Exponential Decay Models
I can write an exponential decay model and explain why its base must be positive.
The problem this sheet solves
Students subtract the rate from zero instead of from one, and the resulting model alternates sign every year: −1,600 after one year, then +128 after two. One odd value looks like a slip; the pair is unarguable. Worse, a negative base has no real value at all at eighteen months, so the model does not merely mislead, it stops existing. This sheet defines the decay factor as what you KEEP, which is the phrase that makes 1 − r obvious, and it evaluates the wrong model at two consecutive times so the alternation is visible rather than described. The decay factor is bounded between 0 and 1 explicitly, giving students a check they can apply before graphing anything.
The key rule
The decay factor is 1 − r, and it is what you KEEP. It lies between 0 and 1. A negative base is not an exponential function at all.
8% lost means 92% kept.
Watch out
The mistake: For 8% decay a student writes y = 20000 · (−0.08)t.
The fix: y = 20000 · (0.92)t. The wrong model gives −$1,600 after one year and +$128 after two, so the value flips sign every year. It also has no value at all at 18 months.
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.