How-To: Synthetic Division

During my Algebra 2 unit on polynomials, I had asked my (support) class if they would like to stick to just using polynomial long division, which works for every single problem, or if they would like to also learn another method (synthetic) that, while far quicker, only works in certain situations.  It was almost unanimous that they favored sticking to polynomial long division, which was fairly surprising to me. I almost figured they would want a quicker method, but their rationale was sound.  They thought that having another method would just trip them up, and they didn’t really see a point if it could only be used for linear binomials.

 

However, a few weeks after our unit on polynomials, we had a bit of down time so I introduced synthetic just for fun.  The students caught on quickly, but still preferred long division since it made more sense to them. (I agree that Synthetic is harder to wrap one’s head around.  It feels a bit more “magic.”)  Unfortunately, most of the class was gone that day due to an optional viewing of the school play being offered for students during the first four periods of the day.

 

As we start moving toward reviewing for finals, I figured I’d make a slideshow for students to view on their phones if they wanted to get a refresher on synthetic division.  Here it is!  I like it because it has a quiz-yourself and work-at-your-own-pace feel to it.

Do you cover both synthetic and long division for polynomials?  Which does your class seem to prefer?

Download a PDF of the slideshow here: synthetic-division-how-to

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Solving Literal Equations “Connect 4” Activity {Student Approved} FREE DOWNLOAD

Recently, I reached out to the MTBoS looking for fun ideas for practicing solving literal equations.  I had searched pretty thoroughly to find any pre-existing activities on the internet, but there wasn’t a lot available.  On top of that, what was there, required way more pre-existing skills (SO MUCH FACTORING!) than my Algebra 1 students currently had a month and a half into the school year.   Unfortunately, the MTBoS and I were pretty stuck. feed

Farther down in this Twitter conversation, however, it was mentioned that someone recently used BetterLesson’s lesson for teaching literal equations.  At that point I had already taught the lesson and most of my students caught onto solving them quite quickly, but I still was looking for a fun way to get a bit more practice in.  While exploring what BetterLesson had, I found this worksheet  that gave me inspiration for a game I could play with my students.  After a little bit of brain-storming, I created what I’m calling a Connect 4 Activity.  Essentially, it’s BINGO, but 4×4 instead of 5×5.

How to play: 

  • Before game: print enough game cards so each student has one, and cut apart the 16 problems.  I fold the problems in half (the problem number to the inside) and put them into a plastic bin.  (When printing from your computer, make sure it says “print double sided, flip on long-edge.”)
  • To start off the game, each student gets a game board, on which they randomly place the numbers 1-16.  Students then pull out a piece of scratch paper, where they will be doing their work.
  • The teacher brings the plastic bin containing the 16 equations around the classroom, letting a student volunteer pick a problem at random. (They LOVE getting to pick!)
  • The teacher then places the problem under the document camera (or writes it on the chalk/white-board if you’re at a low-tech school) for students to solve.
  • After all students have solved the problem, discuss the solution as a class.
  • Once all students are silent, the problem number is revealed for students to cross off on their game card. (The excitement levels usually explode at this point, hence the moments of silence in between.)
  • Repeat for as much class time as you have available, or until all 16 problems have been solved.
  • Each time a student gets 4 in a row, they bring up their card and their work for inspection (they showed their work and corrected any mistakes for each problem), and are allowed to choose a small piece of candy (Jolly Rancher, a Starburst, etc.).

Reasons why I LOVE this game:

  1. It is super easy to set up and is so adaptable for other topics.  This has probably been the lowest prep activity I have made for my students, yet it has been one of the most successful.
  2. Students felt much more confident about their skills and were able to get nearly-instant feedback about how they’re doing.
  3. Students LOVED it. The class begged me to continue letting them play the game through passing time.

