Monthly Archives: September 2012

Sneaky Worksheets!

One of the things I’m trying to do this year is trick students into spending more time working with content. I could lecture for most of the period (definitions, examples, questions, answers, etc.) and then hand out a worksheet for students to start with whatever time remains and finish homework. You probably know about how well this works as you’ve probably done something similar at some point in your teaching career. Replace this structure that barely works if at all with my new paradigm: students watch a video on their own taking notes and complete some sort of activity during class time. Here are my last three activities; you’ll notice each one is basically a worksheet in disguise.

Working with Variables (Combining Like Terms & Distributive Property)

Students match an expression with its simplified form.

Students were much more engaged than they would have been with a traditional worksheet. An important visitor to my classroom seemed to agree!

Properties of Real Numbers

Students assemble a hexagonal Tarsia puzzle by matching equivalent expressions and then color code.

Image

I’m not a 100% sure where I first saw posts about these. I’m going to pretend it was here. (It was a starred entry in my Google Reader about Tarsia and could use an excuse to link to this blog) Students thought this was much harder than simply finding matches – you might notice I was a little vicious and included three instances of “0” and two each of “1” and “x” as simplified forms. We actually spent much more than a single class period on this one.

Solving Easier Linear Equations

Students wager points on their ability to solve certain types of equations. The PowerPoint is set up to automatically advance. Students whiteboard their work, share their work/answers during the blank slide, then the answer is revealed. Students update their score and make their next wager before the next equation pops up.

Some students got into this more than others of course, but it was a hit overall. I’ll definitely use the format again in the future with factoring or something. A few students had a lot of trouble at first with the fact it was ‘automatic’ – they wanted to chat between rounds but didn’t really have time. I told them if you don’t wager before the next equation pops up then your bid is automatically zero. It might have helped some that we were playing for a little candy …

These activities all became a left-hand side entry in my students’ not-quite-interactive notebooks. (At least I’m trying!)

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Small things …

Tonight, two former students randomly contacted me to ask for help with math. They both graduated high school at least five years ago but apparently thought of me when they needed help. One still lives locally wants to meet me at school tomorrow during my planning to help prepare for the GRE math section. The other called me once from the Super Bowl to thank me for helping get him there; he had gotten a job a few years after high school graduation that used basic/minimal math (redesigning the layout of large retail stores or something like that) and was doing well enough that his boss gave him tickets as a reward. Several years later we reconnected again via social networking, he thanked me again for helping him earn his ‘first million;’ he’s now a small business owner (not to mention happily married with kids) in a state far away from here. He’s finally getting around to finishing a degree and wants help in college algebra or a similar class.

 

Now I remember yet again why the daily grind is worth it … I just hope I’m forging the same kind of connections with some of my current students and just need a little more time to pass before I realize it …

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New Blogger Initiation 4

I didn’t get to do as much with this one as I would’ve liked, but I chose:

1. Read another blogger’s post for the Math Blogging Initiation. Write a comment on their post.

Back in round 3, I almost responded to:

5. Statement: “Algebra 2 and Precalculus are a hodgepodge of ideas.” If you agree, what are some unifying and fundamental themes/ideas/concepts that can frame these courses so they can designed to be less of a mess and be something more coherent.

I’ve read a few of the responses to this one in my reader (but I’m getting hopelessly behind so I’m sure I’ve missed many more) and particularly like the responses by Bowditch’s Apprentice and Compact Spaces. I feel like the prompt is a reference to A Mathematician’s Lament by Paul Lockhart. I mostly agree with Lockhart’s main ideas and plan to share the opening analogy, a nightmarish method to teach music and painting, with pretty much all of my math classes in the future despite the fairly harsh criticism of the average/typical math education.

