Factoring Quadratics by Grouping

David Cox teaches “Bottom’s Up” to show how to factor quadratics. (Video here.)

There’s only one thing I don’t like about this method. It has one step which isn’t intuitive, and makes it all seem like magic.

When you have a coefficient in front of the $x^2$ term that isn’t one, you have to divide the factors by that number. And then you do “bottoms up” where the $(x+\frac{1}{2})$ gets converted to $(2x+1)$ . I don’t like that you have to randomly divide by a number, nor that the implicit implication is that $(x+\frac{1}{2})=(2x+1)$ .

A fellow math teacher at my school taught me to teach a very similar method, but that uses factor by grouping.

Given a quadratic, the first part is the same:

$2x^2+7x+3$

Rewrite the $7x$ as $6x+1x$ (numbers from the diamond above)

$2x^2+6x+1x+3$ [1]

Then the problem becomes a “factor by grouping” problem. You group the first two terms and second two terms and factor:

$2x(x+3)+1(x+3)$

Then you see each term has an $(x+3)$ so you factor that out and are left with $(2x+1)$ :

$(x+3)(2x+1)$

It might seem a little more complicated, because you have to factor a few times. But my kids tend to get it after practicing 3 or 4 problems, and it doesn’t involve any knowledge they didn’t already possess. They understand that $7x=6x+1x$ and they understand how to factor $2x^2+6x$ . There is no lingering “why?”

[1] You could write it $2x^2+1x+6x+3$ also and it would work. $2x^2+1x+6x+3=x(2x+1)+3(2x+1)=(2x+1)(x+3)$ .

Favorite Tweets #5

NEWS: You can now access the “Favorite Tweets” series by clicking on the menu on the upper right hand side of the header. (Wow, things are getting FANCY up in here!) This is the last “Favorite Tweets” that I will be posting on the blog itself. All future “Favorite Tweets” will go directly there. I don’t think they’ll show up in your RSS readers, but you’ll know when I post a new one via Twitter. And aren’t you guys on Twitter the only ones who really care, anyway?

Here’s #1 and #2 and #3 and #4

…

samjshah @dcox21 i don’t actually teach my kids in reg calc about sequences. but i think if they get the IDEAS behind it, that won’t matter.

samjshah @dcox21 plus, depending on the sequence, i’ve seen it written both ways (0th term and 1st term)… plus they’ll forget it before 11/12th gr.

dcox21 @samjshah Forget? You underestimate me?

…

k8nowak I hate when my new favorite song turns out to be three years old.

SweenWSweens @k8nowak Has Bieber really been putting out songs for 3 years now?

k8nowak @SweenWSweens You just get me, Sweeney. It’s scary.

…

samjshahARGH!!! #itssundayandireallydontwanttoplanfortomorrowanditsgettinglaterandlaterandimjustnotabletostarttogetittogether

…

ddmeyer This just hit me hard: the same curriculum that pushes our students away from what’s great about math does the same thing to their teachers.

…

mctownsley @samjshah Counting Crows – Mr. Jones! I had a friend in college who mixed up the lyrics. Instead of “Mr. Jones and me…” ..

mctownsley @samjshah (con’t) she thought it was “Mr. Snowman Yee” she’s never lived this down. #nojoke

…

ddmeyer I attended the “Western Caucus” at NCSM. I’m curious how the board will attract new teachers (most of whom caucus online) to that format.

ddmeyer Put another way, “caucusing” (especially in this context) seems more relevant to an age when we couldn’t connect instantly online.

samjshah @ddmeyer on that note of caucusing online, why aren’t you on this map?: http://maps.google.com/maps/ms?hq=http:%2F%2Fmaps.google.com%2Fhelp%2Fmaps%2Fdirections%2Fbiking%2Fmapplet.kml&ie=UTF8&dirflg=b&hl=en&msa=0&msid=102961433061679876855.000484f12a77524f148fe&ll=39.300299,-93.339844&spn=31.685063,79.013672&t=p&z=4

ddmeyer @samjshah What the hell is that?

ddmeyer @samjshah Why haven’t you added me?

ddmeyer @samjshah Hurt.

samjshah @ddmeyer everyone added themselves! it was a bit of a crazy frenzy on twitter yesterday. see my recap at the bottom of https://samjshah.com/2010/04/25/favorite-tweets-4/

cannonsr @ddmeyer Yeah, dude. You were being too cool hanging out at an actual conference. Such a diva.

