Function Composition

I love composing functions.  I always have.  It's one of the topics in algebra I get most excited about.  I don't really know why.  It's cool that functions have this operation that the real numbers don't have.  It's cool that this operation shows up repeatedly in calculus.  But, ultimately, I just think it's really fun.

Every time I go to teach function compositions, I think I finally have it nailed.  I use all the right colors to differentiate between the inside function and outside function; I start out with an application problem to motivate the use; and I'm pretty sure my enthusiasm during this lesson is off the charts.

Still, something was missing.

In the past, I've started with composing at a general expression.  We would start by finding f(g(x)), for example.  I guess I did it that way because that's how the textbooks always did it.

This year, I started with composing at a number.  We started by finding f(g(3)), for example.  I'm not sure why I never thought of this before.  I feel like a total idiot that it took me this long, but it is what it is.  I did 2-3 examples of evaluating at a real number, and the kids took off from there.  As I walked around the room, checking their work, it was clear they nailed it (at least for the day, right?).

Now onto the general case.  But first, a bit of a digression.  I showed them this slide:

We worked these together.  The magic bullet?  f(a+1).  In the past, I would skip from f(a) to f(x+1).  This is where I lost the kids.  x+1 was too much too fast because the original function is in terms of x.  But adding that one little line made all the difference in the world.  From here, we could find f(g(x)) where g(x)=x+1.  No problemo.  Here is my notebook file.

Now if they can just remember that (x+1)^2 isn't x^2+1 I will be a very happy lady.

Kate Nowak wrote this in one of her posts that I've always remembered:

I just find it stunning that you can plan out a lesson 95% correctly and it will miss most of your kids. And you can change one little thing - add three rows to a table - and now all the kids basically get [it]...and tell you why they don't get why this is such a big deal. I feel a little like I have super powers.

They're going to be prepared for calc...so help me God

I've written before about how I feel like a concept we think our students get that they really don't get is the composition of certain functions, specifically trig functions and log functions.  I made a vow to myself to emphasize compositions with my Pre-Calc students a lot this year, so that when they do get to calculus, the Chain Rule and u-substitutions will be two of their best friends, as opposed to worst enemies.

I was reminded of this vow when I asked a student to read an exercise from the book out loud.  The exercise started like this:



And this is how she read it:

"Sin" [as in a transgression, not a trigonometric function] "times pi over two minus x."

I wanted to say, "When have you EVER heard anyone say it like that, girl?"  But, I remained calm.  I ignored the mispronunciation (we have bigger fish to fry here), and focused on the "times" part.

It seems like every time I have this conversation ("It's not 'f times x,' it's 'f of x,' guys."), I feel like the kids are just nodding to get me to shut up.  I can't blame them.  I did the same thing in grad school [way] more than once.  As long as I make the prof think I understand what he's saying, all will be well.

But, inputs.  They're kinda a big deal.  What worries me is that it seems like students often view inputs as some kind of multiplication as opposed to actual arguments, which makes sense as the notation is very similar (parenthesis for both).

I continued to notice this was a problem as we were verifying trig identities.  I don't know if the kids just got so into the proofs that they forgot a few fundamental things...like what sine and cosine are...or what was going through their heads exactly.  But, let me tell you, I saw crap like following slide all. the. time.  So, I made them figure it out:

I would not tell them what was wrong, but I did mention it was subtle.  When they finally started figuring it out, we talked about why we need all those theta's!  Our dear trig functions are meaningless without them!

It's a small step, but if it gets them to remember that these trig functions must have an angle at which they're to be evaluated, even if that angle is arbitrary, well, then, that's a good thing.

Reflections

Somehow summer classes have come and gone.  I know students take summer classes to learn information fast.  I hope that they really do learn.  I do know that I've learned so much more than I thought was possible, so I wanted to take some time to write what I thought went well this semester and what didn't go so well.  This way, if I get to teach calculus again, I can come back and remember what needs to be stressed more.

What I plan to keep:

• Bowman Dickson's method of introducing inflection points through the spread of an epidemic (inflection via infection)!  This worked especially well for my Business Calc students who are not quite as enthralled with pure mathematics as...some.
• Making students find the general antiderivatives for functions and then having them create a chart on the board.  Once a student found an antiderivative, s/he would pass the marker on to another student to find the next antiderivative on the chart.  This was inspired by Prof. Burdette from an article by Faculty Focus.
• Derivative cards.

What I need to change:

• MUCH better review of composite functions before we get too deep into calculus.  The Chain Rule and u-substitutions are just too hard to teach if we can't recognize composite functions as such.
• Critical points.  What are they and why are they important?  I didn't a good enough job emphasizing how important the locations are where the derivative is zero or undefined.  I assumed the geometric representation of critical values would be clear.  Never assume.
• More emphasis on the fact that integration is related to antidifferentiation.  Specifically, Leibniz's elongated s combined with a differential (dx, for example) is the COMMAND to antidifferentiate.  Once you do the command, the elongated s and dx disappear.  Notation, notation, notation.  It's important!  [What worked in private tutoring:  the integral sign is like the capital letter and the differential is like the period at the end of a sentence.  You need both; and combined they tell you to find the antiderivative.]

