Special Products
I can square a binomial and multiply a sum by a difference, and say where each shortcut comes from.
The problem this sheet solves
Squaring gets handed out over a sum as though it distributed, and it does not. It is one of the most persistent errors in the course, and giving students the formula does not shift it, because they already trust a formula — their own. Another formula just competes with it. This sheet argues from a PICTURE instead: (x + 5)² is the area of a square of side x + 5, and cutting the side at the 5 produces four real pieces, two of which are each x by 5. The 2ab stops being a term to remember. It also unifies both special products — the same box, with equal middle cells that double or opposite ones that cancel — so neither has to be memorised.
The key rule
(a + b)² = a² + 2ab + b², never a² + b². The 2ab is the two middle cells, which are equal. In (a + b)(a − b) they are opposite and cancel.
Equal cells double. Opposite cells cancel.
Watch out
The mistake: A student writes (x + 5)² = x2 + 25.
The fix: x2 + 10x + 25. Squaring is not handed out over a sum. The square of side (x + 5) holds two rectangles that are each x by 5, and 5x + 5x = 10x. At x = 1 the square is 36, and x2 + 25 gives 26.
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.