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If you are assigning scores to questions on a test, I used to make hard questions worth more points than easy questions, because I thought that they should carry a greater reward for correct answers. This caused problems because many students who had a decent level of understanding of material were not getting the questions worth the most points on their assessments.

My current approach is the exact opposite, an easy question should be worth lots of points, and a tough questions should be worth a few. The few points are enough to separate out kids who are good from those who are excellent, but students who do well on a test should not be receiving terrible scores. This tends to agree with our intuition about what a student’s total score should be.

I think I figured this out sometime last year, and I’ve been thankful for it ever since.

## Year Plans

One of the things I’d like to do this year is engage more students at their level. Perhaps try to have more class time where there are multiple activities/groupings simultaneously. The main idea behind this is that my one class, one-idea, one-option norm is almost guaranteed to miss the sweet spot for low and high learners. Daily I would think it means that someone in the room is spinning their wheels without traction.

So, I read The Differentiated Math Classroom and I’m picking through Differentiation in Practice: : A Resource Guide for Differentiating Curriculum, Grades 5-9. It was strange to go back to the old bound paper method of professional learning – most of mine usually takes place amongst you all on the internets. But, as far as I can see, the best resources for learning about differentiation are still printed, bound, and sold. They’re not yet offered up freely like so much of the great stuff folks are writing and sharing.

The first outcome from this is two simple documents that I drew up to provide a general overview of the units I plan to teach and the time frame in which I hope they’ll take place. The unit planning stage is where much of the thinking behind differentiation actually takes place, but this is a starting point.

Anyway, here are my year plans for Algebra and Geometry.

Scribd bug fix These documents showed up at lousy resolution for a long time. I wrote an email to scribd and had a response back in less than 24 hours (for a free service I’m impressed). Turns out that I had two lines in the latex code that were making the pdf tougher for the scribd reader to read. After deleting

\usepackage{emerald} %a package of free fonts
\usepackage[T1]{fontenc} %font encoding to go along with the free fonts

The pdfs are back to high resolution, easiliy readable, scribd documents. To the other person in the world, if you’re out there, appreciating this solution: you’re welcome.

## A principle for promoting good learning habits

Unless student exploration and problem solving are the task at hand, give the kids access to correct answers for every activity they complete.

I find that one of the biggest differences between the way that I complete problems and the way most of my students do is that I check the answer using the back of the back, my teacher’s edition, or software1. Students will complete a problem and go on to the next. Once they’ve written an answer for the last problem, the work is done.

I’ve been trying to change this for a while. Last year I started assigning only problems with answers in the back of the text for homework, and this year I’m requiring all problems that can be checked to be checked.

I also write activities to use in class. Up until last year I almost never had an answer key ready for them when I’d write them. This year I’m trying to be better. After writing a worksheet, I print it out, solve it, and translate my work into the answer key. Not only does this make it much easier for me to hand a reference to students in class, it makes it that much easier for students to verify their process and thinking.

Here’s a sheet I’m going to use tomorrow in Algebra

This answer key isn’t meant to be the only source of truth, and I’ll try to give student solutions the main stage, but I think providing a complete model (and having it ready) helps students to gain confidence in their own thinking as they work. It’s also a great way to answer the “am I doing this correctly?” when you are needed elsewhere. Very few students will try copy answers, and it’s always clear when they do.

There are times when I would withhold the correct answer/solution for students almost endlessly to force them to think their way through, and times when I want them to practice applying a new skill with answers close by. This is one of those times.

A few new files for equation solving practice are here. Click on a file to download it. The box is friendly and doesn’t add any pop-up junk.

## A little help with dimensions

As we finish off the year, in Pre-Algebra, I have students working on dimensions. Here’s a sheet I just finished to help them apply formulas for finding

• Area and Perimeter of 2-D shapes
• Volume and Surface Area of 3-D shapes

## Book Suggestion

I ordered, on suggestion of a former teacher of mine, a set of books called The Art of Problem Solving which comes in the basic and advanced varieties.

I’m planning on using them as supplements to class as well as required work for kids on the math team. Concise, full of examples, and easy to read, I think it will be a helpful addition in many ways.

## Reflecting, rebuilding, improving

If I were to take notes, I mean, since I took notes this entire year on everything I would like to do differently next year. It will be hard easy to correct all my mistakes. This should be done over the summer, when there’s not so much going on every day, erg, I think I need to be doing this right now.

First, a few parameters need to be set. I’ve developed a style in setting up procedures, like how to solve systems of equations by substitution, or how to apply the quadratic formula and find lines of symmetry. I need to add a coherent plan of inquiry to go along with the regurgitation of the procedures. A layer cake unit plan, that’s what I’m going for. This might be something like plotting a unit on systems of equations and listing out the raw procedures to be learned, the vocabulary to apply, trying to nail down the concepts and representations, plotting opportunities for inquiry on real problems, problems that beg for multiple representations and engagement. I’ll let you know how it goes.

