Thursday, June 24, 2010

Absolute Value Both Rigorous and in Context

I know I said I was done for the year. SORRY. I am literally sitting around school twiddling my thumbs today. I am ripping this idea off from Dan, but trying to extend it to be appropriate for Algebra 2. Absolute value is one of the first lessons of the year, and in the past my students neither understand it conceptually nor remember an algorithm for solving equations and inequalities with anything like reliability. This feels more like an Algebra 1 lesson to me, but I think it will be necessary.

This is my version... peanut M&Ms were the cheapest/most voluminous things I could find. There are about 230 in a large bag, by the way. Yesterday I polled 50 faculty and staff. In the fall I am going to have to get my butt into overdrive within a day or two to collect at least as many data from students.



I have yet to nail down the details, but the flow will go something like this:

Preliminaries
Put up a picture like this.


Ask how far away the houses are from school. Get a few volunteers to describe the mental procedure they used to determine distance from school. Point out that everyone naturally used a difference and absolute value to express distance. And that further, if we can represent distance as absolute value with an equation, we will be able to use it to ask and answer more interesting and difficult problems than our intuition can handle alone. Graph by hand y = |x| by making a table of values. Note the characteristic V shape.

Questions to Answer
Bust out laptops and distribute excel file. As per Dan's original plan, kids will have some choices about what questions to explore and time to flail.

- Who won?
- Rank everybody.
- Top 10 Guessers.
- Any ties?
- Worst guesser?
- Which grade guessed best?
- Which job guessed best?
- Calculate percent error.
(Maybe some/all kids can present aspects of the results on posters we can display?)

Once that's all squared away, I want everyone to explore:
- On average, how good were the guesses?
- Create the scatterplot that displays the characteristic V shape.
- What is the equation of the connected graph of that plot and what do the variables represent?

(This popping up on my screen should not have been, but was, the best part of my day yesterday:)

Follow-on problems once equation is achieved. Solutions using both the graph and the equation.

- What guess corresponds to the average distance from the correct guess?
- What did the worst guesser guess? The best?
- In what range did the better-than-average guessers guess?
- In what range did the worse-than-average guessers guess?

New problems and generalization:
Write an equation/inequality that models the scenario. Make sure to define your variables.
- Today’s temperature is 10 degrees off from the usual temperature.
- Today’s temperature will be within 10 degrees of the usual temperature.
- Today’s temperature will be more than 10 degrees off from the usual temperature.
- If the usual temperature is 68, find values for the three forecasts above using algebra. Show all work at every step.
- Graph the scenario. Indicate the three different forecasts on the graph.

- Write a general expression for the distance between a changing value and a known value. Define your variables.

- Put this equation into words: |x – 10| = 3
- Solve it, showing all work at every step.
- Write down/discuss a procedure for solving any absolute value equation.

- Put this inequality into words: |x – 10| < 3
- Solve it, showing all work at every step.
- Put this inequality into words: |x – 10| > 3
- Solve it, showing all work at every step.
- Write down/discuss a procedure for solving any absolute value inequality.

Feel free to poke holes in this or let me know how you would implement it differently. Also I need to get them solving and graphing more complicated equations and inequalities like say 10 = 2 |3x - 4| + 7, so I'd love to hear if you see any natural ways to make that happen. I haven't been able to think of any yet.

Wednesday, June 16, 2010

Where Mah Physics Peeps At

I usually talk about vector forces by pushing desks around. You know like 2 people push in the same direction, 2 people push in opposite directions, then they push on adjacent sides and the desk moves diagonally. But then we just make up forces and do practice problems.

Would it work to get 3 bathroom scales to measure the two component forces and the resultant force at the same time? Is there an easier way to do it than trying to balance a scale against a corner of a desk?

I would ask the Physics teachers at my school but I'd have to walk all the way down two hallways and I am very, very lazy.

