I’ve been thinking a lot lately about how I plan lessons and units, particularly lessons and units about things I’ve never taught before. (Partly inspired by this.) Here’s an anecdote describing how I planned a recent unit (that’s still going on) for my 6th grade classes. I like how I planned this and how my planning played-out, so I thought I’d share it with you.
Several weeks ago, I thought to myself, “Hmmm, I think it’s about time to do some work with circles in my 6th grade classes.” But, the crazy thing is, I’d never taught an introduction to pi before. Sure, I’d taught graphing circles to Algebra 2 students. And I’d taught quite a variety of things related to circles and pi to trigonometry students. But, in both of those situations, I’d been able to assume that my students had a working understanding of many important concepts surrounding pi and circles.
This blog may be called, “Recipes for Pi,” but I’d never before taught a group of students what pi is.
Wrapped up in pi are a number of ideas and skills: measuring circles, finding ratios, understanding what ratios of quantities in shapes tell us about the idea of “shape,” understanding irrational numbers, believing in limit processes, understanding the nature of infinity… This list encompasses some very complex and deep mathematical ideas that the most thoughtful mathematicians and philosophers throughout history have had immense trouble wrapping their heads around. (Consider Zeno, for instance.) But, once I’d realized that all of these things were related to pi, and that no understanding of pi would be complete without them, I couldn’t just say to my students, “So, there’s this number, pi. It’s something like 3.14159. And here’s how you use it to find the area of a circle.”
How was I going to get my students to really think about these ideas mathematically?
I knew that I wanted my students to somehow derive a value for pi. I also knew that I wanted them to come to an understanding of the significance of this number, and of ratios of lengths in shapes in general, through hands-on experience. I had no idea how I wanted to deal with the irrational-number thing, but I knew that it had to come up. And with that, the idea of an infinite process limiting to a finite quantity.
So, here’s how I planned our attack on pi: First, I made a list of the large questions that I wanted us to discuss at some point during our investigations. Here are some of my questions:
– How do you find the perimeter and area of a circle?
– What is the importance of pi?
– What kinds of criteria do we use to distinguish among shapes, and what measurements and ratios of measurements in shapes can be used to distinguish among the criteria?
– What is an irrational number?
– How can an infinite process produce something finite?
Then, I picked a place to start – deriving pi and working with quantities that distinguish among shapes – and decided what I wanted my students to do to achieve this. Meaning, I made a worksheet.
When I plan a class, I always start by making a worksheet. The plans that I make for lessons or discussions almost always come after I’ve made the worksheet – because I feel like I only really know what I want my students to learn after I’ve thought about problems through which they’ll be doing the learning.
The worksheet I made had four different regular polygons on it – a square, a pentagon, a hexagon, and an octagon. These polygons all had the same “diameter,” or distance between opposite vertices (for the pentagon, I used the distance between a vertex and the opposite side). I asked my students to use a ruler to measure each polygon’s diameter and side-length, and then to calculate its perimeter and area. Finally, I asked them to compute the ratio of the perimeter to the length of the diameter.
I decided to have them measure the side-lengths and diameter with a ruler instead of giving values because I wanted them to produce the data themselves. I didn’t want them to have the lingering suspicion that I’d rigged the data so that they would all get similar ratios – I wanted this to be a property of the shapes themselves, not the problems I’d created.
For homework, I made them a second sheet of polygons. Some of the polygons had double the “diameter,” others half. Before the students began measuring, I asked them to predict how the perimeter, area, and ratio of perimeter-to-diameter would change if the diameter was doubled or cut in half. They then were asked to repeat the measurements we did in class.
After they completed both of those sheets, I planned for us to spend time in class tabulating our data and making observations about patterns and trends. I hoped that they would find that the ratios were very close across the shapes, while the other quantities they were measuring and calculating were not.
Once I’d made these worksheets and thought about the discussions I wanted us to have about them, I spend a little time listing other activities we could do and discussions we might have. But, I basically stopped planning. That’s because I didn’t really know what my students were going to take away from these activities – and I didn’t want to plan anything that didn’t follow their trains of thought and interests. I try, as much as I can, after the initial raising of a topic, to let our investigations in class be guided by student interest. I thought about where they might want to take the topic and about how I’d guide them based on where they went, but I basically left it at that and trusted to the curiosity and intelligence of my students.
Briefly, here’s how all of this played out in class. The decision to make measuring part of the activity totally worked. Their perimeter and “diameter” measurements were all wildly different – because kids had used different units of measurement – but the ratios were all about the same, across polygons. This made the ratio-consistency all the more poignant.
The wildly different measurements also sparked the question that I’d most of all wanted them to ask: “Which measurement is a property of regularity of shape, not size or necessarily the exact shape – the perimeter, or the ratio between the perimeter and the ‘diameter’?” We had a very focused, deep discussion – generated by the students – about size, shape, and regularity, and the importance of proportion to each quality of a geometric object.
When they saw the ratio, in the polygons and in the circles we measured next, students who already knew something about pi said things like, “Hey – that’s pi!” and then rattled off a ton of decimal places. So we didn’t really spend time deriving a good approximation for pi. But, they did come to understand the importance of pi for circles and the relationship between ratios of measurements and the nature of shape.
And now questions about ratios and infinite-finite quantities and types of numbers have been cropping up all over the place… More about that to come!