Unit 8, Lesson 8
Not Always Ideal
Algebra 2
8.1
Warm-up
Standards Alignment
Building On
Addressing
HSS-IC.A.2
Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability . Would a result of tails in a row cause you to question the model?
Instructional Routines
NoneMaterials
NoneActivity Narrative
The mathematical purpose of this activity 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. If necessary, explain that a free throw is a way of shooting a basketball at the goal from a specific place on the basketball court.
Activity
NoneLin, Kiran, and Diego are each going to shoot 100 free throws 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 all 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
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for students to support or oppose Lin’s mathematical claim using evidence. Invite 3–4 students to share their responses. Here are 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 are Kiran's and Diego’s chances of making a shot?” (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 Jada took 100 free throws and made 61 of them, would that be consistent with the claim that Jada has a 60% chance of making each shot?” (I think it would be consistent with the claim because 61% is really close to 60%. Predictions are estimates, so there should be some room for error.)
8.2
Activity
Instructional Routines
Poll the ClassMaterials
Activity Narrative
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 from the expected 10 heads that are predicted for 20 flips is likely, 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
8.3
Activity
Materials
Activity Narrative
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 the group was likely selected using a biased method or a random method.
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Lesson Synthesis
The purpose of the discussion is for students to recognize that probability gives an expectation for future results, but that reality may slightly differ from that expectation.
Here are some questions for discussion:
- “What was the purpose of flipping the coins and selecting the pieces of paper?” (This allowed us to get data to estimate how likely it is that a particular outcome would happen if the selection methods had been fair.)
- “What do the results of the simulations help you to understand?” (The results help me understand that there can be some variability when an entire class flips a coin or selects the pieces of paper. That variability helps me to determine how likely it is to get a certain number of heads or a certain number of students from the graduating class.)
- “How does this lesson relate to the concepts of measures of center and variability?” (Since we collected data, we could find measures of center and variability from the distribution, 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, which provides more confidence in an expected 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.)
Student Lesson Summary
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 if 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 , we might expect 50 of the flips to land heads up. This is a good expectation, but it should not be surprising if 45 or 57 of the flips are heads. On the other hand, if 95 of the flips are heads, we might become suspicious of the probability applying to this coin. Or, if the process of flipping the coin 100 times is repeated 100,000 times and the mean number of heads 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.