Writing Exponential Functions from Data
I can write y = a · bˣ from two points or a table by counting the steps between them.
What students learn
Students learn to count the steps between two points, divide one output by the other to get b to that power, find b and a, and test the model on every point.
See inside
The key rule
Divide the later y by the earlier y, then take the root that matches the steps; points one step apart give b directly.
y = a · bˣ
Every b on this sheet is positive, so b² = 9 means b = 3.
Watch out
From (0, 5) and (2, 45), a student divides 45 ÷ 5 = 9 and writes y = 5 · 9ˣ.
The mistake: 45 ÷ 5 = 9, so y = 5 · 9ˣAt x = 2 it gives 5 · 81 = 405, not 45.
The fix: b² = 9, so b = 3: y = 5 · 3ˣThe points are 2 steps apart; 5 · 3² = 45.
The mistakes it targets
- Calls the ratio of outputs b when the points are several steps apart
- Takes the first y-value as a with no point at x = 0
- Stops after testing only the two points used
The interactive Teacher Deck (32 slides, PowerPoint)
- The Learn page taught on screen: 8 fill-in blanks, each filled on its own click as students write it
- The worked example one slide per step (four steps), each with the reason for the step
- Six practice and three apply problems: a your-turn slide with a think-time bar, then the solution one line per click
- Jump buttons on every slide: go straight to any problem, the answer bank or the exit ticket
- Speaker notes on each problem: the question to ask, the worked answer and the common error to listen for
- A completed deck PDF (19 pages) with every slide fully built, to print or post
The student packet (9 pages)
- Learn: guided notes with four definitions, the key rule, a worked example and a Watch Out error pair
- Practice: two warm-up, two standard and two stretch problems, with an answer bank for self-checking that hides three common-mistake decoys
- Apply: one word problem, a reasoning prompt and one error analysis
- Assess: an exit ticket of four items, two per page, with an I-can self-rating
- Homework (page 5): eight new problems, two warm-up, four standard and two stretch
- Answer key (pages 6–7): every problem worked step by step, three common mistakes with what to do about each, and a grading guide, plus the homework key (page 8)
- Color and ink-saver editions of the whole packet
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.