## Things I Thought I Knew

Some days this job can be a bit ho-hum. Teach a lesson you’ve taught 37 times before, grade a few tests, make some copies, go home. And then some days you get to learn math that you totally thought you knew, but you had no idea. Friday was one of those days.  I’ve been teaching long enough that I probably should have known this stuff, but I didn’t. So here I am to enlighten you as well. Just be prepared for the following to occur:

##### Topic 1: Classification of Systems

Am I the only one that is really bad at reading directions? But then the students actually read it word for word and ask questions that I didn’t see coming because knew what the question was really asking without having to closely read it. This week, we had a question about making an independent and inconsistent system, which I read as “inconsistent” and knew what the goal was, but then that pesky “independent” thing was brought up by students.

So I always thought there were three types of systems:
Independent: One Solution
Dependent: Infinite Solutions
Inconsistent:  No Solutions

TURNS OUT I was mushing two different categories together:
Category One: Is there at least one solution?
Yes: Consistent  No: Inconsistent
Category Two: Are they the same line? (Technically, “can one be formed from the other with algebraic manipulation?”)
Yes: Dependent No: Independent

So the correct categorization would be:
Independent and Consistent: One Solution
Dependent and Consistent: Infinite Solutions
Independent and Inconsistent: No Solutions

When you realize you’ve been teaching something wrong for your entire teaching career:

###### Topic 2: Logs grow really really slowly.

“Yes, Meg, we know that.” No, I don’t think you do. The wonderful MTBoS Search Engine led me to this GREAT activity from @Johnberray. I modified it a bit, first of all by making some pretty 1-inch graph paper:

(file here) Each group got a sheet in a dry erase pocket and used their calculator to find the decimal values to graph y = log x. Then I asked them to calculate how many inches it would take for it to function to reach 2, 4, 6, and 8 inches high. (At least one group each period said 20, 40, 60, 80 so we may not have the idea of logs down pat just yet). Then I asked them to describe each distance in some way–is it the size of a pencil? classroom? from here to the front office? as tall as basketball goal?  I let them use the internet and a student showed me that if you put a measurement into Wolfram Alpha, it will give you real world comparisons! Pretty neat!

You really need to try it for yourself!  What fun we had when we started talking about how long the paper would be for a height 6, 7, and 8 inches (oh, plus go ahead and try 9!) using this draw-a-circle-on-a-map site that John had linked to. Then it turned into, how high will it be if the paper reaches the moon? The sun? Pluto? I won’t give it away, but guys, logs grow really really really really really slowly.

Yet they still go to infinity.

###### One raised to any power is equal to one, oh, unless that power is infinity.

So I’m pretty good with limits. Then my Calculus teacher friend mentioned, hey, isn’t it weird that the limit of (1 + 1/r) ^ r is e, not 1?  Wait, that doesn’t make sense.  But it turns out 1^infinity is indeterminate. WHAT? But I thought 1 to any power is 1!  WHAT IS HAPPENING TO EVERYTHING I THOUGHT I KNEW?  The best answer I saw was that “we don’t know if the one is really a one”, which ok, I guess, since 1 + 1/r is never exactly one. But what if the number IS actually 1? Not like “kind sorta” close to one but ONE. If 1^infinity isn’t one, at what number does it STOP being equal to one?

I was discussing all this mind-blowing math with my friend right as class was starting, complete with dramatic re-enactments of how high the graph of log will be on Pluto and what was happening to my brain. When I came into class, a student asked what we were talking about and I said “Math!” and she said, “You really do like math a lot, don’t you?”

Yes, yes I do. 🙂

Category: Alg II, Precal | Tags: , ,

## The Limit of Limits

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We finished up our limits chapter and the test results were….underwhelming? There were no HORRIBLE grades (not even a lot of F’s) but there were only a few outstanding grades.  On the other hand, the majority of them did great with graphical limits and algebraic limits except for one-sided limits approaching an asymptote.  Which, in retrospect, I may take out doing algebraically next year (or at least if I teach regular precal).  I mean, we end up just making a mini-graph anyway, so why not just keep it in the graphing section?  And at least no one gave this as answer:

Dude, that may be my favorite math equation pun, right after (sin x) / n = 6.

Also, the ant analogy (see my last limits post) worked a bit too well. At least 10 students “explained” that the limit was equal to the y-value of a hole “because the ants can still reach across” or “the ants will still be at the same place.” So if you’re an AP reader next year and see a lot of talk about ants, you’ll know why.

Here’s what the last half of the chapter looked like:

1) I redid the piecewise function worksheet to have more exciting piecewise functions.  This took them the better part of a 47-minute period. File here.

I have to admit I feel like I must not be doing my best as a math teacher when at least three students ask me “since 2x +10 doesn’t fit on the graph, can I graph x + 5 instead?”

2) Then we jumped into finding limits algebraically, with a chart that I think is fabulous (but I may be biased).  File here, and some homework just cause I like you.

(also, check out that amazing multiply-by-the-common-small-denominator action happening in #9).

3) I’m continually amazed at the fact that no matter how often I told them you need to SHOW me algebraically all three rules when I ask about continuity, I still had maybe 5 that were like, “yep, looks good to me; you can draw it without picking up your pencil!”  Here’s the file and some homework. Yes, the last two on the homework end up being beginning-of-the-next-class-period discussion questions.  Also, cut down the first part of the homework by half.

I also enjoy it when students put smiley or sad faces in the middle of problem because I do it.

4) To Infinity….and BEYOND! (Aw, sorry you got cut off on my scan, Buzz). . I think highlighting the biggest power really helped. This is also the first year we “plugged in” infinity and I think that helped, too. And trust me, just omit #20. Save that sort of heartache for Calculus.

(file here)They finished the rest for homework.  Be sure to talk about #43-45 the next day–how .02 difference affects the limit at infinity!  Also, this was funny: a student was asking about #40 (37^(1/x)) on the way out of class. So I was writing on the board 1/∞ then she said that was 0, so I wrote 37^0.  A kid from my next class walked in and said, so 1/∞ = 37 degrees? 🙂  I told them we should totally start using 37 degrees as our fallback answer for any question.

That was the last lesson before a couple of days of review.  And just to make sure I’ve covered all limit jokes and puns:

Done.

Category: Precal | Tags: , , ,

## Take it to the Limit!

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Yes, it’s limits time in Precal.  Again, stuff is so hard to find so I thought I’d share mine, even though most of it is cobbled together from other stuff I’ve found (and have no reference for…sorry.  If it’s yours let me know and I’ll give you credit!).

We spent a couple days with rationals and then jumped into one-sided limits. I used to start with regular and then do one-sided (I think that is what my first precal textbook did, but it seems to flow so much better doing one-sided first).

Here’s my introductory powerpoint WITH MOVING ANTS!!

After this, we stopped and talked about how 99% of the time in the real world, the ants coming from the left and right are just going to be the function value. I asked where they think there could be problems.  Then we worked on the corresponding NoteTakerMaker (with homework! and typos corrected!):

Tomorrow another introductory powerpoint to real limits with graphs and tables and piecewise functions (Oh my!) (and with more ants!):

And its corresponding NTMs and practice worksheet.

(I actually think I may redo the practice to include more exciting functions than just linear and quadratic. We’ll see how tomorrow shakes out.)

And that’s my limit on what I have to report on limits!

Category: Precal | Tags: