Started a little planning for next year...the other trig teachers and I have divvied up the units and are each planning three. We are to produce a calendar with daily assignments, assessments, a Smartboard Lesson for each day, and keys for all.
Was working today on Unit 8: Rational Expressions and Equations.
I have the calendar and assignments done and have started looking through the lessons I used this year.
They pretty much suck. This statement might preclude me finding employment for next year but I don't care: I am at a loss for how to make this concept interesting or relevant to 16 year olds. All we are doing is simplifying and solving algebraic expressions and equations. With no context. It's going to be a brutal 10 days of instruction if I don't get it together.
I can make it extrinsically rewarding by including lots of partner/group practice structures.
I can attempt to start off with some word problems of the type that one could, if one wished, write and solve a rational equation. Your standard Tom can paint a house in 3 days and Dick can paint a house in 5 days etc etc. I find that these are either easy enough to solve without an equation, or too difficult for this level.
I really don't want to get into the harmonic mean.
Ideas? Help?
18 comments:
This isn't trig, right?
I think of rational expressions as an Algebra II topic (called Intermediate Algebra when done in a community college).
There was a problem in our text that involved relativity. I wrote something explaining how to do it, but it doesn't have the original problem. I don't know how to post it online, so I'll email it to you, and if you like it, you can tell me how to post it. ;> (And if it's useful, maybe I can dig up the original problem.)
Thanks Sue I got your email. I really need to go to sleep but I'll read through it tomorrow.
You're right it's not trig, of course. The class is called "Algebra 2 with Trigonometry", we often call it "trig" for short.
Hello Kate,
I suggest a look to Dan Greene's Rational Functions Unit: http://exponentialcurve.blogspot.com/search/label/rational%20functions
Most of the examples are about Rational Graphs (the game called "Rational Review" in particular is quite well done), so they may not apply to your unit. BUT, perhaps you can find some inspiration or at least some other extrinsic motivations.
I'm stuck thinking about WCYDWT and rationals... I can't narrow it down to one photo with a simple question. Two hoses in a pool, a couple people mowing a law, it's too forced.
Then again, you could show a tub with a faucet on and drain open and ask how long until the tub overflows. You would need to be ready to provide some extra info, but the set-up is right. (I wonder if it's answer-able as is though).
Anyway, there's my 2¢.
By way of introduction, I make a big deal of how the graph of 1/x finally makes it really clear why dividing by zero is not possible. The students seem surprisingly pleased with seeing that old elementary school rule explained in terms of a graph that they agree "looks like college math," and just sketching graphs of 1/(x+a) and confirming that the asymptote is wherever the denominator is zero keeps us busy for a while. Not sure how that would work for your students; my experience is from the weaker Algebra 2 students who spend 1.5 semesters on the course.
As an application, what about a spaceship placed a distance of r from one planet and D-r from another, with the task of finding out where the ship must be to feel exactly the same gravitational pull from either? Could spin some story around that, maybe.
adding rational expressions is basically 1) converting one or both things so that they are both of the same sort of stuff 2) pooling that stuff together. It certainly seems like there has to be a great application in there, somewhere.
(how about instead of work/rate problems, total discount over a variety of items that are different percentages off?)
I also discuss limits in this unit in Precal, and talk about how the rational functions rate of change is the ratio of the two polynomials rate of change. Types of change is a theme for us throughout the semester, and connecting to that type of essential question can be motivating, on it's own, for some of my students
My unit that is referenced above was what I did when I taught algebra 2 honors. Now that we don't have that course anymore, I wasn't able to do it the same way, unfortunately. This past year, instead of making this a separate unit, we did operations on rational fractions as part of the polynomials and factoring unit. I figured, since we weren't going to be able to discuss graphs, asymptotes, domains, and so forth, I might as well just present these problems as practice and extensions of factoring techniques. This is kind of lame, but it's all I came up with.
So far, I have this for the first day (reviewing operations on polynomials).
Still exploring for the rest.
I would like to get into graphs when they are solving rational equations and inequalities, at least a little bit. I might end up just making a guided investigation worksheet thing.
So the problem with word problems is that no-one *really* cares *exactly* how long it takes a bath to overflow - certainly not enough to do complicated algebra. The problem with relevance is that, let's face it, this unit is a stepping stone to more advanced mathematics and actually is *not* really relevant in peoples everyday lives.
