Monday, May 6, 2013

Inside-Out

There is nothing better than noticing something new. Except, perhaps, understanding it.

Points A and B are stuck where they are on the circle, while point C can move around freely, affecting the placement of points D and E naturally. This creates two arcs (well, really four, but that's alright): Blue Arc, and Orange.

Now let's add some personality. Imagine point C is deathly afraid of touching the boundary of the circle. Point C wants to be as far from the boundary as it can be — do you know where C should stay? And if C does stay there, how will Orange and Blue Arc compare?

If you're looking for a loophole in my question, allow me to fill one! Consider two cases: one where C is stuck inside the circle, and the other where C is outside the circle. What happens to Blue Arc and Orange when C reaches its desired location?

Some sketches to help visualize: Inside Case, Outside Case.

Friday, April 26, 2013

Biggest Smallest Logarithm Challenge

Today was a good day. I stepped in for my mentor teacher for her College Algebra course and had a fun time introducing the logarithm problems commonly found on the standardized tests where two approximations are given (e.g., $\log 3 \approx 0.477,\; \log 2 \approx 0.301$) and they ask for a third approximation (e.g., what is $\log 6$). My planned hook actually captured their interest (very easy challenge: using only 2 and 3, make for me: 27, 6, and 3/4), they were excited to hear the junior would have the honor of graduating on the 400th anniversary of the invention (discovery?) of the logarithm, and they were able to do the practice problems with very minimal help from me! I felt like I was teaching how I want to be teaching, and I was very happy with the level of understanding the students were able to show me.

Here's a fun challenge that popped into my head as I was teaching the lesson. I didn't get a chance to share it with the students, but I might later on:

The Biggest Smallest Logarithm Challenge

While keeping $x$ as small as possible, make $\log x$ as large as possible (in the positive direction).

Now do the same, but with $\ln x$. Then check out $\log_7 x$.

Of course, depending on how you think about the challenge, you might have already figured out what $x$ to use for any base $b$ — keep $x$ small, but make $\log_b x$ big!

Sunday, April 14, 2013

Two Challenges for You

First, a simple warm-up:

I love this problem. There are just so many ways to decide what x is! I would have loved to have taken a day in groups to talk about this problem, then discuss as a full class what approaches we used/liked/thought best or worst. But alas, no time...

The first challenge: Find the measurements of as many arcs, angles, and segments as you can in this image.

I did in fact use this as a full-day review coming back from spring break. Of course, not in this format... my students would have immediately shut down! What I gave them was this image without the two slanted lines, and without the intersection points marked. They were asked to identify as many of the circle vocabulary words as they could in the diagram, then use the circle theorems we had learned to measure arcs and angles. They had a word bank for these, and we did a few together.

The second challenge:

My favorite problem, but unfortunately I felt it was too difficult to give to my students. (As a multiple-choice question, I suspect about half of them would get it right, but I'm not confident any of them would have correct reasoning for it.) Again, there are so many ways to argue what x is!

Let me know how you fare!

Tuesday, March 26, 2013

Me

I like my technology setup. There is a smartboard in the classroom, but I don't use it as a smartboard, just as a projector whose screen I can write on (the software / hardware doesn't play nice with my computer [also, the projector: the resolution is off, and I can't fix it!]). Everyday when the students walk into the classroom, they see two things projected side-by-side (winkey+Left/Right is awesome!): their Math Journal prompt, and whatever information they will need for the day / prompt.
 (I'm pretty proud of that GeoGebra applet. Students were impressed, too.)

After they've had a chance to write, we move into the activity+notes phase of the day. I have to be pretty loose with that word "activity" — sometimes it is students actively engaged with some math tool/concept (like measuring with a protractor an inscribed angle, guessing the measure of the inscribed arc, then measuring the inscribed arc with a degree ring and trying to describe the general pattern) but other times it is just me showing them something neat (like this awesome tangents+arcs visual by user griffinp). If I use the computer for that, it's generally full-screen, and I may write on the real whiteboard to highlight parts of the screen.

The notes I complete using Windows Journal Writer. I love that program, even if it does make my computer breathe heavily (I really should clean my fan...). Every teacher in our department uses the same set of Cornell Notes, so I've printed them as a Windows Journal file and I can write directly on them with whatever colors I want. I can highlight, add flags (although I never do that...), add textboxes, select and move what I write to a different location while changing its size if I want, delete entire selections or just strokes (or little square areas like a normal computer eraser), convert my horrible drawings to perfect lines or rectangles or ellipses... It's pretty much awesome, and it's included with (the tablet features of) Windows 7! Actually, I think Vista has it, too. Don't know about Windows 8, I don't plan to switch.
 Except choosing 'square' doesn't produce a square. 'Circle/Ellipse' is labeled better!

