Dividing Polynomials Puzzle Using The Box Method
Saturday, August 17, 2019
Activities,
Algebra 2,
Box Method,
Dividing Polynomials,
Dry Erase Pockets,
Group Work,
lessons,
Long Division,
Polynomials,
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After taking a 2 yr hiatus from teaching Algebra 2 to dabble inwards teaching physical scientific discipline as well as chemistry, I'm dorsum at it this year. One of the things I really missed most teaching Algebra 2 was getting the chance to innovate my students to the super versatile box method for working amongst both multiplying as well as dividing polynomials.
This year, some students receive got been quick to comprehend the box method for multiplying polynomials. Others receive got been resistant as well as insist on FOILing everything. This is okay. My goal is non to force my students toward a unmarried method. Instead, my goal should survive to bring out students to dissimilar methods as well as allow them select which method plant best for them. Exposure to the box method for multiplying polynomials is ample to allow them to occupation the box method for division.
Let's halt for a minute as well as beak most other methods for polynomial division.
Long Division. Sigh. This is how I taught my Algebra 2 students to split upwards polynomials equally a starting fourth dimension yr teacher. This was how I learned to split upwards polynomials when I was an Algebra 2 educatee myself. I had to pass precious shape fourth dimension reminding my students how to create long partitioning of just numbers earlier nosotros could ever delve into the globe of long partitioning of polynomials. It was rattling algorithmic. Then, at that topographic point were those pesky sign changes that students ALWAYS forgot to perform. It was incredibly hard to examine a student's travel apace as well as notice their mistakes.
Synthetic Division. Confession Time: I tin never recollect how synthetic partitioning works. I've tutored many a pre-calculus educatee who has come upwards to me amongst a query most synthetic division. The starting fourth dimension affair I e'er receive got to create is hold off at an illustration inwards the mass as well as remind myself how it works. There's something most this algorithm that keeps me from retaining it over a long catamenia of time. My other electrical load amongst this method is that it's application is limited. It ONLY plant if y'all are dividing a polynomial past times a binomial inwards the shape (x - a) or (x + a).
So, it is my disenchantment amongst these methods that has led me to comprehend the box method, grid method, expanse method, or whatever else y'all desire to telephone outcry upwards it for polynomial division. I love that this method does non experience equally algorithmic equally the other 2 methods.
After tweaking my approach to introducing dividing polynomials over the past times few years, I receive got finally arrived at something I'm pretty happy with.
This year, nosotros started amongst some basic notes. The alone existent guidance I gave to my students to start was that nosotros were dividing instead of multiplying which agency nosotros receive got to laid our box slightly differently. When nosotros multiply polynomials, nosotros laid our box amongst the polynomials nosotros are multiplying on the exterior as well as travel toward finding our respond on the within of the box. With dividing, nosotros know the finally respond (inside of the box) as well as i of the factors (side of the box). Our goal is to notice the missing side of the box.
With this express guidance of where to set what, my students are usually able to figure out the ease of the procedure themselves. It's genuinely a beautiful affair to picket unfold.
I oftentimes acquire asked how the box method works, hence I desire to walk y'all pace past times pace through a solution using this method. I'll survive using a laid of newspaper manipulatives that I created for my students to give them some much needed scaffolding betwixt the problems nosotros did together equally notes as well as the problems they receive got to create independently on their own.
Students were given a dry out erase template, a dividing polynomials problem, as well as a laid of all of the necessary cards to consummate the partitioning process. I similar to think of these equally mini dividing polynomial puzzles for students to solve.
Since nosotros are dividing past times (x-4), nosotros are considering that (x - 4) is a constituent of the master copy polynomial. If it is, nosotros volition terminate upwards amongst ease 0. If it was non really a factor, nosotros volition terminate upwards amongst a remainder. To exhibit that (x - 4) is a factor, nosotros house it on the side of our box.
The polynomial that is existence divided past times (x - 4) represents the expanse of the box. The starting fourth dimension term of this polynomial (when written inwards measure form, of course) volition e'er become inwards the transcend left box. This is usually the indicate where I halt explicitly guiding students inwards how to laid a partitioning job as well as allow them receive got over amongst their suggestions.
x times what equals x^3? x^2. Now, nosotros receive got the starting fourth dimension role of our solution. I tin too multiply the x^2 I just constitute past times the -4 to acquire -4x^2.
At this point, I e'er inquire my students if nosotros desire -4x^2. Yes nosotros do! So, I demand a similar term to add together to -4x^2 that volition non alter its value. The alone affair I tin add together to -4x^2 as well as non alter its value is 0x^2.
x times what equals 0x^2? 0x.