Download the game here:2-8-literal-equations-connect-4-activity-page-001connect 4 problem cards for blog post picsconnect 4 problem cards for blog post pics2More Literal Equations Activities:
(Updated September 2017)
This year I wanted to find more ways to practice literal equations with my Algebra 1 students.  We teach literal equations the week before Halloween, so I wanted to make something really fun and “Halloween-y.”  I made a Carving Pumpkins activity that’s self-checking and SUPER fun!  I couldn’t wait to try it out, so I gave it to my Algebra 2 students mid-September (patience never was my virtue) since they review literal equations in their first unit.  Students though it was fun, and they also found it really comforting that it’s self-checking.  To quote a group of boys, “this is super dope, we should do this for all of the holidays!”

Students are given 12 literal equations to solve for a specific variable.  Depending on what their answer was, they “carve” color the corresponding pumpkin in a particular way. In the end, each of the pictures should end up looking the same, as far as the color and carvings go.

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I’ll be making more activities, and will update the post!

My Favorite Resources #MTBoSBLAUGUST #Made4Math

Over the last year or so, I’ve done a lot of work with very low-end students.  Between teaching summer school for two years straight in the inner city, and teaching support classes in my regular semi-rural school, I’ve really been pushed to find other ways to convey information that work for my students.

One thing that I found is that no matter how small and bite-sized of steps I could break a process down to in our notes, many of my ELL students and students with IEPs for processing disabilities just couldn’t follow along and rework through the steps to get themselves “unstuck” on a problem.  Working toward self-sufficiency is really big for me.  I strongly believe that the purpose for high school is to prepare students to be productive once they enter the “real world,” whatever that means for them (school, workforce, military, etc.).  Being self-sufficient and being able to problem-solve on their own is a big part of being able to reach this point.  So, I kept searching and trying new things until I made my first flowchart graphic organizer.  It was a game changer for my class!

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Students were able to easily follow along.  Using the graphic organizer, they were forced to read and do only one small chunk at a time and they had enough space to do their work right on the flowchart (it’s hard for some students to go back and forth between where the steps are written and where they’re doing a problem on a separate page of paper).  Students were able to use the flowcharts as long as they wanted.  As soon as they felt comfortable enough without it, they stopped using it.  I have also laminated a class set that we used for practice early on.

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I’ve also found that these have been very successful with my older students to jog their memories about a method they haven’t used in a while (such as solving systems by elimination).  For a lot of my seniors, I’m not the only math class that they are taking–many of them are also taking a class called Math Skills that gives them opportunities to take more Work Samples, which are needed for graduation.  Work Samples are an animal of their own and the topics on them can vary widely, so students find themselves needing review on topics that they may have not seen for a couple of years.  I’ve had a lot of these students specifically ask if I had a flowchart for topic _______ that they could look over to remind themselves of the details of how to do ________. 6
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With my younger classes, the first time we learn a method, I have a student working at the document camera as our class’ scribe, and the class (no help from me) discusses their way through the problem.  They determine which path they need to go down (the “yes” path, or the “no” path), and then work in pairs to do that step.  Then, they compare their work for that step as a class, and then move onto the next part of the flowchart and repeat the process.  I love, love, LOVE how student and discussion centered this makes my lessons!  Seriously! LOVE!  It’s almost as if I’m not needed (shh! don’t tell anyone that, because I still want my job).
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From there, we do a few examples that we glue into our INBs, and do some practice with dry-erase pens on the laminated copies of the flowcharts.  I find that starting slow and having them work their way through a problem as a class, without me, helps them remember the ins and outs of the process a bit better, since they had to struggle together as a class.
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Although I don’t have students referring to their notes quite as much as I would like, I have found that they go back to these flowchart examples in their INBs more than anything.  When I ask my students why they like these so much, a lot of what they say comes back to the fact that they have the steps on the paper, and the space to do the work on the paper, and the flowchart really forces them to go one step at a time.  A lot of them know that they have a tendency to rush through steps, and using the flowchart makes that very difficult to do.  Students then self-wean off of the flowcharts at their own pace, which is great in my books!  They are taking accountability for their knowledge.  If they can do their work straight away, they do so.  If they need a bit more help to get through a problem, they don’t just give up–rather, they walk to where I keep extra copies of the flowcharts, grab one, and work through the problem.  This has really helped develop the no opt-out culture in my classroom.  If students want to learn, there are tools to help them learn.  For my classes, the flowchart has been an instrumental tool for their development, both in math skills as well as self-motivation and persistence.

