Taking π out of the equation: The five percent solution. Dave Coulson, July 2012
Schoolteachers, how much fun is a geometry class? Speaking for myself, Ireally like geometry, because it‟s the part of the syllabus in which I cangrab a carpenter‟s measuring tape and go outside. I like objects. I likemeasuring things and determining how heavy they are, or would be ifthey were changed in some way, and how costly it would be to rebuildsomething in a particular way.
My enthusiasm for life gets challenged, however, when it comes tomeasuring circles. Whereas most calculations are done on-the-fly, finding the area of a circle, or the volume of a sphere, or the surfacearea of a sphere, or simply the distance around the outside of a wheelrequires this thing called π, and whether I like it or not, it‟s an awkwardnumber.
Now awkward numbers present you with a dilemma: Do you spend thenext ten minutes reminding your audience how to multiply a number bya three-digit decimal number (i.e., 3.14)? Or do you cop out and use acalculator? The first alternative means the process of multiplication hashijacked your lesson. By the time you have multiplied 3.14 into thebundle, your audience has likely forgotten what the bundle represents; anarea, a distance or a volume? And how long is it before the lesson‟sfinished, Sir?
The other alternative - letting the kids use a calculator - inherentlymeans telling kids that some calculations are too tough for the humanmind, and from there it‟s an easy step to believing that all calculations aretoo tough for the human mind, and the calculator comes out even whenkids are asked to multiply by ten. That‟s not something I want toencourage.
For maths to be valuable, it has to be something so easy and so fast thatkids will not even stop to consider why they‟re learning it. It‟ll be likelearning the words of a song, or the way to the supermarket. It‟s justsomething they pick up along the way. So here‟s what I propose doingwith π: while we‟re teaching kids that π is equal to 3.14-blah-blah-blah, why don‟t we also tell them that π is very close to 3.15?
A long time ago, before calculators we‟re even thought of, I was told bymy teacher that π was close to 22/7 , and that this fraction was a veryconvenient number to use instead of the real thing. Forty years later, I‟dlike to challenge my teacher on that because I don‟t think multiplying anumber by something with a 7 on the denominator is particularly easy.But the idea that approximations are acceptable and normal is significant andsomething we should be sharing with our students.
The measurements that are given to us before we do a calculation are notperfect; we live in a 3-sig-fig world, and so believing we need as manydecimal places as the calculator can give us is leading us to depend oncalculators more than we really need to.
So why 3.15 and not something else? Well, 3.15 is 3 x 1.05. If wetranslate this little expansion back into human-speak, multiplying by3.15 means first multiplying by 3 and then enlarging the answer by 5percent. Hence the title of my article, the five percent solution.
Have a look at some examples to see how it works:Suppose we‟re outside and I see a circular garden plot that I‟d like toprotect with a string cordon mounted on a ring of posts. The garden is 6metres across. I say to the kids, “I‟d like to know how long that piece ofstring has to be to go right round.”Of course it‟s a π calculation. Multiply 3.14 by the diameter. But I tell thekids that the distance around the outside of the circle is about 3 times thedistance through the middle. So we begin by multiplying the 6m by 3 toget 18m.
That‟s already a good approximation to the distance. But then I say thecorrect value is actually about five percent bigger. And so then we get intoa discussion about what percentages are and how to get them. Fivepercent is half of ten percent, and ten percent is a tenth of whatevernumber we‟re working with. Ten percent of 18m is 1.8m, so five percentwould be 0.9m. Add that to the number we have so far and we get thedistance around the garden correct to three significant figures! I‟m within ahandwidth of the correct answer, and I didn‟t use a calculator, nor did Ineed a piece of paper. 6m
Next example: I walk a bit further around the garden and see that mycircular swimming pool (sigh, I‟m rich) needs a cover to keep the leavesout. I‟d like to know how much tarpaulin I need to buy to do that.That‟s an area problem of course. I should measure the diameter of thepool and halve that number and then square that number and thenmultiply through by 3.14. That‟s what we all do in the classroom, exceptfor the naïve students who decide to multiply by 3.14 first and then haveto lug these awkward decimal numbers, line by line, through the rest oftheir calculation.
But I say to the kids, “Imagine putting a square cloth over this pool, onethat‟s just wide enough and just long enough to cover the pool. The bitthat‟s actually over the pool is about three-quarters of the size of thesquare.”So now we‟re started on the process of finding three-quarters of the areaof a square. If the cloth is 4m wide and 4m long then the area is 16m2. Aquarter of that is 4m2 and three-quarters of that is 12m2.
