Tag Archives: calculus

The Dance

I have the odd little habit of finding the prime factorizations of numbers that come across my path (mile markers, page numbers, etc.). I regularly factor whatever page number I’ve told my students to turn to in their textbooks. For some reason, this is awe-inspiring to them.

Yesterday I was speaking with my calculus class about a particular number being prime, and I explained why I stopped checking factors once I got to the number’s square root. “Wow,” they said. (Note that my calculus class is comprised of three students, all of whom have gifts in mathematics, but only one of whom would say he likes the subject.)

I replied, “Aren’t numbers beautiful?”

They asked if I had considered decorating my two-year-old daughter’s room in numbers. While I acknowledged the awesomeness of that suggestion, I corrected them: “Not numerals. Numbers are beautiful…the mathematics, the relationships between the numbers, the way they work with one another.”

D said, “But it’s a cold beauty.”

No, I said, it’s not cold at all; it’s a dance.

He countered that language is a dance.

I replied, “But what is mathematics if not a way of expressing ideas? Isn’t that what language is, too?”

D answered, “If only there were a way to test the temperature of a dance!”

They may, and likely will, graduate without having a love for mathematics. But they’ll know that it’s possible to have such a love, because they’ve seen it from me…and maybe they’ll be a little more likely to look for the beauty, to watch the numbers dance.

Puzzling Problems Preferred

Yesterday morning I determined that I would find something from my day to blog about, and hey! I did! :)

My geometry kids are learning about quadrilaterals. The prior day’s lesson was about properties of rectangles, rhombi, and squares. Here’s the warm-up problem I gave them:

In rhombus QRST, the diagonals QS and RT meet at point U. If QS=12 and RT=16, find the perimeter of the rhombus.

I think it’s a strong problem, because they have to recognize all of these things:

  1. that since the rhombus is a parallelogram, the diagonals bisect one another;
  2. that the diagonals are perpendicular;
  3. that the four small triangles are right triangles;
  4. that the Pythagorean theorem will allow them to find the length of one side of the rhombus; and
  5. that all four sides of the rhombus are congruent.

Almost all of my students struggled with the problem. I gave some hints – “What do you know about the diagonals of a rhombus? Okay, now think about triangles.” One student asked me if it was a “trick question;” I told her that it’s not a trick, just a puzzle.

But once they solved the problem, whether they needed hints or not, they felt proud of their work. That was awesome to me. Some of them told me that they really liked the problem. It was a challenge to them, so they had a sense of accomplishment once they had solved it. One student asked if I could give them more problems like that for their homework; the book’s problems are pretty simplistic and generally don’t require them to make connections across multiple concepts. I really need to get better about giving them more challenging problems for practice.

In other news, I am ordering sample texts for almost every mathematics course we offer for grades 6-12. The math department (all 3 of us) will be meeting at the end of March for a day of articulating who we are, and then using those decisions about curriculum. I’m really excited about this. :)

As part of that process, I’m reviewing the calculus materials available from The Worldwide Center of Math, and so far I am loving their model for the modern textbook. Check it out.

Historical Intro to the Tangent Line

I mentioned when I started this blog that this is my first year teaching math, but that I spent the last seven years teaching history. So it’s probably not a surprise that I find the history of mathematics interesting, though I’ll admit I have a LOT to learn in that area. I’ve pulled a few lessons from this CD-ROM, called Historical Modules for the Teaching and Learning of Mathematics, which I highly recommend. And then I developed a lesson for my calculus students as an introduction to The Derivative.

For full disclosure, you guys are actually getting the edited version. What I gave the students was overly ambitious and overlooked the fact that ellipses and hyperbolas have more complicated derivatives than parabolas, and I was really just going for something simple to ease them into things. I made some on-the-spot changes during the presentation of the lesson, and now I’ve come back to the lesson and changed it to what I really ended up doing.

All that said, here’s the lesson.

