math class 678 4-in-1 skill sheets
Modeling with Quadratic Functions, Algebra 1 4-in-1 Skill Sheet (HSF.BF.A.1a)
HSF.BF.A.1a Algebra 1 Unit 12: Quadratic Functions

Modeling with Quadratic Functions

I can use second differences to decide whether data is quadratic and to build the model.

View on TPT Back to Unit 12

The problem this sheet solves

A non-constant first difference has always meant no rule before now, so students have shown the data is not linear — a true and useful finding — and treated it as the end. The fix is one more row of subtraction: 1, 3, 5, 7 becomes 2, 2, 2, and constant SECOND differences mean quadratic. The second difference is not only a yes-or-no answer: it equals 2a, so for a second difference of 4 the leading coefficient is 2. That turns a test into a construction, which is why this sheet can BUILD the model and not only recognise one. It closes the unit because every other sheet starts from a quadratic and asks what it does; this one starts from data and asks whether a quadratic is the right thing at all.

The key rule

Constant first differences mean linear. If not, difference again: a constant SECOND difference means quadratic, and it equals 2a.

Not constant? Go again.

Watch out

The mistake: For 0, 1, 4, 9, 16 a student says the differences are 1, 3, 5, 7, so there is no pattern.

The fix: Their differences are right, and they have proved it is not LINEAR. Difference those: 2, 2, 2. Constant, so quadratic, and 2 = 2a gives a = 1.

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.