The vertical intercept appears to be approximately 105 decibels, but the origin is not shown on the graph.

Display the scatter plot, then show the best-fit line.

Tell students to keep this data and scatter plot for a future lesson.

Ask:

• “How would the scatter plot and linear model change if the box itself was heavier?” (The dots and line would shift up.)
• “How would the scatter plot and linear model change if larger grapefruits were used instead of oranges?” (The weight would be increased for each point, and the slope of the line would be greater.)
• “How many oranges did we measure?” (10)
• “Since we measured 10 oranges, how does that affect your confidence in the estimate for 50 oranges?” (It makes me very skeptical that the estimate will be accurate. Although the data may look like a line for this section, there may be very different things happening farther away.)
• “Can you think of reasons why the real weight of 50 oranges might be different from the answer we get by using an estimate from the linear model?” (50 oranges may not fit in one box, so additional boxes may be needed, which will change the linear pattern in the data. Many of the additional oranges may be very small or very large and not fit the general trend seen with these 10 oranges. While the estimate from the line is interesting and may be better than a wild guess, it should not be considered a very good estimate.)
• “In this case, there might be some meaning attached to the y-intercept. That is not always the case. Why might interpreting the y-intercept not make sense in some situations? Can you think of a situation in which an interpretation of the y-intercept does not make sense?” (As we get farther from the collected data, the linear model may not make sense any more. Especially in situations in which the amount of something is approaching 0, very different things can be happening. For example, if we have data about water at temperatures around 60 to 70 degrees, that will be very different from what happens when the temperature is at 0 degrees.)

Although the grid lines in the graph do not appear to make squares, each line in both the horizontal and vertical directions are 0.5 apart. Tell students who are confused to look at the values listed on the axes and identify the values attached to a few of the grid lines.

The purpose of this discussion is for students to describe where to find the slope and vertical intercept from a scatter plot and interpret the values in terms of the context.

Ask previously selected groups to share. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “Are the values obtained from looking at the graph close to the values calculated using the equation?” (Yes, the values are close.)
• “Which method will you use to predict values based on the model?” (Maybe a mixture of both methods. Using the equation produces a more precise value from the model, but it is also important to look at the graph to get a sense of how well the model fits the data where I am predicting.)
• “Why do you think the intercept for the model is not ?” (Companies may still try to charge money for items that cost nothing to produce. They need to pay for the salaries of their workers, research and development, as well as other things that require the company to make money. It may also be the case that it does not make sense to interpret the -intercept since we have no evidence to believe that the same linear relationship will hold there.)

The purpose of this discussion is for students to describe the rate of change and the vertical intercept using the context in each graph.

For each question, give students time to think individually and then share their response with their partner, then select a student or pair of students to respond to the question. Ask students:

• “Why is the intercept for the bananas not ?” (A linear model is not exact even for the data it is based on. It is an approximation based on the data represented by the scatter plot. It is possible that it represents the weight of the bag that bananas were placed in. It is also possible that this value does not make sense since there is no evidence to believe the same linear relationship will hold near 0.)
• “How do you interpret the slope for each equation?” (The slope is the change in divided by the change in , so look at the labels on the scatter plot, and describe how for each increase of 1 of the -variable, on average, there is a decrease (if the slope is negative) or increase (if the slope is positive) in the variable represented by the -variable.)
• “When might it make sense to interpret the -intercept for a linear model?” (It makes sense when -values around 0 are in the range of the data used to create the model. In other cases, care should be taken to put too much faith in the answer since the linear trend may not continue to hold farther from the collected data.)

Students may struggle with estimating a slope when the scale on the - and -axes are different. Ask students to find the coordinates for a couple of points on or near the line and find the slope between those points.