Lesson 8

Not Always Ideal

8.1: When Does It Get Weird? (5 minutes)

Warm-up

The mathematical purpose of this lesson is for students to use data to support or oppose a mathematical claim. Students are presented with a model and asked to determine whether the model applies to the situation when the actual data differs from the model.

Launch

Tell students they will use data to support or oppose a mathematical claim.

Student Facing

Lin, Kiran, and Diego are going to shoot 100 free throws each for practice. Based on their shooting in the past, Lin thinks they are all of similar ability, and Lin estimates that they each have a 60% chance of making each shot. They each shoot their shots.

  • Lin makes 63 of the 100 shots.
  • Kiran makes 75 of the 100 shots.
  • Diego makes 35 of the 100 shots.

From the results, do you agree with Lin’s estimate for the chance of each person making each shot? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of this discussion is for students to support or oppose Lin’s mathematical claim using evidence. Some questions for discussion:

  • “Is there evidence to support Lin’s claim that they each have a 60% chance of making each shot? Explain your reasoning.” (There appears to be some evidence from Lin’s observations, presumably of their previous performance, that they have about the same ability as she does. However, the data shows that Kiran made 75% of the free throws and Diego made about 35% of the free throws which is not consistent with Lin’s claim.)
  • “What would you estimate that Kiran and Diego’s chances of make a shot is?” (I would estimate that Kiran has about a 75% chance and Diego has a 35% chance.)
  • “How could you test your estimate?” (I could ask Kiran and Diego to shoot another 100 free throws and test my prediction.)
  • “If Diego took another 100 free throws and made 61 of them, would that provide evidence for the claim that Diego has a 60% chance of making each shot?” (I think it would provide evidence for the claim because 61% is really close to 60%. Predictions are estimates so there should be some room for error when talking about evidence to support predictions.)

8.2: What is Reasonable? (15 minutes)

Activity

The mathematical purpose of this activity is to make, critique, and justify claims using the data-generating process. Students flip a coin to determine the number of heads that show up for several groups of 20 flips. Students should recognize that some variability is expected from the expected 10 heads that are predicted for 20 flips, but that some results are very unlikely even if they are possible. Students should begin to recognize when a statistical model is consistent with results and when the data is so unusual that a new model might be more appropriate.

In later lessons, students will need to recognize when a normal curve is an appropriate model for data so that the area under a normal curve can be used to get information about the data.

Launch

Tell students that they will be flipping coins, recording the data, and reporting their results so that a dot plot can be created from class data. Monitor students as they flip their coins in the applet and collect data from the class to be included in a dot plot.

Action and Expression: Internalize Executive Functions. Provide students with a template so they can keep track of the data for each trial of coin flipping.
Supports accessibility for: Language; Organization

Student Facing

  1. What is the probability that you will flip heads when using the coin in the applet?
  2. Estimate the number of heads you will get when you flip the coin 20 times.
  3. Flip your coin 20 times and record the number of heads you get. Repeat this process 4 more times.

     
    trial number 1 2 3 4 5
    number of heads          
  4. Create a dot plot that shows the number of heads in 20 flips using data from the class.
  5. What is the least number of heads flipped by the class in 20 flips? What is the greatest number of heads flipped by the class in 20 flips?
  6. Based on the class dot plot, describe a range of values that represent a reasonable number of heads to flip when flipping 20 times.
  7. Priya flips her coin 20 times and it lands showing heads twice.
    1. Is it possible for this to happen with a fair coin?
    2. Based on the class distribution, should she be suspicious of this being an unfair coin? What can she do to provide evidence that it’s not a fair coin?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Tell students that they will be flipping coins, recording the data, and reporting their results so that a dot plot can be created from class data. Monitor students as they flip their coins and collect data from the class to be included in a dot plot.

Action and Expression: Internalize Executive Functions. Provide students with a template so they can keep track of the data for each trial of coin flipping.
Supports accessibility for: Language; Organization

Student Facing

  1. What is the probability that you will flip heads when using the coin you have?
  2. Estimate the number of heads you will get when you flip the coin 20 times.
  3. Flip your coin 20 times and record the number of heads you get. Repeat this process 4 more times.

    trial number 1 2 3 4 5
    number of heads          
  4. Create a dot plot that shows the number of heads in 20 flips using data from the class.
  5. What is the fewest number of heads flipped by the class in 20 flips? What is the greatest number of heads flipped by the class in 20 flips?
  6. Based on the class dot plot, describe a range of values that represent a reasonable number of heads to flip when flipping 20 times.
  7. Priya flips her coin 20 times and it lands showing heads twice.
    1. Is it possible for this to happen with a fair coin?
    2. Based on the class distribution, should she be suspicious of this being an unfair coin? What can she do to provide evidence that it’s not a fair coin?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

This time instead of flipping the coin 20 times and counting how many heads result, consider how many coin flips it takes to get a heads.

