Two Book Sections in 40 Minutes

Obligatory apology for rarity in posting goes here.  The end of the soccer season finally allowed me to breath… for a day.  Basketball workouts started almost immediately and I am still juggling 4 preps.  The juggling has eased up a bit – my 8th grade computers class is a quarter-long class that has started the second rotation.  That’s lightened my prep load, though just a bit.

For our standard “Algebra II” class, we use Discovering Advanced Algebra by Murdoch, Kamischke and Kamischke.  I love this book.  After a year of getting used to it and finding all the subtleties in the way the material is presented, I feel like I’m making almost all of those little connections.

The fourth chapter is on functions, transformations and the basic families of functions.  The first section of the chapter is basically this:

4.1 Interpreting Graphs
– Graphing a story
– Relationship between an independent and dependent variable
– Identifying features of graphs by describing the above relationship

Instead of writing a bunch of vocab words on the board and doing “math book” problems (after 8 seconds, how high is the balloon?), I simply showed about 5 graphing stories:

Just a picture... Click to head over to the amazing

Just a picture… Click to head over to the amazing

That was it.  That was the entire section.  I then spent 20 minutes discussing function notation and 5 minutes doing if “f(x) = 2x-4, then f(3) = ?” skill practice.

Bam.  We delved deep into the major concept of a function and truly examined the relationship between an independent and dependent variable all in one class period.

Two closing thoughts:

1.  This is one of the amazing results of Dan Meyer’s decision to take the “Graph a Story” textbook problem and put it on video.  (Though the overlaying of the graph on the video is crucial as well).

2. What other concepts do textbooks take days to develop using the “old fashioned” ways that could be examined deeper and quicker through other media?


The Walking Dead – Exponentially

The little things that make teaching math fun…

I also think this is a decent example of forced context vs. fun context. If you’re dead-set on putting math “in a context” that students understand, you get the forced contexts that fill our textbooks. But if you open your eyes to the world around you and get a little creative, then that context isn’t as forced.

Cells reproducing in biology or zombies and America’s hottest show? You tell me what is more engaging.

PS: Now, if there were a nice video prompt for this instead of simply a screen-grab from The Walking Dead, then that would be the cherry on the top.

The Reflection: Recursion with Philip J. Fry

Previously: The Plan, Back to School Night

In order to succeed, your desire for success should be greater than your fear of failure.

Bill Cosby

To say I had a desire to succeed with this lesson would be an understatement. I showed the Futurama clip to the parents and got all geeked up about how it would enlighten their children to the power of mathematics!!!  In the end, it wasn’t the lesson that fell flat, but the sequencing of several lessons. That’s exactly why I’m blogging – so this gets recorded and I (perhaps some others as well) can make that enlightening lesson, you know, enlighten.

I slotted this as the second day of our Recursion unit. Why? Because that’s exactly where the book had the question: If Lucy puts $2000 in a savings account that accrues 7% interest yearly, how much money will the account have after one year?”  It is the Futurama question. So, it seemed like it would be the perfect place for it.  The prior day we had discussed what recursive definitions were, how to set up tables for sequences, the difference between an arithmetic sequence and a geometric sequence. However, when I wanted to turn to this lesson, we needed more work with these newfangled recursive formulas, what with their subscripts and all.

After spending the needed time to review those finer points, I went to the Fry lesson, but with just 20ish minutes left in class. The kids liked the clip, of course, but I had provided too much help: they were uninterested in finding the result because they knew they would have to push the button 1000 times. In each class I had one person willing to crank out that work. In one class I had someone trying to adjust the problem to an explicit formula.

In the end, I decided I did not properly set up the problem and gave up actually expecting them to find the answer. I showed them a picture of when I pushed the button 1000 times, talked about exponential growth a bit (without actually referring to exponentiation) and how the more money you invest initially, the faster your account can grow.

Next Year

I think the right way to do this lesson is to not provide any kind of structure for the kids. Show it before we start recursive formulas, do a better job of establishing group norms, and be willing to sacrifice a day to the lesson. If I let my students try to find an answer on their own – without pushing the recursive method – maybe (probably!) they come up with some interesting methods. Maybe they are all completely wrong. And that would be great!  Two days later we revisit Fry and find the correct answer. We can still have the discussion of why recursion can by bad and how it would be great to have a faster way to do this.

