Good Math Problems

The Origin of Life on Earth and Logarithms

Today in Algebra II I went off the beaten track. I wanted to make logarithms useful to them. Yeah, I could talk about the Richter Scale, or pH scale, or decibels, but when it comes down to it, logarithms really only become intuitive, natural, and beautiful once you reach calculus. Plus, these examples seem like such cop outs. If the are only good for weird measuring systems, then they aren’t really worth teaching in math class (not that I would be opposed in, say, a chemistry or physics class).

When I was in North Carolina for a math teacher conference, I went to a talk on logarithms, and the speaker reminded me that one great use of logarithms is for displaying data (either on a log scale, or a semi-log plot).

So today, I talked about my students being science journalists and representing data (confession: I cribbed this idea from the NC conference too): namely, I wanted them to create a timeline of major events in the evolution of life, from the existence of prokaryotes (3,000,000,000 years ago) to the emergence of homo sapiens, to the advent of writing (6,000 years ago).

Plotting all the relevant moments on a standard timeline yields a major problem: the events that happened closer to now (e.g. emergence of homo sapiens, taming of fire, writing) all overlap on the timeline — because the scale (of now to 3,000,000,000) is so large. The difference between something happening 6,000 years ago and 15,000 years ago on a scale this large is negligible.

So I taught them how to plot on a logarithmic scale: the events all become spread out, but you lose the ease of pulling off the data immediately from the graph. It’s harder to interpret the data, but it all becomes visible.

I think they learned something from this activity. If I had planned it better, I would have asked them to each find their own set of data to plot on a log scale.

Paper folding and exponential functions

I am teaching exponential functions in my Algebra II classes this week. And I just came back from this teaching conference, where one of the sessions included a few handouts of the types of problems that this one charter school uses. And lucky for me, one was on exponential growth and decay.

I wholesale retyped this activity-based lesson up and gave it to my students. I can’t say it was the “most awesome thing ever,” but I can say that it got students to think for themselves instead of being spoon fed everything. What it had students do is to:

Fold a piece of paper in half and record: (1) the number of folds made, (2) the number of regions the paper is divided into, and (3) the fractional area of each region. Then fold the paper again and record those numbers again. (So after 1 fold, there are 2 regions, each with fractional area 1/2; after 2 folds, there are 4 regions, each with fractional area 1/4; …).

What students discovered was exponential growth and decay. What was interesting i&s that when I had them try to come up with a function relating fold number to the number of regions (y=2^x), many of them couldn’t do it. They would try thinks like y=2x, or y=x^2, but it wasn’t until I reminded them that the number of regions (2, 4, 8, 16, 32, …) could be re-written as (2^1, 2^2, 2^3, 2^4, 2^5, …) that the majority of them could figure it out.

In any case, it took a good 20 – 30 minutes for them to finish the activity (which included some plotting, and some discussion of independent and dependent variables), but overall, I’d like to think they got more out of it than me simply explaining in words what an exponential function is.

Not that I have the time to come up with a bunch more of these, nor the classtime to implement them, but I think having one or two per chapter up my sleeve would be perfect.

"Someone told me…"

I told my calculus class, in the last 15 minutes of class on Monday, that:

“Somebody told me something, and I don’t know if it’s true. They said that if there’s a cubic equation that hits the x-axis three times, then there’s a point of inflection, and it will be the average of these three x-intercepts. I don’t know if I believe them. It sounds plausible, but I’m skeptical. Because why would the zeros be related to the inflection point? And why would they be related to it in such an elegant way? Crazy talk, I said.”

Of course, they called me on it, saying that of course I knew if it was true or not. So I chuckled, and said that they got me, and of course I know if it’s true or not. But I wanted them to figure it out. So I asked them to figure out what the problem is, and guess how we would solve it.

By the end of the class, we together (but letting them lead the discussion) determined that our general equation for the cubic would be f(x)=k(x-a)(x-b)(x-c), where the x-intercepts are at a, b, and c.

Then for homework I let them loose on figuring out if what I was told was true.

