Dividing Polynomials Using The Box Method

Today, I desire to verbalize nearly ane of my favorite things inwards the world.  Dividing polynomials.  I used to dread teaching this because it meant long division.  And, my kids couldn't fifty-fifty holler back how to create regular long division.  In high school, I learned to create synthetic division, but that exclusively plant if the polynomial y'all are dividing yesteryear meets for certain requirements.

Somewhere along the way, I was introduced to the box method.  And, it seriously changed my life.  If you're non familiar amongst the box method, y'all should likely start amongst this post service on multiplying polynomials.

Here's the steps I wrote out for my students inwards instance they forgot the process.  They're non perfect.  But, considering the fact that I didn't give my students whatsoever written out steps final year, they are a vast improvement over nothing.  Feel complimentary to tweak them as well as brand them better!

Then, nosotros did 4 illustration problems together.  The kids gave me grief for non printing this on white re-create paper.  Apparently peach newspaper as well as highlighting don't acquire good together.  Note to self: purpose white newspaper side yesteryear side year!

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

I made for certain I gave them 2 problems amongst remainders as well as 2 without.

See how I receive got them write an R inwards the infinite next the constant inwards the answer?  This is a play tricks I learned from a pupil final year.  She invented that strategy on her own.  As shortly every bit I saw it, I knew I was going to bag it as well as implement it the side yesteryear side year.  I did, as well as I'm then glad I did!

I also made for certain I exposed to them to an "application" occupation that they mightiness reckon on their end-of-instruction exam.

After doing 4 illustration problems together, I could say that they were actually frustrated.  I tried explaining that this was the tardily method.  But, they didn't believe me.  So, I decided to devise some dividing polynomial jigsaw puzzles for my students to pose together.  I took orangish painter's record as well as made boxes similar this on 4 of the desks.  

Students were all abuzz amongst questions nearly what inwards the Blue Planet was on the desks.  My Algebra 1 kids were peculiarly offended when I explained it was for my Algebra 2 classes.  Several students claimed that they wanted to acquire to separate polynomials then they could purpose it.  

My Stats students were a chip miffed, too, because I didn't create this for them final twelvemonth when they were inwards Algebra 2.  

Each station had a pre-built box, a dividing polynomial problem, as well as the necessary pieces to fill upwardly out the box.  Students were placed inwards groups of 4.  They had to circulate to a greater extent than or less each of the 4 stations inwards the room.  Due to small-scale bird sizes, I exclusively had iii groups circulating to a greater extent than or less the 4 stations.

At each station, exclusively ane individual is allowed to touching on the cards.  The ease of the students tin ship away rider vehicle as well as give assistance, but exclusively ane individual gets to pose the puzzle together.  The individual who puts the puzzle together changes at each station.  So, everyone gets a conduct chances to pose ane entire puzzle together.  This was a practiced construction because kids knew that they would endure inwards accuse of putting the puzzle together ane fourth dimension which meant it was of import to pay attention!  And, it also meant that kids had to create a lot of math verbalize because they weren't ever allowed to touching on the cards.  

The stations looked similar this:

And this:

And this:

And this:

If you're confused nearly how this works, you're inwards luck because I'm nearly to exhibit you.

Lay out all of your pieces.

Take the polynomial y'all are dividing yesteryear as well as pose it on the exterior border of the box.  I didn't invention rattling good as well as pose this grid also unopen to the border of the desk.  Some of my students realized that they could start edifice the box from the reverse side!

The start diagonal is made upwardly of but the x^4 term of the term beingness divided.  So, nosotros tin ship away but acquire ahead as well as fill upwardly it inwards the box.

Now, nosotros receive got to inquire ourselves x^2 times what equals 4x^4.  That reply goes on top of the box.

Once 4x^2 is on top of the box, nosotros tin ship away multiply 4x^2 yesteryear 2x as well as -4 to fill upwardly inwards the ease of that column.

Now, nosotros receive got our x^4 diagonal taken aid of.  Our side yesteryear side diagonal needs to amount to brand 3x^3.  We already receive got 8x^3.  We demand -5x^3 to consummate the diagonal.

Then, nosotros inquire what x^2 needs to endure multiplied yesteryear to endure -5x^3.  That reply goes on the top.

Having the term on top lets us fill upwardly inwards the ease of that column yesteryear multiplication.

Now, it's fourth dimension to brand our tertiary diagonal amount to 0x^2.  This sometimes takes a piece for students to acquire used to.  -16x^2 as well as -10x^2 amount to -26x^2.  So, nosotros demand a +26x^2 to complete out that diagonal.

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

Once nosotros figure that out, nosotros tin ship away purpose multiplication to consummate that column.

Once y'all write a constant on top of the box, it is ever followed yesteryear an R for remainder.  The residual gives us a conduct chances to brand for certain the diagonals amount to precisely what nosotros demand them to amount to.  Sometimes, they volition already amount to the right values which way y'all receive got no remainder.

In this case, 20x as well as 52x needed a -74 x to amount to -2x.  And, -104 needed +105 to amount to 1.

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

Does it nonetheless receive got a long fourth dimension to do?  Yes.  Does it brand way to a greater extent than feel than long partition or synthetic division?  Yes.  

One of my students came inwards afterward schoolhouse that twenty-four hr catamenia as well as asked if she could occur ane of the puzzles.  I said sure.  After she finished, she told me that if nosotros had someway to color code the diagonals, it would actually assist her.  So, nosotros started brainstorming ways to create this.  Our start stance was some form of reckon through colored paper.  But, I don't receive got whatsoever of that.  Then, she said nosotros could purpose bingo chips.  I tried this out amongst some other bird the side yesteryear side day, as well as it worked actually well.    

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

Here's the grouping that decided to move from the right side instead of the left side.

My takeaways:

Start using these puzzles amongst multiplying polynomials.  Then, I think they volition demand less explanation amongst dividing polynomials.  

Bingo chip color coding is awesome.  

Next twelvemonth - brand extra cards that don't belong inwards the puzzle to challenge students who demand it.  Could differentiate inwards this way.  Give them choices of each bird - tardily as well as hard.  

After completing the jigsaw puzzles, students were much to a greater extent than confident nearly dividing on their own.  Still demand to occur the best sequencing of when to innovate the puzzle.  

Also - the same pupil who suggested the bingo chips for color-coding diagonals suggested that I give them problems that were already worked out but contained an mistake for them to detect as well as correct.  I actually similar this idea!  I told this pupil that she should definitely consider teaching every bit a career because she had a knack for this type of thing!  

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