Here’s the table of contents for an Algebra 2 book from a major publisher (you won’t be able to tell which one; they’re all about the same):

  1. Expressions, Equations, and Inequalities
  2. Functions, Equations, and Graphs
  3. Linear Systems (includes a little bit of matrices despite the chapter title not mentioning it)
  4. Quadratic Functions and Equations
  5. Polynomials and Polynomial Functions
  6. Radical Functions and Rational Exponents (also includes some advanced function concepts: composition and inverses)
  7. Exponential and Logarithmic Functions
  8. Rational Functions
  9. Sequences and Series
  10. Quadratic Relations and Conic Sections
  11. Probability and Statistics
  12. Matrices (why isn’t the lone section of matrices from Ch. 3 included in here?)
  13. Periodic Functions and Trigonometry
  14. Trigonometric Identities and Equations

After chapter six, you may prefer some other shuffling of the chapters (I’d probably go something like 1-7, 9, 13, 11 as ‘musts’ for Common Core Algebra 2 … then use any remaining time on 10 {even though I don’t care much for conics for some reason}, 8, 12, 14 (harder trigonometry is definitely more ‘Pre-Calculus’ than ‘Algebra 2’ at that point in my opinion) in that order, but I would love to have this book as I’m currently book-less. I basically refuse to use our Algebra 2 book; it’s a ‘classic edition’ that was adopted like 7 years ago and was already old then – a teacher who was new to the district got to pick and chose what they knew they liked and then left two years later. He wanted ‘no tree frogs’ (his version of dogs in bandannas) but left us with a book that was completely visually unappealing (no color in the book except grayscale and lesson objectives in blue text) to use with students who are digital natives.

Every chapter title is a reference to the underlying theme of sets, structures, relationships, and functions in my opinion. Maybe I’m cheating and just being too broad with my unifying or fundamental ideas though. Obvious and not-as-obvious explanations (by the way, obvious is an extremely dangerous word in mathematics – I personally detest it almost as much as variations upon “the proof is left as an exercise for the reader” – thanks scumbag mathematics PhD):

  • Chapter 1: the “algebra 1” they’ve probably forgotten – emphasize solution sets
  • Chapter 2-8, 10, 13: basically have graph or function in the title … I’m only worrying about a lot of the conic section stuff because I’m theoretically legally obligated to include it in Algebra 2 – I really don’t feel like I ever really learned about ellipses and hyperbolas as the high school teachers never got to it and the college professors assumed I knew it (sad as it might be to admit that). I can complete the square but don’t know all the forms and focus business by heart.
  • Chapter 9: A sequence is a function from the natural numbers to the terms of the sequence (in case you’ve forgotten)
  • Chapter 11: Probability is a function of a random variable (the notation P(event) made so much more sense after I started teaching this stuff), and statistics is concerned with sets of data.
  • Chapter 12: Matrices and vectors are probably the easiest algebraic structure for students to consider besides the real numbers (or subsets of the real numbers)
  • Chapter 13: Trigonometric relationships abound … this is the chapter I personally would always run out of time before ever getting to in Algebra 2; too easy to put off until Pre-Calculus.

By the way, now that I’m basically finished with the post – I can merge my themes …. sets are a type of structure, and I was only including relationships to get at non-function conics. Thus, Algebra is the study of structures and relationships. Now I just need to check my work against a few more of the other newbie blog posts that I haven’t gotten to read yet …

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New Blogger Initiation 3

or “Why Two and Two Makes Fish”

Almost out of time yet again (so glad this was timed to coincide with the start of school), but I suppose part of the point is to see if you can handle reflecting as you go during the school year. My choice for week 3:

1. Introduce and show the solution to a math problem that you particularly like.

When I read this prompt, I thought of one of my favorite random teaching tangents (finally worked that in!); I have a few things I try to work in to each class whenever I need to kill some time a student asks a question that I can address with this idea/concept/problem. Unfortunately, these favorite mini-lessons are often not exactly tied to any particular required content, but I’ll have students a year or two later remember these and not much else.

So here’s a somewhat paraphrased, somewhat fictional account of one of my favorite problems and its solution:

Student: I liked math when it was easier; 2 + 2 = 4 is always true … why can’t algebra always work the same way?