What I plan to focus on more with my algebra and pre-calc students to get them ready for calculus:

• Compositions.  I want to start early with this idea.  For example, what does √ mean?  Well, nothing, unless it has an argument.  Similarly, sin, ln, and ( )^2 are all meaningless without an input. 
• Exponents, baby.  My Calc I students seem to be fine on this, but my Business Calc students were way behind (in general).  I need students to know that  can be rewritten as x^(-1/4).  And I need them to be able to recognize this quickly.  Furthermore,  as crazy as this sounds, I need them to be able to add and subtract fractions so we can milk the Power Rule for all it's worth.  Please, for the love of all that is good and holy, don't pull out your calculator to figure out what (1/2)-1 is.  Please.
• I want to pound in the idea that the slope of a horizontal line is zero.  
• How do we find the change in a quantity? (Subtraction.)  Seems simple, but it's something I need to emphasize more.
• Difference quotients.  Less emphasis on computing a bunch of them, more emphasis on how it's just slope.  I don't want my students to memorize a formula, I want the formula to flow out of an understanding for how to mathematically write "How fast does y change with respect to x?"

More thoughts on the Chain Rule

I posted about how teaching the Chain Rule was a lot harder than I thought it would be.  I'm still convinced this is at least partially due my students' lack of understanding/recognizing a composite function.  For example, just the other day we needed to simplify the expression



And a student (one of my top students, I might add) suggested we "divide out an ln."

Hold up.

Let's ignore the fact that dividing by any number other than 1 would change the expression.

Dividing by ln?  So...somehow there's not a connection that ln is meaningless without an argument.  "Dividing by ln" is akin to "dividing by √ " or "dividing by cos."  An empty square root or an empty cosine doesn't have any kind of value, and really doesn't mean a thing.

I was further disturbed when I gave my Business Calculus students a function like



and was told that in order to find the derivative, we should use the Product Rule.

Wha...?

Do students view ln as some sort of constant?  Like e?  It seems maybe so if the function above is thought of as a product and if we can indeed "divide out ln."

I decided we needed to revisit the Chain Rule.

I started by showing a slide that had a composite function at the top and four expressions beneath it such as:

I asked my students to tell me why we needed to use the Chain Rule and then to identify the derivative of the outside function (holding the inside) and the derivative of the inside function.  The next slide highlighted the former in red and the latter in blue:

We did several of these.  Then we concluded with other types of functions to test if they knew when to use what rules.

I think I will start with this type of presentation the next time I teach the Chain Rule.  Giving the students a limited amount of options to start out with seemed to worked fairly nicely.

There are still definitely some issues.  But I now have a better understanding of what needs to be emphasized  in terms of composite functions.  I will try to make them more densely populated in my algebra and pre-calculus classes from now on.

If you're interested, here's the slideshow we worked through:

Test 3 Review

The Chain Rule was harder than I thought

Last week I taught the Chain Rule to both my Calculus I and my Business Calc class.  Let me rephrase that.  Last week I was supposed to teach the Chain Rule to both my classes.  I'm pretty sure I didn't quite get there.  At least not yet.  I was really looking forward to teaching it, because it shows up everywhere.  Also, I never remember thinking the Chain Rule was a particularly hard concept.  But maybe I'm romanticizing my beginning calculus experience.

This is my first go at teaching either class, so I'm sticking pretty close to what the books say.  I figure the authors are the experts on both the subject and the audience, so it's a good starting point.  Both books teach the Chain Rule quite differently, so I was excited to try both and compare and contrast.

Unfortunately, one thing I found across the board was that many of my students don't have a firm grasp on composition functions. Sure, they can compute fog, but ask them to go the other way--to decompose a function--and all of a sudden at least half of them look at you like you've asked them to please go swim across the Atlantic Ocean.  It was a frustrating moment as a teacher because I couldn't find a way to explain decompositions without using the typical vague words like "inside" and "outside" functions. I tried saying that the "inside" function is what has parenthesis around it, or the expression you could put parenthesis around without changing anything.  Yeah...that works for functions like

f(x)=sin(3x−5)

And

f(x)=4x2+2−−−−−−√

But when we got to

f(x)=e5x−1

they told me the inside function was e.

Not e to some power. Just e.

FAIL.

On me, not my students.

Note to self: learn how to teach the decomposition of functions.

The Chain Rule via Leibniz notation did go a bit better. We talked about how if a company that produces video games wants to know how much it is making per minute, it could take how much it makes per game sold and multiply that by how many games it sells per minute:

Similarly, if y=f(u) changes 1/2 as fast as u, and u=g(x) changes 3 times as fast as x, then we can conclude that y changes 1/2 times 3, or 1.5, times as fast as x:

y=(3x2−4)2

Or...

Another thing I tried that I stole from the Business Calc book was beginning with a "guess" for the derivative of a function such as

f(x)=(x3+5)2

For the "guess" for f'(x) we applied the Power Rule to the "inside" function and got

f′(x)=2(x3+5)

Then we found the actual derivative for f(x) by expanding it and using the Sum Rule.  We found that the derivative was the same as our guess but multiplied by 3x^2!

I thought this would be a great "AH-HA!" moment.  Alas.  It was not.

I just got a bunch of, "So, what was the guess for?"  "What's the final answer?"  "How would you enter that into MyMathLab?"

Sigh.

I think the "guess" thing really could have been powerful.  I just need to ponder how to present it better.

So, that's the Chain Rule.  Some things worked.  Some didn't.  Most didn't.  But this is one of the amazing parts of teaching mathematics--learning how others learn math.

y′=2(3x2−4)