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.

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.

Fun on the First Day Back

I’ve decided to do something new for the first day back after break.  Instead of doing the typical “hand back the exam, go over the difficult questions,” I’ve decided to shake things up.

For my algebra 1 and algebra 2 classes, I’m going to follow the following game plan for the first days back.

  • Brain CRUSHER (30 minutes)
  • Hand back exams so students can go over (10 min)
  • “Exam Redux” homework

Nowhere in there will I “go over” an exam problem on the board.  Instead, the problems I think we need to review will be built into homework.

My main goal is to build in a fun, inquisitive and engaging return to math class without having to hold the students accountable to much aside from thinking about a perplexing topic.

For algebra 1, I’ll give the “1=2” proof.  One of my greatest life regrets was not taking a picture of this proof when I saw it written on the bathroom stall in the math department building in college.  Oh, what could have been :)

     a = x            [true for some a's and x's]
   a+a = a+x          [add a to both sides]
    2a = a+x          [a+a = 2a]
 2a-2x = a+x-2x       [subtract 2x from both sides]
2(a-x) = a+x-2x       [2a-2x = 2(a-x)]
2(a-x) = a-x          [x-2x = -x]
     2 = 1            [divide both sides by a-x]

For algebra 2, I’ll pose a simpler question but one I expect to produce some interesting discussion:

0.9999999999999999… = 1

True of false?

I’ll be able to finish this discussion by showing Vi Hart’s video, too!

Much better than your normal “lets go over exam questions,” don’t you think?  Any other good brainteasers/perplexing “math only” questions?  By math only, I’m meaning just purely mathematical questions.  I’ve been inspired by Dan’s post on engaging fake world math and am looking for ways to inject more of it into my classes.

Graphing Stories and Vine*: A Match Made in Heaven

*Except we won’t be using Vine.  The prevalence of explicit adult material on Vine and the lack of privacy controls have led me to look for alternatives, even though I love the simplicity of Vine.  We’re using Viddy, which, although not perfect, does a better job with privacy.

Here’s the beauty of the age we live in.  We can have students record and edit video in our classrooms with the device in our pocket.  This can be done in a matter of minutesThere’s no need to have bulky video cameras, transfer onto a computer, deal with Windows Movie Maker or iMovie, struggle with what file types to save as, etc. etc.  Simply an app that lets you record with a little control over some editing techniques.

In short: I had students create their own Graphing Stories using iPads (well, iTouches to be exact) and the app Viddy.  Yet another shout-out to Dan Meyer’s Graphing Stories.  I played several of the Graphing Stories in class and now am heading up the taxonomy to have students actually create their own.  Here’s one of my examples from the students:

Although Vine is much simpler to use, it didn’t take much of me playing around with it before realizing it would be too risky in the classroom (as an aside: this is very disappointing.  My quick research found articles that alluded to Vine’s pornography problem dating back to January.  What a waste of what could be an amazing tool.)  Viddy gets the job done.  We had three iTouches available to us and the kids were busy enough with recording that they didn’t get into the notion that they were using a social video app.  When you add in making the account private, Viddy almost turns into a video editing app without the pitfalls of public, social sharing.

The ease in which this allows students to create is AMAZING.  We get rid of all stuff we used to have to deal with, as described above, and suddenly the time, focus and energy is on the mathematics.  Example:

Some of the students wanted to record a group member riding one of those wheely scooters down a ramp.  They wanted to graph the person’s height off the ground.  But wait!  What would that graph look like?  Of course, it would be constant if they are referring to height off the floor, but decreasing if they are talking elevation.  That conversation happened in my class.  That is deep mathematical thought in an Algebra I classroom.

This is some exciting stuff.

Below is the project handout.  Each group was required to story-board their videos before they started shooting.  This actually helped with the notion of step functions greatly, in addition to streamlining the logistics of shooting video.  To sum it up: It was an amazing project and easily the best thing I’ve pulled off this year.  I hope it sparks some ideas out there in the MathTwitterBlogoSphere.

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?

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:

If_You_Give_A_Mouse_A_Cookie

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!)

[SCBG] Concept Exploration: “What’s Infinity Times Infinity”

This year I’m going with a Skills and Concept based grading system.  The skills will be like your typical SBG system.  The concept portion will allow us to dig a little deeper into the conceptual side of algebra.  Here’s the first concept lab/project/exploration/whateva you wanna call it.

Overview

This will be in the second week of the school year for algebra I.  After we have talked about operations and evaluating expressions.  The exploration’s main focus is the real number system and the number line.  This exploration will take two class periods, plus a short period of time for a small individual summative assessment.

1. Watch the Clip

2. Individual Reflection

Give the students 10 minutes to reflect on the video in writing.  They may answer the prompts or write their own thoughts, but they must write in complete sentences.  Prompts include:

  • Is infinity a number?
  • Can we count to infinity?
  • How big is infinity?
  • Does “infinity plus infinity” make sense?
  • Which group of numbers is bigger: whole numbers or natural numbers?

3. Socratic-ish Discussion

At this point, we should have about 30 minutes left in our 45-minute period.  I would aim to spend 10-15 minutes discussing what students wrote.  Let them guide the discussion.  Eventually, steer it towards that last question so we can talk about the real number system.

4. Vocab & Day One Wrap-Up

Flip to page two of the exploration and define the various real number classifications for the students.  Ideally, the structure would have been brought up in our discussion.  The cool thing here is we’re talking about cardinality in algebra as a way to set up the real number system.  That’s some high-level stuff!

5. Day Two: Individual Work

Day two is a chance for students to work through some exercises based on the real number system, the number line as well as basic integer and fraction operations.  Really, this is a chance for me to get around to each student and see where they are with their abilities to perform arithmetic with negative numbers, multiply and divide fractions, etc.  However, there will be some gems thrown in there for the students who do have their skills down.  This one is my favorite:

True or False: When you multiply two irrational numbers, the result is always an irrational number.

I am really looking forward to seeing how my Algebra I students grapple with that.

6. Day Three: Short Assessment

On the third day, I anticipate a short, low pressure assessment to follow up the lab.  Something like 2-4 typical problems related to classification of numbers and the number line, perhaps 1-2 arithmetic problems and maybe an essay question just to get the kids used to the idea of writing in math class.  This would be worth no more than 20% of the entire lab.