Airplane Method for Factoring

I mentioned the "airplane" method for factoring in a recent post. Someone asked me what that was, so I thought I'd share.

I have seen a lot of methods for factoring a quadratic with a leading coefficient. Out of the ones I've tried, this is my favorite. The analogy to an airplane is a bit of a stretch, but students seem to remember it pretty well. So I'll take it.

I should also mention that, before I show this to students, I always spend some time letting them work on these by trial and error. I figure a process like this is worthless if they don't actually understand what they are doing. Once I feel like students understand the concept but they are still struggling to get every problem to work, I show them this. We treat it like a shortcut, and boy do they appreciate it!

Here is an example:

First, my students know they will need two binomials, so I start with two sets of parenthesis. Then I put the leading coefficient in each parenthesis. Hopefully, the students have a problem with this. We talk about why it is a problem, and I promise them that we will get rid of the extra 2 before we're all done.

Then, multiply a and c. (See the airplane wings? Use your imagination.)

Look for two numbers with product ac and sum b. (Propeller?  I know this is really a stretch.)

Put those numbers in the parenthesis.

Divide the extra 2. (The landing? Maybe.) It is pretty cheesy, but when students are having trouble I can say something like "you forgot the landing", and they know what I mean.

Done.

For something like this, you may need to divide both binomials. I point out how dividing by 3 and by 2 is the same as dividing by 6. We just choose the division that will keep integers.

Happy factoring!

Figuring Out Factoring

I have been thinking a lot about factoring lately. My algebra 2 students really struggle with it, and we have only factored quadratics (no sum/difference of cubes or grouping yet). I am worried because our first unit after winter break is rational expressions/equations. This unit is challenging when you CAN factor well, and almost impossible if you can't.

I am sure it will be necessary to take a few days to work on factoring before we start rationals. I need to review quadratics, teach cubes and grouping, and put them all together. I am working on how to structure that.

I have been doing some reading and searching for ideas. It seems that almost every method out there requires students to memorize some sort of process. (Except for this, and I am not sure my students are ready for that). As far as algorithms go, I am happy with what I've been using.

There are two issues that I want to address:

1. Helping students make a connection between what they SEE (trinomial, difference of squares, four terms, cubes, etc.) and what they DO (write two binomials using your preferred method, grouping, use a formula, or whatever).

2. Helping students get through problems where more than one method is required. (Like factoring out a GCF and then factoring the trinomial, or using grouping and then recognizing that one of the binomials is a difference of squares).

Here is a flow-chart I sketched out this morning:

I am picturing this on the white board, with an answer box next to it. You could 'drop' a polynomial into the top box and it comes out factored. If there was a GCF, you could put it in the answer box and then examine the remaining part and possibly drop it into another box. You could then take the pieces that come out of the second box and either add them to the answer box or drop them into another box. This all seems a little elementary, but I want it to be very obvious and very clear . . .

I am still refining this and hope to use it for the beginning of second semester. I will write about the final product. In the mean time, suggestions welcome. :)

Fraction Exponents. Easy.

Have you ever found yourself teaching a certain thing a certain way for years, and then one day you think about changing your explanation just a teeny tiny bit? And the new way makes infinite more sense to students, and the thing that used to be impossibly hard is now easy? And then you wonder what took you so long to find that more easy/obvious way of explaining something?

That happened to me today with fractions as exponents.

I won't bother to mention how I used to teach it. It was bad. Very bad.

Today, I started by showing them this, and they all shouted several people went "x squared!".

Then I asked them HOW they knew it was x squared. Somebody pointed out that three x-squareds multiply to equal x^6. Someone else said that you divide 6 by 3.

I made a big deal about writing x^(6/3) before writing the answer as x squared. And we did several more of these including some square roots, so they could see that you divide by two.

Then I showed them this, and gave some time to think about it and write down what they thought the answer should be.

Almost every single student wrote down x^(2/3)! And there were angels singing.

Then we worked on going backwards, which was no biggie at all.  Given x^(1/2), students could easily rewrite as square root of x and so on.

And then I didn't know what to do, because it used to take me a full class period to teach that. And some students would still be sitting there going, "Huh?". But today they just got it in, like, five minutes.

I realize that there is nothing earth-shattering about this method. The thing is, I've taught it this way before. Only I didn't lead with this part. I ended with this part. All I did today was change the order.

Oh, I love these moments of finding the tiniest little change that makes a huge difference.