Some things to note about the execution of it…

  • I didn’t actually say the word “derivative” until they got to the very end. Their textbook doesn’t give them the word until the next chapter, but hey, they found it! Why not tell them that’s what they were doing?
  • Your students need to be familiar with GeoGebra in order to make this thing work. Mine ended up having to use the Van Schooten worksheet to find their tangent line slopes because they couldn’t actually do what I’d asked them to – this was the first time I’d told them to do something other than graph a function on GeoGebra, and it was over their heads. At the same time, they did learn a lot about what it can do and what doesn’t work! :) I’m hoping to have next year’s group prepared by working with GeoGebra a good bit while I have them in trig/analyt this year.
  • Don’t tell your students you just ordered a book on the history of calculus unless you want them to look at you like you’re insane.
  • My students didn’t know what “analytic” meant or how it was different from “geometric.” My own understanding of geometric solutions to problems has been greatly increased as I’ve done my thesis (on Descartes’ La Geometrie), so hopefully I was able to explain it to them well. I am much more of an analytic thinker than a geometric one.

My goal in using this was to have the historical setting provide a context for studying the mathematical concept. I think it worked pretty well. In any case, staring at the board where they’d done a lot of the work together and where I’d circled an expression and written “derivative,” one student said,

You know, I really do think I get this. It looks complicated, but it’s really not that hard – it’s just slope.

Wait, what?

I hate it when kids seem to really get something, it works great for them, they can do the problems…and then they ask me a question that shows me they missed the point of the whole thing.

My calc kids are taking a quiz over the precalc review stuff. One of the questions asks them to find a natural logarithm regression equation for a set of data. Should be no problem – they’ve been doing great with that.

But one of them just came up and said, “I found the equation, but when I look back at the x-values from the data we were given, it doesn’t have the right y-values.”

Now, I know it’s not the stats class, but still, I didn’t realize that they didn’t know what a regression equation is all about. I’m glad to realize it now, but I hate that I was just having them find regression equations without understanding what they were doing. Sigh.

So, what to do about it? I think I need to be more careful, more deliberate, about making sure they understand concepts that I think should be prior knowledge for them. I need to stop assuming that they know something because they can execute an algorithm; that doesn’t help them learn, and it will end up causing me frustration down the line when I want them to build on a concept they never had to start with.

Learning Targets and Brainstorming

Status on Learning Targets…

  • Calculus: done
  • Statistics: not all the way through the text, but I’ve gotten through everything we’ve done in the class I’m taking, and I can add to it later
  • Trigonometry & Analytic Geometry: done, unless I get through things and have time for an extra chapter, at which point I’ll add in those LTs
  • Geometry: just opened the book (I’ll spare you a link to an empty document ;))

If anyone feels like giving me feedback, please do so. This has been a bigger task than I anticipated!

Now for some brainstorming…I have an idea for how I want to structure my units. For each unit, I want to put one or more real problems (of the WCYDWT variety) up on the bulletin board. The idea is that I will choose these Big Problems (don’t I come up with creative names?) as things that can be found using what they will learn in the unit. We’ll keep looking at them as we go through the unit, evaluating how our new knowledge can help us reach new understandings about the Big Problems. (I totally need a better name than Big Problems.)

What I want my kids to recognize is that math can be useful. I think it will also push them in the right direction for completing each semester’s final assessment, assuming I stick with that idea.

So once I get my Learning Targets finished, I think I will tackle the first couple of units for each class and figure out…

  1. Big Problems
  2. Assessments
  3. Instruction
  4. Homework/practice sets

I think doing that will help me feel ready to start the year, though the more units I can prepare ahead of time, the better. Oh, and I also need to work on my Policies & Procedures sheet (to explain things like my crazy new grading idea).

Starting to make LTs

Today has not been particularly productive; Little Precious had a fever of 104.7 at 2:30 this morning, so I have spent the day (after my class) taking her to the doctor and subsequently hanging out with a sick and clingy toddler.

Yesterday, though, I got through 3 chapters in the calculus textbook and set up Learning Targets. It feels like I have too many of them, but I don’t know that I really want to combine them, because I do want students to learn each item on the list. I borrowed heavily from one of the sample syllabi (pdf alert) that the College Board has up for AP Calc AB to create the Learning Targets; the course outline is aligned with the text I’ll be teaching from, so I used that as a starting point and just adapted a bit.

So take a look and let me know your thoughts on the first part of my LT list. I am thinking I’ll end up using problems that address multiple LTs. And…there was something else I was going to say here, but a few interruptions from Little Precious have made me forget what it was. Oh well.