Taking Students’ Money for Fun(ction) and Profit

Inspired by this post, an in the vein of painless ways to improve instruction, I thought I’d share how I introduce functions. I throw up a slide of some function, usually represented by one of those circley-arrowy thingies, and tell students we’re going to play a game. I pick a domain value, and they have to tell me which range value I’m thinking of. If they guess right, I’ll give them $10, but if they guess wrong, they pay me $10. No money ever changes hands, of course, but it raises the stakes nicely. One brave student volunteers to play, and in short order, takes all my money. “Just a practice round! Just showing you the rules! Doesn’t count!” Next slide goes up, and just like that, they are hooked.

Yellow flies work better on sophomores, but go with a nice shadrap for juniors

“Who wants to play next?”

Next slide is a relation with one domain going to twelve different ranges and I try my best to wheedle some student into playing, but they know something’s up. I go through each domain value, asking “Do you know what range I’m thinking of?”  I usually get someone with a relation that has a bunch of one-to-one values and a single domain that goes to two range pairs. The student often justifies playing by saying the odds are in their favor, only to have me break character and pick that same value time and time again — “oooh, this time I was thinking of the 5, not the 2! You almost got me! Let’s keep playing!” After taking all that student’s money (again: not really), everyone is more wary. Each new slide prompts a burst of hurried strategizing, followed by a decision to play with me or stay away. The first slide that has multiple domains going to a single range value always elicits some good discussion, as students initially suspect a trap, but then figure out that they will rob me blind.
After they are okay on maps, we move on to ordered pairs, tables, points on a coordinate grid, continuous graphs (where brief mention is made of the Vertical Line Test), and eventually equations. Students eventually realize that in theory, they should be able to clean house on ANY equation.

No calculators, and x = 2^7^8^12. Good luck, suckers!

Theory, meet Practice

I usually save the equation part for a few weeks later, because once students realize that the game is no fun with equations, we start playing backwards. Now I give them range and they have to determine which domain I used, and all of a sudden, we’re talking about which functions are invertible and which are not. We can push it further and put up equations that have values which “break” the game like radical or rational functions, and now we’re talking about restricted domains.

The advantage to this approach is that it lets students engage functions using their already well-honed intuitions about fair vs. unfair games. Since adopting this approach, students have a much better (if at times more casual) language for what makes a function a function (not that “oh, there’s a repeated x-value, so it’s not a function” BS), and are much more able to explain why a single “problematic” value is all it takes to lose function-hood, and why it’s totally okay to have multiple domains going to the same range value.

And it sticks. First year I did the lesson like this, I had a student mis-identify a function literally MONTHS later, and another kid chimed in with: “Nooooooo, he’s gonna take your money!”


Teaching Absolute Value

Sometimes making teaching work is all about quality materials, innovative lessons, and all that jazz. Other times, it’s the small stuff that counts. In Algebra 2, I like to dive right into linear equations for the first few weeks, then skip back and start talking absolute value. This year, I did the same stuff I usually do, but with one tiny change in language, and the kids went from:









Here it is, and I realize probably everybody already does this, but I didn’t, and it is SO MUCH better.

Start with one-variable equations:    |x + 5| = 8

I do everything on a number line and make students interpret the meaning of this as “the distance from x to -5 is equal to 8.” We solve some problems just like that, talking through them on a number line, but when I go to show the algebra “trick”, I teach them to drop the absolute value and set it equal to 8, or drop the absolute value and set it equal to … (here it comes)…. (seriously this is awesome)…. the OPPOSITE of 8.

I know, I know, everybody does this already. Shut up, I’m excited.

So when the right-hand side has two terms, like  |x – 2| = 2x – 12, the very first example of that type we look at, everyone already knows that either x – 2 = 2x + 12 or x – 2 = -2x + 12 and when I called on a student to explain why, she was like: “Duuuhhh, it’s the OPPOSITE.” Didn’t have to mention distributive property or anything and 100% of the class was ahead of me.

