New Job

I got a new job this year at a Catholic, all-boys school on the block system. So, in addition to teaching new subjects (geometry, statistics/probability, and algebra I), I'll have to adjust to all those things. If any of you have tips on any piece of it, feel free to share here. I'll try to post when more specific problems present themselves.

Matrix Multiplication

Another quick tip. This time how to multiply matrices.

Begin with the matrix multiplication problem:

Then move the first matrix down. [Note: Since matrix multiplication is not commutative, this is important. Although it should be noted that the same effect can be accomplished by moving the second matrix up. But under no circumstances should the reverse be tried.] The answer will go in the new space you have created in the bottom right corner. Immediately you can see (if the product is possible) the shape of the answer. In this example it is a 2x2 matrix.

Pick a position in the answer matrix and follow across from the left and vertically from above to figure out which numbers you will use. Multiply pairs beginning with the outermost numbers (the blue 1 and 7 in the example) and sum with the product of the next pair in until you run out of pairs. The answer will go in the position where the arrows meet.

Remember not to use numbers from your answer when computing other spaces. For example, the 58 was not used to find the 64 below.

Continue with each position until the answer matrix is complete!

But what if the matrices in question are not able to be multiplied?

Consider the following case. Although it initially looks like our answer will be a 2x2 matrix, we see that the 3 does not have a pair, so these matrices cannot be multiplied in this order.

Factoring cubic binomials

To be finished later, but here is a quick idea that came up:

Factoring binomials of cubes (x^3 + y^3) or (x^3 - y^3)

My mnemonic is "SQuiggy CHases Many Purple SQuirrels." It stands for:

• SQuare (the first term)
• CHange (the sign)
• Multiply (the two terms)
• Plus
• SQuare (the second term)

So, (x^3 + y^3) = (x+y)(x^2 - xy + y^2) and (x^3 - y^3) = (x-y)(x^2 + xy + y^2)

Hunger Games Sequence

I have a 45 minute drive to school each way. So, I've been picking up audiobooks to keep me company on the drive. Many of you have suggested reading The Hunger Games, so that was my latest acquisition. The whole thing is about 11 hours, I think, so I should finish it in about a week.

Anyways, getting through chapter one this morning, I listened to this bit:

The reaping system is unfair, with the poor getting the worst of it. You become eligible for the reaping the day you turn twelve. That year, your name is entered once. At thirteen, twice. And so on and so on until you reach the age of eighteen, the final year of eligibility, when your name goes into the pool seven times. That’s true for every citizen in all twelve districts in the entire country of Panem.

But here’s the catch. Say you are poor and starving as we were. You can opt to add your name more times in exchange for tesserae. Each tessera is worth a meager year’s supply of grain and oil for one person. You may do this for each of your family members as well. So, at the age of twelve, I had my name entered four times. Once, because I had to, and three times for tesserae for grain and oil for myself, Prim, and my mother. In fact, every year I have needed to do this. And the entries are cumulative. So now, at the age of sixteen, my name will be in the reaping twenty times. Gale, who is eighteen and has been either helping or single-handedly feeding a family of five for seven years, will have his name in forty-two times.

The Hunger Games by Suzanne Collins.
Chapter 1 as found here

Being a math teacher about to begin a unit on sequences and series in my Algebra II class (as well as a deeper study in Precalculus), I thought this would be an interesting problem to use. They are also all reading this book and seem to like it.

Teaching Resolution: Make More Mistakes

I have a problem with how I work in my classroom: I don't often make mistakes. I don't mean for that to sound conceited, but I generally like for people with authority to show few weaknesses and be both precise and accurate in their communication of concepts. It irks me when the principal (a former English teacher and currently moonlighting as a college professor for English education courses) writes an e-mail to the school that says something like, "We are starting this business that will be ran by..." or when our superintendent sends an e-mail to the entire district full of spelling and grammar mistakes.

I thought about it harder this past weekend, though and have come to a conclusion that this may not be the best strategy in the classroom. By limiting my own mistakes it sends a few false messages to students.

• It implies that it is bad to make mistakes.
• It implies that the concept or problem is (or should be) easy.
• It puts me on another plane than my students so that they think I am way above their level and they will never be able to attain that level of understanding of the subject.

