# [Productive Struggle] Precalculus Logarithms

I’m trying to keep a positive spin on my Precal lesson from Wednesday (a few weeks ago, now), but it really, really flopped.  To be fair, it wasn’t entirely my fault: I had lesson plans using the internet, and the internet was totally out.  So it was only mostly my fault because I didn’t have backup plans

So I decided to try a 3-acts on the spot.  Note to self: do not try this unless you’ve done the specific 3-acts before and you’re very, very experienced at 3-acts.  I have been doing about a 3-act lesson every week or every other week or so, but this little amount of “experience” does not make up for the lack of planning and preparation.

I saw this chart shooting around the Twitterverse:

TECHNOLOGY Price of 1gb of storage over time:
1981 \$300000
1987 \$50000
1990 \$10000
1994 \$1000
1997 \$100
2000 \$10
2004 \$1
2012 \$0.10

After asking “What questions do you have?”, and discussing “how much a GB is”, they got to work plotting this on a whiteboard.

My first goal for students was to graph this and quickly realize that a “normal” scale wouldn’t work here because over half the points just sit on the x-axis, not really telling you anything.  A few creative students decided to make the “squiggles” and represent a significant change on the scale of the y-axis, but these students did not realize that (a) you really shouldn’t do that between data points and (b) you really, really shouldn’t do that multiple times on the same scale.  So they saw the need for a logarithmic scale, but even after they graphed the points on Desmos, they had no way of making the data scale that way.  Mistake #1.

The next mistake that I made was thinking that “because the data merits a logarithmic scale, then the best-fitting function must be logarithmic”.  I didn’t tell students to choose a specific function, but I hinted that since we had been working with “logarithmic functions recently, it’s probably a good place to look”.  I need to get better at Dan Meyer’s slogan of “be less helpful”.  I might as well have required them to use logarithmic functions with that kind of hint.  Even 2 minutes of playing around with the data before class and I would have realized that it is definitely not a logarithmic function.  Instead the students struggled for a good 10-15 minutes before I realized what was going on.  Mistake #2 (at least).

So I decided to give students a “break” while I regrouped and gathered together my thoughts.  Since my school has regular classes on Monday, Tuesday, and Friday, and block periods on Wednesday and Thursday, most teachers give students a break partway through the long periods for students to use the bathroom, get water, and just regroup mentally.  Until this class, I hadn’t given my Precal students a break because they’d been busy with the 3 Acts lesson we were doing.  However, my own fumbles demanded a break.

When the students got back, I explained to them my mistake and pointed out what kind of function they should have been looking for.  They jumped back into groups and started working on Desmos to find an exponential function that fits the data.  Once groups started getting an appropriate equation, I asked them more probing questions about the domain, range, and other specific questions (“How much will a GB cost in 2020?”).  However, I didn’t have one specific goal for all the groups to come back together and discuss, so I lost their focus unless I was standing over their group shooting questions at them.  Mistake # too-many-I-lost-count.

There are tons of other mistakes with this lesson that I’d like to point out:

• I didn’t have a good “hook”, or even a good idea where to take the students after they got their graph.  If I had spent some planning time before to come up with that, then it would have vastly improved their experience.
• I didn’t have any good ideas for how to view the data-that-should-be-on-a-logarithmic-scale.  I’ve never learned how to put data onto a logarithmic scale accurately, so I wouldn’t feel comfortable showing students how to do it.
• A “hook” isn’t just a bunch of questions, but you do need questions before you get a good hook, and I had neither.  I didn’t record the students’ questions beforehand, like I almost always do, and therefore I certainly didn’t come back to them at the end of the lessons, which I also almost always try to do.  In short, I just killed some of my students’ trust in asking me questions in the future.
• I didn’t have any sequels ready for students.
• The information, by itself, wasn’t particularly compelling.  I can imagine making a slide-show of the cost and amount of data on a slide with a picture of an object with that storage capacity.  To actually see it go from several buildings down to the size of less than a thumbnail would leave an impression and provide some other sequel questions.  Missed opportunities.

There were a few positives: students felt the need to create and use a logarithmic scale (however fleeting that feeling was), students practiced fitting an exponential curve to data (they’re getting quite good at fitting all kinds of functions lately), and they learned what a GB is (super-important in today’s world, in my opinion).  However, it felt worse than a wasted class period–it felt like a wasted block period.  Even though this is my 4th year teaching, I’ve got to have back-up plans for technology failures and I’ve got to get better at putting time into these kinds of tricky lessons.