10.1: Which One Doesn’t Belong: Data Correlations (5 minutes)
This warm-up prompts students to compare four representations of data. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Arrange students in groups of 2–4. Display the data representations for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn’t belong.
Which one doesn’t belong?
Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as association, correlation, linear model, strong relationship, and weak relationship. Also, press students on unsubstantiated claims.
10.2: Electric Power (15 minutes)
The mathematical purpose of this activity is for students to practice synthesizing the component skills needed to interpret data. Students are presented with a scatter plot, table, and some concluding statements about the data. Students critique the concluding statements, and explain their reasoning (MP3).
Try to gauge students’ current understanding of the context of utility bills. Ask them, “What type of relationship do you think makes sense for energy consumption and electric bill prices?” Students should understand that the more energy is used, the higher the utility bill. A sample response to this question is, “I think there’s a positive relationship between the two because I think as the energy consumption increases, then the electric bill increases also.”
Here are Elena’s representations of the data set.
|energy (kwh)||electric bill price (dollars)|
|energy (kwh)||electric bill price (dollars)|
After analyzing the data, Elena concludes:
- An estimate for the correlation coefficient for the line of best fit is \(r = \text-0.98\).
- Energy consumption and the price of electric bills have a positive relationship.
- Energy consumption and the price of electric bills have a weak relationship.
- Using the linear model, the electric bill is $260 when 1,200 kWh are consumed.
What parts of Elena’s interpretation of the data do you agree with and what parts do you disagree with? Explain your reasoning.
The purpose of this discussion is to highlight students’ reasoning about synthesizing the various components. Here are sample questions to promote class discussion.
- “Why does a positive relationship make sense for the two variables in the question?” (This type of relationship makes sense for the variables because the utility bill is higher the more energy that is consumed.)
- “Could you critique Elena’s claim of a weak, positive relationship without the \(r\) value?” (Yes, the scatter plot and linear model provide information about the type of relationship that the two variables have. Since the points appear to lay really close to a straight line, I can conclude that there is a strong relationship, not a weak one.)
- “How can you tell that Elena’s prediction using the linear model is incorrect?” (In order to use the linear model to predict an outcome, you have to find the coordinates for a specific point on the linear model. I see when \(x\) is 1,200 kWh, the \(y\) value is $270 not $260.)
10.3: Confident Players (20 minutes)
The mathematical purpose of this activity is for students to practice synthesizing the component skills needed to interpret data. Students are given a data set presented in a table and they practice creating a scatter plot, calculating the correlation coefficient with technology, and drawing conclusions about the data.
Ensure that students have access to appropriate technology to input data, create a scatter plot, and calculate the correlation coefficient.
Before Diego’s game, his coach asked each of his players, “On a scale of 1–10, how confident are you in the team winning the game?” Here is the data he collected from the team.
|players||confidence in winning (1–10)||number of points scored in a game|
- Use technology to create a scatter plot, a line of best fit, and the correlation coefficient.
- Is there a relationship between players’ level of confidence in winning and the amount of points they score in a game? Explain your reasoning.
- How many points does the linear model predict a player will score when his or her confidence is at a 4?
- Which players performed worse than the model predicted?
- Did Diego score better or worse than the linear model predicts?
The purpose of this discussion is for students to share their reasoning about interpreting data. Here are sample questions to promote class discussion:
- “Can you determine what type of relationship two variables have using a table only?” (You could determine if there is a negative or positive relationship, but the strength of the relationship is harder to determine using only a table. For both type and strength of the relationship, it is easier to use a scatter plot.)
- “What are other ways to ask about the relationship between players’ confidence and the amount of points they score?” (The question can ask about a correlation between the two. The question can also be worded using “variables” instead of explicitly asking about the specific variables in the problem. Another way to ask this question is to ask whether a variable increases or decreases in relation to the other.)
- “How can you determine if players performed worse or better than what the model predicts?” (Look at the linear model and where the points lay in relation to the line. If points are above the line, I know they scored more points than predicted and that is better than the prediction. If points are below the line, I know the player scored fewer points than predicted and that is worse than the prediction.)