Thursday, June 10, 2010

I Kind of Hate the Stupidly Ubiquitous Video Cameras

So in the last ten minutes of fifth period, awards had been given out, instructions about what to bring and not bring to the Regents exam had been given, and D plugged his ipod into my computer speakers to entertain us...fine, cute. He started dancing, so did another student, they were very talented, well-practiced, adorable, etc. So then D asks if I want to learn The Stanky Leg and I'm like "Sure! I hope I don't hurt myself! I am comically uncoordinated!" He starts trying to show me, and I start trying to imitate him, and I'm sure it was hilariously awful. But after like a minute, I look up and there are at least three cameras pointed at us. And I stopped. I couldn't make myself continue. I don't know how to feel about that. It would have been fun to continue, and I was fine embarrassing myself in front of these 20 people I've spent so much time with this year, but I wasn't fine embarrassing myself in front of the universe.

Tuesday, June 8, 2010

The Envelope Please

Riley thinks we should inject a little ceremony and gravitas into the last day of school. And I agree! He was lamenting the departure of his kids before he was able to properly see them off, but "luckily", my last day of classes is Thursday. I thought it would be cute to think of an award for everyone.

Awards I might give out if I can think of a few more:
  • The Heat Seeking Missile - for the Best Pattern-Noticer
  • The Bulldog - for the Most Tenacious Problem Solver
  • The Honorary TA - for the most prolific/effective peer tutors
  • The Librarian - "Sshhh! You guys! I'm trying to LEARN!"
  • The Scion of Pythagoras - for the most beautiful compass and straight-edge constructions
  • The Helpdesk - for the angels I can put on tech support duty on computer days
  • The Up Up Down Down Left Right Left Right B A - for the nerdiest t-shirts

I am still working on it. I have two whole days. I'll get there. But I had lots of other ideas.

Awards it would be impolitic to give but amuse me nonetheless:
  • The Your Tutor Gets an A+ Award - for the highest discrepancy between grades on in-class and take-home assignments
  • The Draco Malfoy - for s/he with the parent that made the most threats
  • This is Not the Phone You're Looking For - for the stealthiest texter
  • The Deanna Troi - for the Cleverest Hans
  • You Make Me Die Inside A Little - for the kid who tells me things like 0 is a solution to (x+3)/2x = (x-2)/x because undefined = undefined.
  • Are You Taking a Class Called 'Advanced Field Trip'? - for the most excused absences
  • The I Love You Man - For the girl who writes something like MISSNOWEEZIE IS THE BOMBDIGGITY on the whiteboard every freaking day.

Monday, June 7, 2010

The Personal Invitation

I have a trick for recruiting students for voluntary activities. For example, an enriching day of mathematics, or to contribute writing for community outreach, or to mentor some freshmen. And it's not to make announcements to whole classes to say, come talk to me if you are interested. That doesn't work.

I know people might object to this, because maybe it seems unfair, like opportunities are being limited. (Even though, as I said, the open invitation never works anyway.) But I think of which students would be good candidates. Who has appropriate talents and who will benefit. Then I ask them to take a lap with me around the building (the corridors make a giant rectangle), and I explain what I want them to do, why I think they're the right person to do it, how it will benefit them. And I ask if they would be interested.

It always works. Nobody has ever said no. They usually say something like "I am totally into that." And, they follow through. They jump headlong into these projects with enthusiasm and grace.

It's powerful, the personal invitation. To know your teacher sees something special in you, despite your maybe not being an academic superstar, despite whatever flaws your fears tell you are evident. It's hard to talk back to that.

But, I've been thinking, wow. I need to invite them to learn some math. Frequently. Not as a group - charismatic lecturing is not my forte. Not necessarily every kid every lesson every day. In a way that appeals to their individual talents. Because once it's given a chance, this stuff is startling, beautiful, descriptive. Once they know they bring something to it, and it can benefit them. I have no idea how to pull it off. But I need to find a way.