So, I am resolving to make more mistakes in my classroom. Some will be intentional; some may not be. Just today in precalculus, for example, I was trying to number our examples as we went and I put up numbers 1, 2, 3, 6, and 7, then I asked them to work through those seven problems. Those students who were paying attention got confused and corrected me. Yay!

Vocabulary or Jargon?

While we're entering a new era or spelling in our culture, the debate reigns whether to join the revolution or fight for the original ways. It's a debate on which I can argue both sides and about one in which I have a hard time deciding my own position. While "kids these days" are writing things like "lol ur rong" does seem somewhat uneducated, it also gets the point across and isn't the point of language to communicate ideas?

While that example may fall more squarely into the English teachers' realm, I have an internal debate about it in my own classroom, too. How important is it really that I use the words "denominator" and "numerator," when it's so much quicker (and more easily understood) to say "bottom" and "top?" When a line with negative slope is changed to get more vertical, how wrong is it for me to say that the slope gets "more negative?"

I realize that there are some words that need to be defined and used correctly so as to avoid confusion. I just finished a section of Algebra 2 where we talked extensively about the difference between permutations and combinations. The words here are important to the understanding of the logic and the formulation of the mathematical expressions needed to solve these problems.

On the flip side, though, in precalculus we are discussing vectors. What is the real difference between calling two vectors orthogonal versus just the familiar "perpendicular" word they already know? Later we introduce the term "normal" for this same concept. Do we really need three words to express the same concept? (Or are they not the same concept and I'm thinking about it incorrectly?)

I realize that even the words "denominator" and "numerator" are important when we involve more complex expressions. But even they get confusing when we discuss things like "multiply both the numerator and denominator of the entire fraction by least common denominator of the terms in both the denominator and the numerator" to simplify an expression like

So, what say you? When teaching my students math, how important is it to indoctrinate them to the traditional language of mathematics and somewhat confusing vocabulary? Is it possible to communicate some concepts using more colloquial language?

iTextbook Thought

I have been intrigued by Euclid's Elements ever since I saw this website: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html. I'd always thought it would be a cool way to explore geometry. Begin with the basics and then see what you could do with those things.

So, my idea for an iTextbook is based around those ideas. You begin with some "simple" definitions and postulates as in Euclid's book. Maybe give some reasoning behind the postulates by asking questions like, "Why do we need 2 points to draw a line? Challenge: I am thinking of a line in this window. See if you can guess it with zero hints. (Student draws random line and it won't match what's in the computer.) Here. I'll give you one hint: This point is on the line (point appears, kid draws another line, it won't match). You missed because here are some example lines that it could have been (show a bunch of lines through the point). Here is a second hint: This point is also on the line (connect the dots = winner)."

Expand by asking students to create some tools needed. For example, show a picture of a pile of items that you might find in a junk drawer. These include, but are not limited to, some string, some push-pins, a pen, tape, scissors, a straightedge of some sort, etc. Ask students "How could you create a perfect circle (see definitions 15-16) from these items?" Once they arrive at a correct answer, they "level up." A screen appears with something like, "You have acquired the circle tool!" and the glittery circle button appears in their tool-bar at the top which works like it would in any drawing program (click for the center point, then click again for a point on the circle to fix the size).

Basically, you build geogebra tools by finding out how to construct each thing. You have the line-segment tool and circle tool, so Challenge 1 is available to you: construct an equilateral triangle using the tools you have.

Once you show you can bisect a given line segment, you get a new "midpoint" tool. This unlocks new challenges in which you need to use the midpoint for other constructions.

After you construct an object, you get follow-up questions like: In constructing the midpoint, do the two circles need to be the same size? Is the bisecting segment always perpendicular to the original segment? Does the original segment bisect the segment you created? etc.

It'd be dynamic as with geogebra or sketchpad or anything like that, but you can only use the tools when you "level up" by figuring out how to get build them from more basic tools.

There's a "sandbox mode" where you can play around with tools you have or preview what will become available later. Challenges available are based on the tools you have acquired so far, so they are somewhat ordered, but not necessarily linear. They could be labeled with a difficulty level. Unlocking new tools unlocks new challenges. Hints can be available for the tougher constructions/proofs, like in most games, too.

Anyways, what do you think? The idea just struck me this afternoon, so I haven't thought through all the angles (pun intended), but it seemed like an interesting way to introduce parts of geometry.