Author: samjshah

Pendulum Lab, Reprise

This post refers to the pendulum lab I recently posted about. I had my students collect data, and this is what they got:

Length (inches)

Period (sec): Group I

Period (sec):
Group II

Period (sec):
Group III

Period (sec):
Group IV

60	2.57	2.57	2.49	2.50
54	2.36	2.38	2.34	2.50
48	2.31	2.29	2.17	2.23
42	2.16	2.06	2.03	2.09
36	2.07	1.96	1.86	2.00
30	1.82	1.89	1.77	1.79
24	1.64	1.67	1.64	1.63
18	1.44	1.43	1.42	1.39
12	1.12	1.14	1.12	1.16
6	0.86	0.83	0.87	0.78

The data was pretty consistent among the various groups. Remember we did this lab in the context of parabolas. However, when the groups plotted their results, they were getting:

What?! The data doesn’t look quadratic. If anything, it looked to us like a square root or a quadratic with a negative $x^2$ coefficient. We used our calculators to do a quadratic regression, and got (for one set of data):

I tricked my class into believing this was a good model. I mean, look at it! The parabola fits the data so well!

But then we looked at the $x^2$ coefficient and saw it was negative and nearly zero. And then when we expanded our domain, we got:

It was at that point that students saw how our model sucked. Because they said that if we increase the length of the pendulum, the period should increase too.

So we went back to the drawing board. I suggested that we plot period versus length, instead of length versus period. (Next year I’m going to have us discuss this idea more — the swapping of x and y coordinates, and how something that looks like a square root might look like a parabola if we do that. Because of timing issues in this class, I just told them that was what we were going to try. Sigh.)

And we did, and found the quadratic that modeled it, and saw:

And then we extended it to see that if we increased our period, if the length would increase too…

It does! It makes conceptual sense too! (We also talked about whether it should hit the origin and why our model does or does not hit the origin.)

With our newfound analysis, I had students answer the following question based on their “good” quadratic model (in our case above: $y=10.57x^2-4.62x+2.85$ ):

(a) If your pendulum has a period of 1.5 seconds, estimate the length of the pendulum.
(b) If your pendulum has a period of 20 seconds, estimate the length of the pendulum.
(c) If your pendulum has a length of 10 inches, estimate the period.
(d) If your pendulum has a length of 1,200 inches, estimate the period.

What is nice is that (a) and (b) just involve students plugging in $x=1.5$ and $x=20$ into their model. And all the groups got very similar answers for the first length, and really different answers for the second length. So we got to have a short (I wish it could have been longer) discussion of why that is so. (We talked about interpolation versus extrapolation.)

And then (c) and (d) involved students solving a messy, real world quadratic because they’re setting $y=10$ and $y=1200$ . The same thing that happened in part (a) and part (b) happened in part (c) and part (d); all the groups got very similar answers for the first period, and really different answers for the second period.

What we didn’t get to talk about, unfortunately, is the theoretical answers, based on physics. The formula for the period of a pendulum is $T=2\pi\sqrt{\frac{L}{g}}$ where $T$ , $L$ , and $g$ are in standard metric units. So I was hoping we’d get a chance to do some unit conversions to see how our experimental data relates to to theoretical data.

I did get to show my students how their values compared with the theoretical data:

Length of String	Group 1	Group 2	Group 3	Group 4	Theoretical
60	2.57	2.57	2.49	2.50	2.48
54	2.36	2.38	2.34	2.5o	2.35
48	2.31	2.29	2.17	2.23	2.21
42	2.16	2.06	2.03	2.09	2.07
36	2.07	1.96	1.86	2.00	1.92
30	1.82	1.89	1.77	1.79	1.75
24	1.64	1.67	1.64	1.63	1.57
18	1.44	1.43	1.42	1.39	1.36
12	1.12	1.14	1.12	1.16	1.11
6	0.86	0.83	0.87	0.78	0.78

I think they were impressed, though I didn’t get the ooohs and aaahs I was hoping for. I’ve plotted the theoretical (purple) with the actual data (yellow) so you can see how good the experiment was. I am not plotting it on a period versus length graph, though if I were to show my students, I would do that because that’s the way we analyzed the data (we got a parabola).

