Writing Systems of Quadratics to Solve by Graphing

This week I was making up systems of quadratic equations for my kids to solve by graphing. In my experience this is the sort of task that seems like it should be easy but which I have spent a number of years doing poorly, carelessly, and at length!
Here's my new trick:
1. Choose two binomial factors whose products will have integer roots, i.e. (x + 3)(2x - 4)
2. Multiply those factors.
3. Set the product equal to zero.
4. Use inverse operations, your own creativity and the principle of equality to move some part of each term to the other side.
5. Those are your equations, and the solutions will be {-3, 2}, the roots of your original quadratic.
Voila!

At this point you get to play with how nice the roots and vertices of your quadratics are, but it doesn't matter much to me for solving systems by graphing. Of course then you can plug your equations into Wolfram Alpha to check yourself.

Hope this helps!

New York Math Circle Summer Workshop 2010!

If you look at my first posts on this blog, you'll see that many of them are my notes on the PD I received at the New York Math Circle Summer Workshop I went to last summer. This summer's workshop proposes to be even better, with a week of study of the Pythagorean Theorem on Bard College's beautiful campus.
The NYMC instructors and Bard Professors that are leading workshops are tremendous mathematicians and experienced teachers and I highly recommend attending.
They have recently extended registration to April 30, which reminded me to let you know about it.
Go here for more information and registration.
There's also an information sheet.

Go and enjoy!

Go to the Kaplans' Math Circle Institute!

After reading Out of the Labyrinth and meeting Bob and Ellen Kaplan last year, I had the immeasurable pleasure of attending their week long intensive at Notre Dame last summer. It was the best professional development I had attended up till that point: I learned a ton of math, I got to teach a bunch of math circles, and I built professional relationships that continue to sustain my enthusiasm for teaching today. I am so inspired by the work that the Kaplans do, and highly encourage anyone interested in cultivating more enthusiasm and fascination in themselves and their students to read their books and go to their intensive this summer.

"Since 1994 The Math Circle at Harvard and Northeastern Universities has made math a source of intense delight and collegial enjoyment for students from 4 to 70. Our approach is to have the students do the discovering and proving for themselves, in friendly conversation.

Branches have now opened elsewhere in America, and abroad, thanks to the Summer Institute we hold for a week on the campus of Notre Dame. This summer it meets from July 5th to 11th. The cost is $800 for room, board and all expenses (except travel). If you are interested in running a Math Circle yourself or using its approach in your classroom, please contact Bob and Ellen Kaplan: kaplan@math.harvard.edu."

Math for America deadlines approaching...

Having benefited immensely from Math for America's financial, educational and professional community support for 5 years, I am delighted to have the opportunity to share the info about two of their amazing fellowships for NYC public school teachers.

Spread the word and check out their website.

"Math for America has an exciting new Early Career Fellowship beginning the 2010-2011 school year for first, second and third year math teachers of secondary mathematics in New York City public schools. The Fellowship provides four years of professional support and growth opportunities for teachers early in their careers. The deadline is May 7, 2010. Benefits, eligibility requirements and application deadlines are available at Math for America.

Math for America's Master Teacher Fellowship rewards exceptional New York City public secondary school math teachers (with over four years of experience) with a four-year Fellowship. The deadline is May 21, 2010. Further details on the program, including stipends and professional development opportunities, can be found at Math for America."

Inappropriate Models

We're trying to come up with the generalized process for solving equations. Now that we've got combining like terms and distribution and variables on both sides and sometimes no numbers, just a bunch of different letters and all the combinations of that stuff. What is it that we're really doing when we solve equations?

The supervising mentor that observes and works with our student teachers is a cool old guy, used to teach math, was a principle, still works in schools, and has no shame and no hesitation. He's smart and cheesy in the most compelling way and he fascinates and weirds out the kids when he comes. It's great. He's awesome. Last time he was here, I overheard him talking to a kid about how when we solve equations, we are trying to get x naked. That's why it makes sense to do inverse operations in the opposite order you use to evaluate, because when you're undressing you take off your shoes before your socks.

So I was trying to go with that, and here's what I came up with:

1- Who do you want to get naked? What do you want to solve for?
2- Focus their attention on you. Simplify the environment. Simplify the sides of the equation.
3- Get 'em in a room. Get all the variables (you're into) on one side of the equation.
4- Take off their clothes (shoes before socks, remember!) Use inverse operations to get variable alone.
5- Do they look good naked? Check your answer: does it make the equation true?


I almost told my kids this today, and ended up garbling it in an attempt to make it somewhat appropriate and talking about changing baby's diapers. I actually said "OK, wait, I'm thinking naked kids. No, I mean little kids," out loud. It was hilarious and memorable but pretty gauche.

I thought you guys might have some good ideas, either from experience or improving on this one. I want to laugh this much at school everyday.