Teacher: Well actually, you really need to remember math is completely made up by humans. I mean 2 + 2 = FISH is honestly just as valid if you know what you’re doing …

Student: Stop joking around …

Teacher: Let me show you – but you have to be willing to let me bend the rules and change the meaning of a few things. Math is a game; when you know the rules well enough, you know how to bend, break, or even make up your own rules.

I’m going to almost use normal addition, but you’re limited to combining four symbols: 1, 2, 3, and FISH (Greek alpha). First, I need to tell you that FISH is sort of like zero but not exactly. Also, I’m going to use ‘circle-plus’ since this isn’t quite normal addition …

<scribbling on board>

 

 

Student: … um, isn’t that the exact same thing as usual???

Teacher: so far yes, but how can we complete the table without using any new symbols (only FISH, 1, 2, and/or 3), and the table be consistent – it has to make sense

<student suggestions, teacher prompting/questioning, more scribbling>

 

Student: are you just making this up?

Teacher: I already said I’m just making it up, but it has to make sense! Okay, let’s try another one … with less numbers and multiplication instead of addition … yeah, this should do it … we’re going to use the symbol i; there’s a rule that i^2=-1 by the way {yes, I can sort of use LaTex}

<scribbling, questions, more scribbling>

Student: Okay, you can play games and move around squiggles on a piece of paper just so … what’s the big deal?

Teacher: You happen to skateboard, right?

Student: Yeah, so?

Teacher: Come here … face the class; this is position zero. Show me a 180° … good, reset then show me a 90° … okay, same thing but 270° … fine, 360° … wait that’s the same as the starting position?

Student: duh

Teacher: Let’s make it interesting then … let’s start build tricks or turns on top of one another … show me a 90° followed by another 90° without a reset.

Student: 180° of course

Teacher: Reset, then show me a 180° followed by a 270° … you might want to sit there and actually work through the turns.

<student attempts, teacher helps, asks for a few other examples if necessary>

Teacher: So tell me what we just figured out …

Student: Well, you can kinda sorta add angles together but if you get to 360° you start over – it’s the same as 0° in a circle.

Teacher: Great, now consider a square in the coordinate plane … it’s basically like the skateboarding stuff we just discussed?

<scribbling>

 

Teacher: We’re just adding angles of rotation together, so let’s make a table … you should be getting the hang of this by now … work with a partner

<work, work, work>

Student:

Teacher: Great, notice anything yet?

Student: Not sure … this table starts over like the FISH stuff?

Teacher: Yeah! You’re getting there – can’t we really just think of these angles as multiples of 90, though?

Student: a 180 is two 90s … 270 is three … 360 resets to zero …

Teacher: You’ve just about got it … look at all three tables … color code them if you have to …

<student looks it over>

Student: The pattern is the same on each table isn’t it !?!!?!?!!?

Teacher: Awesome! The relationships are the same even though the ideas seem completely unrelated. On the first one, I used FISH instead of zero, because I wanted you focusing on relationships to get started. This problem demonstrates the basic concept of something called a group. You were just working on college-level math by the way – the kind for math majors even. We’re going genius-level in here …

Student: You’re kidding me – that seems easier than what we’re working on now!

Teacher: It turns out once you make it past calculus – math is mostly things you already know how to do but more abstract and even more interested in the ‘why’ than the ‘how’ … too bad so many people get turned off by the tedious calculation bits along the way

Student: This was the best thing we’ve done all year – it makes more sense than a lot of the other stuff …

Teacher: Remind me to show you the one about the empty trashcan, empty bag, and empty bottle not being empty anymore sometime … we should probably get back to factoring quadratics before class is over

Student: Yuck

Teacher: I know, I know … but this is a topic guaranteed to be on the state-sponsored high-stakes exit exam.

[Partially due to internet issues this is now over an hour ‘late’]

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