And with that, we finished our lab.

Moore’s Law

The Technology Review magazine has an arresting photo essay on Moore’s Law — as told through a bunch of stunning pictures of computer chips. Click on the link above to see all the other circuits. For those who don’t know, Moore’s Law says that about every two years (some say 18 months), the number of transistors that can fit on a circuit doubles (for Wikipedia article, click here).

The only thing I wish about the photo essay is that there was some sense of scale for each picture. Regardless, the captions tell the year each circuit was created, and the number of transistors on each circuit. The data are:

1958	1
1959	1
1961	4
1974	5000
1979	68000
1978	29000
1985	275000
1991	200000
1993	3100000
1993	2800000
2000	42000000
2007	410000000
2009	758000000

So of course, even though this data isn’t perfect nor complete, I thought I’d see how it’d look graphed.

Ohhh, it looks like it could be exponential… Let’s plot it on a log-scale. If it’s exponential, we should get a straight line:

Ohhh, this looks pretty linear! I wasn’t sure that it was going to work out.

The exponential line of best fit is: $Transistors=e^{0.397*\text{Year}-777.29}$ . When I plot the data (pink) and the exponential line of best fit (blue) on the log-scale graph, you’ll see that Moore’s Law looks like it has some serious bite to it.

Doing a little algebra with the exponential model we came up with, it appears that the number of transitors doubles about every 1.75 years.

And if you cared, Wikipedia gives their own following graphical illustration of Moore’s Law:

I’m going to be teaching exponential functions in a bit. I hope we’ll have time to do regressions. If so, I’ll probably make a 2-day investigation out of Moore’s Law.

Other posts I’ve made about logarithmic and exponential functions:

Logarithmic Graph in the News
Earthquakes, Richter Scale, and Logarithms
The Supreme Court, Linear and Exponential Growth, and Racial Segregation
The Origin of Life on Earth and Logarithms
Paper Folding and Exponential Functions

Pendulum Lab

This week I’ve had one and a half Algebra II classes to “kill” because I’m ahead of the other teacher and we need to sync up again. Since we’re working on parabolas, I thought we could do something fun.

A while ago, I watched this video:

And I decided, perfect! I’m going do the pendulum thing in class. I got some string and washers, and put masking tape on the string every 6 inches. And I had student calculate the period of the pendulum when the length of the string was 6 inches, 12 inches, 18 inches, … , 60 inches.

To minimize the error generated by a student not exactly stopping the stopwatch when pendulum swung back and forth, I had students have the pendulum swing three times. That way any reaction time error of the person operating the stopwatch gets reduced by a third! And I had students do 3 trials for each length of string, to further minimize error.

It took all 50 minutes for students to collect all their data, plot it on a graph, and enter the data in their calculators.

Tomorrow we get to have fun. To warm up, we’re going to talk about sources of error. Then each group will get to share their graphs and talk about their findings. Then we’re going to perform a quadratic regression on our calculators, talk about if we have a good or bad model and ways to decide, and then use our model to make some predictions. (If we know the period, can we find the length of the pendulum, and vice versa.) Then, I’m going to conclude by showing students the theoretical formula for the period of a pendulum ( $T=2\pi\sqrt{\frac{L}{g}}$ ) and we’re going to see if their collected data matches up with the theoretical predictions.

The best part about this is that all the groups data seems in line with each other, and in fact, they are all really close to the theoretical predictions. I can’t wait to see if there are oohs and aahs about how accurate these data points they got are to the theoretical predictions.

I’m excited for tomorrow!

UPDATE: my data collection sheet (PDF), my lab debrief sheet (PDF).

Digital Artifacts

Dan Meyer has recently been asking teachers to consider (a) the problems with textbook application problems and (b) finding or creating compelling content for the classroom.

