Students practice using precise wording (MP6) to describe the positive or negative association between two variables given scatter plots of data.

The purpose of the discussion is to talk about trends in data based on the representations in scatter plots.

Consider asking some of the following questions:

• “Is it surprising that the price decreases as mileage increases? Why doesn’t mileage predict price perfectly?” (There are other factors, like whether the car has any damage or has any extra features.)
• “Is it surprising that devices with a longer initial battery life have a longer later battery life? Why doesn’t the initial battery life predict later battery life perfectly?” (It is not surprising. There are other factors, like how much the devices are used or the materials used in the batteries.)
• “Is it surprising that energy consumption goes up as the temperature increases?” (No, because of air conditioning.)

For the last scatter plot, highlight the outliers by asking:

• “What might cause more energy consumption on a cool day?” (laundry, using power tools, using an electric heater)
• “What might cause less energy consumption on a hot day?” (being gone from home, using fans instead of air conditioning, raising the thermostat temperature)

Students may notice that the association between high temperature and energy consumed is more variable than the other situations. There is still a positive association or positive trend, but we would describe the association as “weaker.”

In the previous activity, students noticed trends in the data from the scatter plot. In this activity, the association is made more precise by looking at equations and graphs of linear models for the data to determine the slope. The numerical value of the slope is then interpreted in the context of the problem (MP2). 

The purpose of this discussion is to develop a quantitative sense of trends based on linear models of the data.

Consider asking some of the following questions:

• “What was the easiest way to find the slope for each situation?” (The coefficient of the -coordinate in the equation.)
• “Do the answers for the meaning of the slopes make sense in the contexts of the problem?” (Yes, battery life probably decreases over time as a percentage of its initial battery life, so 1 hour increase initially could only give an increase of 0.85 hours later. A 1 kilogram difference in the weight of a car is not very much, so it may lower the gas mileage, but not by much. A 1 degree temperature increase is not significant, so it should not need a lot more energy to cool off a building, but some would be needed.)
• “What is the difference between a positive slope and negative slope in your interpretations?” (A positive slope means both variables increase together. A negative slope means that when one variable increases, the other decreases.)
• “The model for energy consumption and temperature predicted a 0.59 kilowatt hour increase in energy consumption for every 1 degree increase in temperature. Estimate how much the temperature would need to increase to raise energy consumption by 6 kilowatt hours.” (A little more than 10 degrees.)

This activity returns to scatter plots without linear models given. Students determine whether the data seems to have a linear association or not. If it does, students are asked to decide whether the variables have a positive or negative association (MP4). 

The purpose of this discussion is to solidify understanding of trends in scatter plots and look for associations in the data.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the question “How can you tell when a linear model is a good fit for data and what associations the data might have?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback: 

• “_____ makes sense, but what do you mean when you say . . . ?”
• “Can you describe that another way?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

After Stronger and Clearer Each Time, ask:

• “How would you begin if you had a table of data?” (Rearranging the table so that one of the variables increases as you read down the table might help show how the other variable is trending, but it is difficult to notice whether there is an association in a table, so I would first create a scatter plot.)
• “How does the slope of a good linear model relate to data that has a negative association? A positive association?” (If the data has a negative association, the slope of a good linear model should also be negative. Similarly, if the data has a positive association, a good linear model should have a positive slope.)