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Polynomial Puzzler

  • Lesson
Terry Johanson
Langham, SK, Canada

In this lesson, students explore polynomials by solving puzzlers. To solve the puzzlers, students factor polynomials and multiply monomials and binomials. The lesson includes ideas on how this format can be applied to other mathematical concepts.

This lesson will have students reinforce their knowledge about multiplying monomials and binomials, and factoring trinomials. The puzzler structure is simple, and allows for differentiation of the activity sheet. Begin by explaining how the puzzler works by having students look at the completed puzzler on the Polynomial Puzzler Overhead.

overhead Polynomial Puzzler Overhead 

2938 example 

What is the structure of this puzzler? The puzzler is structured so that the four numbers on the top left corner are multiplied in rows and columns with the products in the right hand column and bottom row. This is illustrated below:

 2938 explanation down 

2938 explanation across 


Have the class try a puzzler on the overhead. This overhead will allow students to solve simple number puzzlers, and then move into puzzlers with monomials, binomials, and trinomials. You might think about asking your class these questions as they work through the examples:

  • In Question 2, where are the only two spaces you can begin? Why?
    [You must start in one of the spaces highlighted in pink:

    2938 start 

    This is because you need to have 2 pieces of information in a row or column in order to figure out an unknown space.]

  • If you were to make the mistake below when solving Question 1, what would happen to your final solution?

    2938 mistake 

    [As you continued to work through the rest of the puzzler, you would notice that there are 2 answers for the bottom right hand space, depending on whether you use the right column or bottom row to find the value for that space. This is one way to know that you have made a mistake filling in a puzzler.]
  • Was there a difference in how difficult the puzzlers were in Questions 1 and 2? Why do think this might be?
    [Students will probably think Question 1 is less difficult than Question 2. This is because Question 1 only requires multiplication, while Question 2 involves both multiplication and division. Some students may find division to be a more difficult operation than multiplication.]
  • Before solving the puzzlers in Questions 3 and 4, which puzzle do you expect to be easier? Why?
    [Students will probably expect Question 3 to be easier because it only involves multiplication of monomials and biniomials. Question 4 will require both multiplication (expansion) and division (factoring). Some students may feel that expanding monomials and binomials is easier than factoring.]
  • What is special about the bottom right space in a puzzler?
    [The bottom right space is like a self-check. If the right column and bottom row both multiply to give you the answer in the bottom right space, you know that your puzzler solution is correct.]
  • In Questions 3 and 4, the bottom row and right column each contain 2 binomials. What must be true of these 2 sets of binomials?
    [When multiplied, these two sets of binomials must be equal. In fact, these two sets of spaces in the bottom row and right column will always be equal.]

The second page of the overhead is the solutions. You can either display the answers on the overhead and have students ask questions or have students share their answers with each other.

Students are now ready to solve puzzlers on their own. Students should work individually or in pairs to complete the Polynomial Puzzler Activity Sheet. If students struggle with how to fill in the puzzler, remind them of the strategies used to solve the examples on the overhead. How could these strategies be applied to more complicated polynomial puzzlers?

pdficon Polynomial Puzzler Activity Sheet 

When students have completed their puzzlers, allow them to share their answers and thinking with the class. Here are some ideas to help you structure this:

  • Don’t simply put up the answer key. Have students write their solutions to the puzzlers on the board or fill them in on an overhead copy of the activity sheet. As they fill in the spaces, ask them to explain verbally or in writing how they approached the puzzle.
  • If students worked in pairs, allow them to present the solutions in pairs.
  • As students are reflecting, you may wish to ask them questions such as the suggested Questions for Students below.

pdficon Polynomial Puzzler Answer Key 


“Developing Mathematical Fluency: Activities for Grades 5 - 8” by Dr. Grayson H.Wheatley

Assessment Options 

  1. Have students submit their Polynomial Puzzler activity sheet as an assessment.
  2. Include a puzzler on a quiz or exam, such as one created from the completed puzzler below:

    2938 assessment 

    To create the assessment, delete entries from 5 spaces. To make an easier question, delete entries from the right column and bottom row. To make a more difficult question, delete some entries from the four spaces in the top left hand corner. Note: There are combinations of entries that are not solvable, such as puzzlers like the ones below if only the spaces highlighted in pink have entries:

     2938 comboleft 

    The only space that can be filled is the center.

     2938 comboright 

    Again, only the center space can be filled. Any other combination that fills one line and one other space will be unsolvable.

    To build a puzzler, begin with any 4 values in the top left corner of a blank puzzler. Multiply the rows and columns to figure out the right hand column and bottom row. Then, delete 5 entries from the puzzler to finish.

  3. Use page 2 of the Blank Polynomial Puzzler Overhead to create puzzlers as warm-up exercises.
    overhead Blank Polynomial Puzzler Overhead 


  1. The basic structure of a Polynomial Puzzler can be used for almost any math concept involving multiplication or addition. For example:
    • integer addition
    • exponent multiplication, including fractional and negative
    • complex number multiplication
  2. Students can build Polynomial Puzzlers to exchange. Give them a Blank Polynomial Puzzler Overhead and directions on how to build their puzzlers. See the instructions in Assessment Option 2.
    overhead Blank Polynomial Puzzler Overhead

    Caution students that if they don’t want to go beyond degree 3 polynomials, they must be careful of the degrees of the expressions they put in the 4 spaces in the top right. Since all 4 expressions will eventually be multiplied together, the sum of the 4 expressions will tell them the degree of the polynomial in the bottom right space.

    Look at this completed puzzler:

    2938 assessment 

    Of the 4 spaces in the top left, two are degree 0 and two are degree 1. Without looking at the bottom right space, we know the degree will be 0 + 0 + 1 + 1 = 2.


Questions for Students 

1. Did you try to expand first, and factor only if the spaces couldn't be fill in otherwise? Did you seek out the spaces that required factoring first?

[Answers will vary.]

2. Did you use a traditional method to expand and factor, such as FOIL, or did you develop your own strategies as you worked?

[Answers will vary.]

3. Were there certain paths to solving the polynomial puzzlers that were easier than others? Why?

[The pathways that allowed you to find the bottom row and right hand column entries by multiplying (expanding) rather than dividing (factoring) are easier.]

4. What is the mathematical relationship between expanding and factoring?

[Expanding and factoring are inverses of one another. Students may also talk about the fact that expanding is multiplying and factoring is dividing.]

Teacher Reflection 

  • What effect did the puzzler format have on the level of student enthusiasm for this topic?
  • What type of interaction did students have with their partners while working together? 
  • What makes this activity better suited for individual versus partner work and vice versa?   
  • Were there any misconceptions regarding expanding or factoring that this activity revealed?  What were the misconceptions and how can they be addressed?

Learning Objectives

Students will:

  • Understand the relationship between expanding and factoring polynomials.
  • Reinforce their knowledge of factoring trinomials.
  • Reinforce their knowledge of multiplying monomials and binomials.

NCTM Standards and Expectations

  • Understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions.
  • Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.