I’ve been thinking about this since he’s started posting, and I’m trying to think of where I can find — but not create — the interesting digital media that would satisfy his criteria.

In the meanwhile, I decided to look on youtube for some videos that would naturally lead to a particular discussion/topic for some high school math class. Some are better than others. Without further ado…

Inequalities and Quadratics

In Algebra II,, we’ve recently been delving into quadratics. I recently blogged about how I taught completing the square and the quadratic formula, and put up a bunch of resources. Since then, we’ve moved on to graphing quadratics, followed by inequalities.

The complete topic list for inequalities is:

I’ve been trying something new, which is creating packets for students to work on. In essence, I’m creating my own textbook for these sorts of questions. I thought I’d share them with you in case they prove useful [1]. I’m pretty proud of them — and they way they fit together and build up understanding, not just providing a method to solving problems.

1. PACKET 1: Linear and Quadratic Inequalities on the Number Line (PDF version)
2. Additional Homework on Quadratic Inequalities (PDF version)
3. PACKET II: Linear and Quadratic Inequalities on the Coordinate Plane (PDF version)

4. PACKET III: Systems of Inequalities (Linear and Linear-Quadratic) (PDF version)
5. Additional Homework on Systems of Inequalities (PDF version)
6. Pop Quiz on Inequalities and Quadratics (PDF version)

Hopefully they’ll be useful to someone else out there!

[1] The formatting might be a bit off for you… It looks slightly off (meaning the pages don’t end where I intended them to end) on my mac but fine on my PC. I think you need to make sure that on a Mac you select all and convert the font to “Gill Sans” (on a PC, I think it’s called “Gill Sans MT”, which is creating the problem).

UPDATE: PDFs posted, without typographic weirdnesses.

On Time

Last week I was rushing to get to one of my classes, after not being able to get away from one of those hallway conversations with a fellow teacher. I made it down the four flights of stairs and as I walked into the room panting, the students — looking at the clock — hollered “just 5 more seconds Mr. Shah… if you came in just 5 seconds later you would have been late!” Indeed, they were right.

This moment struck me deeply.

Why? This is the moment that tells me I’ve succeeded. I have clear expectations with my students regarding certain things, including being on time to class. And I make damned well sure that if I give myself permission to call out those who come in late, I’m going to be in the classroom every day on time.

I like that they’ve internalized it. This moment could never have happened otherwise.

Another failed AMC 10/12

So my school offered the AMC 10 and the AMC 12 (math competitions) last week. And although we have a number of pretty strong math students, none of them broke the 100 point mark. No student in recent history — apparently — at my school has done that. The fact is that to do well on these competitions, you have to be familiar with the types of questions and methods to solving them. There are techniques to doing well, tricks that any student who has seen enough of these can put away in their mathematical arsenal. These contests require a different way of thinking, a different way of approaching problems.

I am the faculty adviser for my school’s math club, and I just go with the flow. I listen to what the students want and we do it. Sometimes students bring an interesting problem or an extension of a problem. Sometimes I bring a problem. Sometimes we watch a video. Sometimes we work on contest problems.

Last year and this year, the students haven’t wanted to dedicate time outside of math club to doing math. We have only 25 minutes a week to meet. Well, frankly my dear, you can’t get through much in that time — especially if students don’t want to concertedly work on problems outside of math club, and then use math club to present solutions or failed methods of attack. When it comes down to it, to do well on these contests, you need to practice.

I’m okay with students not wanting to spend time outside of mathclub working on math problems. They are all busy and well-rounded and are juggling a ton of different things.

Still… I am waiting for the day when someone in mathclub says: “Everyone, I am going to solve every one of these 25 competitions problems by next week” and goes at it. Whether or not they succeed, it’s irrelevant to me. That’s the kid I want to take under my wing.

PS. I finally got around to taking the AMC12 under testing conditions (75 minutes, no calculator). I scored a 108. Which is around the same score I got last year and when I was in high school.

Continuous Everywhere but Differentiable Nowhere

I have no idea why I picked this blog name, but there's no turning back now