Conjectures are more Powerful than Facts in the ClassroomNovember 12, 2020
There’s a place for facts in math class. They end experiences.
Think of a fact as a snapshot you take from the top of the mountain you just climbed. It reminds to of the journey, and helps you conclude and summarize it in a satisfying way.
But if you begin a lesson with a fact, you’ve ended the journey before it began. It’s like showing the photo to a kid who’s just starting to climb and telling them that they already climbed the mountain. The fact needs the experience, the understanding, behind it in order to be meaningful, and to inform future learning. Seeing photos of mountaintops doesn’t make you a mountain climber, and memorizing a bunch of facts doesn’t make you a mathematician.
If facts end experiences, conjectures extend them. Consider these approaches to a lesson.
Approach 1. State the fact
Fact. Any rational number (i.e., a number of the form a/b where a and b are integers) can be written as either a finite or repeating decimal.
Starting here just sucks. If this fact is true, it should be shocking. But we couldn’t even check that it’s true without doing a lot of work.
In fact, how much work is required? You could test a bunch of rational numbers by doing the long division (or just using a calculator) to see if a/b is a repeating or finite decimal, but there are always more rational numbers to check. How can you check an infinite number of them? How could something like this even be known?
In fact, if I plug 1/47 into my calculator, I get 1/47 = 0.021276595744681… Why should I believe that decimal repeats? How long do I need to go before I should be satisfied that it repeats?
That statements which describe infinite collections and ascribe unexpected properties to them can be proven true (or false) is extraordinary. Stating them as facts and sweeping the miracle of their existence out of sight is backwards. Don’t end before you’ve begun! Don’t start with the fact.
Approach 2. Explore first, then state a conjecture.
What if I started by looking at the decimal form of a few fractions instead?
1/2 = 0.5
1/3 = 0.33333…
1/4 = 0.25
1/5 = 0.2
1/6 = 0.16666666…
Interesting. It looks like they either end, or involve a repeating number. Rather than presume I know everything, I’ll just take a crack at guessing what might be going on.
Conjecture. Rational numbers are either finite or end with a repeating digit.
Seems reasonable. What happens next?
1/7 = 0.1428571429… This is what my calculator says. What’s going on?? Is the conjecture wrong? This is one of those moments where long division let’s us see what’s happening more clearly.
The possible remainders after we divide by 7 cycle through all the options from 1 to 6. Once that starts repeating, we know we’ll get the whole cycle of digits again. And again and again and again. (Digest that argument if you haven’t seen it before. It’s not obvious.) If you actually do the long division, you see that you’re going to be doing the same thing over and over. So the digits will repeat, but they’ll repeat in a six-digit interval.
Revised Conjecture. Rational numbers are either finite or end with a repeating block of digits.
Question. How long could the block of repeating digits be?
That question is interesting on its own, but it’s also especially relevant if I’m doing the long division by hand. Once again, if I put 1/47 into the calculator I get 0.021276595744681… How long should I expect to look before I can be reasonably certain that this decimal repeats (or doesn’t)? I need some kind of insight from the process of actually exploring these things.
Let’s look at a few more.
Now this is weird. They’re all finite or repeating so far, but why are some finite and some repeating? Can you predict which will be which?
Also, Some have just 1 repeating digit, while others have 2, or 6. What are the possible lengths of repeating blocks of digits? I have a wild conjecture, just based on the sequence 1, 2, 6, … but I feel like it’s unlikely that it’s true.
And I’m always putting 1 in the numerator. What will change when I put other numbers up there?
Also, here’s something weird: not only is the repeating block for 1/7 and 1/14 the same size (6 digits long), they also consist of the same digits. They’re even in the same order, with the 7 moved from the back of the block to the front! What’s up with that?
By allowing an exploration to lead to conjectures, we have a much richer sense of the known and unknown, of the mysterious and the meaningful. I could present proofs for whatever I thought was vital kids know as facts now, and it would be more likely to stick. And if I wanted to end this experience, that’s what I would do. Codify what we’ve found as fact.
Or, if I wanted to extend it, I’d keep focus on the questions and conjectures. And wouldn’t that be fun?
- What are the possible lengths of repeating digits in the decimal expansion of a rational number?
Wild Conjecture: The lengths must be n! E.g., 1, 2, 6, 24, 120, etc.
[This feels like it’s probably false. Is there a counterexample?]
- How can we predict which fractions will stop and which will repeat?
- How can we prove that all rational numbers are either finite or repeating?
- What is the longest a repeating block of digits could be for a given rational number a/b?
- Why do 1/7 and 1/14 have the same digits in their repeating blocks? Will 1/21 and 1/28 have the same digits too?
Conjecture: 1/(7n) will always have the same 6 digits repeating in some order.
Not only is the focus on questions and conjecture much more alive than math, students who have this kind of experience will retain a much deeper and robust understanding of what’s actually true. We haven’t proven that all rational numbers are either finite or repeating, but with a little more exploring, we could probably put a nice argument together! (I don’t want to ruin it for you, though.)
Even more, we’re actually doing mathematics now. We’re climbing the mountain. There are questions we could pursue on this topic that are currently unsolved in mathematics. This is the kind of experience that can change lives, as long as students get intrigued enough to want to do the work and have the experience.
I’d like to collect a good list of conjectures, along with the space you’d be exploring them. This conjecture was about converting rational numbers to decimal form, and needs long division to explore. If you have favorite conjectures, I’d love to make a list available so everyone can feel like offering conjectures and questions—not just facts—is feasible in whatever context they’re teaching.
Midpoint parallelogram on a quadrilateral?
Just heard this one… 1111…111 can only be prime if the number of digits is prime.
Those are both great!
Area seems to be related to anti-derivative.
Multiples of 3 seem to have a digit sum of 3,6, or 9.
It seems we can approximate trig functions and exponential functions with polynomials.
The digit sum one is really nice because it’s so natural to ask it for ANY digit. And then there are just a couple that it works for.
Approximating functions with polynomials is interesting… you almost need an intuition or definition of what approximating means in that context. (Though of course, that’s precisely where calculus goes with Taylor series).
I suppose an earlier version of that is, we can get as close as we want to any number with rational / irrational numbers.
I love this article but I have a question – how do you get the curiosity back after it has been beaten out of them by years of poor curriculum choices? The culture of my school has totally deemphasized the interest in learning over grades.
Frankly, I think it’s hard. But I don’t think it’s impossible. It takes being realistic about how much time and energy it will take to reengage students, and then trying a bunch of different stuff to see what they’ll respond to. But fundamentally, it’s what we have to try to do!
A few years ago, my student in grade 3 made a conjecture that in regular polygons opposite sides are parallel. This was later refined to “in regular polygons with even number of sides”. This conversation lead to a search of more properties that are specific to different polygons and symmetry. Since the learning outcome was classifying polygons based on the number of sides, we’ve got a lot of practice with both skills and vocabulary as well. The prompt for the lesson was WODB.
That’s really nice. I can imagine that going all kinds of places.
I had a series of classes at a similar age where the conjecture/question was (this is in adult lanuage): for any n, there is an n-gon that tessellates the plane. Super fun to try to find families of polygons with various numbers of sides that could tessellate!