Tag Archives: polynomials

It Really Is All About Lines.

I’m sure this has never happened to you, my wonderful smart readers, but sometimes I read/hear something about math and outside I’m:

But inside I’m really:

Case in point: Glenn’s (@gwaddellnvhs) obsession about how everything is just made up of lines (It’s somehow comforting to know that Dylan struggled with this idea as well.).

Well guess what, y’all? NOW I GET IT!

It was all thanks to a presentation by John Abby Khalilian that I attended at the Alabama CTM Fall Forum, where he not only presented an activity that involved multiplying lines, but actually had us do the activity. And I was fortunate enough to be seated next to another attendee (but I don’t remember his name!) who was super excited about joining me in a conversation about extending it past the Algebra I level at which it was intended.

I modified the original file slightly (with permission from Dr Khalilian):
files from megcraig.org

files from megcraig.org(File here) I urge you to work through the first page–I promise you this whole made-up-of-lines-thing will finally make sense!  Note: I would make some modifications for next year. First, Dr Khalilian gave us two versions and I used the one we did in class. However, his first version used two positive-sloped lines, which I think would work better as the first example (Just use the lines from #10) Second, instead of giving them the equations of the lines in #10 and 11, I would have them already graphed and ask them to sketch the result?

Also, I know what you’re thinking, “Man, those are some Algebra I level questions on the first sheet.” I agree, and yet it was definitely a review that my Precal students needed.

After the worked on this, we then reviewed graphing polynomials, tying it together with the idea that we were just adding another line and what that would do. I was also made sure to focus on the fact that, “Oh, we have two negative lines and a positive, that would make it positive” because I knew what was upcoming:

Polynomial and Rational Inequalities.

Well, it went stunningly this year. All we had to talk about was the fact that sometimes drawing all those lines could be tedious and messy, when all we really care about are the places where a line is positive and negative. And if the line has a positive slope, it will be negative before its x-intercept and positive after. Like most math things, easier to see than to tell:

files from megcraig.org

The red represents the sign of the red line, blue of the blue line, and black is the final result. So now sign charts are SUPER EASY.  We don’t plug things in! Who wants to do that? MATH BABIES, that’s who. We are going to use our BRAINS instead:

files from megcraig.org

note the two rows needed for 0 in #5, since it is a double root, meaning there were TWO lines that had zero as an intercept

files from megcraig.org

(file here) Note: I started them on the first page with making sure that we always got rid of a negative leading coefficient (you have to “flip the sign”–get it??). This made it easier on the back when we had -x-5 hanging around; yes we could have pictured a negative line and write the chart as + – – but I felt like they would forget that one was negative. Easier just to take care of it than remember it!  They all did fantastic with this on their test!

I know, could this whole “everything is lines” thing get any more exciting and awesome? Yes. Hold on to your hats, I’m going to change your Rational Function Graphing World.

Maybe your world use to be like this: Find your horizontal and vertical asymptotes (and holes) using these rules that I told you. Then make a t-chart of all the values on either side of the vertical asymptotes and plug in to find points to graph:

files from megcraig.org

See, I hated doing the arithmetic so much I made it into a powerpoint. (But the powerpoint is pretty awesome for introducing horizontal asymptotes).

But then I used Kate’s wonderful introduction to rational functions worksheet that I always admired but never quite saw how it fit we how I graph rationals, but NOW I GET IT! It’s just like polynomials, except dividing! (modified only because I did not want to pay scribd to download it, plus one page, amirite?)

files from megcraig.org

(File here)  I’m going to modify some more for next year by (a) replacing trash panda (I have a serious problem where I can’t leave blank space) with the questions “what is the equation/zeros of f(x)/g(x)?” and (b) adding some more graphs, mostly stealing from being inspired by Sam’s worksheet. Instead we spent a few minutes the next day talking about what would happen if we were dividing a line by itself in order to lead into the idea of a hole.

So NOW! we can graph rationals THE SUPER EASY AWESOME WAY.

H: Find the horizontal aysmptote

F: Factor and find holes

Z: Find zeros of the numerator (x-intercepts) and denominator (vertical asymptotes)

S: Make a sign chart with all the zeros you found in the previous step and then graph!

When I asked the students to come up with a mnemonic, the most memorable was, “Horses Fart, Zebras Smell.”

So here it is in all its glory:files from megcraig.org

(file here, with more practice on page 2!) More modifications to make: Now that I don’t need all that space for a sign chart, I can add another example to the front!  The one we did in class involved two factors in both the numerator and denominator. Also, I would change #2 so the horizontal asymptote isn’t 1/3! If you notice, in #1, I did the sign chart below the graph, but that got super confusing in #2 with everything so close together, so we started drawing it separately. We almost finished up the front page (including introducing horizontal asymptotes with the above-mentioned powerpoint) in one day, then did a couple more examples and practice on the back the following day.  When we get back from spring break, we’ll do another day involving Weird Things That Can Happen like 1/x^2, but I think the sign chart is going to work nicely again!

I hope this helped you figure out the power of understanding the whole “everything is lines” idea and how it can carry through to so many things!

Precal Files: Quads and Polys

Ok, technically, “Quadrilaterals and Polynomials,” but doesn’t “Quads and Polys” sound more fun? Also, this is my second-to-last unit for my Precal Files so my goal of having them up before school starts may actually happen! (See more of the files and FAQs here).

