The Box Method for Factoring Trinomials

I love using the box method (area model without appropriately sized side lengths) to help students learn how to multiply polynomials. I love, love, love to use the box method to divide polynomials. But I only started using it to factor non-monic trinomials last year and I did a horrible job of it. Really horrible. I'm sure none of my students understood it because I didn't really get it. I am happy to report that I now LOVE using the box method for factoring non-monic trinomials. I shared this with the other math teachers at my school (I send out a weekly "Math Minute" - a link to a cool activity, a blog post, an idea that is worth sharing... and this was what I sent this week). I tried to colour code it to make it easier to follow and made a second box to better show the steps.

Here is an attempt at an explanation in case the example is not clear. Start by finding two numbers that multiply to the product of 'a' and 'c' (here -120) and add to 'b' (here -2). In this case the number are 10 and -12. The box represents the area (trinomial) and we are looking for the length and width (binomials). I always put the x^2-term top left, the constant term bottom right and the x-terms along the remaining diagonal. The number we found are used as the coefficients of x so 10x and -12x go in the boxes along the diagonal. Then I common factor the first row and the first column (that's where the 2x and 4x come from). This would normally all happen with one box, but now jump down to the second box. Figure out what multiplied by 2x will produce 10x and what multiplied by 4x will produce -12x. Those complete the factors and you can check that it all works out with the constant term (does -3 times 5 equal -15?). There you go. I love this because it is not a trick - it makes sense and has built-in error checking. I find it really fast, too.

I should note that I also show my students how to factor using decomposition and give them the choice of which method to use. So far more are choosing to use the box method. I can't wait to show this crew how to divide polynomials in a couple of years!

Rethinking Factoring Special Quadratics

Have you ever had a day when a lesson you took the time to rethink actually worked noticeably better? Let's be honest - I don't have the time (and sometimes not the motivation either) to rethink all my lessons. "What worked well enough last year is good enough for this year" happens far more frequently than I'd like to admit. I try to make notes if something really doesn't work or if I have a brilliant idea after the fact. And I do my best to act on those notes to my future self. Occasionally, if I teach more than one section of a course, I will make changes on the fly as I teach the second class. But the reality of teaching full-time and raising a family is that every lesson may not be as good as it could be. This is a difficult reality for me.

The change I made to yesterday's lesson was a simple one - I did the opposite of what has been done in the past. Let me back up for a minute (and I apologize if you've heard this all before)... The math teachers at my school all share lessons for all courses. I am the renegade who sometimes does things differently. I have been spiralling my grade 10 applied classes for several years and I spiralled my grade 10 academic class for the first time last year. I put a lot of thought into the order of topics and how each would be approached and blogged daily. This year I am tweaking what I did last year - the biggest change being that I am introducing more quadratics concepts earlier in the course. I am trying to be intentional when I look at past lessons and ask myself whether this is the best way to approach the topic. I looked at the "department lesson" on factoring special quadratics (at this point I have no idea who created it - it could have been me???) and just wasn't happy with it.

The old:

... followed by exclusively difference of squares practice questions. Then:

... followed by exclusively perfect square trinomial practice questions.

The new:

As I wrote above, I got students to do the opposite of the old lesson. Instead of expanding, they factored. This was good practice for them and they could see that there was a shortcut within the patterns. We had a whole-class discussion, talk with the people at your table, test your conjecture(s), come back to whole-class discussion kind of thing going, but we got there. They came up with the patterns (I didn't tell them) and they saw the value in what we were doing (I think). I think the old lesson tended to fall flat because they didn't see a need for more ways to factor - it was just confusing. They didn't see these special cases as being helpful. I hope this year's students do. They also know that they can also successfully factor them as they would any other trinomial if they don't notice that they are dealing with a special case. (Confession: Until I started teaching grade 10 applied, I did not think of a difference of squares being a trinomial where the x-term has a coefficient of 0. Factoring these with algebra tiles was a revelation!)

One of the things I love about spiralling is that it freed me from common test days. When my students need more time on a topic, I give them that time. So tomorrow we are factoring a little more. A few students are really solid with all types of factoring, but most have a more tenuous grasp of what to do when. My room is currently all set up for some factoring speed dating. Tomorrow should be a fun-filled day of factoring!

