8th Grade Calculus

graphThere’s a blog post from Reflections in the Why out there in which the blogger asks “Which graph best represents the importance of teacher knowledge of mathematical content as a function of grade level taught?”  

At the time (a year ago) I answered graph E.  My reasoning:

The more I learn about high school math (second year teacher, now teaching Alg I, Alg II, Pre-Calc), the more I realize how nuanced upper level topics are. I sat in on a Calculus class and was blown away at the difficulty of it (coming from a math major!) – we’re not just cranking out derivatives here. While TEACHING each grade level requires specific knowledge of HOW students learn each topic, I think the complexity of the math itself increases. Probably not exponentially, but faster than linearly.

Now in my third year, I think I’m pretty much in the same place.  I’d like to move that initial value in (E) way up though.

Of course, in a perfect world we’re all 4.0 math majors from highly reputable universities, no matter what level we teach.  We all aced our Analysis/Theory of Calculus class (HA!) and can drop some sigma-epsilon proofs on 3rd graders if necessary.

Okay, maybe not.

CALCHowever, it’s still crucial we have that background.  Because, then when you’re teaching 8th grade algebra and introduce slope, you can ask the question:

What is the slope right now?  What is the speed right now?

You can reinforce the idea of average speed over a period of time vs. instantaneous speed at an instance.  And you can then expect students to understand slope as an average rate of change, especially when you’re looking at data and lines of best fit and not the pretty, everything-works-out-to-a-round-integer-answer examples in the text book.

This was one of my more fun discussions of the year.  Sure it was probably over the head of a third of the class.  That’s okay.  I’m sure even those that weren’t ready for it could see that we needed two points to calculate slope and this magical calculus I spoke of let us get slope with one point.  Now, I’m primarily an 9-12 math teacher that teaches an 8th grade class.  However, it is so, so important that any math teacher, in any grade, can make that calculus connection to rate-of-change, even if they’ve left behind the chain rule and related rates years ago.

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Zombies and Math: There are no coincidences!

This is my intro to exponential functions in algebra 2:

zombie-exponentialThe 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…

28days

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.

GraphingStories.com: 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 GraphingStories.com

Just a picture… Click to head over to the amazing GraphingStories.com

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?