So most of this should be a review for Precal students so we booked through quite a lot of it. Starting with a quick review of parabolas:

Polynomial Files from megcraig.orgPolynomial Files from megcraig.org(file here, modified from unknown source)

Some homework for the chapter:

Polynomial Files from megcraig.org(file here)

Then we did a really cool NMSI activity about concavity. I added this to the end of it for a little derivative preview:

Polynomial Files from megcraig.org(file here)

Review graphing polys:

Polynomial Files from megcraig.org(file here)

And dividing polys:

Polynomial Files from megcraig.orgAnd solving polys!

Polynomial Files from megcraig.orgAnd solving polynomial/rational inequalities:

Polynomial Files from megcraig.org

(file here) I did a factor-sign-row chart and we also did a mini-graph on some to determine the signs. If you’d like to see more about the factor-cool-way to do sign charts, here’s a showme video of me doing a quick explanation.

A pretty intense group-work day on these inequalities:

Polynomial Files from megcraig.org(file here) And then wham, bam, time for the study guide!

Polynomial Files from megcraig.org(file here) And if you’re superinterested (or want to use the study guide and not make a video yourself), here is the showme video key.

Only one more unit to go!  Woot woot!


Alg II Files: Polynomials

(see more files and FAQs here) This is one of my favorite chapters in Algebra II because it’s the first time we discover that:

We Can Do Hard Things Poster(poster file here)

I used to start with this page:

Polynomial files from megcraig.org(file here)

But then my co-teacher made this page that I liked and used this year:

Polynomial files from megcraig.org(file here) and I just did the bottom part of the first page in class on the board. And during the same class period, we jump into this:

Polynomial files from megcraig.org(this is the second page of the previous file). Then it’s time for some graphing!

If you want to do some discovery, I’ve had success with this page:

Polynomial files from megcraig.org(file here) but for the sake of time, I’ve been doing the same thing with this desmos file. Then we put all of our conjectures together and practice:

Polynomial files from megcraig.org(file here) after teaching this about a bazillion times, I now really like how it goes.  The only thing I may change next year (and maybe more so in precal) is talking about how x^3 has three roots at 0, with (x-2)(x-3)(x+1) we just translate those three roots to 2, 3, and -1 just like we translated (x +2)^2.  Is this even a thing or am I just seeing transformations everywhere? Also, yes, we do call cubic functions and triple roots “John Travoltas” (I stole it from someone on the #MTBoS) because:

Then we spend a day practicing:

Polynomial files from megcraig.org(file here).  I usually go around and stamp each row when they have completed it successfully, and then can only turn it in once it has all four stamps.

At this point I throw in solving sum/difference of cubics and quartic trinomials:

Polynomial files from megcraig.org(file here)  S.O.A.P is a handy mnemonic that I learned from my coteacher. It tells you the signs of the sum/difference formula: Same, Opposite, Always Positive.  It becomes a bit of a chant: “Cube root; cube root, square, multiply, square; same sign, opposite sign, always positive.”

Some homework:

Polynomial files from megcraig.org(file here) Because of scheduling, it was a good time to throw in complex numbers for a day or two:

Polynomial files from megcraig.org(file here) Ugh, now there’s something that can be taken out of Algebra II if you ask me (but no one ever asks).

At this point I usually take a break and quiz:

Polynomial files from megcraig.org(file here).  Yup, there’s a review powerpoint as well:

Polynomial files from megcraig.org(file here).

Then it’s time to really get our hands dirty with some division:

Polynomial files from megcraig.orgBut I really want to try the box method next year as promoted by @TypeAMathland (especially since I can probably get a tutoring session since Anna is going to be my #TMC15 roomie!).  But with just a bit of modification I can still use the same homework:

Polynomial files from megcraig.org(File here)

So the Algebra II book that we use likes to spend a section on “I give you a factor, you find all the rest” but that seemed like a waste of a day, instead I go with “I give you a factor, find all the zeros” as a lead-in for when “I give you no factor”:

Polynomial files from megcraig.orgI learned a while back that it’s handy to have them figure out how many answers there should be and write out that many blanks. Otherwise many would forget that the original given factor also told you about a zero.

Here’s another day that I’m not a fan of:

Polynomial files from megcraig.org(file here).  I finally took a stand and stopped teaching the “what are the possible number of real/imaginary roots this could have?” because WHY?  I almost want to take a stand on “hey, I’m only going to give you 2/3 of the answers and one of them happens to be imaginary so do you think you could figure out the third?” because WHY? but I’m pretty sure that is specifically in our course of study. At least it’s a nice breather after all the heavy lifting we’ve been doing.

Then finally the moment we’ve all been waiting for!  Let’s solve some polynomials!

Polynomial files from megcraig.org

After doing a couple without the calculator, we start using the graphing calculator to find the first zero (or the first two if it’s a quartic).

Then let’s wrap it up:

Polynomial files from megcraig.org(file here).  And of course a review powerpoint:

Polynomial files from megcraig.org(file here).

Are polynomials one of your favorite things?  Do your kids know who John Travolta is or do you have to do the dance for them? Wait, am I the only one doing the dance?