MPM2D - Day 59: Stuck on Factoring

I planned on looking at perfect square trinomials and moving on to completing the square today, but that did not happen. I had several students tell me that they just weren't getting factoring. I know that this is in part due to some of the "ugly" questions they faced for homework last night, but also that they don't have enough experience to recognize how to factor. I had that internal debate between "we need to move on" and "they don't get it so we need to slow down" and the vibe in the classroom told me I didn't actually have a choice. 

We started by finishing yesterday's work.

Then we factored. A lot. They all may have nightmares tonight of me saying "Is there a common factor?"! Here is a sample:

See how I sneaked in talking about perfect square trinomials? We did another one later and when I said "I know that's a perfect square trinomial", several students asked "How can you tell?" - setting things up for tomorrow! Back to #11 for a minute. Can someone tell me if there is a way of factoring that with tiles without first factoring out -1? I can't figure it out and it's bugging me.

Update: thanks to Hélène Matte for showing me what I couldn't see yesterday (so tired...)

Now I know I lost many of them when we did this one (which I love because I'm weird like that):

but I still think they are ahead of where they were at the beginning of class. The types of quadratics they need to factor to solve problems for this course are not like this. The most complicated are ones where they have to take out a common factor and are left with a complex trinomial. I'm not entirely sure why we ask them to factor all kinds of crazy expressions beyond that we had to do them and some of us really like factoring them.

I feel like I need a separate blog post about what I have done in terms of teaching my class factoring this semester. That is, what hasn't worked well and how I plan on fixing it for next time. However, with a new puppy and the craziness of December (which includes 4 family birthdays in my case), that may have to wait a bit. Feel free to bug me about it though :)

Here is today's homework - I told them they may not be able to do question 1. Thankfully the rest was all lagging homework.

MPM2D - Day 58: Solving Quadratic Equations by Factoring

I was planning on giving my students today's class to practice more factoring, but changed my mind part way through first period (it's at times like this that I am really glad that I'm the only one teaching this course in this way!). Instead, we solved quadratic equations by factoring - this gave them the factoring practice, but also moved us forward.

We spent a bit of time talking about how great 0 is. We looked at a product like a * b = 12 and determined, after a long list of possibilities, that there was an infinite number of combinations of values for a and b that would make this true. However, if a * b = 0, we know that a or b must be 0. Many students have been struggling with finding the zeros of questions like these:

I think they now understand why you can just take the opposite of the constant term in every case.

We consolidated the process of solving a quadratic equation that can be factored and then followed up with a lot of practice questions.

The next few questions were a little more challenging as some students have yet to master factoring complex trinomials. I used both the box and decomposition. I did an informal poll of the class and they are pretty much split half and half between the methods.

Then we hit one with a common factor and I, of course, did what my students suggested and factored it without taking the common factor out first.

We talked a lot about this one. If you are factoring, but not solving, you must take the common factor out either at the beginning or the end. They saw pretty quickly that taking it out at the beginning made their solution much easier. We talked about why you could divide by 3 when you are solving, but not when you are just factoring. After class a student asked me about using the box method for example 2a. I had not tried one that contained a common factor and it is not as obvious as you might think (well, it was not obvious to me, anyway). I will go over it with him tomorrow, stressing that taking the common factor out first will make it all work much more nicely.

Part (c) gave us the opportunity to look at a difference of squares. I drew the corresponding tiles on the whiteboard and they all remembered factoring these types of quadratics. Part (d) was gave them the tools to deal with a -1 coefficient of the variable squared. We finished with this one:

Today's homework was the second box of the handout from yesterday.

MPM2D - Day 57: Factoring & Speed Dating

It is interesting to watch a class of teenagers arrange the desks given only verbal instructions. They eventually got them into two rows, facing each other. I told the class that they would be speed dating this morning, which caused a few raised eyebrows. Then I said factoring speed dating. I gave each student a piece of paper (half of an 8.5" by 11" sheet) upon which was written a quadratic expression. Their first job was to factor that, below the fold line. They also had to check their work by expanding and simplifying their answer. They were now each the expert on factoring their quadratic.

Once everyone was ready (and I took care of the one quadratic that didn't factor - oops) they got ready. They each factored the quadratic opposite them. I started the timer at 1 minute which turned out to be too short - we settled on 1:30 for each round, then cut back to 1 minute once they were all sure of what they were doing. I also got some background music going as it seemed too quiet. They each ended up factoring 13 quadratics in a much more fun way than just giving them a worksheet and saying factor.

