your group’s distribution?” (The class distribution is more bell-shaped and more symmetric.)

The mathematical purpose of this activity is for students to practice finding the mean and standard deviation of sample means and to estimate an associated margin of error. Students are presented with several situations, each with several sample means, and asked to estimate the true mean and provide a margin of error.

Identify students who talk about the relationship between sample proportions, sample means, and margin of error.

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

The goal of this discussion is for students to see that the margin of error can be estimated using the standard deviation of the mean of the sample means analogously to using the standard deviation of the mean of the sample proportions. Ask previously identified students, “How is the relationship between sample proportions and margin of error similar to the relationship between sample means and margin of error?” (In both situations you calculate the mean of the sample statistics and find the standard deviation. Once you get the standard deviation you double it to get the margin of error.)

Here are some questions for discussion:

• “What does the margin of error tell you?” (The margin of error tells you that the population mean has a 95% chance of being in the interval from the mean minus the margin of error to the mean plus the margin of error.)
• “What interval is likely to contain the population mean for the UFO question?” (Between 303 and 854 sightings.)
• “Can you think of a situation where sample means would be used to estimate the population mean?” (I think that they could be used to test that products, like boxes of cereal, weigh the correct amount. A box of cereal probably does not weigh exactly what the box says, but probably within a certain margin of error of that amount.)
• “Do you think that increasing the size of a sample would reduce the margin of error? Explain your reasoning.” (I think it would reduce the margin of error because a larger sample should be more representative of the entire sample so it would likely display less variation.)