Common Core Standards: Math
3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
As we've already seen, a scatter plot can help us see what data is doing. And if the data follows (or almost follows) the equation of a straight line, we can tell a lot about the data. Not only can we see what it's done so far, we can also anticipate what it will probably do in the future. (Basically, we're fortunetellers minus the crystal balls.)
On one hand, students know about scatter plots and the line of best fit. On the other hand, they know about the y-intercepts and slopes of lines. Time to put those hands together.
Students should apply their knowledge of linear equations to the context of lines of best fit, understanding what the slope and y-intercept mean in terms of the axes of the scatter plot.
For example, let's say we've found that the relationship between Justin Beiber's high notes and capuchin monkey births has a line of best fit.
Students should be able to interpret the slope of as 1 capuchin monkey birth for every additional 150 Hz in the frequency of the Beib's voice. Reinforcing the concept of slope as "rise over run" and pointing out which variable is on which axis might be helpful for the students. (Remember that x is the independent variable, while y is the dependent variable.)
Additionally, we can interpret the y-intercept of -1 as being the number of monkeys born when Beiber is singing at 0 Hz (or, uh, not singing at all?). But how can we have negative births? Does that mean Beiber is killing baby monkeys whenever he isn't singing? Our guess is probably not.
Before blindly accepting the line of best fit's offerings, students should double-check that they make sense within the experiment, within the scatter plot, and whether or not it's a feasible possibility in general. For example, our graph only goes down to 300 Hz, so we don't have actual data about any lower frequencies. Besides, how can J.B. sing at 50 Hz when the human vocal register doesn't even go below 80 Hz?
Students should remember to interpret the line of best fit within the context of the data provided by the scatter plot. The line of best fit is meant to understand the general trend of data, but it might not be able to explain everything about it, but that's okay. We've got our brains to do that.