Math = Love: Dividing Polynomials Using the Box Method

Tuesday, October 27, 2015

Dividing Polynomials Using the Box Method

Today, I want to talk about one of my favorite things in the world.  Dividing polynomials.  I used to dread teaching this because it meant long division.  And, my kids couldn't even remember how to do regular long division.  In high school, I learned to do synthetic division, but that only works if the polynomial you are dividing by meets certain requirements.

Somewhere along the way, I was introduced to the box method.  And, it seriously changed my life.  If you're not familiar with the box method, you should probably start with this post on multiplying polynomials.

Here's the steps I wrote out for my students in case they forgot the process.  They're not perfect.  But, considering the fact that I didn't give my students any written out steps last year, they are a vast improvement over nothing.  Feel free to tweak them and make them better!

Then, we did four example problems together.  The kids gave me grief for not printing this on white copy paper.  Apparently peach paper and highlighting don't go well together.  Note to self: use white paper next year!

And, here's a close-up of each problem.

I made sure I gave them two problems with remainders and two without.

See how I have them write an R in the space following the constant in the answer?  This is a trick I learned from a student last year.  She invented that strategy on her own.  As soon as I saw it, I knew I was going to steal it and implement it the next year.  I did, and I'm so glad I did!

I also made sure I exposed to them to an "application" problem that they might see on their end-of-instruction exam.

After doing 4 example problems together, I could tell that they were really frustrated.  I tried explaining that this was the easy method.  But, they didn't believe me.  So, I decided to devise some dividing polynomial jigsaw puzzles for my students to put together.  I took orange painter's tape and made boxes like this on four of the desks.  

Students were all abuzz with questions about what in the world was on the desks.  My Algebra 1 kids were especially offended when I explained it was for my Algebra 2 classes.  Several students claimed that they wanted to learn to divide polynomials so they could use it.  

My Stats students were a bit miffed, too, because I didn't do this for them last year when they were in Algebra 2.  

Each station had a pre-built box, a dividing polynomial problem, and the necessary pieces to fill out the box.  Students were placed in groups of 4.  They had to circulate around each of the 4 stations in the room.  Due to small class sizes, I only had 3 groups circulating around the 4 stations.

At each station, only one person is allowed to touch the cards.  The rest of the students can coach and give assistance, but only one person gets to put the puzzle together.  The person who puts the puzzle together changes at each station.  So, everyone gets a chance to put one entire puzzle together.  This was a good structure because kids knew that they would be in charge of putting the puzzle together one time which meant it was important to pay attention!  And, it also meant that kids had to do a lot of math talk because they weren't always allowed to touch the cards.  

The stations looked like this:

And this:

And this:

And this:

If you're confused about how this works, you're in luck because I'm about to show you.

Lay out all of your pieces.

Take the polynomial you are dividing by and put it on the outside edge of the box.  I didn't plan very well and put this grid too close to the edge of the desk.  Some of my students realized that they could start building the box from the opposite side!

The first diagonal is made up of just the x^4 term of the term being divided.  So, we can just go ahead and fill it in the box.

Now, we have to ask ourselves x^2 times what equals 4x^4.  That answer goes on top of the box.

Once 4x^2 is on top of the box, we can multiply 4x^2 by 2x and -4 to fill in the rest of that column.

Now, we have our x^4 diagonal taken care of.  Our next diagonal needs to sum to make 3x^3.  We already have 8x^3.  We need -5x^3 to complete the diagonal.

Then, we ask what x^2 needs to be multiplied by to be -5x^3.  That answer goes on the top.

Having the term on top lets us fill in the rest of that column by multiplication.

Now, it's time to make our third diagonal sum to 0x^2.  This sometimes takes a while for students to get used to.  -16x^2 and -10x^2 sum to -26x^2.  So, we need a +26x^2 to finish out that diagonal.

Getting the hang of this yet?  We need to figure out what x^2 needs to be multiplied by to equal 26x^2.  This goes on top.

Once we figure that out, we can use multiplication to complete that column.

Once you write a constant on top of the box, it is always followed by an R for remainder.  The remainder gives us a chance to make sure the diagonals sum to exactly what we need them to sum to.  Sometimes, they will already sum to the correct values which means you have no remainder.

In this case, 20x and 52x needed a -74 x to sum to -2x.  And, -104 needed +105 to sum to 1.

And, we're done!  4x^2 - 5x + 26 R (-74x + 105) is our answer.  

Does it still take a long time to do?  Yes.  Does it make way more sense than long division or synthetic division?  Yes.  

One of my students came in after school that day and asked if she could work on one of the puzzles.  I said sure.  After she finished, she told me that if we had someway to color code the diagonals, it would really help her.  So, we started brainstorming ways to do this.  Our first idea was some sort of see through colored paper.  But, I don't have any of that.  Then, she said we could use bingo chips.  I tried this out with another class the next day, and it worked really well.    

Here's some pictures of the bingo chips in action.  

Here's the group that decided to work from the right side instead of the left side.

My takeaways:

Start using these puzzles with multiplying polynomials.  Then, I think they will need less explanation with dividing polynomials.  

Bingo chip color coding is awesome.  

Next year - make extra cards that don't belong in the puzzle to challenge students who need it.  Could differentiate in this way.  Give them choices of each level - easy and hard.  

After completing the jigsaw puzzles, students were much more confident about dividing on their own.  Still need to work on the best sequencing of when to introduce the puzzle.  

Also - the same student who suggested the bingo chips for color-coding diagonals suggested that I give them problems that were already worked out but contained an error for them to find and correct.  I really like this idea!  I told this student that she should definitely consider teaching as a career because she had a knack for this type of thing!  

Download files here.  


  1. you could also not use the index cards for anything except the original problem and have them use expo makers (for ones that don't need cards). Expo wipes off desks incredibly easily.

  2. Ahh - thank you for the step-by-step photos!

    Actually... I had an Algebra 2 student transfer to my school in Missouri last year, and one day I was looking for some foldables on your website, not realizing that your blog was showing on the Smartboard. As I scrolled and your picture showed, the student said, "Hey, that's my teacher from Oklahoma!" My students made fun of me for the entire year because I went crazy telling him about how his previous math teacher is a famous blogger :-)

    Anyway, he showed me the box method that you taught toward the end of the year, but I didn't take the time to really get to know it well enough to teach it. Now I know! Thank you!! And thanks for everything you do - your blog has inspired me to do so much more in my classroom!

  3. Thank you for this! I love it. What great low-tech ideas. I have always struggled with finding a creative way to get students to understand this concept. It's a dry topic, and these strategies give it life!

    I had the students make Google Slide presentations showing the step-by-step process for solving. Here is a link to one that a student made:

    Thanks for sharing!

  4. On the problem on the top left of the notes, along the left side of the box you put a positive one at the bottom and the problem says negative one? Do you always change the sign?

    1. I was wondering the same thing. Looking at the other examples, to me, it should be a negative one on the side - which changes many of the boxes and the remainder.

  5. First, I must say, you are amazing.
    Second, thank you for sharing your lessons. I am using your posts for past 3 years.
    Third, THANK YOU again!

    P.S. Your post “Dividing Polynomials Using the Box Method” has a mistake in first example. It starts in first column last row: 4x times -1 is -4x and it continues in second column last row: -2 times -1 is 2. Which changes the remainder to 14x-11.

    1. Thanks for catching my mistake, and thanks for reading my blog!