Multiplying Polynomials *with boxes*

If you can’t tell… I LOVE PROVIDING STRUCTURE (!!!) in math class.  I use boxes and circles to combine like terms, squiggly lines to indicate equal signs, and tables to work through transformations of functions.  While these structures are not necessary for the honors or highest level students to achieve success in math, I’ve found that my struggling English Learners and Special Education Students truly benefit from the organization.  If we can help them to independently decide on the structure to apply in a specific scenario, then they have an added visual clue to help them understand how to complete a problem.  And if these are best practices for our struggling learners, can’t we apply them in the general education classroom as well? 

I HATE FOIL (First, Outer, Inner, Last… for those of you unfamiliar with the term):

Oops, did I just type that “aloud”?  And hate is such a strong word….  Side note: I think I like the slide-and-divide method of factoring even less…   My IB students learned to FOIL in middle school, so of course, I won’t rain on their parade.  As we engage in class discussions, I refer to “FOILing” because I prefer to teach in the context of their prior knowledge.  It’s what they know, and it’s not their fault.


  • Teachers often instruct students to draw moons to represent multiplying the “First, Outer, Inner, and Last” terms.  Struggling students often can’t figure out which one of the four they forgot.  Or they multiply the same two terms… twice.
  • It’s old school.  My IB (Honors Pre-Calc) students have used FOIL since middle school because their teachers taught them to FOIL when multiplying binomials.  And it’s the acronym their teachers learned in high school back in the day.
  • FOIL becomes all parts of speech.  A noun (This is FOIL).  A verb (FOIL it!).  An adjective (The FOILED polynomial).  An adverb?  (We simplified this expression FOILly?  haha.  JK.)
  • What happens when we multiply a binomial times a trinomial?  It’s no longer technically FOILing.
  • I’d almost prefer to have a student break (5x – 2) (3x + 4) into 5x(3x+4) and -2(3x + 4), applying the distributive property to each.  But then they mix up their multiplication and exponent rules, and it’s more writing.

The more you can provide structure without requiring tons of extra writing, the more likely students will actually implement the structure.


Also known as “Bartley Boxes” (in honor of my 9th grade math teacher), I swear that these multiplication boxes work wonders.  They provide organization, ensuring that students multiply each term in the first parentheses times all terms in the second parentheses.

Because we’re using a different structure to multiply polynomials, it also indicates that different rules apply from when we’re adding and subtracting polynomials.  With addition and subtraction, *the name (exponent) stays the same* and students just combine the coefficients.  By implementing multiplication boxes, students begin to realize that they have to:

(i) multiply the coefficients and

(ii) count up the total number of xs being multiplied (or add the variable’s exponents together).

If those reasons aren’t enough to convince you, check out this AWESOME work by two of my Algebra Part 1 students.  The student that completed the 6 x 6 polynomial multiplication was one of the lowest achieving students in my class at the beginning of the year, due to his lack of number sense (adding/subtracting/multiplying/dividing integers).  If they can handle these types of questions, your classes can too!

Example 1: IMG_2505

(1)  Create a Multiplication Box

This is a binomial times a binomial, which means we need 2 spaces or boxes on the left and 2 spaces on top.  (We have already spent time identifying polynomials by the number of terms – before adding and subtracting polynomials, and definitely before multiplying polynomials.)

Look out (!!) – students often forget to write the negative sign in front of the 4, which results in all positive terms.


(2)  ♪♪ “MULTIPLY to get the MIDDLE” ♫

We multiply our coefficients and count the number of xs that are multiplied together.


(3)  ♪♪ ADD to get the ANSWER” ♫

I have students circle their like terms and remind them that “the NAME stays the SAME” when we add terms.  The exponent doesn’t change.


Side note:  I really do repeat ♪♪ “MULTIPLY to get the MIDDLE, ADD to get the  ANSWER”♫, in a sing-songy voice whenever the students forget a particular step.  And I make them write it on their papers at the top when we begin this process.

Example 2:img_2512-e1541525539603.jpg

(1)  Create a Multiplication Box

This is a binomial times a trinomial, which means we need 2 spaces or boxes on the left and 3 spaces on top.

(2)  ♪♪ “MULTIPLY to get the MIDDLE” ♫

Sometimes I’ll have my struggling students write out 2x^2 as 2xx so they can actually see the total number of xs for each square.

(3)  ♪♪ ADD to get the ANSWER” ♫

In this case, there are two different sets of like terms, so we circle both sets.



  • 9th graders often take Algebra 1 and Biology simultaneously and recognize the connections between multiplication boxes and Punnett Squares, which describe the possibilities for dominant and recessive genes. (Yay for cross-curricular teaching!)IMG_2516
  • This structure supports and simplifies students’ multiplication of square root sums and differences, if this topic is included in your Algebra 1 curriculum.


  • This organization helps them for Algebra 2 with simplifying the product of complex numbers.


  • You can use the exact opposite process to teach students how to factorI learned how to factor with these Bartley Boxes as well.  For more on factoring, see these posts:  Part 1 (GCF), Part 2 (Factor by Grouping), and Part 3 (Factoring Trinomials)
  • Side note:  I find that the process of combining like terms for the final answer reinforces integer addition and subtraction, which many of them struggle with from day 1 in Algebra 1.  (And you were WORRIED when I said I let my students use their calculators early on…)


To get a head start on Multiplying Polynomials with boxes, check out the following resources on Teachers Pay Teachers.

*This post has been brought to you by today’s issue of The Washington Post (if you’ve noticed some funky coloring in the background of my pictures…).  Hey, when inspiration strikes, you use what you can!* 


Miss Elsie

4 thoughts on “Multiplying Polynomials *with boxes*

  1. Pingback: Factoring Part 0: Factoring Prep! | Eat, Pray, Solve for X

  2. Pingback: Factoring Part 1: GCF | Eat, Pray, Solve for X

  3. Pingback: Factoring Part 2: Grouping | Eat, Pray, Solve for X

  4. Pingback: Teaching in a Pandemic (Part 1: Algebra) | Eat, Pray, Solve for X

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