The Importance of Change

I was at a car parts store the other day, and I saw a sun shade for your windshield that advertised this:

I had to double-take.

A change of 44F is the same as a change of 7C?  Now, I'm not as savvy at converting from Celsius to Fahrenheit as Kate Nowak, but even I knew this was some faulty converting by a marketing department.

No.  I take that back.  I admit, they did the conversion fairly well.  44F is indeed (approximately) equal to 7C.  But a change of 44F and a change of 7C?  That's a little different.

Couldn't they just have tried an example?  One example.  That's all I ask.  Like, take two numbers whose change is 7, say 0C and 7C.  Covert those to Fahrenheit (32F and 45F, respectively), and you'd see right away--that's not a 44 degree change.

Check for the reasonableness of your answer, as I tell my students.

So, what is a change of 7C equal to in Fahrenheit?  Let's do a little algebra!

We know to get from Celsius to Fahrenheit we can use:  

So, let's take two temperatures, in Celsius and call them C1 and C2.  Then their change, in Fahrenheit, can be calculated through the following:

In other words, the change in Fahrenheit, is equal to 9/5 times the change in Celsius.[1]  Which means that a change of 7C is only equal to a change of 12.6F, a far cry from 44F, in my opinion.  And this verifies the consistency of our example that we tried earlier.  Which is good since this is a linear function we're talking about.  No change in slope here.

So, which is it?  Does this sun shade keep our vehicles up to 7C (12.6F) cooler, or 44F cooler?  Maybe we just get to choose.


[1]  This can be changed from an algebra problem to a calculus problem rather nicely.  I think this would be a lovely introduction to using differentials as approximations for actual change (or in this case, since the function is linear, the differential will be equal to the true  change).

Step 1:  Give the kids the function for F with respect C.
Step 2:  Have them calculate dF when dC=7. [dF=F'*dC=12.6]
Step 3:  Show them the  picture at the top.
Step 4:  Watch them pee their pants (they're math nerds too, right?).

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.

Math History: We're all learning here

Like every other math teacher I know, I love me some math history.  Mathematicians are just some of the craziest people to read about and study, I'm sure of it.  I'm also sure that putting historical context into what you're studying helps you remember it more.

In Pre-Calc this year we've had a new mathematician for each unit we've covered.  The mathematician correlates to what we're studying (sometimes I do a better of job of picking the mathematician than other times).  If students write a paragraph about said mathematician and then write a fact on the white board, they get bonus points for that unit's test.  The catch is that they have to write something new on the board.  So, the longer they wait to do the assignment, the harder it gets, because the dude's birth and death dates go pretty quick.

One unexpected outcome of this has actually been my Algebra II classes' response to this board.  They'll see something they think is interesting and it starts a conservation about our spotlight mathematician.  [Read: kids are excited about math!]  I've also heard them say things like, "Mrs. Peterson!  Wasn't that a super smart answer?! Maybe my picture and name should be on that white board!"  To which I usually reply something snarky.  But, the point is that they're finding mathematicians cool, and that makes my heart happy.

Here are the mathematicians we've done so far, in chronological order.  I think the kids are getting the hang of it.

Archimedes

Pythagoras

de Moivre

I still have a long ways to go in incorporating math history; there's much more that can be done here.  But, I'm learning.  And so are the kids.

Dear Mom, I know something you don't know

This is something I did in the very beginning of the year that I meant to blog about and just never did.

At our school, students are introduced to the graphing calculator in Algebra II, which means I've gotten to show kids lots of awesome techniques to problem-solving that they haven't been exposed to yet.

One of our first tasks of the year was solving systems of linear equations by graphing, specifically using the graphing calculator (TI-84).  I asked the kids if they knew when the graphing calculator was invented and when it became standard use in high school mathematics courses.  I asked if their parents would know how to use such a device.

The students assured me their parents would have no idea how to use a graphing calculator, so I asked them to write their parents a letter describing how to solve a system using the TI-84.  I was somewhat surprised--every student (I have about 100 young algebraists) sat quietly and wrote.  And wrote.  And wrote.

This was my favorite:

Yo momma, it’s yo boy ________ and I’m telling you how to solve for y.  Add x on both sides then you will end up with y=x+0.  On the other problem you are going to subtract 8x on both sides and then divide by 4 on both sides.  You should end up with y=-2x-6.  Then you want to put this in your graphing calculator and press graph.  BOOM!  There’s the graph.  Then will want to press 2^nd then calc, then press enter three times and you should have your coordinates.