Linear vs Exponential Growth
I can tell a linear model from an exponential one and say which wins in the long run.
The problem this sheet solves
Nothing in the first several years reveals what happens later. The gap is still WIDENING at t = 5, and it is not until t = 10 that the curve passes 1,000 against 1,024 and never falls behind again. This is why exponential dominance is its own standard: it cannot be seen from a short table, so it has to be argued. This sheet treats the misconception as REASONABLE and then extends the table until the crossover happens, which is the only honest way to make the point. Checkpoints deliberately straddle the crossing so no item can be answered from a single value, and constant difference against constant ratio is the test students take away.
The key rule
Constant difference means linear; constant ratio means exponential. An exponential ALWAYS overtakes a linear one eventually, however far ahead the line starts.
Behind now is not behind forever.
Watch out
The mistake: At t = 5 the line gives 500 and the curve 32, so a student says the line always wins.
The fix: At t = 9 it is 900 against 512, still the line. At t = 10 it is 1,000 against 1,024 and the curve is past it — and stays past it forever after.
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.