Then the end of class I'm trying to reserve for checking homework and some spiral review questions. I've used different platforms for that, sometimes Windows Journal for hand-sketches, SumatraPDF (so much better/resource-light than Adobe!) if I've created a document of them (which I would do with either LibreOffice's Export as PDF option, or through LaTeX), but my favorite is Notepad++. It's just a text editor, so it doesn't work for pictures, but it is amazing for text.
 The left document was my "to do" list for last week. Well, Monday through Friday, at least.

I say "trying to reserve" because it's really hard to actually do that... I feel so much pressure to "get through the notes" that I feel I'm forced to sacrifice practice time already, let alone 10 minutes at the end of class of basically un-directed checking (because really, 10 minutes before the end of class and the teacher is done talking — are you going to do work? They won't even work when they come back an hour later for a 20-minute 'tutorial' section!!). One tool I recently downloaded that I hope will help me with this is called "Compact Timer", which I found from Classroomtech. It lets me create timers of custom lengths, and even start multiple of them at the same time. It can count down for a specified length of time, count down to a specified time (i.e., a date&time of day); when it reaches 0, you can set it to sound the default alarm (which I find ugly) xor play a music file from your computer (I downloaded a chime track from SoundJay), or display a pop-up message, or even open/run a file you have saved. One day I tried setting it to open up my homework assignment list when ten minutes were left in class, but I forgot to set the window to open at maximum size, so it didn't actually take over the board like I had planned, and I ended up forgetting! But I've really liked it so far, and the option to play your own alarm music is a necessity, since I have a student who will definitely freak out if an alarm tone goes off without warning in class! (He has gotten acclimated to the bell schedule, though, so that's alright).

Paid tools (site license): Kuta software (Infinite Geometry) and a test generator that came with the textbooks (I think... I don't really play around with it, since we all use the same assessments, and the more experienced teachers create them).

My Students

The school currently has a "no device" policy for the classroom, which is interesting given that they will have 1-1 laptops next year for the students...

But a lot of the teachers are pretty lax about enforcing it. I was also willing to be lax about it, but that ended up backfiring when students chose texting over listening in math class. Now I'm extra strict: no non-math supplies can be out or on their desks, and if I see anything (phone, music, homework from another class... mirror being used to check teeth for nuts [really? In math class?]) I take it and hold on to it until the end of the period.

Before I was forced into this strict mode, I let students use their phones as calculators. One of the students had an app called MyScript Calculator, which is part of a broader MyScript platform which parses handwriting through the power of the internet (how awesome is that?? I would love to work for this company... combining linguistics, computer science, and mathematics! Three of my top five! [missing philosophy and religion]) The free app is able to understand what you type, and (almost) instantly compute the appropriate answer — throw in a ?-mark for a variable, and it will solve for the variable! Check out the promo over on youtube, but be warned: not everything is mathematically precise!

Oh, and did I mention next year each student will be given a laptop? I love the future! But a lot of the math teachers don't think it will be beneficial... and they are somewhat right, it will be hard to do algebraic manipulations and plot points on graphs and draw geometric figures on a laptop. Except, Kill Math, bust out Desmos and whyarewedrawingifwehaveGeoGebra???? (Caveats: I don't fully agree with the kill math ideology, and I actually do think it's important to know how to plot points by hand and use physical tools to do geometric constructions (rebuttal: the way these kids are pushed through algebra, they'd be better off being given a scrubbable calculator for the Geometry problems so they don't miss a new concept, and this school doesn't have physical tools for constructions, anyway.))

Future (for me to invest time in learning/figuring out)

The MyScribe stuff looks really interesting, and I see there's the potential I can get it for just my laptop+smartboard.

Similar to it is FluidMath, which also looks crazy awesome. "Simple gestures can quickly produce graphs, tables, computations, and more..." I was going to request a free trial over the Winter break, but I was sufficiently underwhelmed by the online version that I didn't feel like going through that effort. But the advertisements look amazing, so I do want to eventually try it out.

A couple of the teachers in the department have figured out the Haiku Learning Management System for their students; that's something I feel I should already know about, but for whatever reason it just hasn't come across my radar until now, and I've just been too busy to try and explore it. I definitely should do that, though...