That‟s already a good approximation to the area of the circle. But then Isay, “The true answer is actually about five percent bigger,” which meanswe go through that percentage process all over again.We start with an estimate of 12m2 and raise it by half of tenpercent, which is 0.6m2. It‟s as simple as that. And we‟re out by just 0.2percent, which is the size of a handkerchief.
How about a volume calculation? Let‟s imagine a ball 2m in diameter –perhaps one of those supersized bouncy balls that people climb into androll down hillsides. I might be interested in knowing how much oxygen isin one of those things, which is related to how much air is inside, whichis another way of saying „volume‟.
Okay, so imagine a box fitting tightly around the ball, a box which hasthe same width in all directions as the ball itself. The ball occupies abouthalf of the volume of the box. That‟s half of 2x2x2 m3, which is 4m3.That‟s already pretty good, but guess what? Adding five percent will makethe answer accurate to within one part in 400. That‟s about as much airas would fit into an egg carton.
All these calculations are so easy I can do them midstride as I walkaround the garden, The maths is so inobtrusive I barely notice it, whichmeans I can think more about what the numbers are describing. Thenumbers add to the discussion instead of distracting us from it.
Tackling circle geometry in this way is likely to raise a few questions inthe minds of other educators, so I‟d like to finish this article withsomething like an FAQ, a list of Foremost Anticipated Questions:• Isn’t there something inherently dirty about approximating π ?The idea of rounding π off to some convenient close-by number is notnew. In pre-calculator days we were encouraged to treat π as 22/7, whichis out by 0.04%. That approximation lost relevance in the calculatorera, when kids learned to key in 3.14, which is out by 0.05%. Myapproximation is out by 0.25%, which in comparison is substantiallygreater but nevertheless ensures accuracy to within one part in400, absolutely acceptable for most purposes. I think what‟s important inthis question is whether or not the students remember that π is actually3.141592655etc and that 3.15 is just a convenient simplification. Ifthey‟re measuring crystals in a laboratory or tracking the stars in anobservatory someday, and really do need those extra decimal places, theywill be able to go and get them. No-one‟s throwing calculators away.
• What’s wrong with showing the kids the formulae?Nothing dramatically wrong with that. However, in my experience, I‟vefound that kids are better at learning scripts or step-by-step proceduresthan learning formulae. If I say “Do this and then do that,” theyunderstand. If I show them a formula, they often do not. A formula is asummary of a procedure, a way of writing it down in a book. Formulaemake more sense to adults than they do to kids - which is not to say thatall kids have difficulty with formulae, but a great many do.• Will the kids ever see the proper formulae?The answer to this has to be yes. This is not an either-or debate where wehave to choose one approach at the expense of the other, but ratherallowing kids to have access to both approaches. Maths comes in twoflavours: theoretical and practical. We teach the theory so that kids canadvance into science if they choose. But we also teach practical numberskills (or should), so that kids can juggle numbers in their heads or onpaper easily and quickly. It‟s not one or the other but both.
• Wouldn’t we be encouraging kids to be sloppy with their calculations?Not really. All measurements are inherently approximate, so we should beteaching kids to be aware of the level of accuracy in the measurementsthey are given at the start of the problem, and to round all othernumbers (like π) accordingly. We live in a 3-sig-fig world, meaning thatmost measurements we encounter are accurate to that level. Thereforerounding π to 3sf, especially when it offers this great simplification, is notonly permissible, it‟s a good idea.• Teaching kids this way doesn’t prepare them well for assessments, whichultimately have to use formulae. So where’s the advantage?This opens up the great debate of how well exams reflect learning.Maybe we should be changing the way we assess kids so that we rewardthem for how well they apply arithmetic skills in the field, as well as whatthey achieve in the classroom. That‟s a big ask, but I think it‟s a betterproposal than maintaining the skewed perspective that learning issomething that comes off one piece of paper and goes onto another.
• Aren’t the kids going to lose something by learning geometry this way?I don‟t think so. When I‟m teaching kids this material, I find myselftalking a lot about percentages, decimal numbers, and demonstrating afair bit of on-the-fly number-processing. When these kids leaveschool, most of them will encounter percentages and decimals far moreoften than they will encounter circles and spheres. So I think that byassociating π with these other numbers, I am broadening theireducation, and showing how it all ties together.-Dave C, July 2012.