Tag Archives: rational functions

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!

Alg II Files: Rationals

My second-to-last post for my Algebra II files! (See more files and FAQs here)You can tell by how long I’ve procrastinated that I’m not really a fan of what I do for this chapter. Each year seems to get a little bit better, but I would never classify put it in the “win” column. In other words, feel free to take any of this and make it awesome. Then let me use it. 🙂

The first few sections don’t have note taker makers, because the problems need space to be neat and tidy:

Rational Functions from megcraig.orgRational Functions from megcraig.org Some notes on my notes: At the very top, it’s hard to see, but we start off talking about how we reduce 12/15. We don’t subtract 10 from each and say it’s 2/5! We talk about invisible ones.  This year, I want to focus on the fact that in the expressions, I can substitute any value for x and you will get the same value in the beginning expression and in the final expression. Magic!

Also, I talk about restrictions as “warning the villagers.” What if someone came along and put 5 into x/(x-5)? THE WORLD WOULD ERUPT INTO FLAMES.  And sure, I know *you* wouldn’t do that, but what if some innocent villager came in off the street and started putting numbers in? We need to warn them! This year, I also want to pull in the graph and talk about how the original has a hole at 5, so in order to be equivalent, the reduced one also needs a hole, which we make by excluding 5 from the domain. (I *just* realized this fact in the last few weeks.)

Some homework:

Rational Functions from megcraig.org(file here)

Then…can you hear the jaws music….

Adding and subtracting!

Rational Functions from megcraig.org Rational Functions from megcraig.org

Then some practice with every operation as either a worksheet

Rational Functions from megcraig.org(file here)

or as a step-by-step powerpoint (great for whiteboard practice)

Rational Functions from megcraig.org(file here)

Let’s solve some now!

8_3 8_3 pg 2I’m not *in love* with just crossing those denominators out, but haven’t figured out anything better yet.

Some practice:

8_3 HW

(file here) And a study guide for these three sections:

8_3 SG(file here)

Some inverse graphing:

Rational Functions from megcraig.org(file here). Hint: make a t-chart by listing pairs that will equal k. Super easy!

Inverse/Direct/Compound variation:

Rational Functions from megcraig.org(file here)

Powerpoint to go along with NTM:

Rational Functions from megcraig.org

Rational Functions from megcraig.org(file here)  Yes, I’d like to do a lot more with this, but this chapter is always rushed since it’s the last one before we have to start trig at the end of the year.


Rational Functions from megcraig.org(file here)

Then it’s time for graphing rational functions. I like what I do, but I think I want to streamline the process a bit–we don’t have to do all of these by hand!- and bring in some of Sam’s stuff.

We spend the first day talking about end behavior and points of discontinuity:

Rational Functions from megcraig.org(file here) Why, of course there’s a powerpoint!

Rational Functions from megcraig.org Rational Functions from megcraig.org(file here)

Then the next day we review the first two steps, then jump into graphing them. This goes a lot better than just throwing everything at them in the first problem!

Rational Functions from megcraig.org(file here)

And yes, another powerpoint:

Rational Functions from megcraig.org Rational Functions from megcraig.org

(File here) There’s even a powerpoint to check the homework that’s at the bottom of the NTM-file here!

Then it’s finally time for the study guide (note: this does repeat problems from the first study guide!)

Rational Functions from megcraig.org(file here)

So what are some of YOUR magical ways of staying rational while teaching rationals?!? Leave a comment or tweet me your secrets!