Writing Exponential Functions from Data
I can write an exponential model from two data points by counting the steps between them.
The problem this sheet solves
From (1, 6) and (4, 48), dividing gives 8 — but the points are THREE steps apart, so 8 is b cubed and b is 2. The division was right and the counting was wrong, and the resulting model looks perfectly tidy while giving 3,072 where the data says 48. Nothing about it is visibly broken. This sheet makes counting the steps between points the first move, then takes the nth root, so the gap between rows can never be assumed to be one. Every model is TESTED against a second data point, which is the only check that catches this particular error, and tables are built with uneven spacing on purpose.
The key rule
Two y-values n steps apart divide to give bn, not b. Take the nth root. Then recover a by dividing a known y by bt.
Count the steps, then root it.
Watch out
The mistake: From (1, 6) and (4, 48) a student divides 48 ÷ 6 = 8 and writes b = 8.
The fix: They are THREE steps apart, so 8 is b3 and b = 2. With b = 8 the model becomes y = 0.75 · 8t, which gives 3,072 at t = 4 instead of 48.
What is included
- Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
- Practice page — six problems, building from foundational to multi-step
- Apply page — a word problem, a reasoning prompt, and an error-analysis task
- Assess — a half-page exit ticket, two to a sheet, with a score box
- Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
- Ink-saver edition of all 7 pages
- 14-slide teacher deck (PowerPoint) with presenter notes on every slide
Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.