Zombies and Math: There are no coincidences!

This is my intro to exponential functions in algebra 2:

The Walking Dead is still a smash hit.  Today, we got to solving logarithmic functions and we went back to that final question (which I purposefully left hanging back at the beginning) and solved it algebraically…

And when you solve that function for x… Oh.. Oh dear God, it can’t be….

28 days!!!

As in the zombie movie 28 Days Later!!!

Math.  Zombies.  It’s all connected.  It all makes sense.  The Zombie apocalypse is coming, and math may be our only hope.

Oh hey there, how’s it going? Let’s read “If You Give a Mouse a Cookie”

Yeesh.  Been a while.  Back on the first day of school I started taking notes for what was supposed to be my annual “First Day of School” blog post.  And then the day just kept getting crazier and crazier.  Here’s my day, from a very foggy memory, in neat, numbered-list form:

1. Copiers explode (or something like that)
2. All-School assembly
3. A period (Algebra II)
4. B period (Foundations of Computer Science)
5. C period… is missing?  Apparently, they had locker day the first day (8th grade computers)
6. D period… why are the eighth graders all missing? (Algebra I)
7. D period… wait, why are the eighth graders at lunch and not in class- oh… oh, dear god no…
8. D period… 9th grade goes to lunch and…
9. D period… 8th grade shows up.  I get no lunch 😦
10. E period (Algebra II)
11. F period…. PREP!  FINALLY!  I don’t remember for sure, but I’d bet everything that I got out of the building for lunch, even if I had brought my own.
12. G period…. PREP!  But not really!  We have an away soccer game!
13. Soccer away game!  We lose! 😥
14. I get back to the school around 7pm, home by 7:30pm.

Things of note:

I have five classes, four preps this year.  They are the first five periods of the day.  Algebra II (twice), Algebra I, Foundations of Computer Science, 8th grade computers.  The Algebra 1 class is split; half 8th graders and half 9th graders.  This led to the crazy scheduling fluke described above.  Basically, at our school D-period is the lunch period which unfortunately and inadvertently was when this class was scheduled.  Normally, E period is the lunch period, but on some special days, it is bumped to D period.  Basically, I had D period twice and I lost my lunch in the process.  Foundations of Computer Science is a new class to our school.  It’s AP CS lite for now and it is just a semester long class.  The students are learning Java and hopefully the foundational concepts behind all computer programming.  8th grade computers is also a new class to our school.  Students are learning the introductory ideas behind programming.  We’re “coding” in Scratch.  The 8th graders are creating computer games.  This is fun.  It’s a quarter-long (8 week) class that will get repeated 4 times so that each 8th grader in the school takes it.  Our school is one step (AP CS) away from a fully comprehensive, 6-12 computer science curriculum.  This is awesome.  In a time that most schools have cut CS, ours has added it and gone all-in.  This is very awesome.

Coaching a varsity sport during fall semester is tough, even if I’m only an assistant coach.  Soccer’s final regular season week is upon us, however it’s likely we’ll go several games deep into the playoffs, so there’s still a few weeks left.

I am loving teaching algebra II.  I really have nothing that’s all too amazing to add here.  My students are awesome and I’m finally getting the chance to teach a course two years in a row and it is so much fun.

My parent-teachers conferences were da bomb.  I couldn’t fit my head through the door after all my students’ parents told me how much of a baller I was.  Seriously, though, they went so smooth and mostly because we have awesome kids and awesome parents.  I’m very lucky!

The Coolest Thing I’ve Done in the Class So Far

In my 8th grade computer class we took a day, left the classroom, sat in a circle in the hallway and had story time.  The book:

The reasoning behind it was an introduction to control statements: if-thens.  I cannot emphasize this enough:

Teenagers LOVE reading children’s books!

I think it’s a change to look back at how simple things were for them just a few years ago.  They truly enjoy it.  I always start by reading If you give a mouse a cookie.  I don’t tell them why at first.  They love it.  They demand I provide ample time to look at the pictures.  They sit silently, fully attentive, as if this was the most enjoyable thing they’ve done in a classroom in years (and maybe it is, which says…).  I then ask for volunteers to read the other three stories in the series.  This takes about 35 minutes total, leaving time to talk about If-then statements.

This works great for geometry classes, but also it worked very well for my 8th grade CPU class.  I’ll probably bust it out for my Foundations of Computer Science class as well.  I strongly, strongly recommend every math teacher buy the book that has all four stories in it, one of the best purchases I’ve made:

Mouse Cookies and More: A Treasury

It comes with the four original stories and a bunch of songs and recipes that I ignore.  I bought it for $25 at a Barnes and Nobel (before educator discount) but you can get it cheap on Amazon at the link above.

Hope everyone’s first month of school has gone well (and maybe a little less busy than mine!)

(More Of) The Most Beautiful Math I Have Ever Seen

Check out the first post on what my students came up with while doing Sam Shah’s Families of Functions project.  Here’s some more awesomeness:

Click to bigify

Click to bigify

Paper Plate Unit Circles, Done Efficiently

I like the idea of doing paper plate unit circles and today was my third go at it.  However, in the past it had always been more of an exercise in using a protractor or compass than actually focusing on the unit circle and trig.  So today, I decided I would take the time and do the grunt work for the kids so they could focus on the math.

