Quadratic RAFTs

I felt like my students were not yet ready to test on quadratics again (all methods of solving), so on Friday we took a day to recap what they've learned so far by writing RAFTs.

When I first heard about RAFTs, I was pretty excited, but I wasn't sure how juniors and seniors would respond.  To be honest, we do some pretty cheese-ball stuff in my classes, and I think this qualifies as such.  But, cheese-ball can be hilarious.  Here's evidence of the hilarity:

Role: Discriminant

Audience: America

Format: Campaign Ad[1]

Topic: The usefulness of the discriminant

Dear people of America,

They drew this on their paper,
but I'm too lazy to scan it

Are you tired of solving quadratic equations and wondering what the answer should be?  You waste minute upon upon trying to figure out when to stop solving.  Not with the discriminant.  With the discriminant you can instantly know what to look for while solving.  The problem under the square rot becomes hardly a problem at all if you vote to keep discriminant around.  Do yourself a favor, and check 'Yes" for "Vision D"!

Sincerely,

Board of Discriminants  

*****

Role:  i

Audience: Negativity

Format: Letter

Topic: How i and the negative numbers work together

Dear Negativity,

Your square root is always bringing us together.  At first we had a problem because you were always being fake, but then I came around and made being a real a possibility.  I know that sometimes your square root makes you feel imaginary but I'm always there to rescue you when he does that.  Many, many years ago you were a problem to everyone and no one knew how to fix you.  When I came along things changed and your negativity no longer was a problem.  I love you and your square root.

Sincerely,

i

*****

Role:  i, The Illusionist

Audience: Potential magic show-goers

Format: Ad

Topic: The coolness of i

Hi, my name is "i."  Some call me "Imaginary," and some call me "Illusion."  If you come out to this amazing show you won't regret it!  There are many fascinating things about me that I would like to show you!  Depending on when you catch me at the show, depends on my reality.  Let's just say there is a certain pattern to me.  Sometimes I am just imaginary when I feel like being myself, but I can also be in the form of -1, -i, and 1.  Do you think you can figure me out?  Come to the show and you will see!  Or will you...?

*****

I let my students work in "groups" of ones, twos, or threes.  They brainstormed on whiteboards, and then wrote their final product on a clean sheet of paper.  The activity took about thirty minutes.  A few of my students had written RAFTs before, but most of them had not.  So there was a lot of "I don't get what you want us to do."  And there were a few kids who just sat there for the first few minutes, which I'm pretty ok with.

I definitely had to encourage some students more than I did others.  But, reading the final drafts was both fun and enlightening for me.  It's clear that there are some topics that the students really understand, and some that we really need to discuss further.  This is what I love about writing in math class:  it's incredibly revealing, is it not?

[1] Sounds more like an infomercial to me, but to each his own.

Warm Up for i

Sometimes we have to relish in the little things, right?

This is a warm up I gave to my Algebra II students, just a couple days after they had first been introduced to i:

While I do like the warm up, what I'm really quite proud of is how I implemented/graded it.  When students felt like they had finished the warm up, I had them let me know.  I checked their work quickly.  If I liked what I saw, they were given the day's assignment (and a 100% for the warm up).  If not, they were given some verbal questions from myself, such as...

"You say i is imaginary, but you also say it's equal to -1?  Are you saying it's impossible (not real) to lose a dollar (-1)?"

"i is equal to the square root of 1?  But the square root of 1 is...?  Oh, so we need two symbols for the multiplicative identity now?"

"i is equal to i?  Try again.  This time tell me something."

Yes, I was harsh and sarcastic.  But this is an important concept.

Eventually, everyone had true sentences on his/her paper (which means everyone who came to class got a 100).

Each day I've been doing an "EOI Preview" as a warm up and I've been taking the highest 3-4 grades for the week.  This warm up gave everyone a chance to get an excellent grade in for the week, and I didn't let students move on until they could articulate the truth.

I know, I know...I really need to switch to Standards-Based Grading.  Sigh...

Is mathematics invented or discovered?

We've been solving some quadratic equations in Algebra II currently, and I've had an ulterior motive this whole unit.

Quadratic equations seem to lend themselves particularly well to math history lessons (or "math commercials" as my principal calls them--love that).  For example, when we talked about the Square Root Principle, I gave a mini-lesson on Christoph Rudolf (names that rhyme are the best, aren't they?) and the introduction of the square root symbol and how it's supposed to resemble a lowercase r, etc.

I asked them innocently here, "So, do you think mathematics is invented or discovered?"

If they said, "invented," I said something like, "So, the square root of two didn't exist until Rudolf came up with a name for it?"

If they said, "discovered," I said something like, "So, you're just going to ignore the contributions people like Rudolf made to mathematics?"

I let them hash it out a little, playing devil's advocate all the way.  And then I ended with, "Well, interesting conversation, guys," and proceeded to my next slide.  Which, inevitably had the effect of "WAIT!  Aren't you going to tell us?"

"No."

Which then had the effect of, "I'm going to Google it!"

"Go for it."[1]

The next day, Day 2, we continued with the square root principle, but now we tried to solve equations like x^2=-1.  I let them try to convince each other that there is no real solution to this equation (though their multiplication skills are still lacking, so...sigh).  Here's where we talked about imaginary numbers and a mini-lesson on Euler ensued.  I told them Euler couldn't stand not having an answer to this problem, as it--along with other problems like it--had been appearing in mathematics for nearly two thousand years.  So, Euler made his own solution, and called one of the solutions i.