The Question Reveals . . .

I saw this quote somewhere a while ago, I don't remember where:

"It is better to solve one problem five ways than to solve five problems one way."

Recently, for bell work, I asked my students to give a response to that quote. I was kind of proud of myself, because it might be the first time my bell work has been something other than a problem to solve. The lesson for the day was solving quadratic inequalities both algebraically and graphically (they have done both, but I wanted students to see them side-by-side), so it seemed to fit.

The responses were limited to just a few thoughts:

You can use one method to check another method.
Sometimes one of the methods may not work.
You might forget one method so you could use another.

I am not sure what I was looking for. Maybe I was hoping that someone would think about how solving a problem multiple ways helps you better understand the concepts and how they all fit together?

What I think is interesting, is how much these answers reveal about where my students are at in their understanding of math. To my students, math is still a bunch of procedures to remember and repeat.

And here's the thing:  I think I am still mostly teaching that way.

Sorting Out Quadratic Methods

Now that my students can solve quadratics in five different ways, I wanted them to weigh the pros and cons of each method. I wanted them to be able to look at a quadratic equation and choose an efficient method for solving. Maybe it is just me, but watching someone pull out the quadratic formula when the equation can be factored kinda makes me cringe. I also wanted to review all the methods at the same time.

First, I gave them this sheet. It has the bell work and the practice problems.

For bell work, students worked out an example of each method in the first column as a review. Then we had a class discussion about the strengths and weaknesses of each method. We talked about how factoring may be the shortest method, but you can only use it if the quadratic isn't prime. And so on. We also talked about why you might choose one method over another. (Like how complete the square is so much nicer when the coefficient of x-squared is one and the coefficient of x is even.)

Next, I gave them this set of 16 cards* and a piece of card stock divided into four sections. I told them to try to put exactly four quadratic equations in each section. They needed to choose carefully, because next they will use that method to solve that problem. I just walked around and coached them a bit as they worked, and mostly I heard some good discussions going on. A few groups struggled, mostly because they had trouble figuring out how to tell if the equation could be factored or not. That is one problem I was hoping to correct with this activity, and it was pretty easy to identify who needed some help with that.

Then, students worked out the practice problems using the method they chose.

The next day I used a similar set of four quadratic equations for a quiz. Students could solve using any method they chose, but they could get bonus points for using each method only once. Students did fabulously with this. Yay!

*I also put the 16 problems on the practice sheet. Whenever I do an activity like this, I try to create the sheet so that someone who was absent could so something similar outside of class.

Happy Halloween To Me

A former student met me in the hallway today. She heard we had just learned the quadratic formula in algebra 2, and she took it upon herself to choreograph a dance to go with the song I teach. With illustrations.

She offered to come to class and perform it for my current students . . .

. . . but only if I joined in, which I did. Except for the slap the butt part.

Sometimes, high school students are just adorable.

Today's Million Dollar Question

I have been struggling a bit this year with getting a student or two to show steps/process/work/setups (or whatever you call it when you say that it isn't okay to give a lonely answer with no justification). The silver lining is that this struggle forces me to think about WHY students should show work.  Here are a few reasons I have:

Showing steps . . .
Puts the focus on the process, rather than the solution.
Communicates your solution to others.
Makes it possible for you (or someone helping you) to locate your mistakes.
Slows you down, so fewer careless mistakes happen.
Gives evidence your answer is right.
Demonstrates your understanding.
Helps reduce cheating. (Some might still copy, but at least they must copy the work too.)
Finally, in my class an answer bank is given. Showing steps keeps practice from becoming nothing more than a matching game.

I am also asking myself some questions like . . . WHY do some students struggle with showing work?

Maybe because . . .
It takes too much time.
They can do it mentally.
They don't know how to show work.
They are bored.
They don't believe in its value.
They are cheating.

Writing this, I realized that when a student is repeatedly refusing to show their process, I tend to go straight to negative assumptions. I assume they are being stubborn and uncooperative, or that they must be cheating.

I am going to try to put the whole issue in a more positive light and see where the students are coming from. Maybe these students think that showing steps is just for the teacher's sake, and has no benefit to them personally.

Or, maybe they genuinely don't know how to express how they got the answer.

It also has me thinking about the types of questions I am asking. If someone can calculate the answer mentally, maybe the question wasn't challenging enough?

How do you motivate students to show their thoughts?