  1. Before flipping estimate the average number of flips it will take to get your fist heads.
  2. Flip! Record how many flips it takes to get your first heads. Then start again and record how many flips it takes to get your next heads. Keep flipping and recording how many flips it takes to get successive heads until you get 30 heads. Create a dot plot that shows how many flips it took each time to get a heads and compute the mean number of times it took.
  3. Based on the dot plot, describe a range of values that represent a reasonable number of coin flips it takes to get a heads.
Blank number line, 8 tick marks, 1 through 8.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may be uncomfortable with the less precise answers for the work in this unit and this activity in particular. Remind students that the actual world is messy and the work of statistics is to help make sense of that messiness. Math is not proving Priya has a fair coin or not, but can give us evidence to suspect that it is not and a decision can be made based on that evidence and anything else that is known about the situation.

Activity Synthesis

The purpose of this discussion is to talk about making and justifying claims based on data. Here are some questions for discussion:

  • “You performed 20 coin flips 5 times. Based on your data alone, what range of values would you predict for the number of times the coin would land on heads if it was flipped 20 times?” (I got 8, 9, 10, 12, 13 heads so I would predict that a coin would land on heads between 7 and 13 times. I included seven in my estimate because 13 is 3 above ten and 7 is 3 less than 10. The coin should land on heads an average of 10 times.)
  • “How does your prediction compare to the range of values you described from the class's data? Why do you think it was the same or different? Explain your reasoning.” (My range was smaller than the range from the whole class data. Since there was much more data for the whole class, I expected there to be a broader range of values since there was likely to be more variability.)
  • “If 1,000 people flipped a coin 20 times, how might that change your prediction from the rage for the class's data?” (If we had 1,000 data values, I would feel really confident about the range of values that I would expect from 20 coin flips. I know that it is possible to get between 0 and 20 heads, but I wonder how unlikely it really is to get 0 or 20. I think the 1,000 data values would help me to know a typical range of values.)

8.3: Is That Fair? (15 minutes)

Activity

The mathematical purpose of this activity is to make, critique, and justify claims using the data-generating process. The situation involves questioning whether the apparent disproportionate representation of a sample is due to bias or random selection. Students perform a simulation to determine how often the given scenario is likely to happen through random selection. Finally, students determine whether it is likely the group was selected using a biased method or a random method.

Launch

Tell students that they are going to perform a simulation to determine if the results of an experiment are reasonable. Monitor students as they select names and collect data from the class to be included in a dot plot. Identify students who use measures of center or variability to support mathematical claims.

Reading, Listening, Conversing: MLR6 Three Reads. Use this routine to support reading comprehension of this word problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (Choosing students to interview for a news station). After the second read, students list any quantities that can be counted or measured, without focusing on specific values (the number of students in the graduating class and the number of students in other classes). After the third read, invite students to brainstorm possible strategies to answer the questions. This helps students connect the language of the questions with the reasoning needed to solve the problem.
Design Principle(s): Support sense-making
Action and Expression: Internalize Executive Functions. Provide students with the 25 equal-sized pieces of paper, which they can label “GC” for graduating class and “OC” for other classes, as well as a template, so they can keep track of the data for each trial.
Supports accessibility for: Language; Organization

Student Facing

The local news station wants to interview 8 students from a school. There are 25 students on the student council. Ten of the students are from the graduating class and 15 are from the other classes. The principal has a difficult time deciding which students from the council will get interviewed, so she tells the group of students that she will put all of the names in a bowl, mix the names, then the first 8 names who are selected from the bowl will get to be interviewed.

The next day, the principal returns with the names selected. It turns out that 5 of the students who get to be interviewed are in the graduating class and only 3 of the students selected are from other classes. The students who are not in the graduating class complain that this doesn’t seem fair. They suspect that the principal chose the group rather than selecting at random.

  1. Do you think the principal could have chosen this group of students at random like she promised? Explain your reasoning.
  2. Simulate the drawing many times to find some possible results.

