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.
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:
Have the class try a puzzlers 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
- In Question 2, where are the only two spaces you can begin? Why?
[You must start in one of the spaces highlighted in pink:
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?
[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 in individually or in pairs to complete the Polynomial Puzzler
activity sheet. If student 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
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 at the suggested Questions for Students below.
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
2. Did you use a traditional method to expand and factor, such as FOIL, or did you develop your own strategies as you worked?
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.]
- Did the puzzler format increase the level of student enthusiasm for this topic?
- Did students discuss their solutions as they were working through them with their partners?
- Do you think this is an activity better suited to individual or partner work?
- Were there any misconceptions regarding expanding or factoring that this activity revealed?