Lesson 18

The purpose of this warm-up is for students to interpret the meaning of a single point in a scatter plot by looking at the point's coordinates and the graph's axis labels (MP2). 

Display the graph for all to see. Give students 1 minute of quiet think time followed by a whole-class discussion. 

Ask students to share their responses. Record and display their responses for all to see. If all of the students responds, “At 36 months of age, the panda weighed 82 kilograms,” ask them the following questions to clarify where they found that information in the graph:

• "What did the first coordinate of the point tell you?" (The first coordinate shows how far to go on the horizontal axis.)
• "What did the second coordinate of the point tell you?" (The second coordinate shows how far to go on the vertical axis.)
• "How did you know the meaning of each of the numbers?" (You can read the meaning of the numbers by looking at the labels on the axes.)
• "What specific information did each axis give you?" (The labels on the axes indicate attribute, age or weight, and the units in which each attribute is measured, months or kilograms.)

The analysis of scatter plots continues with data about the mass of automobiles and their fuel efficiency. Again, students connect points in the scatter plot with rows of data in a table, but now there is a third column that gives a name to each pair.

After picking out information about points in the scatter plot and table, students make their first foray into thinking about how scatter plots can help them make predictions. Specifically, when looking at two particular points in the scatter plot, students are asked if the results are surprising given the overall trend of the data. In later lessons, students develop more sophisticated tools for answering questions like this (specifically, by fitting lines to the data), so a heuristic discussion is all that is expected at this point.

The purpose of the discussion is for students to compare pairs of data in both representations and to begin thinking about trends in the data. Ask students to share their thinking about the last two questions. Draw out the fact that cars N and O seem to fit the overall trend that an increase in mass typically corresponds to a decrease in fuel efficiency but cars F and G don't fit that trend when compared with each other. Mention that we will see many examples of trends that don't necessarily apply to individuals in a population.

Consider asking some of the following questions to conclude the discussion:

• “How did you find the automobile with the greatest weight?” (I looked for the point farthest to the right.)
• “How did you find the automobile with the greatest fuel efficiency?” (I looked for the point that is highest.)
• “The data in this table is from automobiles that run solely on gasoline. Where do you think a hybrid car would appear on the graph?” (A hybrid car would use less gas for the same weight, so it's point would be below the other points with a similar weight.)
• “Did the scatter plot match your prediction?”

In this activity, students continue to identify points on a scatter plot as representatives of a single month when two variables are measured. Students are also asked to explain the meaning of some abstract points in the context of the problem and identify when it does not make sense to extrapolate information from the graph based on the context (MP2). 

Monitor for students who understand that 60 degrees Celsius is unreasonable for an average monthly temperature in most parts of the world, as well as those who identify the trend to predict negative sales which does not make sense in this context either.

Arrange students in groups of 2. Allow students 5 minutes quiet work time followed by 5 minutes partner discussion and whole-class discussion.

Ask students to predict what a scatter plot might look like when the temperature is along the \(x\)-axis and the sales of coats at a store is along the \(y\)-axis. 

​The purpose of the discussion is for students to understand that each point in the graph represents a single month in which two measurements are made. Select students to share their responses, including those identified for the last problem. 

Consider asking some of the following questions:

• “When discussing functions, what is a point of the form \((0,A)\) called? What about a point of the form \((B,0)\)?” (The \(y\)-intercept. The \(x\)-intercept.)
• “In a scatter plot is there a \(y\)-intercept?” (It depends on the context. For some situations, it may not make sense for the variable represented along the \(x\)-axis to be zero. For example if the variable is the height of a building. On other situations, there may be more than one point along the \(y\)-axis. For example, in a situation like the one in the task, there may be multiple months where the average temperature is 0 degree Celsius.)
• “When the average monthly temperature goes up, what seems to be happening to sales?” (They are decreasing.)