Then when we get to inequalities, it’s all very natural that something like |x – 2| > 7 will split into one branch of x – 2 > 7, and the other branch should be x – 2 < -7 because duuuhh… it’s the OPPOSITE!

And now we’re on to absolute value in two variables, and after some good old table of values graphing to convince them that they really didn’t want to have to do that all the time, I went back and talked about splitting up the absolute value equations into 2 linear equations, and wow… did not feel like I was earning my paycheck.

“So when we look at the second branch of y = |x + 4| – 2, how do you think we should write that? Should it be y = -x + 4 – 2 or y = -x – 4 – 2 or y = -x – 4 + 2?”
Usually questions like this spark an intense debate, or at least a lot of students with that scared “I ain’t picking anything” look.
But this year, all I got was eyerolls as 25 kids all picked the middle option. WHY?
“Because x +4 is in the absolute value, so you just take the opposite.”
“And why not take the opposite of -2?”
“Well, it’s not in the absolute value, so it doesn’t represent a distance where the answer could be in either direction. You’re just subtracting 2, like it says.”

So, 3 days of absolute value and we hit equations, inequalities, graphing two variables with tables and linear equation, including writing out interpretations referring back to the number line, no solution & all real number solution situations, and everyone’s on board. So if anyone out there is still teaching absolute value without using THE OPPOSITE, come give it a try. The most remarkable thing about this profession is how those tiny, tiny changes can have such an impact on students.






Socratic Questioning

To introduce points, lines, planes and just the general notion of undefined terms in Geometry, I like to start with a game of Socratic questioning. Usually students give the background on Socrates, and we talk a little about his life and how he operated.

So annoying, an entire city voted to make him drink poison


Then I divide the class into pairs, one as “the explainer” and one as “the questioner” and write a question on the board such as: “What is a rectangle?”

Here are the rules:
1) the explainer must make a good-faith attempt to explain the concept at hand

2) the questioner must ask a question designed to expose a flaw in the definition or to frustrate the explainer

3) Neither player can give up. No matter how good the explanation, the questioner must find something to question, and no matter how annoying the question, the explainer must try to answer it.

After a few minutes, I have the class vote on which role they thought did a better job, then we share some of the more frustrating questions that were asked. This is usually awesome, as the questions will range from the completely off-the-wall to the remarkably-apposite-to-the-lesson. “What is space?” “What is a point?” “What is a thing?” “What is a shape?”

We usually play four rounds, switching roles each time. The first two rounds I use actual defined objects (rectangle, circle, square, etc.), while the last two rounds I go with “What is a line?” and “What is a point?” The proportion of the class who thinks the explainer “won” goes sharply down in rounds 3 and 4.

I use this to talk about the four basic types of mathematical statement: Undefined Term, Definition, Axiom/Postulate, and Theorem, and show how definitions are based on undefined terms.

I played a game with a student where I asked: “What is a point?” and he said, “Don’t make me answer that, give me something else.” I let him pick what he wanted to explain, and showed how in a few questions, I could get him right back to “What is a point?”

Try this out with your classes — loads of fun, starts the year off well, and really gets students thinking about definitions in a more precise way.

Getting Students to “Get” SBG

I’m starting my 3rd year of Standards-Based Grading (SBG), and in short, I freaking love it. SBG has allowed me to increase the rigor of my assessments and increase student accountability at the same time. My parent-teacher conferences have laser-like focus, my feedback is more timely and effective, and my students demonstrate more of a growth mentality.

For the purposes of transparency to students/parents and ease of calculation, I use a 10-point scale, but I don’t write any numbers on the student’s paper the first time around, just qualitative feedback, and if on the second time, their score is lower than a 6, I just write an “R” (the actual numbers are viewable through our online grade report). It’s great, but students aren’t used to it, and some of their strategies that have made them “successful” in other classes no longer work in mine.

