Lesson Title: Solving by Completing the Square
Date: _____________ Teacher(s): ____________________
Course: Common Core Algebra 1, Unit 3
Start/end times: __________________
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed?
A.REI.4 Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x –
p)2 = q that has the same solutions. Derive the quadratic formula from this form.
MP1:
MP2:
MP3:
MP4:
MP5:
MP6:
MP7:
MP8:
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Algebra I, Unit 3
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Think-Pair-Share
Have students solve the following equations
individually, share their solutions and methods with
their partners, and then discuss the solutions with the
entire class.
1. x2 = 9
2. (x+1)2 = 9
3. (x+1)2 -5 = 4
The class should discuss how their methods for each
problem differed. In addition, students should
recognize that all answers are to include  and why.
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and provide
a foreshadowing of tomorrow? List the questions.
Solve by completing the square using algebra tiles, if
necessary. Show all work that leads to your solution.
Include a sketch of the tiles, if necessary.
6 = x2 – 4x + 3
Collect as an exit ticket or discuss as a class.
As a class, challenge students to figure out how they
might complete the square if a > 1. Allow students a few
minutes to discuss possible methods.
(Look for evidence of MP2, MP4, MP5, and MP7.)
(Look for evidence of MP2, MP3, and MP7.)

Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson?
1. Have pairs of students model each of the following functions using algebra tiles. Have partners try to form a
square with their tiles, and discuss why they cannot. Ask the partners to try to add the fewest tiles possible in
order to create a square for each equation.
1. 0 = x2 + 6x + 5
2. 0 = x2 + 4x
3. 0 = x2 - x + 3
2. As a class, discuss each equation. Using the tiles, have students model their methods under the document
camera. As one of the partner’s models with the tiles, have the other partner write the corresponding algebraic
representations on the chalkboard or on chart paper. As a class, have students generate general rules about how
to complete the square. Students should be able to see that the constant must equal one-half of the b-value,
squared. Discuss how this would change each equation and how they could ensure they equation remain
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Solving by Completing the Square
Course: Common Core Algebra 1, Unit 3
Date: _____________ Teacher(s): ____________________
Start/end times: __________________
balanced. Add to the work on the board, if necessary.
3. Students should be familiar with algebra tiles at this point. Ask them to think about how the resulting expression
(the “square”) could be rewritten in order to help them solve the equation. Refer to the warm-up for
reinforcement. Using the dimensions created by the square, students should be able to rewrite their equations
and solve for the zeros. As a class, come up with a general description of what must occur to solve an equation
by completing the square.
4. Have students form partners once the class has discussed generally agreed-upon methods of solving by
completing the square. Have the pairs solve the following by completing the square. They may choose to use
algebra tiles or not.
1. x2 – 2x – 12 = 0
2. x2 +6x = 5
3. x2 – 3x – 4 = 6
(Look for evidence of MP1, MP2, MP3, MP4, MP5, MP6, MP7, and MP8.)
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student mastery? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened (and conceptual) understanding.
Students should be able to complete the square using algebra tiles.
They should then be able to solve for x by isolating the binomial squared, and square rooting each side.
Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.
Once students have mastered completing the square, have them use the techniques to develop the quadratic formula.
There is a pencast on the Algebra Wikispace that shows the quadratic formula development.
If students are struggling with the algebraic representation, encourage them to model using tiles and write each
algebraic representation as they work. Verbally explaining the effects of the changes to the tiles and equation is also
helpful.
There is at least one problem that has no real/imaginary solutions. Students should discuss what this would look like
graphically.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Algebra tiles
Document camera
List some general rules for completing the square for an
equation where a > 1. Explain your rules in words or by
showing your method using a given equation.
Lesson Reflections: What questions, connected to the lesson objectives and evidence of success, will you use to
reflect on the effectiveness of this lesson?
Are students able to complete the square?
Are students able to solve for x by square rooting both sides of the equation?
Are students able to interpret what a function with no real solutions might look like graphically?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.