If you like the flowcharts, you can find them at my TPT store!  Today, they are 19% off when you couple your purchase with the 10% discount code OneDay.

Solving Systems of Linear Equations Flowchart BUNDLE 

Solving Multi-Step Equations Flowchart

Thank you so much for reading!

 

Special Right Triangles Display {FREE Download} #MTBoSBlaugust #Made4Math

A while back I made a display for special right triangles, and realized I never shared the files! You can download the PDF and the editable Publisher files here!  You’ll need to download the free font HVD Comic Serif Pro if you choose to edit the Publisher file yourself.

Here’s a picture of the pre-laminated pieces.  I took a few pieces of the finished product on my walls in the classroom, but each one had a nasty glare from the laminated finish.

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Reference Angles Poster {FREE Download} #MTBoSBlaugust #Made4Math

The last few weeks of this summer have been filled with poster-making to spruce up my classroom and to make the walls more of a resource to my students.  To determine what would be the most useful posters to make, I’ve been doing a lot of reflection about last year.  One topic that I wish I had a public record/anchor chart style poster for was reference angles, which is covered in our Trigonometry unit in Algebra 2. I’ve been contemplating a lot about how I’d want it to look and had considered making an entire unit circle poster, but decided against it since I felt as if that promoted students to memorize all of the values.

I really want to get my students to the idea that they can derive as much or as little of the values as they desire.  If they want, they can derive them all from scratch, and we discuss how to do this in class.  We then talk about how if they feel more comfortable with memorizing them all so they can work quicker on a test/quiz, then they are free to do so as well.  We then have the conversation about my own personal preferences:  I memorize the sine and cosine values for the first quadrant.  From there we can easily get tangent by dividing the sine value by the cosine value, and we can think about the properties on the coordinate plane to get the appropriate signs for any other angles.  Most students choose to take my approach as well since it is a nice middle ground.

reference angles posterHere’s the PDF File – Reference Angles Poster.  I sent mine over to my school’s print shop to be printed on 24″x36″ paper. You can print it on standard 8.5×11 paper if you select “Fit” or “Shrink Oversized Pages” on your prniter’s menu.

My last project for the summer will be making labels for 0, 90, 180, and 270 degrees with their corresponding sine and cosine values to be put around the wall clock in my classroom.

Happy Sunday!  This marks 3 more weeks until I go back to school.  How much time do you have?

Trig Ratio Posters for Geometry and Algebra 2

This summer I’ve been busy making posters to spice up my very blandly decorated classroom. This is what my room looked like for my first year of teaching:

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I don’t have a lot of wall space (the other two “walls” of my classroom are just windows), but I think I could definitely better utilize the space and make it much more of a usable resource for my students.  In the back (L-R, top to bottom) I had a poster on adding polynomials, the 8 mathematical practices, naming polynomials by degree, our bell schedules, naming polynomials by number of terms, adding polynomials, factoring trinomials, and the mathematical practices of habit and mind.  Some of it was very useful for a while, but didn’t need to stay up the whole time. Definitely more of a unit-specific anchor chart, than anything. Buuuuut, my walls were really blank, so I left them up for the rest of the year.

This summer, however, I’ve been making tons of posters to put up on my wall.  Well, tons of Algebra 1 and Algebra 2 posters, that is.  Geometry somehow hadn’t gotten any love, so I decided to remedy that by making a trig ratio poster.

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I originally was just going to do the “big three” trig ratios since those apply for the geometry class, but I thought I’d add their reciprocals as well, seeing as they get used in Algebra 2.  I hope having them up at the beginning of the year will somehow help this information sink into their minds before we ever get to the actual trig units during second semester.

If you want to use this in your own classroom, you can download a PDF version here.

Q: What do you put up on your walls for students to use as resources throughout the year?

Roots and Radical Expressions Unit Notebook Pages (Algebra 2)

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Polynomials, Factoring Polynomials, Polynomial Division, and more! Notebook Pages (Algebra 2)

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