Completing the Square to Find the Vertex

What students learn

Students learn to add and subtract (b/2)² on the same side to rewrite y = x² + bx + c in vertex form (factoring a out first when a ≠ 1), verify by expanding, and read the vertex.

See inside

Watch out

A student writes y = x² + 6x + 2 as y = (x + 3)² + 2.

The mistakes it targets

• Adds (b/2)² and never subtracts it
• Loses the sign of half of b
• Halves b but does not square it
• Subtracts the number added inside the parentheses instead of a times it
• Combines the constants with the wrong sign

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, five 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