I'm a fifth grade math teacher in a medium-sized k-5 school and was wondering if you had any thoughts on a recent question that came up in class. We're in the midst of our end-of-year review and last week we went over the main points of geometry. We were discussing circles, (the applicable formulas for area, perimeter, of course the main terms (diameter, radius, the methods of construction, etc.)
The question, which came from a student who is usually shy to raise her hand with a question, preferring to wait until we break for recess when her classmates are out of earshot, was simple:
How can anyone really prove that the formulas are correct or even close to accurate? Specifically, she wanted to know how anyone can say exactly what the area of a circle is, since no matter how much you cut and paste and rearrange a circular piece of paper it seems impossible to get an exact rectangle that you can say is x square inches or square centimeters. I had her go online to research it and come up with a suggestion that she and I could work on together to share with the class. She found an excellent lesson plan that used cans of paint and the information printed on its label about how many square feet it should cover. The idea was for the class to paint circles and then see how their results compare with the square-unit measurement....
But of course those numbers printed on the cans are approximations, depending upon how thick the paint is applied, and depending upon the usual and unavoidable wastage from whatever dries in the bottom of the can as well as whatever remains in the bristles of the brush to be washed down the drain, as well as whatever paint evaporates as you work, etc.
Plus, what a mess!
Rescue me, please...
Dear Mrs. B.,
There is something I've done with classes and have recommended to colleagues with the confidence I only feel when I am as sure as one can ever be about anything in life that if done with even half the enthusiasm students have a right to expect from their teacher, the result will be one of those lessons that students recall as an "experience" and can even be heard describing it to peers during snack time.
That's the 'Good News'.
Now for the 'Bad News':
As is often the case, to set the scene with everything in place to maximize the likelihood that this kind of learning magic occurs, there is a bit more pre-class preparation than many teachers ordinarily find feasible, particularly as they approach the last few days of the school year when most people's energy seems to be decreasing faster that the previous year's experience had prepared them for.
The most sensible compromise, by the way, is NO COMPROMISE. If you're feeling exhausted, then DO THIS ANYWAY.
THE RESULTS WILL REVITALIZE YOU, AND YOU'LL APPROACH SUMMER BUOYANT AS A NEWBORN BUBBLE.
You need a supply of three items.
1. Circles that have been precut. Best would be to have a variety of sizes, i.e. small, medium and large, but a few circles of the same size can be used. These circles can be cut from any material that has some palpable weight to it. Eighth-inch-thick or quarter-inch-thick Masonite, which (I think) is a composite of wood filings, cardboard and glue, are the two materials I have used, and they both work well, but one could just as easily use plywood or a solid wood, such as pine. I have seen teachers use linoleum, sheet metal cut from recycled cans, even recycled heavy-weight cardboard.
2. A supply of "square inches". These are exactly what they sound like: a supply of squares cut from any of the materials listed above, with each square exactly one inch by one inch, or one centimeter by one centimeter, or whatever similar-sized standard unit you would like to reinforce. THERE ARE TWO IMPORTANT THINGS ABOUT THESE SQUARES:
1) YOU WILL NEED TO HAVE AN ABUNDANT SUPPLY OF THEM,
2) THEY NEED TO BE CUT FROM THE EXACT SAME MATERIAL
THAT YOU HAVE SELECTED FOR YOUR CIRCLES,
as described above.
3. Balance scales.
Unless you choose to do it as a demonstration for the class, you will benefit from having at least a few such scales, one for each of the groups into which you have divided students up for this investigation. Most teachers have told me that if their math department does not have balance scales on supply, the science department almost always does and is happy to lend them to their mathematical colleagues. If that presents a challenge, try the preschool or the lower grades. For some reason, kindergarten-students-through-third-graders tend to have more interest in a balanced education (or balancing education) than do older children, who often seem to have lost faith in the idea of achieving any kind of balance in their lives. (little joke, sorry).
Well, introduce the circles, have them take measurements (the diameter, which, of course, should lead to the radius), and reintroduce the constant pi, the formula for a circle's area, and how it is used.
Next, have them figure out the area of the circle. Personally, for this purpose, to make the point, it may be one of the rare occasions when fifth graders should be permitted (or even encouraged!) to use calculators. Remember the goal: We're trying to prove that the formulas work and are accurate, even if it means taking a brief break from the endless opportunities to practice performing basic operations.
They should come up with an answer in the form of square inches, square centimeters, or whatever unit you have chosen to reinforce.
Have them place their circle in the bucket on one side of a balance scale.
Next introduce the square inches (...or square centimeters or whatever).
Then have them count out their answer, which is now a prediction for how many squares it will take to get the scale to tip to the balanced position, and allow plenty of time for observing their procedure and results.
NOTE: Although there may be a little rounding involved, and the scale may not come to rest at a perfect-looking balance, the scale should do an impressive job of coming close. HINT: Try not to get any fancy high-tech balance scales, because this is one of those odd situations in which too much accuracy can be a hindrance to showing a classroom miracle.
HINT: Although I did not have the heart to mention it above, in addition to the previously mentioned squares, it would be great to have some smaller squares, such as half-inch ones, or quarter inch ones, or eighth inch ones. This way, the students are more likely to finish the experience off with an impressive balance.
Remember, this whole thing works because the circle and the square-inches you present are cut from the exact same material, so the weight is a constant. DISCUSS.
The idea is this: When it comes to circles, just introduce the ideas, and then let the students roll with it.
Finally, so far I mention the 'good news' (the lesson ROCKS), and I mention the 'bad news' (there's a fair amount of prep work for the teacher).
And now for the FINAL, FINAL news—which, fortunately, is VERY GOOD NEWS:
The intense prep of cutting squares and circles is a one-time-thing. Next year, and every year thereafter, you can do the lesson without having to obtain exotic materials or cut them into precise squares and circles!
Hope this helps,