A few units I’m currently working on:
Systems of Equations: suggestions for a smart title are warmly requested
Factoring

## Law of Cosines

I just lost a couple of hours to the law of cosines. It kicked my. So now, with my newfound knowledge I feel ready to have a go at class tomorrow. Thanks to the kind folks at the math forum I set it up with geogebra, schemed off about 5 pieces of scrap paper, and have a fool proof, coordinate proof-inspired strategy for explaining it. I also, however, have a bucket of lemons for the text book which presents the law, for the sake of presentation, I assume. I hope that it’s not common knowledge that cos(x)^2 + sin(x)^2 = 1 at this point in their careers. I also cannot think of how any of the other ways of deriving the law would make sense at this point in the game. I imagine that a few years off I will either not teach this or will present it in more of its glory, but for tomorrow, having an explanation is better than nothing at all. Mine looks like this.

Picture the triangle with A at the origin, B at the point (c,0).

Knowing the angle A, b, and c, we can find a (the length of the side opposite the angle at the origin). First we find the coordinates of C by drawing the altitude from C to the x-axis. C = (c cos A, c sin A). Then using the points (c, 0) and (c cos A, c sin A) we apply the distance formula as follows.

$\inline a^2 = (b-c \cos A)^2 + (0 - c \sin A)^2 \\ &= b^2 - 2bc \cos A + c^2\cos A^2 + c^2\sin A^2\\ &= b^2 - 2bc \cos A + c^2(\cos A^2 + \sin A^2)\\ &= b^2 - 2bc \cos A + c^2 \hfill \mbox{ note: } \cos A^2 + \sin A^2 = 1\\ a^2 = b^2 + c^2 - 2bc \cos A$

Having figured out how to use the law of cosines, we have the opportunity for a sweet detour into a little calculator programming. I bring you, the law of cosines in LCOSSSS – which will find all the interior angles, given the sides of a triangle.

The SAS version allows you to find the side opposite an angle given an angle and the two adjacent sides.

And the SAS version – given an angle and the two adjacent sides, find the side opposite the angle.

## Python, Phi, and Portland’s own Kirby Urner

I stayed home yesterday because of and woke up again today with a sore throat. So, being the fish that I am, I swam around on the internets for a while. I don’t remember how I stumbled on it, but I ended up at Kirby Urner‘s 4d solutions and read through quite a bit of his philosophy on teaching math, geometry in particular, and his belief in the necessity of teaching programming (especially in python) with math. I’m impressed, he’s also a Portland local.

Anyway, I was inspired to brush off the ‘ol skillz and see what I could contribute. I imitated some Mathematica code I found to write the python code below. The loop finds the Golden Ratio (phi)

a = 1 b = 1 for i in range(30):

 x = a + b a = b b = x print(i, x/a) 

There were a few ideas in Kirby’s material that struck a chord for me.

So if you really want to be an effective teacher, you’ll not get in the way of students surpassing you, and understanding in ways you simply do not. Let them also teach *you*. Make it a two way street, from the get go. [cite]

That’s progress in his view. The brightest light Urner’s work set off for me is the potential value of programming in the classroom. For example, instead of pulling up the calculator I could pull up a python command line to perform basic arithmetic. Using loops, I could show students how a computer can simplify/automate long processes. ALGEBRA!! Watching computers store values in variables, and generally being able to read code that ends up looking algebraic offers nice options to extend students’ thinking. I can picture a lesson in which code similar to that above was written and then Pre-runtime: asking students the basic CS question: what’s gonna happen?

If you’re not very familiar with python, then we’ve got something in common, but this little tutorial might make it seem within reach. I used camstudio to make the capture.

phi.avi – (slightly) better quality

## Marathon Planning Session

I just finished a bunch of plans and worksheets. Sometimes when you feel out of it, getting a bunch of planning done makes me feel like I’m back on top of the game.

Check out the new goodies in my box.net folders.

and

Hope you enjoy.

## What Can You Do With This: GeoGebra Standard Form

Dan has a great thing going with his What Can You Do With This posts. Here’s my first attempt at joining in. The motivation for this comes from an attempt at getting students to play with and by extension understand, or “have a feel for” the Standard Form of a linear equation (Ax + By = C). I’d probably put this on the board, play around with it, ask a few questions and miss the full potential of this as a tool to reach deeper understanding/application.

I’m looking for a better way to use this. The file allows you to change the values of A, B, and C using the sliders, also you can show and hide the graph. Feel free to download this and the rest of my materials using the box.net window at the top of the page.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)