Algebra 2 Polynomials Unit INB Pages

Yesterday I was tagged in the following tweet, asking for resources for Algebra 2’s Polynomials unit.  I’ve been meaning to get around to posting some of my Algebra 2 inb pages from last year, but never have.  This is finally the kick in the pants I needed, so…without further ado, here’s the pictures.  If you want to know more about what I did in conjunction with the notes, let me know!

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How I Teach Factoring Trinomials

When I teach the unit on polynomials and factoring in Algebra 1, I start off my first lesson on factoring trinomials with a discussion on which has fewer options: multiplying to a number, or adding to the same number?  Students take a couple minutes to list out all pairs of numbers they can think of and then share out to the class.  After doing this twice, quite a few students start catching on to the fact that there’s an infinite amount of ways to add to any given number, but only a handful of ways to multiply to the same number.  Multiplication gives us fewer options, which will allow us to do less work.  This will be really important to what we do in just a moment. (NOTE #1: I provide students with a factor pair chart as an aid to help with identifying factors later on. NOTE #2: We originally began using only whole numbers as a starting point, but students then wanted to branch out further.  Could we extend this question to include integers?! Yes-and we did!)

From there, we go back to something students have just learned a few sections earlier: multiplying two linear binomials of the form (x+A)(x+B).  We do this a few times and then look at a bunch of already expanded examples and I ask students what they notice.  It doesn’t take long before students start realizing that the middle term, the coefficient of the x, always comes from adding the two numbers A+B, and the end term, the constant, always comes from multiplying the two numbers, AB.

Then, I switch the question around.  How can we figure out what someone expanded to create a quadratic expression?  Is there any easy way to figure this out?  Students start to volunteer info that they know: the middle term comes from the addition (A) of the two factors, and the constant term comes from the multiplication of these two numbers (M).

So, then the question is, which number do we look at?  The addition number, or the multiplication number? Technically, it doesn’t matter, BUT mathematicians love to be lazy efficient, so we’ll look at the multiplication number.  Students justify looking at the the multiplication number first because, just a few questions prior,  they determined that there’ll be fewer options with multiplication than for addition.

From there, I ask students to make further generalizations and predictions about the signs of the terms and the signs of the factors and use that information to work both forward (expanding) and backward (factoring) using some diamond puzzles.

The next day, we practice some more with basic factoring when a=1, using the patterns we found from the investigation the day before:

Then, we kick it up a notch.  How the heck do we do this factoring thing when there’s more than one x-squared?!  No problem! GCF to the rescue.

After that, we look at what do we do if a GCF alone isn’t enough to get rid of the a-value, or, even worse, there’s no GCF at all?

This brings us to my FAVORITE part of factoring quadratic trinomials: Slide, Divide, Bottoms Up!  If you are unfamiliar with this method, let me start off by telling you that it’s awesome.  It’s firmly rooted in the same concepts we’ve been using for the last three sections of factoring, and it just makes sense.  Another benefit to the Slide, Divide, Bottoms Up method is that it is efficient.  Doing guess and check (or the box method) can become very frustrating for students when the a-value is larger than, say, 4 or 5.  There’s just too many options and it ends up taking forever, even with a decent intuition about which numbers to test out as factors.  Also, this method even works for special factoring cases like difference of squares!  Students can certainly utilize the factoring shortcut for difference of squares, but, if they forget, Slide, Divide,  Bottoms Up still has their back.

Here’s how Slide, Divide, Bottoms Up works:

Let’s talk through an example:

Like all factoring problems, we check if there is a GCF, first.  If we’re lucky, that will remove the a-value and we will be good to do what we normally do.  However, in this example, we weren’t that lucky.  No GCF, so what to do with the 6?  We certainly don’t want more than 1 n-squared, so we’re going to temporarily transfer it to the constant term by multiplication (we “SLIDE” it over).  At this point, we discuss what “temporarily” means.  It means, “only for a while,” so that tells us that, at some point, we’re going to have to undo it. This should be perfectly “legal” because if we do something but then undo it later, that just cancels out to what we started with.  It might also be worth noting that we transfer the a-value through multiplication because we are factoring, which literally means returning an expression back into a product (multiplication) of two factors.

Now that we’ve gotten rid of the a-value of 6 for a moment, we’re left with a standard trinomial that students know how to factor in their sleep with their eyes shut, at this point.  The only thing they have to remember after factoring it is that our factored form is for our temporary expression, not the one we started with.  So, how to undo what we did?

Well, if we multiplied the a-value into the coefficient, it stands to reason that we should do the inverse operation and just divide it back out (DIVIDE)!  Since we’re dividing, make sure to reduce the fractions!

Lastly, we didn’t start out with any fractions.  Actually, we started out with a number that was a coefficient (our a-value of 6).  To get rid of any fraction(s) that we introduced, we bring the denominator(s) back up in front of the variable to be a coefficient, once again (bring the “BOTTOMS UP“).

Here’s some more examples.  Note example 5 where there’s a GCF but we’re still left with an a-value of 4.

Here’s how it works with difference of squares problems.

After using Slide, Divide, Bottoms Up for the past 3 years, I can’t see myself doing factoring any different.  I’m pretty smitten with this method, and, hopefully, it’s easy to see why.

After doing all of the different factoring, I give students one last reference sheet to use in their notebooks, which can be used at any time to refresh their memory on how to solve ANY quadratic trinomial.

If you are interested in this flowchart, it is available in three different sizes here.

Let me know how you teach factoring quadratic trinomials in the comments below!

TRIGSTER! – A student created review game for trigonometry.

At the end of the school year, I had my Algebra 2 + Support students do a project.  The project was to focus on any unit we had done that semester and create something that reviewed that unit that the rest of the class could also benefit from.  I got a lot of things from a comic book, several board games, a ukulele song, and an amazingly edited YouTube video, but my favorite project was TRIGSTER.

This was the project’s instructions:

Now, I present to you…TRIGSTER!  Two girls made this game as a twist on Twister.  It’s mean to be a very fast-paced game where 4 aspects of trig are reviewed: (1) the trig ratios for the big six functions, (2) abbreviations and connections for big six functions, (3) exact trig values from special right triangles, and (4) reference angles and angles in standard position (as well as a couple of formulas to convert between radians and degrees).  I absolutely was blown away by this project and am so excited to play it in the years to come for my future Algebra 2 classes!  Students can do amazing things, if we let them.

I’ve included their customized instructions, below.

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

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.

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:More 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.

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!

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.

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 ________.

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).

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.

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.