So to get them on board and change their mindsets, on our REAL first day of school (not freshman first day), while other teachers hammer away at syllabus, rules, expectations, consequences, etc., I have my kids break into groups of three and talk their way through four grading scenarios. I preface this by explicitly telling them not to give the answer they think I want, but to really think about what grades would be fair. After their small-group discussions, I bring them together as a class and we compare notes. It’s a great opportunity for me to practice their names (I walk around and introduce myself to everyone when they are in groups, then call on them flawlessly by name during the discussion — thanks, seating chart!), to introduce the norms for discussion without getting them lost in the syllabus OVERLOAD, and for kids to realize that talking in math class doesn’t have to be scary. And of course, I make sure to slot Futurama characters in as the protagonists.

Bender Bending Rodriguez: Ruining school-appropriate images since 1999

There is a lot of disagreement usually, as well as the opportunity to dig deep into some interesting questions — “Does it matter that Fry didn’t learn the material in the same time-frame as Amy?”; “Who would you rather have as your financial advisor, Hermes or Zoidberg?”, “Why might Scruffy’s 92% be artificially high? Why might the 70% be artificially low?”, “For what classes might Hubert’s strategy be more effective than Leela’s?”

So if you’re looking for a way to introduce SBG, here’s a first-day discussion I have found to be effective. I hand the .doc out to students and then summarize the class results on the board using the notebook file (the last page is to show how each situation would actually be graded under my system). Feel free to take and modify for your own purposes.

Grading Scenarios Word DOC: http://bit.ly/gradingdoc
Grading Scenarios Notebook File: http://bit.ly/QYX91S

Freshmen First Day

Our school does a modified first day schedule, for freshman only. No sophomores, no juniors, no seniors — just the frosh. It’s an awesome way to create a supportive environment and ease some of those first-day jitters as our young men and women transition from middle school and high school. Here’s the basic idea:

First thing is an assembly in the gym, and because the size of the school permits it (500-600 students total), we do a whole staff receiving line. Every freshman walks down a snaking line of teachers, introducing themselves, shaking hands, exchanging pleasantries, and at the end, running for the hand sanitizer. We all wear nametags and do our best to be cheerful and friendly without looking like total losers.

Welcome to high school… DUUUUUH!

Then the assembly is short speeches from the principal, VP, activities director, academic dean, resource officer, student council president, etc…. It drags on a bit, but people keep it short and cycle through pretty fast. We teach the freshmen their class cheer (necessary for the all-school assembly the next day), and this year, the faculty entertained them with what I’m told was a heart-breakingly beautiful rendition of the school fight song.

Following the assembly, students are sent off with a staff member (4-5 students per teacher usually) for an hour long “advising” period. We make sure their lockers open, take them on a tour around the school and point out all their classrooms, then break out the student handbooks and give them the highlight reel of “need-to-know” info. Any extra time we fill up with getting-to-know-you stuff, questions about activities, etc.

Then it’s off to an abbreviated class schedule, with 25-minute periods. This is in some ways the weakest part of the schedule, because it’s pointless to start with the usual first day stuff, as you’ll just have to repeat it the next day for the upperclassmen, but everyone else is doing the “get to know each other” routine, and after about 2nd period, the kids are sick of it. I usually play games (Lone Wolf is ideal if you have enough students), show magic tricks (either math, cards, or object manipulation), do puzzles, or show the students cool SmartBoard tricks (if it’s a small enough group that everyone can get involved).

After the last period, there’s a quick 10-minute debrief with the same “advisor” as before, then a return to the gym for the a closing 20-minute assembly, which usually involves singing and some sort of ridiculous Double-Dare style competition that the Student Council dreams up.

Why don’t more schools do this?
Freshman-only first day is a great way to begin the school year. Most first-day issues are freshman issues, so we get those taken care of immediately and on the actual first day, it’s only the upperclassmen who can’t get into lockers, find classrooms, etc. Walking through the school without the usual crush of bodies helps orient our newest students so that they feel comfortable finding their way around once the hallways look like this:

Hi upperclassmen!

It also helps build student-faculty connections and gives the usual anti-bullying initiatives an added boost, because the most vulnerable students are all there together when we go over who to contact, what bullying is, etc.

Anyway, this is one of the things I feel my school really does right, and from talking with other teachers, I do not think it is common practice elsewhere. BUT IT SHOULD BE.