JackieB @cannonsr Hey, I was at the actually conference. /And/ I added myself to the map, don’t let @ddmeyer off the hook that easily.

…

welikesnow Tip: Don’t cut your leg with a chainsaw, even a little bit.

…

misscalcul8 How do I get them to think?

JackieB @misscalcul8 Give them good problems, ask questions, then don’t answer your own question. Wait. And wait. Repeat if necessary.

…

k8nowak “Please don’t call me ‘Mom’. It makes my uterus clench in terror.” – Me to 5th period.

…

SweenWSweens Recently all this evidence is pointing towards me being an adult… but I don’t buy it!

…

RobertTalbert OH: “I love when I walk down this hallway and see all the lights on, because it means that mathematics is open!”

…

approx_normal Student: “I heard our lesson is hard today.” Me: “What’d did you hear was so hard?” Student: “I heard we have to think.”

…

CardsChic Busted. “How did you come up with this idea? Did you just make it up today, or did you steal it from someone on twitter?”

…

hemantmehta I named my iPod “The Titanic,” so when I sync my iPod, it says “The Titanic is Syncing.” (via a student) #fb

…

jimwysocki My seniors have so checked out. I got so annoyed with them today that I walked out of class.

SweenWSweens @jimwysocki One time I got so upset with one of my classes that I fake quit.

jbrtva @jimwysocki that’s how I felt today…with my Alg1 class (fresh/soph) #commisserating

…

samjshah the most fabulous thing, though, was getting a HUG from @k8nowak. yes, you heard it here first. she does hug. you just have to booze her up.

dcox21 @samjshah @k8nowak She’d gonna punch you in the neck for hugging and telling.;-)

dcox21 @jybuell @samjshah @k8nowak I believe it happened. Didn’t you notice her talking about shoes last night? New shoes and hugs? What’s next?

…

samjshah @RobertTalbert i just finished FTC II in calculus. that was … fun.

samjshah @RobertTalbert “WHY DO WE HAVE TO KNOW THIS?” “Because it is FUNDAMENTAL. And the sequel is better than the original.”

…

lpudwell apparently “there are i balls in this box” sounds a lot like “there are eyeballs…” even to a combo class that’s used to balls in boxes. :)

…

CmonMattTHINK In the future, if I ever tweet something about considering giving a take-home test, somebody please smack me upside the head.

…

k8nowak @samjshah this tweet is coming from INSIDE THE HOUSE!

samjshah @k8nowak i see dead people. BRAINS, CLARICE.

cannonsr @samjshah @k8nowak Huh. Wouldn’t have pegged y’all to become a horror movie. But somehow not surprised.

…

JackieB No idea what my husband is watching on the TV, but his yelling “Don’t look at the TV” did not have the intended effect. Gross.

…

dcox21 Sorry, but you don’t prune cotton. #needaredstamp #FarmProject

k8nowak @dcox21 nobody needs that stamp. except you.

…

k8nowak @jreulbach Find a worked out example you want to practice, put a blank page next to it & copy the problem. Cover the example with your hand.

k8nowak @jreulbach Do as much as you can, compare it to the existing work, copy when you need to. Repeat until you can do it without comparing.

…

k8nowak By the way, if @samjshah ever offers you a tour of Brooklyn, take it. Trust me.

…

dcox21 @k8nowak Honestly, while on the train, how many times did the phrase “no sleep till…” enter into your head?

k8nowak @dcox21 Wow. Zero. 7th grade me would be so disappointed in me.

dcox21 @k8nowak Oh, c’mon. I sang it at least 10 times cause I knew you were going.

k8nowak @dcox21 Well, 7th grade me thinks you’re rad.

dcox21 @k8nowak You just earned your 7th grade cred back with an appropriate use of “rad.”

…

ThinkThankThunk @samjshah This twitter thing is totally worth it just for your links. I’m showing all of these videos tomorrow!