While it was a little deflating to be so excited for a lesson only to watch it flounder, I feel like my desire to succeed outweighed my fear of failure. Now the question is, next year, will my fear of failure while sacrificing a day stop me from conducting what could still be an excellent introduction to recursion. Of course it won’t, but it will up the ante that much.

The Plan: Recursion with Philip J. Fry


Previously, students will have been introduced to recursively defined functions.  I’ve thought a lot about the value of introducing recursive functions at the beginning of algebra II, which can be a discussion all by itself.  However, the “Fry Bank Account” problem provides an excellent opportunity for recursive growth functions and the costs of recursively defining functions.

The question is simple: It’s the year 3000 and Fry needs some cash.  Luckily, he left $0.93 in a savings account back in the year 2000.  If there’s 2.25% interest, how much is Fry’s account worth?

The beauty of how this should play out is that my students won’t have the wealth of exponential knowledge to quickly fall on.  I suspect they are unlikely to make the jump right to: Balance = 0.93*(1.0225)^1000.  Instead, they’ll have to define it recursively, which leads to some beneficial complications:

b1 = 0.93

bn = 1.0225*bn-1

This should lead to an interesting and meaningful version of Dan Meyer‘s envisioned plan of the clip found by Timon Piccini.


The opener will give a quick example problem in which students have to recursively define a formula for a geometric sequence.  In preparation for Futurama clip, I’ll also ask how much a $59.99 pair of jeans costs if there is 6% sales tax.  Here’s a good as any place to note that the the Futurama problem assumes interest compounded annually.  This is really not a major concern lesson, but I will be interested to see what kind of questions are asked about how and when the interest rate is applied.  I suspect few will see this at first and use the sales tax question as a model to find Fry’s account balance for year 1, year 2, year 3 and so on.

The Main Act

Students will be in groups of three when I play the clip.  Before I turn them loose, I’ll steal Dan Meyer’s lead-in questions and ask what types of numbers we think are way too high.  What about too low?  Otherwise, the only directive I plan to give is “Use what you’ve learned about recursion to find how much money is in Fry’s bank account.  As they discuss and come up with ideas, I’ll push when needed: “How much money does Fry have after 1 year?  10 years?”  “What is a recursive formula for this instance.”

What I’m really hoping to see is this:

A student feverishly pushing the “Enter” key of their calculator 1000 times.  That’s the type of action and situation that really drives mathematics: “There has to be a better way.”  Of course, there is.  Perhaps some student will write out their work algebraically.  Perhaps, then, they’ll see this common factor popping up.  However, this would be quite a leap from a concrete situation to an abstract one and I’m not expecting any student to make it.  Instead, I want to set the scene for that future abstraction.  “Remember when Mark pushed the enter button 1000 times?  Wasn’t that awful?”   Then we can get to the explicit formula and – whoa – all we need is one push of the button.  But for now, I want the focus to be on recursion; what it can do, how we can use it to define methods of problem solving, and why it is not always the best option.

The great part is that even if a student successfully hits the button 1000 times, we can continue the idea of “you’re doing it the hard way” by asking them to graph the sequence.  If time permits we can go into the “Sequence” graphing mode on the calculator and start to talk about the shape of the graph: “Does anyone remember from Algebra I what type of graph this represents?”  “What does Fry’s bank account have to do with exponential graphs?”


While the goal is modeling growth and decay using recursion, exponential functions is the powerful, underlying concept.  Thus, it’s a natural step for a student who breezes through the recursive aspect of the exercise to try to connect an exponential-looking graph to an exponential function.  Of course, it doesn’t stop there.  We could discuss annual versus monthly compounding interest.  Or, as Dan Meyer suggests, to stay on the objective at hand I could ask when Fry hits the $1,000,000,000,000 mark.  It’s clips like this that I’m constantly looking to bring into my classroom: lots of roads to take from just a short video of pop culture.