Three Acts: Basketbola Preview

Previously: Basketbola

This is probably the biggest lesson/activity I’ve put together this year. Thanks to today’s snow/ice day, what was going to be spread out over 3 days will now have to be done in two. I think it’ll be doable.

First, I’ve made it a contest. There are nine total videos of people shooting a basketball. Students will watch Act 1 for each shooter and make their prediction. Does it go in, or not? Correct predictions are worth 3 points. During Act Two they’ll have the chance to change their prediction, however it will cost them 1 point.  If they are wrong, they receive no points.

screenshot1

Act One

I’ll go through showing the nine Act Ones, as seen below. Students will make their guess. Very low entry bar. Anyone can make a guess.

Act Two

Then we’ll go into Geometer’s Sketchpad and start modeling parabolas. Here is where the competition gets interesting. Once students model their equation, they may decide their original guess was off. That’s fine, but it’ll cost them a point if they want to change. That should provide a little incentive to be as precise as possible in their modeling.

screenshot2This screenshot is in Geogebra, but my students will be working with Sketchpad, which should allow things to be a little more intuitive. I’ll set them up with a standard parabola in vertex form: y = -0.5(x – 2)^2 + 2.  That should be plenty to get them going and graphing.

As they progress through the shooters, they get more difficult. One picture will have a random three balls in the picture. One picture has three balls very close to each other, right after they leave the shooter’s hand. One shooter hits a bank shot which should really throw people off.  I’ll have them record their equations and record whether or not they want to keep their prediction, just to keep them honest.

screenshot3

Act Three

The next day we’ll come back and reveal the results.

It’ll be interesting and fun to see how much they get into it, especially when the bank shot throws the entire class off.  We’ll take a few minutes to discuss the investigation questions:

  1. Why is the value of a negative? What happens if it is positive?
  2. Describe what happened when you changed a. What happened when you made it smaller? Bigger?
  3. Describe how h and k changed. What does the point (h, k) represent? What does the line x = h represent?

As for the contest? Whoever wins gets to add their video to the library for next year.

Sequels

There’s so many other things to do with this! I could use any other form of a quadratic as the focus of the lesson (or all of them).  The sequel I like the best is this:

screenshot4In fact, this is what I hope we get to if the students are able to model the first nine shots quick enough. Had I still had 3 days instead of two, I’m sure we could have. This is a pretty obvious way to show, graphically, that having only two points is not sufficient to graph a quadratic function.

You could also talk about other curves instead of parabolas, too. I’ll be very interested to see what other questions come up in class.

UPDATE

Check out the reaction post.  In short, my students loved it:

The Plan: Recursion with Philip J. Fry

Overview

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.

Opener

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?”

Contingencies

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.

The Right Way to do Smart Boards

As the school year winds down and I give this blogging thing a go, I figured a nice way for me to reflect on the year and hopefully share some things would be to recap some highs and lows.

I’ll be the first to say I haven’t taken full advantage of the SmartBoard in my class.  Basically, I use it as a projector.  I run through slides, pull up interesting websites or pictures, play Vi Hart videos, and occasionally it is used as a whiteboard.  But nothing really “Smart.”  Until this:

A simple download of Angry Birds to my desktop suddenly changed an electronic whiteboard into an interactive exploration of parametric equations.  In my Pre-Calculus text book, parametric equations is a one-section topic, thrown in with vectors but not with parabolic motion (which is studied 4 chapters earlier) and before polar equations.  It’s a little odd and  in order to properly teach it, I thought I needed a solid 3-4 days to do it.  Instead, some sacrifices were made and we added a couple of days to really dig deep into what was going on in Angry Birds, including the creation of our own Angry Birds course. We even measured initial vectors of slingshot-ed birds until the primary concern became “can anyone hit the target?!”

A refresher of modeling one-dimensional motion using one of Dan Meyer’s “Falling Rocks” 3-Act lessons was our start, followed by a discussion of Angry Birds, initial vectors, and some trigonometry and so on.  Suddenly, we had a reason for wanting to know a vector’s angle from the x-axis: will the bird hit the piece of wood I think it will?

I’ve found the beauty of the Smart Board is that kids, for one reason or another, love playing with it.  But they love drawing on my old-fashion white board too.  So, how can I really take this expensive piece of equipment and make it more than a combo projector-white board?  I think Angry Birds is a good start.

Next Year

  • Angry Birds in class, for a whole class period, not just the 5 minutes in between class and the 5 minutes at the start.  Let’s make teams.  Who’s the best player at Angry Birds? Why is she the best player? Why can’t you beat her? How can you beat her?
  • Do I save Angry Birds for when we discuss parametric equations or get right to it when we start vectors?
  • How can I turn a 5 day mini-unit into a 3 day mini-unit?  Or, how do I make this 5 day mini-unit more mathematically rigorous?  I’m a little worried the time taken away from the math to play the game, build a real-life level was too much taken away from practice in class (based on concept tests and the final exam after).
  • How can I fix this guided worksheet on one-dimensional motion?  Does it need fixing? Do I need it at all:  Intro to Parametrics
  • If I’m not going to get to Polar Equations anyway (like this year), can I play with a more thorough connection of trigonometry to parabolic motion prior to getting to Angry Birds?

Next on tap: The failure of my texting in class activity, and why I need to try it again.