Algebra I Lesson

Solving Quadratics: Completing the Square (The Missing Corner)

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8th or 9th GradeBuilding Thinking ClassroomsCCSS.MATH.CONTENT.HSA.REI.B.4, CCSS.MATH.CONTENT.HSA.SSE.B.3b60 min

Learning Objective

Students will solve quadratic equations by completing the square, including cases with fractional half-values and leading coefficients, and connect the algebraic process to the geometric model of a perfect-square area.

Lesson Overview

Students begin by visually discovering the geometric meaning of 'completing the square' using tile diagrams, then thin-slice through algebraic solving — from perfect square equations to leading coefficients and vertex form — building toward deriving the quadratic formula.

Materials

Vertical non-permanent surfaces (whiteboards or chart paper)
Dry-erase markers (one per group)
Erasers or paper towels
Visibly random grouping cards
Four-quadrant note-taking template (half-sheet)
Desmos-enabled device or projector for interactive screen

Scaffolded Task Progression

1Task 1

Solve the equation

(x - 3)^{2} = 25

. (Hint: What operation undoes a square? Remember that both a positive and negative number can be squared to get

25

!). Write down both of your final values for

x

2Task 2

The Visual Bridge. Draw a square on your board representing the area

x^{2} + 6 x

. Start with the

x^{2}

tile in the top-left. Split the

6 x

into two equal halves — place

3 x

along the right side and

3 x

along the bottom. How many unit tiles must fill the empty bottom-right corner to complete the large square? Write the completed square as

(x + ?)^{2}

and expand it to verify.

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