    1. Cut a piece of paper into 25 equal-sized pieces. On 10 on the pieces of paper, write “graduating class.” On the other 15 pieces of paper, write “other classes.” Fold the papers in half and mix them up.
    2. Take turns with your partner to draw 8 pieces of paper and record the number of students chosen that are in the graduating class.
    3. Repeat this process 4 more times.
    drawing number 1 2 3 4 5
    number of students in the graduating class          
  3. Create a dot plot that shows the number of students chosen from the graduating class by all of the students in your class.
  4. Based on the dot plot, do you think it is reasonable that the principal selected the students for the interview at random and still chose 5 of the 8 students who are in the graduating class? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The goal of this discussion is for students to use data to support or oppose a claim. Ask previously identified students “How does using measures of center or variability help you to support or oppose the students’ complaint?” (I calculated the mean, approximately 3.33, and the standard deviation, approximately 1.50. Five is not within one standard deviation from the mean but it was very close so I think it was possible for the principal to select five students from the graduating class using the random method. Five is within two standard deviations of the mean.)

Here are some questions for discussion:

  • “Based on the class data, what is the probability that 5 or more students in the graduating class were chosen?” (Sample response: It happened 15 out of 75 times, or 20%.)
  • “What percentage of trials had fewer than 5 students selected from the graduating class?” (80%)
  • “If the principal had selected 7 students in the graduating class to go on the trip, would there be enough evidence to prove that the principal was not being fair?” (The evidence would suggest that the principal was not being fair, but it would not prove it. It is still possible even if it is unlikely, so the statistical reasoning cannot prove this claim.)
  • “How many students from the graduating class would you typically expect to be drawn from 25 applications if 8 students were selected at random? Explain your reasoning.” (I think 2, 3, or 4 would be drawn typically. Since 10 out of 25 students are in the graduating class, and 8 names are drawn, and \(\frac{10}{25}\) of 8 is 3.2 which is approximately 3. When I look at the data in the dot plot, I see that 2 and 4 are typical as well.)
  • “Do you think a normal curve would be a good model for this data or the data from the previous problem? Explain your reasoning.” (Yes because the data is approximately symmetric and bell-shaped.)
  • “How would fitting a normal curve help in understanding the results?” (I could find out how likely it is that between 2 and 4 students would be selected.)
  • "Have you ever experienced a situation that seemed unfair, but might have been possibly due to random chance? What are some mathematical things you could do to help decide whether it was due to random chance or deliberately unfair?" (Each week, the adults at our house decide who will take the trash out between my brother and I. I've had to do it 4 weeks in a row, which seems unfair. I could flip a coin 52 times to represent a year of taking out the trash and see how often either heads or tails comes up 4 times in a row. Repeating this many times might give me an idea of how often it might happen by chance or whether the adults are picking on me on purpose.) 

Lesson Synthesis

Lesson Synthesis

Here are some questions for discussion:

  • “What was the purpose of actually flipping the coins and actually selecting the pieces of paper?” (This allowed us to get data to estimate how likely it was that a particular outcome would happen.)
  • “What did the results of the simulations help you to understand?” (The results helped me understand that there can be a great deal of variability when an entire class flips a coin or selects the pieces of paper. That variability helped me to determine how likely it was to get a certain number of heads or a certain number of students from the teacher’s class.)
  • “How does this lesson relate to the concepts of measures of center and variability?” (Since we collected data, we could have found measures of center and variability and used those to create a model like the normal curve.)
  • “Why is using the dot plot with the data from the entire class more useful than using just your own data?” (The larger sample size means there is more information to provide more confidence in what to expect for a probability.)
  • “Do you think that you would have come to different conclusions if we made a dot plot using data from an even larger group like multiple schools?” (It is possible, but I think it is more likely that we would just have additional evidence for the conclusions we made from the class data.)

8.4: Cool-down - Suspicious Rolls (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

An important concept is that mathematics can often provide a model for a situation so that estimates and predictions can be made, but it is rare for the actual results to match predictions exactly. A single result that differs from the model slightly should not invalidate the model, but if many results are different from a model or results tend to be drastically different, then the model may not do a good job of approximating the situation.

For example, imagine flipping a coin 100 times. Since the probability of a flipped coin landing heads up is \(\frac{1}{2}\), we might expect 50 of the flips to have landed heads up. This is a good expectation, but it should not be surprising if 45 or 57 of the flips were heads. On the other hand, if 95 of the flips were heads, we might become suspicious of the \(\frac{1}{2}\) probability applying to this coin. Or, if the process of flipping the coin 100 times is repeated 100,000 times and the number of heads is centered around 45, then maybe the assumption that the coin is fairly balanced to result in heads 50% of the time is incorrect and the model should be adjusted accordingly.

A good mathematician will often use data to suggest a model that can approximate a situation, then reevaluate the model by testing it against additional data. The model is then improved with the additional information to more closely mimic reality. This process may need to be revisited several times until its accuracy is satisfactory.