Content Objective
Students will solve quadratic equations by completing the square and justify each algebraic step.
Language Objective
Students will explain completing the square using vocabulary: perfect square trinomial, vertex form, coefficient.
TEKS Connection
A.8(A) — Solve quadratic equations having real solutions by completing the square.
Curriculum Connections
A.7(A) vertex form · A.8(B) quadratic formula · Geometry area models
Materials & Technology
Mini whiteboards
Dry-erase markers
Algebra tile sets
Colored highlighters
Exit ticket slips (tiered)
Anchor chart paper
Desmos Graphing Calculator
GeoGebra Algebra Tiles
Kahoot!
Google Slides
Google Forms (exit ticket)
Lesson Sequence
Opening question on board: "Where have you seen a perfect square before? Can you write one?"
- Students pair-share: What is a perfect square? Can they give an example?
- Call 2–3 pairs to share. Record responses on anchor chart.
- Preview today's goal: "Today we'll learn how to CREATE a perfect square — a method called Completing the Square."
Scaffolding
Below grade — provide a word bank. ELL — allow native language pair-share first.
Above grade
Prompt: "Can you write a perfect square trinomial? What pattern do you notice?"
Launch problem: "A square patio has area x² + 10x. How do we find its side length? What's missing?"
Algebra Tiles — Building x² + 10x
The tiles below represent x² + 10x. Click each question to explore what's missing.
Q
How many x tiles do we have, and what do they represent geometrically?
We have 10 x tiles — they are rectangles with width 1 and length x. They represent the "10x" part of our expression.
+
Q
If we split the 10 x tiles evenly — 5 on the right, 5 below — what shape are we trying to build?
We're trying to build a square! A square with side length (x + 5). The x² goes in the top-left corner, and the x tiles form two arms of equal length.
+
Q
Look at the bottom-right corner. What piece is missing? How big is it?
The missing piece is a 5 × 5 = 25 square. It has area 25, made of 25 unit tiles. This is (b/2)² = (10/2)² = 5² = 25.
+
Q
If we add 25 to complete the square, what expression do we get? Can you factor it?
x² + 10x + 25 = (x + 5)². Adding 25 "completes" the square — the entire shape becomes a perfect square with side length (x + 5).
+
Transition: "Today we learn a method called Completing the Square — it's literally what we just saw geometrically. Watch me first."
Teacher models with think-aloud. Students follow on graphic organizer (6-step template).
Example 1 (a = 1, integer answers): x² + 6x − 7 = 0
- Step 1: Move constant → x² + 6x = 7
- Step 2: Find (b/2)² = (3)² = 9
- Step 3: Add to both sides → x² + 6x + 9 = 16
- Step 4: Factor left side → (x + 3)² = 16
- Step 5: Square root both sides → x + 3 = ±4
- Step 6: Solve → x = 1 or x = −7
Example 2 (irrational answers): x² − 4x − 1 = 0 → x = 2 ± √5
Scaffolding
Color-code each step with a different marker. Below grade — pre-fill Steps 1–2 on graphic organizer.
CFU
Pause after Step 3 and Step 5 — ask students to show work on mini whiteboards before continuing.
Transition: "You just watched me — now let's do one together."
Student pairs work on mini whiteboards. Teacher circulates with observation checklist.
| Problem |
Focus |
Notes |
| x² + 8x + 7 = 0 |
a = 1, whole-number answers |
All students |
| x² − 6x + 2 = 0 |
Irrational answers, simplest radical form |
All students |
| 2x² + 12x − 14 = 0 |
a ≠ 1 (divide through first) |
On grade + above |
- After Problem 2: whole-class pause — 3 students share approaches (Socratic discussion).
- Desmos check: Graph each equation — do x-intercepts match your solutions?
Above grade
Write and solve their own problem — must produce irrational answers.
ELL / SPED
Sentence frames: "I added ___ to both sides because…" · Peer partner support.
Transition: Kahoot! — 5-question rapid-fire review (see Part 5).
5 multiple-choice questions targeting common errors. Live class dashboard reveals top misconceptions.
- Wrong sign when moving constant
- Forgetting the ± when taking square root
- Not simplifying radical form
- Adding (b/2)² to only one side
- Errors when a ≠ 1 (forgetting to divide first)
Teacher move: Identify the top 2 misconceptions from Kahoot data and address them directly before independent practice.
Transition: "Great data — now YOU try independently."
Tiered by readiness — color-coded folders. Students self-select; teacher may redirect.
| Tier |
Problems |
Supports |
| Tier 1 Foundational |
3 problems, a = 1, whole-number answers |
Step frame provided |
| Tier 2 On Grade |
4 problems, a = 1 and a ≠ 1, irrationals included |
Anchor chart available |
| Tier 3 Enrichment |
4 problems + derive the Quadratic Formula from ax² + bx + c = 0 |
Connects to A.8(B) |
- Students work silently for 12 minutes; teacher circulates with clipboard.
- At 10-min mark: peer-check one problem with a partner.
- Last 5 min: selected students share solutions under document camera.
SPED / 504
Extended time option; graph paper for organization; chunked one step at a time.
- Class builds the 6-step anchor chart together — students supply each step.
- Vocabulary wrap-up: each student says one key term to a partner in a sentence.
Exit Ticket (tiered — matches tier folder color):
| Tier | Exit Ticket Task |
|---|
| Tier 1 |
Solve x² + 4x − 5 = 0; label each of your 6 steps. |
| Tier 2 |
Solve 3x² − 6x − 9 = 0; explain in writing why you added the value you did. |
| Tier 3 |
Solve x² + bx + c = 0 in terms of b and c; connect result to vertex form. |
Google Form digital option available. Teacher sorts slips at door:
Got it / Almost / Needs reteach — informs next day warm-up.
Preview: "Tomorrow we connect this to the Quadratic Formula — it's the same method!"
Differentiation Summary
| Strategy |
Below / SPED / ELL |
On Grade |
Above Grade |
| Content |
Pre-filled step frames; a = 1 only; whole-number solutions |
Mixed a values; irrationals; full 6-step process |
Derive quadratic formula; explore vertex form connection |
| Process |
CRA sequence; sentence frames; partner support |
Mini whiteboards; peer check; Desmos verify |
Independent extension; teach-back to peers |
| Product |
Tier 1 exit ticket with step labels |
Tier 2 exit ticket with written justification |
Tier 3: generalize with variables b and c |
Teacher Reflection
What worked well?
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What would I change?
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Students needing reteach
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Connection to next lesson (A.8B)
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