Now that 0x is inwards our solution, I tin multiply it past times -4 to become far at 0x.
Do I really desire 0x? No. My master copy job tells me that I desire -2x. What tin I combine amongst 0x to become far at -2x? -2x.
x times what equals -2x? -2, of course.
Now, what is -2 times -4? Positive 8.
Do I really desire 8? No. I desire 3. At this point, I can't add together whatever to a greater extent than damage to my solution, hence I must resort to creating a ease to acquire my 3. 8 - five is 3, hence my ease must survive -5.
We tin write this solution 2 dissimilar ways - using the R symbol for ease or writing the ease equally a fraction. I ordinarily allow my students write the solution whatever way they wish, but I say them that they demand to recognize both ways of writing the solution as well as survive able to switch betwixt them depending on the context.
I ended upwards creating vi of these polynomial partitioning problems for my students to travel through inwards groups. Each job as well as its pieces were printed on a dissimilar color. These colors serve several purposes. First, if a slice ends upwards on the floor, it's super slowly to reunite it amongst its set. Also, the colors allowed my students to continue runway of which problems they had finished as well as which even hence remained to survive done.
Running this activity amongst my Algebra 2 students iv times over the course of pedagogy of i hateful solar daytime allowed me to brand some tweaks to my instructions/set-up AND think critically most changes to implement inwards the future.
I kinda threw this activity together at the finally infinitesimal (thank y'all starting fourth dimension hr planning period), hence I didn't receive got fourth dimension to brand to a greater extent than than i re-create of each problem. I had vi problems as well as vi groups. This meant that if i grouping finished a job earlier some other grouping finished, they had to sit down as well as wait. Students sitting idle is something y'all definitely desire to avoid inwards the classroom if at all possible. Next fourth dimension I create this, I volition definitely impress 2 or iii sets of each problem. Other times, some other grouping did destination at the same time, but they had a color that the other grouping had already completed. If i grouping spent a long fourth dimension on a unmarried problem, it became an final result because all the other groups even hence needed that problem.
For my starting fourth dimension shape of the day, I asked them to only travel inwards groups to consummate the partitioning puzzle. I apace noticed that i or 2 students tended to create most of the work. For my subsequent classes, I had students select i somebody to handgrip the cards for each problem. The other students were allowed to assist equally much equally they wanted, but alone the i designated educatee per job should survive manipulating the pieces.
This made a HUGE difference. As I circulated, my students didn't just follow my instructions of alone i educatee touching the cards each problem, but I did notice that all of my students tended to survive to a greater extent than engaged inwards the activity since they knew their plow was coming soon.
There was a lot to a greater extent than focus on HOW to solve the job instead of rushing to finish.
Even amongst the colors, I did notice that afterwards my students had finished 4 or five of the problems, they started to forget which colors they had non yet solved. So inwards the future, I think I volition brand a postage sail of sorts for groups to continue runway of which colors of problems they receive got solved as well as which they haven't.
Because I was running brusk on fourth dimension as well as because I even hence was afraid that my activity mightiness receive got a few typos, I didn't laminate these cards. They were a chip worse for the vesture past times the terminate of the day, hence I volition definitely survive laminating whatever sets of these inwards the futurity for durability.
I would too similar to expand the activity to characteristic partitioning past times a trinomial, but that volition receive got to become on this summer's to create listing since nosotros receive got finished as well as moved on from dividing polynomials.
Another possible modification for this activity would survive to innovate some decoy cards that are non role of the solution process. As the activity is currently designed, every unmarried carte du jour volition survive used for every unmarried problem.
I accidentally printed some of the travel mat templates on missive of the alphabet sized paper, as well as my students ended upwards placing them inwards missive of the alphabet sized dry erase pockets (affiliate link) as well as using them equally templates to consummate their Delta Math problems. Students were rattling wretched when they had to describe their ain boxes to consummate the partitioning problems that involved dividing past times a trinomial!
Want a picayune flashback to the past? Here's my endeavour at teaching the same lesson dorsum inwards 2015. It was the same basic idea, but it involved handwritten index cards as well as a lot of fourth dimension spent making boxes out of painters record on our desks.
Before I part the files, I create demand to give some credit where it is due.
The problems that students were solving inwards this activity were non created past times me but past times my husband, Shaun Carter. They are featured inwards his work-in-progress Algebra 2 Practice Book.
So, thanks, love husband, for making my teaching life easier equally I made this activity to occupation amongst my students as well as part amongst other teachers.
Without whatever farther ado, the files for this activity are posted here. Have ideas for making this activity fifty-fifty better? I'd LOVE to take away heed them inwards the comments!
