Divide By Zero

I’ve been still trying to wrap my head around factoring, and I think I’m approaching a much reasoned presentation. The work I gave out today in both Algebra classes was more successful than it’s been before. In one class I gave them a very much scaffolded set of problems and explanations to help them tie together much of what we’ve learned. I haven’t made an answer key.

Ws Factoring Overview All Techniques

Hoping for a similar effect I used this sheet to introduce factoring ‘s by splitting the middle in a different period. The justification I’m giving for splitting the middle is one I think extends beyond some of the simple type expressions. Anyway, this isn’t a magical topic, but I think my students this year will have a more solid ability to reason about factoring than before. That’s a good thing, I suppose.

This starts on the 3rd page ‘cuz that’s where I kicked off the page with the class.

ws-ax2+bx+c-intro

I’ve got one more to share, which may get used later.

ws-ax2+bx+c-practice

In other news, I thought I’d post a picture of how I get this generated. I write the worksheets in LaTeX using Emacs to write the code, and I use Maxima to help simplify and factor expressions so that I know they’ll be correct on the worksheet. Here’s a snapshot of how that all works.

My Geometry class is entering the 3-d part of the year. But instead of watching Avatar, we’re studying Area, Surface Area and Volume. Or at least that’s what the last three chapters in the book are. Thing is, in some ways the book throws a ton of easy pitches to the kids:

Find the surface area of this shape, given only the relevant dimensions and nothing else.

Same thing for volume and everything else. I thought I’d kick them off with a little detour to get them thinking about why we care about things like surface area and some of the problems that arise from surface area calculations as applied to real problems. I told the class that we were going to spend a day doing a little “art appreciation” and with no other introduction started the following slide show of images happily lifted from Christo and Jeanne-Claude’s site.

Christo Project

The slides naturally spark lots of “is this really art??? I could cover this place with stuff!” type sentiment. I took some straw polls like “How many of you think it would be easy to wrap our school building?” a split favoring those who voted “not easy” and then made the easy votes describe how they’d go about getting the school covered. We continued through the projects, discussing issues that seem to come up naturally whether or not they seemed especially mathematical. We stopped to read the press-releases as they were in the order. And by the time we reach the “WWCAJCD?” pyramid slide, they have lots of ideas about a project that might be done and the issues involved in making it happen.

We’ll be working on the project for two more days. Part of my hope is that good questions arise, and that students feel a little more confidence in working out solutions because there isn’t necessarily an answer they can check in the back of their book. This problem has not already been solved.

I haven’t taken the time to write, and I’ve barely been keeping up with all the great posts being written these days. Very big thanks to Dan for summarizing the many sessions at NCTM and NCSM.

I’m enrolled in a class titled Problem Solving for the MS Math Teacher. The class is great. We get three problems for each HW assignment, and they really forced us to justify everything about our solutions. If you set up an equation or a formula it needs to be justified, if you perform a calculation the theory behind it should be clearly explained. The emphasis on reasoning has been a great push, and it’s really valuable to see how so many others reason through the same problems. Here’s a sample of the problems we’ve been assigned

• Saved by Zero. How many zeroes occur at the end of the expanded numeral 999!?
• The Last Straw. Two piles of straws are on a table. A player can remove a straw from either pile, or a straw from both piles. The player who takes the last straw loses. If there are two players how should you play?
• The Case of the Grouchy Customers. Every morning at local cafes sleepy customers stumble in for their morning cup of coffee. One such cafe has a row of 10 seats at the counter. Typically, morning customers do not like to engage in conversation. How many different ways can three customers sit in those 10 seats so that no two customers are sitting adjacent to one another?
• Put Down the Ducky. A man selling ducks sold half his flock and half a duck to Amy. He then sold one third of what was left and 1/3 of a duck to Beth; then 1/4 of the remaining flock and 3/4 of a duck to Cathy, and finally he sold 1/5 of the remaining flock and 1/5 of a duck to Dina. He now has 19 ducks and he never cut a single duck (whew!) What was the size of the man’s original flock?

It’s been a fun challenge to figure out how best to explain myself as I approach these, and some of them have been good problems to have on hand for students.

I’m also taking Intro to Computer Science and Programming for no credit through the MIT OpenCourseWare offerings. As a model for online learning I love that MIT is doing this. I have already found myself applying python to multiple problems beyond this class in a more sophisticated manner than prior to beginning the lectures / readings / problem sets. It’d be great to be able to get credit for this, but you can’t beat the cost.

The entire offering is fantastic and well curated. A lecture I recently watched was entirely re-taped because technical glitches ruined the video of the original class session. The professor entirely re-did his near hour long lecture for the sake of OpenCourseWare. It is lecture based, so not for younger folk, but if you have the patience to really work through the materials provided you really could educate yourself with little more than internet access and time. Take a look at the offerings if you are interested!