Step 0: Materials

Brads, scissors, a Paper-Mate flair fine point marker, the best Google Image search of a unit circle that is blank and has the lines drawn in, paper plates, a spare manilla folder

Step 1: Draw the Circles

Step 2: Mark off Angles

This is the true genius of this method, if I may say so.  I simply cut down the unit circle I printed off so that it was slightly smaller than the circle on the paper plates and drew in the tick marks.  Perfect 30, 45, and 60 degree angles in a matter of seconds.

Step 3: Draw the Axes

The axes were drawn simply with a straight edge – I cut an easy-to-wield piece of the manilla folder to do so.  Piece of cake.

Step 4: Cut Some Terminal Rays

Just sliced some appropriate length pieces out of strips of the manilla folder. Then, I took two snipes at an end to create a point.

Step 5: Teach the Unit Circle, not Protractors/Hand-writing/Neatness

So. Much. Better. Than in the past.  We were able to construct (mathematically speaking here) the unit circle: review degrees to radians, use special right triangles to find the first quadrants, use symmetry and reason to fill in the rest.  Additionally, I still had a good 15 minutes to spare to do some exact value trig problems at the end of class (i.e. tan(30) ).  What has taken twice or three times as long in the past was done in about 25 minutes.

The process of creating 35 “ready-to-go” unit circles, took me 45 minutes. In the past, it would have taken each student about 25 minutes, on average, to do the same for their one paper plate using a protractor.  Much better use of time and we really got to get into the nuance of the unit circle and why it is so simple, elegant, and important.  Students finished it off by using the brad to connect their arrow and then filling in the angle measure (degrees and radians, of course) and the coordinates of the points on the circle.

Voila, here’s your unit circle.  Love it, cherish it, keep it forever and ever and post it in your dorm room freshmen year of college. You will thank me.

Basketbola: The Reaction

Previously: Basketbola, Three Acts: Basketbola Preview

It’s not often I get my kids yelling and hollering about math, but if the Basketbola project was anything, it was fun.

From a math/content/standard perspective, the biggest thing the project did was visualize the result of a negative leading coefficient of a quadratic. While I hoped for a little more, I’ll take it for the first time doing the mini-project. In the future I’ll probably flesh-out some of the other properties of parabolas and address those properties a little more directly. After about 25 minutes, matching the graph to the flight of the ball was a little repetitive, though still interesting enough to hold a sixteen-year-old’s attention.

Monopoly, Anyone?

I don’t know why, but during break I found myself playing Monopoly online. Then, somehow I stumbled onto this site:

How to Win at Monopoly^® – a Surefire Strategy

Since grading my exams, I’ve been thinking of ways to detach my students from their calculators. Their reaction to any kind of arithmetic is to jump, dive, kick and scream for their calculator. I thought “I’d just like them to play Monopoly for a couple hours and not use their calculators to compute the change.”  Now I have another reason:

Source: http://www.amnesta.net/other/monopoly/

The table shows how many opponent rolls it takes, statistically, for a player to break even on their investment in a property. If you go to the site and read the comments, you’ll see many people provide anecdotal confirmation of what the statistics say.  I’m starting to think that this could be a fun way to introduce probability – especially if I can join in on the fun, lay the smackdown, and say “why am I so good at Monopoly?”

The table says that your fastest way to return investment is to throw three houses on the St. James/Tennessee/New York group. This seems to be the kind of thing that students should be able to easily understand why its true:

• It is not the most expensive color group
• It is not the cheapest color group
• It is a second color group (on a board side, meaning it earns higher rents for the same improvement costs)
• It is 6-9 spaces from Jail.

The last point is the one I can see easily transitioning into probability from.  What is the most often roll of two dice?  Why is that important in Monopoly?  The questions here are endless.

Luckily, I have plenty of versions of Monopoly sitting in a closet in my childhood bedroom. Now I just have to find a way to carve out a day to play Monopoly…

Basketbola

This isn’t an original idea.

However, I got a fancy new camcorder for Christmas and decided to put it to use.

What I will try to make original is my implementation and how to really get the students into it.  The video, of course, is paused when the ball is frozen in mid-air. At that point, the conversation begins at a very low entry level. Anyone can take a guess. For my algebra I students, we’ll move into Sketchpad and they’ll start playing with changing the constants to model the flight of the ball, thanks to the nicely provided picture.

For my algebra II students, we can go several routes. What I like most is that we can go tech and use sketchpad, or we can got paper, pencil, and rule and plot three points and use a system (which we’re approaching soon). That’s a nice luxury to have when time becomes a factor.

How I’ll spice up Basketbola, however, is by letting the students be the stars. A side benefit of coaching basketball is I now have a library of videos and pictures of these shots by students.  So, there’ll be ten or so “problems” to solve.  We’ll watch the first acts and everyone will submit their guess. At the end of the project whomever was the best prognosticator will get a chance to take a shot, kick a field goal, or try to spike a volleyball into a bucket and it’ll be added to the library. Should be fun!

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

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

b_n = 1.0225*b_n-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.