…

k8nowak Der Subs: I don’t want to read an aria about how every kid deviated from acting like a perfect angel. Just effing deal with it.

k8nowak “Students asked questions about the yellow packet &the period ended before I started the lesson.” You were just. Supposed. To. Collect. It.

…

samjshah @ddmeyer why are you not yet on this? http://maps.google.com/maps/ms?hq=http:%2F%2Fmaps.google.com%2Fhelp%2Fmaps%2Fdirections%2Fbiking%2Fmapplet.kml&ie=UTF8&lci=bike&dirflg=b&hl=en&msa=0&msid=102961433061679876855.000484f12a77524f148fe&ll=40.713956,-74.003906&spn=61.260704,158.027344&z=3 ADD YOURSELF

ddmeyer @samjshah How do I do that? You and Riley are the only collaborators with editing privileges.

cannonsr @ddmeyer Really? The edit button was enough for the rest of us. (Maybe collaborate first.)

ddmeyer @cannonsr @samjshah Hmph. Edit button needs to be a little bigger if you ask me. I’m good now.

cannonsr @ddmeyer You’re the one with the Google connections to change it.

k8nowak @ddmeyer Agreed. If only we knew someone who worked at Google. We could tell them.

ddmeyer @cannonsr @k8nowak I’d rather blame @samjshah.

…

Fouss @smallesttwine We had a fire drill when I was in hs and taking an AP test. Hmm – maybe that’s what I could blame my score on. :)

samjshah @Fouss i had to take my AP calc test in the autoshop room which smelled STRONGLY of bleach. why? kids earlier in the day pooped in the room.

samjshah @Fouss PLUS a horrible daddy long legs spider crawled on my desk. i am TERRIFIED of spiders. so i had to call the proctor over to remove it.

samjshah @Fouss there MIGHT have been some yelping and gasping and panting involved. you may cue your imagination of this scene in 3… 2… 1…

smallesttwine @samjshah AHHHH!!! I just actually screamed out loud. Not sure what is worse, poop or spider.

Fouss @samjshah Can’t stop LOL… seriously (or not, I guess). And yet you still got a 5, right?

Fouss @samjshah It’s the yelping and gasping and panting that’s killing me.

jreulbach @samjshah OMG, poop and spiders? What the hell goes on in shop class??

samjshah @Fouss obvi. neither autorooms nor bleach nor poop nor spiders shall keep this mathlete from the swift completion of his AP calc exam. 5.

Fouss @samjshah I would expect nothing less.

…

sig225 It’s spelled i-s-o-s-c-e-l-e-s #needaredstamp Otherwise, it’s a comb & perms question given the number of different things students write

…

hemantmehta An AP photographer was in my classroom today. I told the kids to applaud when I enter the room & bow down to me. They refused. Dammit. #fb

…

dcox21 @samjshah But we could saddle one [spider] up and you can take it for a ride to the restaurant. Oh, well, your loss.

SweenWSweens I was thinking about not ebaying the 4th mathblogroadtripmobile seat and saving the room for my prized spider collection instead.

Fouss @samjshah I’ve got kids – I’m ok with shrieking. And as long as the blood is @SweenWSweens and not mine, go for it.

…

samjshah there was almost no positive today, except for one thing, which was huge for me. one kid who i taught in 10th and 12th grade said that…

samjshah… he wanted to take me to college with him, because i was the only person who was able to make math make sense to him. i melted.

…

samjshah you know when you feel like soon you’re going to have too many variables on the board and kids will freak out? http://brizzly.com/pic/2CTX

k8nowak @samjshah what are those green things? turnips?

dcox21 @samjshah @ddmeyer Kinda what I thought too.

park_star @k8nowak I was guessing sad apples.

