Understand that quantities exhibiting exponential growth or decay can be linearized using a log transformation.
Interpret the parameters of exponential regression models and their associated linear regression models after a log transformation.
Quick Lesson Plan
Welcome to the lesson you’ve probably never taught before in your life (I know I haven’t!). What’s a semi-log plot anyway?! It turns out that it’s a pretty cool and often-used statistical tool that also really gets at the heart of logarithms! Let’s dig in.
In today’s activity, students look at data about how the GDP per capita has changed over time. Make sure to print the data set (found under “Additional Materials”) so each group has one copy. First, students use a numerical approach and look for patterns from the tabular data. Students should see that there is not a constant change in the GDP per capita over each 5 year interval, and then decide if there is a constant second difference or a constant ratio. Again, this is great spiraled review of identifying growth patterns of certain function types. They are then provided the scatterplot of the data as well as three different regression models: linear, quadratic, and exponential. In questions 3 and 4, students interpret the parameters of the exponential regression equation. When you get to debriefing this part of the lesson make sure that students are using the word “estimated” or “predicted” in their interpretations. This skill is often assessed on the SAT and students that forget to use “estimated” or “predicted” are making too strong of claims, communicating that something is definitive even though it is not. In question 5, students use the model to make a prediction, which is one of the key reasons for using a regression model.
Note that x represents the years after 1800, not the number of 5 year intervals.
Questions 6-8 get at the heart of the lesson, which is our ability to transform exponential data to make it appear linear. While question 6 is mainly computational (calculate the log), question 7 has students think about why we might want to do this. Make sure they answer this question before turning the page and seeing the transformed graph!
Question 9 mirrors questions 3-5 except now students interpret and use the transformed linear model. It is critical that students understand that every reference to “y” is now referring to the dependent variable of the semi-log plot, which represents the log of GDP per capita! In part c, they evaluate the linear model for x=30, just as they did in question 5, but they will need to recognize that the y hat value represents the log of GDP per capita, and make the necessary adjustments.
Question 10 asks students how the parameters are related and the result is rather surprising! Both parameters are simply the log of the exponential model’s coefficients. Convince yourself why this is true. It is very rare that f(a*b)=f(a)+f(b), except of course with logarithms! The debrief should make explicit this use of logarithm properties.
How do you know there isn’t a constant rate of change in the GDP per capita?
Why is the y-intercept of the model not 2545 (the actual GDP per capita value in 1800 given in the data set)?
How far is your predicted value from the actual value? If you were to look at a residual plot, would the residual for x=30 be negative or positive?
Why does this new plot look linear?
Since this equation is linear, does that mean GDP per capita is now increasing at a constant rate?
Is GDP per capita $3.4741 in 1830? That seems really low…
How does your prediction in 9c compare to your prediction in question 5?
Today’s margin notes will focus on the key features of a semi-log plot and its interpretation. Since all regression models tend to use x and y hat as variables, students may not realize that the y hat in the linear regression represents something different than the y hat in the exponential regression. Help students connect the input and output of the regression model to the variables on the x and y axis of their respective plots.
The key question we are answering today is why a semi-log plot can take data showing exponential growth (or decay) and make it appear linear. The answer to this question is the very essence of logarithmic functions! Since logarithmic functions take inputs that grow multiplicatively and produce outputs that grow additively, we are in essence doing a function composition (You can think of f as representing the original exponential relationship, g as representing the common logarithmic function, and the semi-log plot showing (x, g(f(x))).
Note that a semi-log plot can also take a logarithmic relationship and make it appear linear by transforming the input values and plotting (log x, y). Here’s a great example.