# Tagged: circumference and diameter

# Physics Week 1 – Circles Lab

Last week was our first week with students for the school year. For the first time (I think ever) I’m actually quite pleased with how my classes went. This post will outline my week in Physics I class.

Physics I is class that consists of Juniors and Seniors. All have taken Science 9 (mostly physical, but other as well) and 10th grade biology, some were in Chemistry last year but not all. Mathematically the range is from Algebra 2 to Trigonometry to Pre-Calculus.

This year I’m trying Unit packets (i.e. Kelly O’Shea), so on Day 1 I handed out my Unit 0 Packet. More on Unit 0 later. Using a packet will free students from individual Lab Notebooks and hopefully guide them to focus on lab content rather than organization. The strategy allowed me to by pass all the Lab Notebook setup stuff I’ve done in the past. So I was able to just get started.

I began with a circle drawn on the board. My first question was, “How could you describe this shape, WITHOUT using the word circle.” Things such as ’round,’ ‘curved,’ ‘360 degrees,’ ‘all points equidistant from the center,’ were stated. Good enough.

Since I introduced this lab as and exercise in measuring I next ask the students what could we measure about the circle. The usual things were brought up: radius, diameter, circumference, area, and maybe something else. Next I said we needed to narrow our list down to the two easiest things to measure. We agreed upon circumference and diameter. I have a classroom set of 10 meter tape measures, so from there I let them go.

I made sure to emphasize that this lab was a MEASUREMENT lab, because a few students always try to get away with calculating circumference. One thing I did not mention was whether to measure in inches or centimeters. I got about a half and half return. I set students free to find circles around the school building, with the goal of measuring between 10 and 12. This takes about 30 minutes.

The next day the students returned to class with all of their measurements. From there it is time to graph. In the past I’ve gone directly to LoggerPro, but this year I decided it was important to at least draw a couple of graphs by hand. I gave them the following guidelines:

One your Graphs be sure to:

- Use Pencil
- Label your axes with symbols and units
- Give the graph a title (“[vertical axis variable] vs. [horizontal axis variable]”)
- If the data is linear, draw a Linear Fit line (don’t connect the dots)
- Find the slope of the line using points on the line (not data points)
- Write the equation of the line (EOL) using the variables from your axes (not “x” and “y”); make sure the slope and intercept have the correct units attached to the numbers.
- Put units on numbers, but never on variables

I did NOT tell them which axis which variable need to be on. I again got about a half and half return, which is good for the whtieboard discussion. I did have to do some “teacher talk” regarding the Linear Fit line and the EOL. Since these are things that we will be doing ALL THE TIME I feel it was important to spend a few extra minutes to introduce common terminology and expectations. Hopefully it makes my life easier in the future.

Once graphs are drawn by hand on paper each group transfers their graph to a whiteboard. It doesn’t need to be a perfect graph, but something that shows approximate locations of data points, and most importantly the Linear Fit line and the EOL.

The final piece to the puzzle is the whiteboard discussion. We take our time presenting and discussing things such as methods of measurement, variety of sizes measured, why they decided to put C and D on the axis they did, how they calculated slope, etc. We then compared all groups graphs to each other.

The first thing that was noticed by the students was that graphs that were C vs. D were different than D vs. C. GOOD! Then they realized that the graphs that were done the same all had slopes that were very close to each other.

Me: INTERESTING, why would that be?

Students: Because we all measured circles.

M: Oh, ok, so what does this slope mean then (looking at the C vs. D graph)?

S: Well they are all around 3.

M: Exactly 3?

S: No, just a bit more.

M: Interesting

S: Pi

M: Who had pie? I like pie.

S: No the slope is pi.

M: Cool, what about this one? (D vs. C graph)

S: Inverse of pi

M: What does that mean?

S: 1 over pi

M: Why?

S: The variables are graphed on the opposite axes.

Ok, you get the idea, but more importantly SO DID MY STUDENTS!

Based on all of this discussion we are now able to write and mathematical model for circles. From the C vs. D graph we get **C = πD**. From the D vs. C graph we obtain **D = (1/π)C.** Someone usually notices that these models are one in the same. This is awesome, because most students realize they have seen these equations before in some form or another, but never had the experience of knowing where they come from. Thus I give them one of my favorite sayings: **“Equations come from experiments NOT textbooks!”**

The very last thing I added this year was a quality addition. I took time the next day to do a summary of how models were used in this lab. On the whiteboard I created a summary that looks like this:

The final thing I wanted to do was come up with a definition for the term Model. With a little guidance both of my classes agreed on the fact that a **MODEL IS A REPRESENTATION**. I can now refer to this simple definition anytime students wonder what the heck I’m talking about when I call and equation a model. It simply represents the situation in a mathematical form.

I absolutely love the way this lab introduces my students to just about everything we are going to do in physics this year. Defiantly worth the 3+ days to complete.

WOW! How can something as simple as measuring circles turn into this ridiculously long post?