The next day (today), I asked them how far they got. One person solved it, and a few had the right idea, but got frustrated with the algebra. No one “checked” to see if the inflection point truly was an inflection point (if at that point, the function switched curvature from being concave up to concave down or vice versa). But going over the solution together was awesome because:

1. I got to reinforce that a, b, and c in this equation are constants, not variables (a few were confused about that)
2. I got to show them a quick way to “foil” out (x-a)(x-b)(x-c)
3. I got to remind them how to prove something is an inflection point
4. I got to show them what a formal proof looks like

and most importantly,

I got a few of them to see how cool it was. I basically told them why I loved problems like this… because even though the algebra can get hairy, even though you might make a wrong turn somewhere along the line, we were able to show something that is totally not intuitive. To use the words I used in class, that “the payoff is worth so much more than the work.” And even though only one person solved it on their own, I think a few of my students felt that ownership as we solved it together in class.

In theory and practice it was 30 minutes of class well spent. I should do more of these sorts of problems. Hard things we do together in class.

Math Club

I’m co-advising MathClub with another teacher. And two weeks ago, I presented the students with the idea of continued fractions and continued exponents. After a brief introduction, I gave them a really tough problem. Honestly, I didn’t expect them to solve it the same day, but they did.

One of the problems I used to motivate the more difficult problem we worked on was:

Find x if 2=x^x^x^x^x^x^… (ad infinitum). [problem 1]

Not getting into details, it turns out that x is the square root of 2.

BUT one student, a few days later, noticed that if you say:

find x if 4=x^x^x^x^x^x^… (ad infinitum) [problem 2]

and do the exact same analysis as you do for problem 1, you find that x also equals the square root of 2. Anyone spot the contradiction? (You basically are saying 2=4.) It turns out that problem 1 has a solution, but problem 2 doesn’t.

So it turns out that if you say k=x^x^x^x^x^x^x^… (ad infinitum), there are only certain values of k which have a solution for x. And this student decided this was something he wanted to work on.

So I let him loose… I thought the right way to think about this problem is though sequences, but this student decided to work graphically. And his method is ingenious, and he got the answer.

So next MathClub he’s presenting it.

And that’s amazing. Because he came up with the problem, and he solved it!

EMERGENCY! (lesson plans)

I spent some of today (say, maybe 5 hours) working on finishing up my high school first day of class presentations (by the way, I found this amazing first-day-get-to-know-you sheet on dy/dan which I’m definitely stealing), as well as writing up emergency lesson plans for all my classes. These plans are what the school has on file for the substitute in case I am ever (gasp! God forbid!) out sick.

For my middle school class, I am going to have them play “24” if I’m absent. (Unfortunately, I don’t own the game, so I am going to have to construct my own game cards this weekend.) It seems perfect because it has to be related to what they’re studying, and yet not be tied to the curriculum, because I could be sick the 2st week or the 30th week. And 24 tests something that gets retaught in the first week (order of operations — PEDMAS) so it can work if I’m sick early on, but still is fun if I’m sick later in the year.

Instead of giving my high school students busywork worksheets, I decided to give them genuine math-dork-approved-stamped math puzzles. When looking for a few accessible, non-stupid, non-IMO level puzzle sites, I came across this gem, from which I stole all my puzzles from in one fell swoop. Who’d’ve thunk thank I wouldn’t have to steal bits and pieces from a thousand different sites? This was a great find.

I am pretty confident that the puzzles I chose for the two classes (Algebra IIand Calculus — click to see lesson plans) are age and level appropriate. But maybe not? I used to be a huge math puzzle freak [1], so when it comes to puzzles, I know I have a distorted sense of “easy” and “hard.”

The reason I really like them is because the solutions to these problems can each lead to a wonderful extended discussion of “proofs,” “combinatorics,” and “graph theory” among others. Plus, I think these are the types of problems that kids can really work together on. I’m slightly afraid that the inability to get a solution in a minute or less (how long most students take per homework problem) will lead to great frustration.

Maybe I should give the substitute a hint for each problem, for when the kids get stuck?

(As an aside: I really want my kids — especially my calculus kids — to leave class knowing what a “proof” is and why it’s important. Yeah, I will introduce the epsilon-delta proofs, and I’ll derive some things for them, but I want them to know what makes a proof watertight — not just accept a proof because “the teacher said so.” And the best way for them to know is to do.)

[1] Not only did I attend Mathcamp (more than once), but I also completely of my own volition found and started writing solutions to the USAMTS competition problems when I was in high school.