This year, some students receive got been quick to comprehend the box method for multiplying polynomials. Others receive got been resistant as well as insist on FOILing everything. This is okay. My goal is non to force my students toward a unmarried method. Instead, my goal should survive to bring out students to dissimilar methods as well as allow them select which method plant best for them. Exposure to the box method for multiplying polynomials is ample to allow them to occupation the box method for division.
Let's halt for a minute as well as beak most other methods for polynomial division.
 |
| Image Source: https://en.wikipedia.org/wiki/Polynomial_long_division |
Long Division. Sigh. This is how I taught my Algebra 2 students to split upwards polynomials equally a starting fourth dimension yr teacher. This was how I learned to split upwards polynomials when I was an Algebra 2 educatee myself. I had to pass precious shape fourth dimension reminding my students how to create long partitioning of just numbers earlier nosotros could ever delve into the globe of long partitioning of polynomials. It was rattling algorithmic. Then, at that topographic point were those pesky sign changes that students ALWAYS forgot to perform. It was incredibly hard to examine a student's travel apace as well as notice their mistakes.
 |
| Image Source: https://en.wikipedia.org/wiki/Synthetic_division |
So, it is my disenchantment amongst these methods that has led me to comprehend the box method, grid method, expanse method, or whatever else y'all desire to telephone outcry upwards it for polynomial division. I love that this method does non experience equally algorithmic equally the other 2 methods.
After tweaking my approach to introducing dividing polynomials over the past times few years, I receive got finally arrived at something I'm pretty happy with.
This year, nosotros started amongst some basic notes. The alone existent guidance I gave to my students to start was that nosotros were dividing instead of multiplying which agency nosotros receive got to laid our box slightly differently. When nosotros multiply polynomials, nosotros laid our box amongst the polynomials nosotros are multiplying on the exterior as well as travel toward finding our respond on the within of the box. With dividing, nosotros know the finally respond (inside of the box) as well as i of the factors (side of the box). Our goal is to notice the missing side of the box.
With this express guidance of where to set what, my students are usually able to figure out the ease of the procedure themselves. It's genuinely a beautiful affair to picket unfold.
I oftentimes acquire asked how the box method works, hence I desire to walk y'all pace past times pace through a solution using this method. I'll survive using a laid of newspaper manipulatives that I created for my students to give them some much needed scaffolding betwixt the problems nosotros did together equally notes as well as the problems they receive got to create independently on their own.
Students were given a dry out erase template, a dividing polynomials problem, as well as a laid of all of the necessary cards to consummate the partitioning process. I similar to think of these equally mini dividing polynomial puzzles for students to solve.
Since nosotros are dividing past times (x-4), nosotros are considering that (x - 4) is a constituent of the master copy polynomial. If it is, nosotros volition terminate upwards amongst ease 0. If it was non really a factor, nosotros volition terminate upwards amongst a remainder. To exhibit that (x - 4) is a factor, nosotros house it on the side of our box.
The polynomial that is existence divided past times (x - 4) represents the expanse of the box. The starting fourth dimension term of this polynomial (when written inwards measure form, of course) volition e'er become inwards the transcend left box. This is usually the indicate where I halt explicitly guiding students inwards how to laid a partitioning job as well as allow them receive got over amongst their suggestions.
x times what equals x^3? x^2. Now, nosotros receive got the starting fourth dimension role of our solution. I tin too multiply the x^2 I just constitute past times the -4 to acquire -4x^2.
At this point, I e'er inquire my students if nosotros desire -4x^2. Yes nosotros do! So, I demand a similar term to add together to -4x^2 that volition non alter its value. The alone affair I tin add together to -4x^2 as well as non alter its value is 0x^2.
x times what equals 0x^2? 0x.
Now that 0x is inwards our solution, I tin multiply it past times -4 to become far at 0x.
Do I really desire 0x? No. My master copy job tells me that I desire -2x. What tin I combine amongst 0x to become far at -2x? -2x.
x times what equals -2x? -2, of course.
Now, what is -2 times -4? Positive 8.
Do I really desire 8? No. I desire 3. At this point, I can't add together whatever to a greater extent than damage to my solution, hence I must resort to creating a ease to acquire my 3. 8 - five is 3, hence my ease must survive -5.
We tin write this solution 2 dissimilar ways - using the R symbol for ease or writing the ease equally a fraction. I ordinarily allow my students write the solution whatever way they wish, but I say them that they demand to recognize both ways of writing the solution as well as survive able to switch betwixt them depending on the context.
I ended upwards creating vi of these polynomial partitioning problems for my students to travel through inwards groups. Each job as well as its pieces were printed on a dissimilar color. These colors serve several purposes. First, if a slice ends upwards on the floor, it's super slowly to reunite it amongst its set. Also, the colors allowed my students to continue runway of which problems they had finished as well as which even hence remained to survive done.
Running this activity amongst my Algebra 2 students iv times over the course of pedagogy of i hateful solar daytime allowed me to brand some tweaks to my instructions/set-up AND think critically most changes to implement inwards the future.
I kinda threw this activity together at the finally infinitesimal (thank y'all starting fourth dimension hr planning period), hence I didn't receive got fourth dimension to brand to a greater extent than than i re-create of each problem. I had vi problems as well as vi groups. This meant that if i grouping finished a job earlier some other grouping finished, they had to sit down as well as wait. Students sitting idle is something y'all definitely desire to avoid inwards the classroom if at all possible. Next fourth dimension I create this, I volition definitely impress 2 or iii sets of each problem. Other times, some other grouping did destination at the same time, but they had a color that the other grouping had already completed. If i grouping spent a long fourth dimension on a unmarried problem, it became an final result because all the other groups even hence needed that problem.
For my starting fourth dimension shape of the day, I asked them to only travel inwards groups to consummate the partitioning puzzle. I apace noticed that i or 2 students tended to create most of the work. For my subsequent classes, I had students select i somebody to handgrip the cards for each problem. The other students were allowed to assist equally much equally they wanted, but alone the i designated educatee per job should survive manipulating the pieces.
This made a HUGE difference. As I circulated, my students didn't just follow my instructions of alone i educatee touching the cards each problem, but I did notice that all of my students tended to survive to a greater extent than engaged inwards the activity since they knew their plow was coming soon.
There was a lot to a greater extent than focus on HOW to solve the job instead of rushing to finish.
Even amongst the colors, I did notice that afterwards my students had finished 4 or five of the problems, they started to forget which colors they had non yet solved. So inwards the future, I think I volition brand a postage sail of sorts for groups to continue runway of which colors of problems they receive got solved as well as which they haven't.
Because I was running brusk on fourth dimension as well as because I even hence was afraid that my activity mightiness receive got a few typos, I didn't laminate these cards. They were a chip worse for the vesture past times the terminate of the day, hence I volition definitely survive laminating whatever sets of these inwards the futurity for durability.
I would too similar to expand the activity to characteristic partitioning past times a trinomial, but that volition receive got to become on this summer's to create listing since nosotros receive got finished as well as moved on from dividing polynomials.
Another possible modification for this activity would survive to innovate some decoy cards that are non role of the solution process. As the activity is currently designed, every unmarried carte du jour volition survive used for every unmarried problem.
H5N1 slightly dissimilar twist that I receive got too considered would survive to receive got several cards out of each handbag that students must render themselves via dry out erase marking on the template. I really constitute that some of my groups were writing inwards what the damage should survive inwards dry out erase marking earlier they constitute the corresponding carte du jour to house on the template. To me, this seemed similar they were just making to a greater extent than travel for themselves, but I appreciated that they were thinking critically throughout the process.
I receive got too had thoughts of expanding this activity hence every unmarried educatee is working on their own. I think I could receive got done this successfully at my previous school, but I think amongst my electrical flow shape sizes of thirty Algebra 2 students that I mightiness become mad trying to banking concern check that many students' travel at once!
For this activity, I printed my partitioning templates on 11 x 17 cardstock (affiliate link). This fits perfectly inwards my 11 x 17 dry out erase pockets (affiliate link) that are PERFECT for grouping activities. The colored cards are printed on regular missive of the alphabet sized cardstock. If y'all don't receive got access to xi x 17 newspaper or receive got the powerfulness to impress on it, just impress both PDFs at just about 63% scale. This volition allow y'all to impress on regular, missive of the alphabet sized paper. Of course, the activity volition survive much to a greater extent than appropriately sized as well as hence for private travel than grouping work.
I accidentally printed some of the travel mat templates on missive of the alphabet sized paper, as well as my students ended upwards placing them inwards missive of the alphabet sized dry erase pockets (affiliate link) as well as using them equally templates to consummate their Delta Math problems. Students were rattling wretched when they had to describe their ain boxes to consummate the partitioning problems that involved dividing past times a trinomial!
Before I part the files, I create demand to give some credit where it is due.
The problems that students were solving inwards this activity were non created past times me but past times my husband, Shaun Carter. They are featured inwards his work-in-progress Algebra 2 Practice Book.
So, thanks, love husband, for making my teaching life easier equally I made this activity to occupation amongst my students as well as part amongst other teachers.
Without whatever farther ado, the files for this activity are posted here. Have ideas for making this activity fifty-fifty better? I'd LOVE to take away heed them inwards the comments!
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