CarissaJuneK OMG @samjshah…that log proof is amazing. I like the apples, cherries and bananas :)

samjshah @CarissaJuneK WINNER! she got all the fruits, because she has EYES. seriously, @k8nowak @park_star

k8nowak @park_star maybe the bananas are sad. you know, since they are blue.

park_star @k8nowak good point. I was actually going to say colorabi, but then I wasn’t sure if anyone would know what that was.

dcox21 @CarissaJuneK @samjshah I think those “apples” are tomatillos.;-) @k8nowak

…

samjshah @ddmeyer might you promote @k8nowak‘s binomial expansion contest on your blog? i bet we could get some good fodder for convo from these vids

jybuell @samjshah I’m hurt u didn’t ask me to promote. Over 12% of the people who share half of my genes have read at least one post

jybuell @samjshah I’ll just tell all my subscribers about it next sunday dinner

…

SweenWSweens @hemantmehta @jbrtva Make sure you go out for coffee afterwards. ZING! I’ve been waiting on bringing that one back for months now.

jbrtva @SweenWSweens @hemantmehta We’ve affirmed our “just friends” status long ago…coffee shops unnecessary. :-) well played, though

…

Fouss @samjshah I killed a spider in my closet today and thought of you.

…

misscalcul8 Give time limits. Students don’t pay attention to 5, 10, 15. Try 7. #pippens

…

cannonsr @JackieB @ddmeyer 1 of 3) Been trolling the academic journals. Keep running into conclusion….

cannonsr @JackieB @ddmeyer 2 of 3) that building online communities apart from physical ones doesn’t happen often.

cannonsr @JackieB @ddmeyer 3 of 3) Frustrated by my reality vs research reality. Given my position, that’s not a bad thing.

…

sumidiot getting excited about spending my evening proctoring an exam. wait, what? no, that’s not right. dammit.

A binomial expansion throwdown. You in?

Oh k8, my k8, has thrown down the gauntlet. Or in more modern day kid-speak, she asked you to “BRING IT ON!” (That’s Kate Nowak, for y’all.)

A while ago, she scoured every nook and corner online for videos teaching the binomial expansion, or for some ideas which make the teaching of it… well… not excruciatingly boring. Actual videos that didn’t make her want to stab her eyes out, they didn’t quite exist.

So she’s asking you to: make one. Anything that’s better than what’s out there.

You have weeks to do it (deadline: May 27th). She’s offering some sort of t-shirt prize. I’ll sweeten the pot. If we get 7 or more video submissions, I’ll buy the winner a copy of that Lemov book that the New York Times article featured a few months ago (as long as you don’t live somewhere with crazy shipping costs). And if you own that (or don’t want it), I’ll buy you some insanely cool math book. Yes, this is my own money. No, I don’t know why I’m doing this, since I’m pretty poor.

So when Kate says “BRING IT ON!” I hope you enter so you can say “IT’S ALREADY BEEN BROUGHTEN!”

Also, if you have a math or math teacher blog and want to spread this around, that would be super duper awesome.

I love when kids stump me

So in multivariable calculus, it happens a lot. Most of the time, I can work things out and come back with a cogent response, and occasionally, I turn to you good folk. Today I was stumped, and then worked through it, and felt all proud of myself for about 60 minutes after I was able to figure things out. Here’s the set up and the problem:

Today, I met with a student who is working on an awesome end-of-year project on center of masses. Basically, he’s making a bunch of semi-complex foam figures (with some other materials, like wooden dowels). He’s going to multivariable calculus to find the center of mass for these figures. Then he’s going to paint them black, mark the center of mass with neon orange, and toss them in the air while video taping it.

He was inspired to do so by this video I found online:

He’s going to throw the video he makes of his own crazy figures into LoggerPro (but Tracker would work just fine) and see if he gets a parabola.

One of his objects is going to be something like 2/3 or 3/4 of a foam torus. He was having trouble finding the center of mass of it. The first thing we did was simplify the problem — changing the 3D foam figure of uniform density into a 1D bent wire of uniform density.

For the problem, we assumed 3/4 of a circular wire and we gave it a radius of 1.

Then the question was to find the center of mass for this thing. Clearly it won’t be on the wire [1], so you can think of it as such: if you wrapped the wire in super strong but super duper infinitely light saran wrap, and then you wanted to balance this wire+saran wrap figure on a pencil point, where would you place the pencil point?

So I’ll admit that I struggled — but not as much as I anticipated. I started from first principles when solving the problem (cutting the wire into a finite number of pieces, and then making a Riemann Sum). And then this method allows me to find the center of mass no matter how much of the torus I have, whether it be 2/3 or 3/4 or e/pi.