So, to my suggestion. I say, don't make it relevant: make it interesting. Theme a lesson or two on (for example) a space mission. Include some newspaper cuttings [e.g. on proposed Mars missions] and a colourful portfolio for them to complete. Then you can include all the contrived questions you want, and it won't matter. Why not?
I: Everybody knows that for astronauts - unlike for baths - accuracy matters
II: It might not be relevant to them, but at least they might accept that *somebody* does this stuff
&III: Dealing with space and filling in their colourful portfolios might actually take their mind off the whole maths thing.
Just an idea, I've never used it myself and it could be a lot of work... but I think it has the right ingredients.
Did you come up with anything satisfactory for this unit, Kate? I'm teaching our Intermediate Algebra course, and will explore resistors (in parallel, they add by their reciprocals) and relativity effects on velocity as applications. I'd love other ideas.
Like Dan, I've included it in the same unit with factoring.
Not so much. I did make a guided investigation thing about undefined values and vertical asymptotes. And also one for solving rational inequalities. I think I posted about them but I'll send them to you.
I have a kinda half-idea for an application of rational expressions and equations using the thin lens equation. What I managed to put together the night before class is fairly weak ("Here's something cool. Wanna know how it works? Well, that's complicated. But here's an equation that describes how it works that we can solve.") Link here: http://rationalexpressions.blogspot.com/2010/12/thin-lens-equationrational-equations.html
I don't think that this is great, but it's a tad more accessible than parallel resistors.
It's kinda funny how different things seem easier and harder to us. I would want to know why that's true (the lens equation).
The resistor equation is kind of like the mixed rates problems, I think. (Still thinking about it...) I told my students yesterday that electron flow is often compared to water flow. If you put two resistors in series, you're slowing it down twice. If you put them in parallel, you've got something like two faucets letting water through at once. (!)
Here's my (yes, weak) story: One resistor broke off a circuit board. (I actually used to stuff circuit boards, as a piece-work job while I was in college.) I know the total resistance this circuit is supposed to have (I would not really know this, hmm...), and can figure out the resistance on the resistor that's left by its stripes. I need to figure out what resistor to use to replace the one that broke off.
Sue: and then some smart aleck kid is going to ask you why can't you look at the stripes on the broken one to find out resistance?...
That one fell between the cracks in ___ and I'll never see it again. ;^)
Depending on how comfortable you/your kids are with really open ended questions, I might start with:
What lens should I buy?
Hopefully this will lead to questions like "What do you want to take pictures of?" and "What's the difference between all of these lenses". I'd probably use just a few lenses and remove the extraneous information such as price. You could also decide to only look at prime lenses.
Or you could go a completely different route:
1. Find some fractions between 1/5 and 1/6.
2. Find some fractions between 1/12 and 1/13.
3. Find some fractions between 1/n and 1/(n+1).
4. Repeat the above questions, but try to find a fraction between the two numbers with the smallest denominator.
No real world context, but I think these questions are mathematically interesting.
In math class we're learning about rational expressions. Now like any division problem, the denominator cannot be zero. So:
(x-7)(x+7)
----------
(x-7)
Then x =/= 7, otherwise the denominator is zero.
But you can cancel (x-7) and simplify the expression to just (x+7).
Given the expression (x+7), x can be equal to any real number. But if it is equal to 7, then in the previous unsimplified expression, (x-7) = 0, and thus zero was divided by itself to simplify (x-7) from the equation.
But if that's possible, then it's possible to prove that all real numbers are equal:
0*1 = 0*2
0*1 = 0*2
----------
0
then
1 = 2.
It's fucking with my brain, I tell you. :D
nick, here's the problem: (x+7) and (x^2-49)/(x-7) are not actually the same. The first function is defined (by default) on the entire real line, while the second is defined (by default) on the line except for the point x=7. But where they *are* both defined, they have the same value.
So let's say you're faced with (x^2-49)/(x-7). You see it's got a gap discontinuity at x=7, but the *limit* still exists there. That means there's exactly one way to patch that gap and fill in the hole to make a continuous function. And the algebra tells you how to do that without actually taking a messy limit with epsilons and deltas and all.
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