"Now do you think mathematics is invented or discovered?"  We took a poll[2]:

3rd Hour

5th Hour

At the end of class, I had them write a letter to me defending their answer.  They were instructed to choose only one (invented or discovered).  Here are two really great letters, one from each point of view:

Dear Mrs. Peterson,
Mathematics was discovered, because just because a human didn't know the answer to something doesn't mean it doesn't exist.  Before the Pythagorean Theorem was invented, a right triangle still had an area.  Humans simply put words/letters/numbers and theorems to help us find the answer, and explain math, but the problem they solve, and answers they find, were always there.  Some species of animals haven't been discovered yet, but when someone finds them, they didn't invent the animal, they discovered it.

Dear Mrs. Peterson,
I believe mathematics is invented.  I believe this because invented means to create or design something that has not existed before, or make up an idea, name, story, etc.  You have to create a name for mathematics to exist.  One apple is not one apple unless you give a name to the number or quantity of the apple.

The next day, something happened that I think will go down as one of my favorite teaching moments of all time.  A student, who has said from Day 1 that she's not good at math, came up to me before class and looked at me with her precious, sincere, huge brown eyes:

"Mrs. Peterson?  I really need to ask you something."

"Go for it.  What's up?"

"Can you PLEASE tell me--is mathematics invented or discovered?  I can't stop thinking about it."

Cue burst of emotion and huge cheesy grin on my face.  Why was this so wonderful?  Because she just experienced what makes mathematics so addictive:  the deep longing to solve or to prove, and the pleasure that follows the accomplishment.

Luckily (or maybe unluckily) for her, on this day, Day 3 of our discussion, I wrote a letter to my classes, defending my point of view.  Now, I didn't give myself the same restrictions I gave them, and I typed this up the night before (I know, bad Rebecka), so it's pretty rough around the edges (hey, that's the great thing about teaching--now I have a whole year to make it better).  But, here's what I wrote:
Discovered or Invented

What was so, so cool about this whole discussion is that it really appealed to most of my students.  They were hooked.  They kept asking about it.  They wouldn't let it go.

What more could I ask for?

So, what do you say:  Is mathematics invented or discovered?  I'd really like to know your input...so I can make my letter better for next year.


[1]  When I checked in on these students, they seemed more confused than when they started.  Let's hear it for UnGoogleable Problems!

[2] Don't let the total numbers fool you.  Only 1/2-2/3 of my classes participated (don't want anyone thinking I have a class of 17!).  Not sure if the rest didn't want to commit to a single answer, if they didn't have access to a phone, or if I just didn't quite hook 'em...

Quadratic Regressions with Angry Birds

Before we ended our discussion on quadratic functions, I wanted to introduce my students to quadratic regressions.  What better way to do this than modeling the trajectory of an angry bird?

This is the picture I projected on the screen.  There is more angry bird/quadratic function material out there on the world wide web that is better than this.  However, I couldn't find exactly what I needed with the materials I had (I wanted a screen shot of an angry bird, whose path wasn't yet finished, with a grid on top).  So, I made this one on Word using a picture I found on the internet.  Be nice.  It took much longer than one would think.

We chose some points that the top parabola passes through and then fit a quadratic regression using those points.  I then asked if the students thought the bird would hit the ice block (around (9.5, 4)).  The results for this varied depending on the class and what points they chose to input into their lists (which is fascinating).

Below gives a picture of the points we plotted in one of the classes, the quadratic regression, and the point (9.5,4), which was plotted post finding the regression.

There's more to be done here, I'm sure, but it's a fine start.

Parabolic Path of a Flying Q-Tip

This one I owe to one of my favorite teachers of all time.  It's a game we played in physics.  What you'll need for this is a drinking straw and 3-5 cotton swabs per student.

The lesson is on analyzing graphs of parabolas.  Since we just went over function transformations, we start with parabolas in the form f(x)=a(x-h)^2+k.  In this form, students seem to have very little trouble identifying the vertex.

So we graduate to f(x)=ax^2+bx+c.  Now, where's the vertex?  I used to do an elaborate presentation, completing the square for the general case and thus getting us back to the beautiful form above, and eloquently pointing out what h and k now are.  And then I came upon this beauty:

"Where is the x-coordinate of the vertex in relation to the two zeros?"  I ask.

From here, we have a little discussion, and eventually decide to average the two zeros to come up with the x-coordinate for the vertex, which turns out to be -b/2a (horrah!).

Now we practice.  I give the students a few quadratic functions and ask them to tell me all sorts of things about its graph.  The point I try to emphasize is that once you know the vertex and the direction the the parabola opens, you know a whole host of other information (such as the axis of symmetry, the min/max value and where it occurs, where the function is increasing/decreasing, and its range).

Now for the game.  I tell the students to take out the straws and Q-tips I passed out in the beginning of class.  I play this video and tell them whoever can shoot the cat with a Q-tip gets extra credit:

I gave out three Q-tips per students, but will most likely up that to four next semester, depending on how many students I have.  The students love this part of the lesson--the athletes in the room seem to take special pride in hitting the Nyan Cat.

And now back to the math.  We talk about how the path of their Q-tip takes the form a parabola.  We discuss if a should be positive or negative.  I then give them a model such as f(x)=-4x^2+4x+25 and explain that the path of their Q-tip could be modeled by such a function, where x is the time after the Q-tip's launch in seconds and f(x) is the Q-tip's height in inches.  (No, the model isn't perfect, but the numbers are fairly easy to work with and it gets the point across.)  We play with the model a bit, finding the ever-important vertex, among other things.

In the past, I've had students shoot at me during the lecture.  They love this.  Until they have to do the homework that night and realize they caught zilch during the lesson.  So, I tried the cat video this time.  It's probably not quite as entertaining, but in the long run, I think it may work better.