I’m sure there’s an easy way to do this — much easier than reducing the problem to first principles and starting from scratch. But now that I have, I am pretty darn proud of myself. I think I understand the problem, now that I can look back at it, at a much deeper level. I can see symmetry arguments and how they come into play through the algebra, from working it out. I also can see how I can solve this sort of problem given any bent wire (any wire which I can describe parametrically, anyway). So yeah, I got a little bit… glowey.

My favorite part of solving this (which involved discovering the error which confounded me for 5 minutes!) is when I ended up with:

$\int_o^{end} r\cos(\theta)\sqrt{1+\cot^2(\theta)}*-r\sin(\theta)d\theta$

Immediately I wanted to convert $\sqrt{1+\cot^2(\theta)}$ to $1/\sin(\theta)$ and have that cancel with the other $\sin(\theta)$ in the integral. That’s when I realized my huge mistake… after 5 minutes of hunting.

You can’t assume that $\sqrt{1+\cot^2(\theta)}=1/\sin(\theta)$ . In fact, it equals $1/|\sin(\theta)|$ . And this small distinction makes all the difference in the world!

No, there isn’t any advice for you, and this isn’t about things I’m doing in my class, or even me fretting about how I’m not doing an amazing job. This blog also acts as a little digital archive, and I wanted to set aside this little glowey moment.

And if you’re wondering, I’m going to let my student sweat it out, and keep on working at it, until we next meet. If he hasn’t had that moment of insight yet, I’ll help him out.

PS. If you want to work out this problem or any variation, and come up with some beautiful and elegant solution (which y’all are oh so amazing at!!!), feel free to throw your thoughts/approaches/etc. in the comments.

[1] What he’s going to do, in order to throw it, is to put two or three light toothpicks in this partial torus, with a neon orange sticker attached to the toothpicks where the center of mass is calculated to be.

Topic Lists, Reprise: Obvious and yet, I never would have thought of it

This idea totally came from someone else, and I’m awful for not remembering who from the math-teacher-edu-blogosphere came up with it. But it’s just such an awesome idea, and I wanted to spread the love. If this is your idea, just throw the original post down in the comments, and I’ll be sure to add a huge giant link to it so you can have credit.

It could be really useful if you’re trying to help kids get organized for an end-of-year exam.

I wrote a while ago (causing some chafing for a few) about how I give my kids topic lists before major assessments.

They used to look like this:

Now, I’ve added a single image, in order to help students more effectively learn how to study:

So you can see what it looks like in it’s final glory…

View this document on Scribd

It’s a little late in the year to make this effective, but I’m hoping it’s helped a few kids identify where they should focus their (precious and limited) time studying. If a student bombs an assessment, when I meet with them, I can ask them to pull out their topic list with these little boxes filled out, and we can start a conversation correlating their assessment with their filled out topic list.

(Of course, this is after the all important question: “Tell me how you prepared for the assessment. In detail. Don’t leave anything out.”)

Weights! Goldsmiths! Optimization!

I am in a problem solving group at my school, and I took 45 minutes of one of our sessions to lead a mock class. Not really mock, to be fair. I assumed I’d have 3 math teachers and 2 science teachers as my class, and I wanted a problem which would get them to think, work together, and also let me guide without leading (or is it lead without guiding).

The problem I chose was exactly the problem that Brent just wrote about on The Math Less Traveled: the broken weight problem.

A merchant had a forty pound measuring weight that broke into four pieces as the result of a fall. When the pieces were subsequently weighed, it was found that the weight of each piece was a whole number of pounds and that the four pieces could be used to weigh every integral weight between 1 and 40 pounds. What were the weights of the pieces? [I gave the problem with ounces.]

I have to say that I was really thrilled that I was able to get them to a solution, with very little nudging. I let them take their time. I started them out by giving them slips of paper of various sizes with corresponding weights written on them, and asked them to use those weights to be able to weigh something like 10 ozs. I helped them organize their thoughts with observations, and I helped them latch onto key ideas once they emerged. I never gave the key ideas, and I didn’t push. It was awesome to witness them work together.

