## How I teach factoring + Factoring Investigation

If you’ve followed me for a while, you might already know that I’ve taught just about every factoring method under the sun over the years.

Through many trials and errors (along with teaching progressively more advanced classes such as pre-calculus), I have come to the conclusion that factoring by grouping is the best and most flexible method for students to learn.

Not only is it easy to catch onto and has a built-in check halfway through the process, but it’s flexible enough to keep being used in any future math course (something that is certainly not true for all factoring methods).

## Everything You Ever Wanted To Know about Using Exit Tickets in your Math Classroom

Although the idea of exit tickets is well known, figuring out how to master using them in your middle & high school math classes is a different story. From finding time to create them, remembering to actually give them, and teaching your students how to do them, this 5-part blogging series covers it all. At the end, there’s even a bonus installment that’s all about how to turn your exit tickets digital!

## Implementing Exit Tickets in Middle & High School Math – Why I Failed Before & How I Fixed It

This is the fifth installment in the Everything You Ever Wanted To Know about Using Exit Tickets in your Math Classroom blogging series. If you haven’t already, check out the first four posts and then keep reading!

Read Post 1 here – 5 Reasons you Should be Using Exit Tickets in Your Middle & High School Math Class. This post covers what an exit ticket is and why you would want to use one in your math class.

Read Post 2 here – How often should I use an exit ticket? A secondary math teacher explains all. This post discusses how often you should be giving an exit ticket, and ways to save time in creating them so you can actually keep up and make it routine.

Read Post 3 here – How to Implement Exit Tickets like a Math Teacher Pro. This post discusses how to introduce them to your students and tips for actually remembering to give them each day!

Read Post 4 here – What do I do now? What to do with the exit tickets after your students hand them in. Reviewing, feedback, grading, and more! This post goves over all of the logistics and teacher-decisions behind what to do after your students actually complete an exit ticket. This one is jam-packed with easy-to-implement ideas!

## What do I do now? What to do with the exit tickets after your students hand them in. Reviewing, Feedback, Grading, and More!

This is the fourth installment in the Everything You Ever Wanted To Know about Using Exit Tickets in your Math Classroom blogging series. If you haven’t already, check out the first three posts and then keep reading!

Read Post 1 here – 5 Reasons you Should be Using Exit Tickets in Your Middle & High School Math Class. This post covers what an exit ticket is and why you would want to use one in your math class.

Read Post 2 here How often should I use an exit ticket? A secondary math teacher explains all. This post discusses how often you should be giving an exit ticket, and ways to save time in creating them so you can actually keep up and make it routine.

Read Post 3 hereHow to Implement Exit Tickets like a Math Teacher Pro. This post discusses how to introduce them to your students and tips for actually remembering to give them each day!

## How to Implement Exit Tickets like a Math Teacher Pro

This is the third installment in the Everything You Ever Wanted To Know about Using Exit Tickets in your Math Classroom blogging series. If you haven’t already, check out the first two posts and then keep reading!

Read Post 1 here – 5 Reasons you Should be Using Exit Tickets in Your Middle & High School Math Class. This post covers what an exit ticket is and why you would want to use one in your math class.

Read Post 2 here How often should I use an exit ticket? A secondary math teacher explains all. This post discusses how often you should be giving an exit ticket, and ways to save time in creating them so you can actually keep up and make it routine.

## How Often Should I Use an Exit Ticket? A Secondary Math Teacher Explains All

This is the second post in the Everything You Ever Wanted To Know about Using Exit Tickets in your Math Classroom series. If you missed the first one about the 5 Reasons you Should be Using Exit Tickets in Your Middle & High School Math Class, you can catch up and read it here.

Now that you’re familiar with what an exit ticket is and why you should be using them in your math classes, let’s dig into some of the details. If you haven’t already read my first post in this series, make sure to read this first!

## 5 Reasons you Should be Using Exit Tickets in your Middle & High School Math Class

Exit slips, exit tickets, tickets out the door, quick-checks, check-ins, show me what you know’s…whatever you call them, they’re incredible teaching tools that every secondary math teacher should be incorporating into their regular teaching practice. In this first installment of the Everything You Ever Wanted To Know about Using Exit Tickets in your Math Classroom series, and I wanted to dive straight into the reasons why YOU, yes you, will benefit from using exit tickets in your classes. No need to waste any time, let’s get into it!

## 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!

## 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?