Completing the Square Activities and Resources for TeachersCompleting the Square.png

This page is dedicated to various tools to aid in the teaching of completing square. The page includes, in order: diagnostic tools, intro activities, games, and common mistakes. Completing the square is part of the common core standard under High School: Functions:Interpreting Functions:
CCSS.MATH.CONTENT.HSF.IF.C.8.A
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

1. Diagnostic Tools
  • Perfect Square Trinomials: In order to complete the square, students should have an understanding of perfect square trinomials. The first problem on this site, has examples of questions you could ask students to gauge their understanding of perfect square trinomials. It may be helpful to include some with a coefficient in front of the x^2 that is greater than 1 and have students write the trinomial in factored form.

  • Solving for x and Following a Form (1 & 2): The following two links give students the chance to solve equations in a completed square form. Other questions ask for students to complete the square but provide students with the form in which their answer should look. So even if students are unfamiliar with completing the square, they may be able to work toward a solution given this.


2. Intro Activities
  • Completing the Square Exploration Activity: The set-up teacher Kate Nowak has designed allows students the chance to explore individually or in pairs before really introducing the subject. The focus is on students noticing the pattern used when completing the square without being directly told. She also suggests using two days to cover the topic of completing the square and offering a wide variety of examples (coefficients being numbers or variables and examples with i).

  • Algebra Tiles: Algebra tiles are a tool teachers can use to help their students understand factoring as well as completing the square (students could make their own tiles or the teacher illustrate with a set). Using algebra tiles allows students to visualize the process and actually create a completed square. Students will recognize the importance the units play in completing the square and making sure their equation is always balanced.

  • Hip to Be (completing the) Square Introductory Lesson: This activity is a bit long for an introduction, but provides a nice way of teaching completing the square representational through the box method. The box method helps students to see what value they add/subtract to complete the square. The activity moves through stages of learning very well from concrete, to representational, to abstract. Lynn Boyles has many useful insights on how to carry out this activity, and it seems to be very easy to adapt.

  • Steps & 2 Examples:: This Algebra 2 – Trig Lesson Page is a well crafted instructional walk-through of how to complete the square. Clear and succinct, the link below provides a short list of steps for completing the square and follows it up with two distinct examples to help demonstrate the process. The activity seems like a clear way to remind students of the steps if they have forgotten how to complete the square.


3. Completing the Square Great Games
  • Complete the Square Helicopter Game: Here is an interesting, simple game that allows students to test their skills. Players are challenged to select the answer (a, b, c, or d) that corresponds to a number that completes the square of a partial equation that is presented. This is simple in concept, but developed enough to allow students to spend some entertaining time testing themselves to see how quickly they can answer the prompts.

  • Completing the Square Quiz Game This may be a bit dry as far as games go, but it is really useful to help students master the problem solving of completing the square numerically at a variety of difficulty levels. This is a great game to help students check their understanding as it starts at easy, then medium, hard, and lastly extreme. The game is full of practice exercises, hints, tips, and help. Students can see their growth clearly as they are quizzed, and it is great in preparation for a test for numerical understanding.

  • Jeopardy Jeopardy makes for an awesome review game. This game features more than just completing the square and hits on factoring, and solving for the zeros by graphing as well. It would probably be an excellent tool for the end of a unit. I would add in some candy for prizes too.

  • Completing the Square Stations: This activity sets up stations around the classroom with a completing the square problem at each station. Students move around and work together in groups to put the steps of the problem (printed out and cut into separate sheets) in the correct order. Once finished organizing the steps, students detail why each step was done. The activity allows students to focus on thinking through the process and reasons without making arithmetic errors.


4. Common Trip Wires about Completing the Square
  • Mistakes to Avoid: This document contains common mistakes involving quadratics, but specifically the third page discusses how to complete the square and common mistakes. The page does not explain much about how to help students avoid the mistakes. Rather, the author discusses the correct process in place of the mistake.

  • Solving Quadratic Equations by Completing the Square Trip ups: The document explains some common mistakes involved in completing the square. The document also provides steps for completing the square. Another cool feature is that it actually states the properties involved. A teacher could elaborate on these properties for deeper student understanding while helping students avoid mistakes.

Additional Resources and Activities for Completing the Square

CCSS Book of Worksheets:
  • The following link is a preview of a text titled “Supplemental Worksheets for the Common Core: Completing the Square to Solve Quadratic Equations” and provides examples for teaching students through quadratic equations. The worksheets allow for approaching the building blocks that lead up to completing the square from a variety of perspectives.

Module Lesson w/text and video:
  • This lesson by Purplemath can provide a more in depth teaching guide for students to work through after a basic understanding of completing the square is attained.


Completing the square calculator:
  • Although some students may abuse technology that solves an equation for them but this completing the square calculator goes well beyond giving an answer. The square calculator solves an equation but not before giving a thorough explanation of the steps it took to get to the final answer. Students who are having difficulties in completing the square may be able to turn to this resource to diagnose the problems that they are having since the steps are laid out for them prior to seeing the answer.


Loads of CCSS materials
  • Here are a wide range of different activities that can help students practice their skills in completing the square. There are proofs, mini-lessons, examples, digital activities, and a variety of links that help build the ability and knowledge of students in the subject of completing the square and the other quadratic skills surrounding the subject.

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Page created by Joshua Gates, Brittany Bordewyk, and Kevin Forster