Math 1 · Unit 10 · Lesson 4

Exponential Regression

Adds or multiplies, even when the data is messy. When the ratios from one output to the next are roughly the same and the scatter plot bends, fit an exponential, y = a times b to the x. Technology finds a and b, and you read them in the story: a is the starting value and b is the growth or decay factor. The gnats grow about 63% an hour. The pennies lose about half with every pour, and the model comes out near the story's one half, not exactly on it, because real data wobbles.

About 45 minutes Ratios roughly the same

The big idea

Adds or multiplies is still the first question, even when the data is messy. If the differences between outputs are roughly the same, try a line. If the ratios are roughly the same and the scatter plot bends, try an exponential, y = a times b to the x. Exponential regression finds the best-fitting curve, and technology gives a and b. Read them in the story: a is the starting value and b is the growth or decay factor, which is 1 plus or minus the rate. The gnats fit about 12.48 times 1.63 to the x, about 12 at the start and growing about 63% an hour, and a table or a horizontal line on the graph shows 200 gnats at about 5.7 hours. The pennies fit about 348 times 0.49 to the x, close to the story's 300 times one half to the x but not exactly, because coin flips are random and real data wobbles. You support an exponential model with the bending graph and the roughly constant ratios; you are not asked to prove it.

The parts of this lesson

Each part is its own short page. Go in order, or pick the one you need.

1The Words We UseThe vocabulary for this lesson, in one place.Start here 2Adds or multiplies? The gnatsAdds or multiplies, even when the data is messy. 3Decay: the pennies, and why real data wobblesNow decay. 4Practice ProblemsTry them, get help one step at a time, then print how it went.

Printables

One page with the family check, the gnats model and what a and b mean, the 200-gnats question, the pennies and why real data wobbles. Plus a worksheet with an answer key and a short lesson plan.

Study guide (PDF) Worksheet and answer key (PDF) Lesson plan (PDF)

Where this goes next

Previous lesson: Residuals, Checking the FitActual minus predicted. Next lesson: Association Is Not CausationWhy a strong pattern never proves that one thing causes another. Back to the Math 1 mapEvery Math 1 lesson, in course order. Tell us what you thoughtA few quick questions about this lesson. It takes a minute, and it is how the lesson gets better.

NC Math 1 standards in this lesson: NC.M1.S-ID.6c (fit an exponential function to data with technology and use it), NC.M1.F-LE.1 (linear or exponential from equal steps), NC.M1.F-LE.5 (interpret the parameters in context), NC.M1.F-IF.8b (growth and decay factors).