We spent the remainder of the period working through examples together. 

Beyond just getting more practice, we focused on noticing. Noticing when there is a common factor and how dealing with that simplifies the resulting trinomial. Noticing that if the product is negative the numbers must be positive and negative, and if the sum is negative, then the absolute value of the negative number must be bigger than the positive number (that's really not how we said it though!). Noticing what is special about a difference of squares. I still don't have a good feel whether they "get it", but once I look at yesterday's homework, I should have a better sense of what we need to do tomorrow.

Here is the homework for the next few days. I asked them to do the first box tonight.

MPM2D - Day 56: More Factoring

I felt the most unsure that I have this entire semester going into today's class. I had a really hard time predicting where my students would take the lesson. I presented them with all the quadratics they factored using algebra tiles on Friday. I asked them what they noticed and whether they could organize the quadratics in some way. Had I given them more time and given them the quadratics on slips of paper they could move around, they may have done more toward organizing them, but I decided against that as I didn't think it was worth the time. I don't think I would do this differently next time, but I will have to think on it some more.

They suggested separating them based on the signs within the factors and they said that some had a coefficient in front of the x^2, and others didn't (we clarified that they all had a coefficient). We went with the second suggestion. Here is the organization that we came up with, as a class, with definite prompting from me:

This is what they looked like moved into groups:

We then focused on monic trinomials. What was special about these?

It did not take long for them to say that the two numbers in the factors multiplied to something in the trinomial and added to something else in the trinomial. This would have gone more smoothly had I factored the first one correctly (ugh!). We fixed it and moved on, tying this back to how they have been factoring in homework for weeks and to the box method.

Then we looked at complex trinomials using the box method and decomposition.

We had time for a few examples and got help from Desmos for the one that did not factor.

We will need to continue to explore factoring different types of quadratics tomorrow and consolidate what we did today. I don't think I did a particularly good job today, so I will have to figure out what to do differently next time.

Here is today's homework.

MPM2D - Day 55: Factoring with Algebra Tiles

Today's entire class was devoted to factoring with algebra tiles. My students had two pages of expressions to factor and they did a really good job.

In some cases they ran out of the right colour tile, so they improved instead of getting another set.

Here's a difference of squares:

The goal today was to begin noticing some patterns emerging. We will talk about those on Monday.

Here is today's homework.

MPM2D - Day 54: Factoring, Day 2

The algebra tiles were waiting on students' desks as they walked in this morning, but as we had not finished what we started yesterday, the tiles had to wait. My students were not impressed with common factoring binomials. Even with pumpkins and happy faces.

And then I really made things worse with this:

We did a little work on the whiteboard to try to figure out how to deal with this case. They game me numbers, we worked out x - y and y - x. We talked about what was the same and what was different and then I asked how I could get the value of x - y given the value of y - x.

Of course multiplying by -1 will change the value, so how can we overcome that? Multiply by -1 again! Yeah, I lost a whole lot of them at this point. I know I did. I decided to let it sit with them for now and come back to it next week for a second try...

But then (I am so cruel) I made them factor by grouping. This is all leading somewhere, I told them. I'm not sure they believe me!

I think they were all really thankful to move on to "playing" with the tiles. We expanded, we factored; all was good again. I emphasized that they need to make a rectangle and that the x^2 tiles will be in one corner, while the unit tiles will be in the opposite corner.

A couple of examples for them to try:

Then we added negatives to the mix. Here is the link to the Gizmos activity. I click on "new" until I get a trinomial with negatives. I find this quicker for introducing the concept of adding zero pairs. And then they tried a few that were not monic.

I think they understand the how and the why of factoring trinomials. They will spend tomorrow practicing many more questions with the tiles and (hopefully) will start to notice some patterns emerging.

Here is today's homework.

MFM2P - Day 50 (Factoring, continued)

Today was another lesson study observation day so I did not get to teacher my 2P class. My colleague did, but sadly, as the wifi was down, they did not do the planned Would You Rather. They moved on to talking about different representations of algebraic expressions and started working on this matching activity from the Shell Centre. I LOVE this activity - it is great. There are linear and quadratic expressions, they need to expand and factor, write mathematical expressions in words, match and fill in tables of values and match area models.

We will finish it up tomorrow and I will post some pictures then.