It was also surprising in two other ways:

1. I had the pathway in mind that I thought they were going to take — basically a recursive approach. They did not go that way, and it was afterwards — when examining the problem once they had the solution – that they saw the recursion.

2. I had prepared two “hint cards.” They were written on origami paper and folded up — because, why not? I told ’em that if they all agreed, they could take the first hint card, and if they felt they really needed it, they could have the second hint card. They didn’t take any of ’em. I thought they would. In fact, I predicted that they would get frustrated and take the first one pretty quickly, so I put on the first hint card: “YOU CAN DO IT! Keep working at least for another 5 minutes.” It wasn’t a hint, but a “work through frustration” note. The second card had a hint leading them to recursion (saying something like “What if you only had any 2 weights… what would they be so that you can weigh the most: 1 oz? 2 ozs? 3ozs? 4 ozs? …”)

As a result of watching them operate, and places they struggled (including understanding the problem!), I wanted to challenge myself.

How could I create a formal lesson plan for this? A lesson plan that guides without leading.

Here’s my first crack at it (PDF here):

View this document on Scribd

PS. Yes, I know there’s a typo in question 1.

Solution to the “what curve is this?” problem

So a while ago I posted a problem that me and another teacher worked on in our problem solving group. We didn’t have the most elegant solution (that honor goes to Jake), But I think it is slightly qualitatively different than the solutions posed in the comments of the original post. Our solution involved systems of equations and parametric equations and L’Hopital’s rule. Yup, believe it or not, L’Hopital arose naturally in the wild, and when I was coming up with my plan of attack, I suspected it would if things were going right.

To remind you, I wanted to find the equation for this blue curve:

(If you want more details, just check out the original problem.)

So here it goes.

The crucial question we asked ourselves is: if we drew all the red lines, where would the blue line come from?

The answer, which was fundamental for our solution, was: if we drew two red lines which were infinitessimally close to each other, their intersection would give us one point on the blue curve. Think about that. That is the key insight. The rest is algebra. If we could find all these intersection points, they form the line.

So we picked two points close to each other: one with endpoints $(a,0)$ and $(0,5-a)$ and the other with endpoints $(a+\epsilon,0)$ and $(0,5-a-\epsilon)$ .

Notice that as we bring $\epsilon$ closer and closer to 0, these two lines are getting closer and closer to being identical. But right now, $\epsilon$ is just any number.

So the first line is (in slope-intercept form): $y=-\frac{5-a}{a}x+5-a$ (any of the red lines)
And the second line is: $y=-\frac{5-a-\epsilon}{a+\epsilon}x+5-a-\epsilon$ (any of the other red lines)

We want to find the point of intersection. So setting the $y$ s equal to each other and solving for $x$ , we get:

$x=\frac{\epsilon}{\frac{5-a}{a}-\frac{5-a-\epsilon}{a+\epsilon}}$

Of course now we want to see what happens to the intersection point as we bring the two lines infinitely close together. So we are going to take the limit as $\epsilon$ approaches 0.

$x_{blue}=\lim_{\epsilon \to 0} \frac{\epsilon}{\frac{5-a}{a}-\frac{5-a-\epsilon}{a+\epsilon}}$

Notice you’ll see that we get a $0/0$ form if we just plug in $\epsilon=0$ , so we must L’Hopital it!

When we do that (remember we take the derivative of the numerator and denominator with respect to $\epsilon$ ), we find that:

$x_{blue}=\frac{a^2}{5}$ .

And plugging that into our equation for the first line, we find that the $y_{blue}$ coordinate is:

$y_{blue}=\frac{(a-5)^2}{5}$

At this point, we rejoyce and do the DANCE OF JOY!

GAAAK! Almost. You silly fools. You’re like my kids, who get so proud when they do the hard part of a problem, that they forget what the question is asking and move on to the next problem. We still don’t have an equation. And what does $(x_{blue},y_{blue})$ mean anyway?

To start, that point represents the intersection point of two lines infinitesimally close to each other in our family of red lines above. But this $a$ business? It’s confusing. I like to think of it like a parameter! As I move $a$ between $0$ and $5$ , I am going to get out all the points on the blue curve.