FOIL = FML and Using Cell Phones on Tests

FOIL = FML

So my Calculus class made a discovery the other day when they tried to reference the distributive property:  I don't like the term FOIL.

The gutteral reaction here is the same as the one that I have to the non-word "guesstimate"*.  The kid was talking and I couldn't help but interrupt with "that's not a thing." After a bit of confusion, I clarified that FOIL is a trick for one specific case of distribution that doesn't apply to anything else, logically, so why do you call it that?  He tried to counter it, but ultimately I tossed up a binomial x trinomial on the board and told him to apply FOIL to it.  It went something like this:

Kids: Okay, so multiply the first ones.

Me: Sure. (draws a line between terms and labels it as "F")

Kids: Then...uh....the middle?

Me: uh-huh... (draws a line between terms and labels it as "M")

Kids: Then last!

Me: (seeing this coming, drawing the line and labeling it "L") and there. Now you know how math feels about FOIL


They had a good laugh and got the point in a very non-PG way, but oh well.


Cell Phones on Tests

I gave my 7th graders a test a few days ago with a slight caveat: I required that they look stuff up to answer questions.  I'm pretty sure I'm going to keep doing this, and here's why:

If I'm promoting realistic thinking skills, why not include the one resource that they'll always have?

The thought is that instead of providing the exact conversion statistic necessary, or rate of whatever, why not put it in their hands to think about what it is that they need?  My question on the test was as simple as:  If a star is 3.412 x 10^7 km away, how long will it take until we know that it has burned out?

Part of taking a test has (hopefully) been knowing what information to use... so why not structure the test so that they need to find the information necessary to solve it?  If we want to promote kids wanting to figure things out outside of class, then why not structure test questions in the same light?

Obviously, this has its drawbacks.  Not every math concept has this kind of possibility attached.  There are lots of resources online to solve straight problems for them.  There are lots of places that have answered questions online that we've all asked, so this isn't totally applicable across the board.  I'm not saying that everything should be like that, but it's an interesting possibility.

I'll have to do some more analysis on the matter and run a few more experimental questions, but so far I'm pleased with the results.

Like a Bandaid. RIGHT OFF

Calculus Blurb:

So I'm about to drop my instructional experiment on Calculus. For years I've been really stuck on a few things that haven't made me happy.  Considering that I'm starting at a school where apparently the AP Calculus results had been less than impressive, I figure this is my chance to experiment.

Here's the breakdown:

1) Early transcendentals.  I know, nothing mind-blowing.  I just have felt kind of gross introducing natural log and e so late that it's well beyond when they actually would have cared. It also makes them seem like something special, which they aren't.  They're about as special as trig, but I don't want to get off on that tangent.  (Sorry)

2) u-notation from the beginning.  I've always hated how stupidly difficult substitution has seemed to kids later.  I figure if we get it flowing at the start, it would help with SO many things (namely derivatives of composite functions).   Again, not mind-blowing, but definitely a step in the right direction.


My thinking is this:  Much like Sam Shah, I have always had a weird problem with Chain Rule.  It's always been just a kooky rule explained with some analogy that has, in my mind, always been classified as a magic trick to teach the kids that robs them of something.  I hate that.  In addition, it's even more of a ridiculous moment when u-substitution rears its head.

"Here, kids, learn a thing that's supposed to make integration of a formerly composite function easier."
"This is hard."
"Hmm... Yeeees....It is."

So u-subs to start sets the stage, introduced under the heading of transformations and their effect on derivatives.  Chain rule becomes "old hat" after a while, and integrating with u's is a footnote as it's just going backwards.  It becomes incredibly simple with natural logs and e's, so there's no need to make a special introduction later.


7th Grade Blurb:

I don't understand Pre-Algebra curriculum layout.  It's the same few skills over and over again, but what drives me nuts is the insistence in so many places to teach the skills in isolation.  Why do you have to learn how to deal with exponents so long before you actually use exponents?  It made way more sense to intro scales of the universe using scientific notation, have that naturally segue to multiplying massive scales by massive scales, and BOOM exponent rules.  Not only that? Exponent rules when coupled with a coefficient.  Turn that 10 into a variable? Same rules.  Yay.

I think my real concern is that I don't feel like there's a sequence to things.  It's just algebraic reasoning explorations coupled with occasional geometry.  I have to be wrong, but I don't know where.  There is so much to be done with more rudimentary skills, like speed in processing order of operations, etc., but I just don't know what my endgame goal is.  Everything else I've taught has had a pretty clear endgame goal.  Algebra is being able to reason with unknowns and interpret various stimuli algebraically.  Algebra 2 is the same but with far more complex relationships.  Pre-Calc is about reasoning with angular movement/forces/observation data.  Calculus is interpreting instantaneous and accumulated changes from various phases of data/functions.

And 7th grade is.....maaaaaaaaaaaaath?  Blargh.

Calculus Limit Introduction

One problem I've had in years passed is kids getting a good grasp of what limits are and how to approach them. A few years back, I had a moment of clarity as a particularly confused set of seniors were staring back at me.  In that moment of frustration, I opened up Google and pulled up a map of the school.

I asked them this: If I had never been here before, how would you tell me to get to the school?  Kids jumped on this, giving me the address, the landmarks nearby and how to proceed from them, etc.  One kids went so far as to give me cardinal directions based on nearby freeways (and Houston has enough of those).

I then posed this: pretend that, unbeknownst to you, a bomb blew up underneath the school (please don't tell anyone I said that) and the whole thing flew up into the air and landed about 20 miles that way... were your directions wrong according to your best knowledge?

This sparks some debate.  Yes, because the school isn't there anymore.  No, because you didn't know it blew up.  Yes, because how could you not hear the explosion you live like a mile away.  In the end, they focus on the "to your best knowledge" part and say "No, the directions where right".  Good.

"Why?"

And how quickly they hit this phrase: "Because that's where the school was supposed to be"

From there, it's a quick jump to graphs and values and all that good stuff.

I've since refined that practice.  I start with that stuff now.  I give them a series of addresses/coordinates in the city and ask them to look them up and tell me what's there [f(c)] or what should be there[limit as x approaches c of f(x)].  Some are actually there (continuous) some are construction sites with a "coming soon" (it's a bit hard to find these on google maps unless you remember that something is a new building).  I'll also include two streets that don't connect properly.

"What's on the corner of Wilcrest and Barryknoll?"  "Which one?"  Bam, discussion of one-sided limits and limits failing to exist.   

Conceptually, this hits home pretty quickly.  Number crunching comes later, but by then it's easy since they just know what they're looking for.  The next step is to have them submit screenshots of places they find on maps and refer to them using limit notation.

Example:  Chuy's is located on 9350 Westheimer Rd.

Limit as x approaches 9350 of Westheimer equals Chuy's  (in actual limit notation, with arrow, etc.)

Westheimer(9350) = Chuy's 

Westheimer is continuous

Example:  They tore down a Starbucks on 1655 Voss Rd. Thanks, Obama! (not really, it's still there.)

Limit as x approaches 1655 of Voss Rd. equals Starbucks  (in actual limit notation, with arrow, etc.)

VossRd(1655) = Rubble 

VossRd has a removeable discontinuity at 1655

Thanks, Google.