Modeling with Quadratic Functions
I can use second differences to tell when data is quadratic and to build its model.
What students learn
Students learn to take first and second differences of a table with equal x-steps, decide whether it is linear or quadratic, write y = ax² + bx + c from the differences, and use the model in a context.
See inside
The key rule
Not linear does not mean no pattern: take the second differences. If they are all the same (not 0), a quadratic fits.
2nd difference = 2a (x-steps of 1)
Table from x = 0 in steps of 1: c is the y at 0, and b = (y at 1 − y at 0) − a.
Watch out
At x = 0, 1, 2, 3, 4 the y-values are 2, 4, 8, 14, 22. A student says the differences 2, 4, 6, 8 are not all the same, so there is no pattern.
The mistake: no patternNot linear is not the same as no pattern.
The fix: second differences 2, 2, 2: quadratic, a = 1Take the next row: the jumps grow by 2 each time.
The mistakes it targets
- Stops when the first differences change and says there is no pattern
- Takes a as the whole second difference instead of half of it
- Calls a table with second differences of 0 quadratic
- Uses the first difference as b
- Subtracts each y-value from the one before it
- Reads c as the first y when the table starts after x = 0
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, seven 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.