High School: Statistics and Probability
Interpreting Categorical and Quantitative Data HSS-ID.C.7
7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in context of the data.
The fun part about being a statistician or math teacher is getting to say things like, "I love working with models," and, "My model is better than yours." Let other people assume what they will about your "models," but your students should know that a model is a way to use math to approximate a data set. After all, if statistics had beautiful men and women walking around, we bet it would be a much more popular subject.
One of the most common models used in statistics is the linear model. You may know this better as a linear equation or a best fit line. We can call it whatever we want, but it still makes estimates of the dependent variable, which means it's a model.
Students should know that all linear models take the form y = mx + b where m is the slope and b is the y-intercept. Hopefully it looks familiar—too familiar—to them. In fact, they probably think there's nothing more they could possibly learn about linear equations.
They might be right, but it's worth repeating.
The important part in using linear equations as models is the context. Students should be able to associate the independent variable with x and the dependent variable with y. Students should also be aware of what the slope and y-intercept represent in terms of the context and whether these values make sense or not. For example, it wouldn't make sense for a model describing a tree's yearly growth to have a negative slope.
Graphically, the slope of the line affects how "steep" the line is. The following plot shows the effect of varying slopes.
Graphically, the y-intercept of the line affects how "high" the line is. The following plot shows the effect of varying intercepts.