## Conjectures and Counter-Examples

Without question, introducing the terms “Conjectures” and “Counter-examples” into my classrooms has been one of the most impactful changes to student engagement in my classes over the past few years. And just as exciting–it’s super easy for you to put it in your classroom too!

Although I’ve used these terms day-in-and-day-out while I was studying mathematics, using them in K-12 classrooms was inspired by Math for Love–they’ve got a great write-up about it here. I’ve presented about this framework at the National Council of Teachers of Mathematics conferences in Boston and Salt Lake City this fall (slides here), and now I’m putting it online to share with anyone and everyone.

Rather than defining the words, let’s just dive right in:

Conjecture: Everything that has legs can walk.

I’m sure you’ve already thought of a handful of counter-examples (tables, chairs, . Of course, you only need to tell me one in order to convince me that my conjecture was false. Students are super eager to share counter-examples when they find them–and it’s precisely this eager engagement that is largely absent in most math classrooms across the world.

When you put rich math tasks and rich math problems in front of students, they naturally notice patterns, and they are eager to talk about those patterns. And when they’re primed with the language of Conjectures and Counter-examples, that eagerness to talk becomes eagerness to pose conjectures and to find counter-examples. The talk becomes rigorous mathematical discourse using the exact same vocabulary that professional mathematicians use.

Let’s look at one of my favorite problems to make this a bit more concrete. It dates back at least as far as Pythagoras and seems to engage learners at all levels from 4th grade and up:

Choose a positive integer.
List its factors.
Add up all the factors, including 1 but not including itself.
Repeat the process with the number you found.

Example:
Start with 16. The factors are 1, 2, 4, 8, and 16.
We add 1 + 2 + 4 + 8 and get 15.
Now, we continue from 15: the factors are 1, 3, 5, and 15.
We add them up and get 9.
Continuing, we see the factors of 9 add to 4.
And the factors of 4 add to 3.
And the factors of 3 add to 1.
And then we stop, because adding an empty list of numbers seems problematic.

To summarize that example, we went from 16 to 15 to 9 to 4 to 3 to 1. If you’re a curious creature, all sorts of problems are likely popping into your head.

Do the numbers always get smaller?
16 and 4 both went down by 1–is there something to that?
The pattern went even, odd, odd, even, odd, odd–is there something to that?
We ended up hitting every square number along the way from 16 to 1… does that always happen?

Although I’ve paraphrased them a bit, these questions are all directly out of the minds and mouths of my 4th and 5th graders. And 4th and 5th graders are quite capable of turning these questions into conjectures!

Numbers always get smaller.
Powers of 2 always go down by 1.
The pattern will always go even, odd, odd, even, odd, odd.
If you start with a square number, you’ll hit every square number smaller than the one you started with.

Students find counter-examples to most of these conjectures within a few minutes of playing around. Wait–I found a counter-example! 30 gets bigger!

The classroom bubbles with genuine engagement with student-posed ideas about one of the most oft-discussed math problems of human history.

The way this operates in my classroom is really quite simple: I have a stack of Conjecture Pages and Counter Example Pages available at all times. A bit later, I also introduce Definition Pages, but let’s keep it simple for the time being. Whenever a student wants to, they grab a Conjecture page, write down their idea, and use a magnet to post it on the whiteboard. At this point, we say the student has entered a conjecture into the classroom conversation. And it really is part of the classroom conversation–when a student can’t make sense of someone’s conjecture or thinks they have a counter-example, peer-initiated, peer-moderated, and peer-resolved conversations about rigorous mathematical ideas arise naturally in the classroom. And if you play your cards right, they’ll continue outside the classroom too!

And amidst all this intense curiosity about deep mathematics, students are working on grade-level content skills: they’re working with peers to make sure they find all the factors and adding long lists of numbers. They’re relying on peers rather than an instructor or an answer key to check their work, and as an added twist, I’ve tasked my students with drawing scale rectangles to represent the factor pairs they find. And because I happen to know just how surprising it ends up, I’ve asked them to draw rectangles for each step of the process starting with 30. It turns out that from 30 to 42 to 54 and then to 66! (But Mr. Gafni, how can we draw a scale 1 by 66 rectangle on graph paper? There’s not enough boxes!) I’ll let you play around with it yourself to see what’s going on with this pattern. The rectangles might shed some light on it: 30, 42, 54, and 66 all yield a very similar collection of rectangles… Maybe prime factorization would help?

Here are some Student Posed Conjectures from a high school Number Theory course I taught in the Summer of 2018. I’d like to put up some 4th and 5th grade conjectures too, but I’m having scanner issues at the moment… Feel free to ping me about it if they’re not up by the end of October!

## [Laser Cutting] Intro to Rhino and Grasshopper

I’m excited to share a brief, hands-on introduction to Rhino and Grasshopper–powerful software for 2D and 3D design work. Rhino is perfect for designing all sorts of objects–art, tools, toys, prototypes, etc. Rhino creates files that are ready-to-go on laser cutters and 3D printers. Rhino has a 90 day trial, and you’ll find that any skills you pick up in Rhino will transfer reasonably well to AutoCAD or other CAD softwares(Computer Assisted Design). Grasshopper is free.

Grasshopper is an easy-to-use add-on that allows you to easily create much more complex designs very easily–empowering users to make explicit use of mathematics within the design process. (More technically speaking, Grasshopper is a drag-and-drop functional scripting interface for Rhino.)

This intro is written to take you from 0 to On Your Way: if you follow along in the software, you should be able to precisely reproduce the pieces below, and you should understand the techniques well enough to get started making your own [insert your ideas here]!  ## What is an Algebra? (Part III)

In our next few Algebras, we broaden our sense of elements. Until now, we’ve looked at different types of numbers as our elements.

Now, consider matrices as elements. Given any two 2×2 matrices, $A$ and $B$, we can certainly computer $A+B$ and $A\cdot B$. And crucially, this sum and this product are still 2×2 matrices. So, we see that the set of 2×2 matrices forms an Algebra. It shouldn’t be too surprising that the set of 3×3 matrices also forms an Algebra, and more generally the set of $n$x $n$ matrices do as well.

We could also use functions (say, ones that turn a real number into another real number) as our elements. In that case, we have addition and multiplication of functions as operations. And we have another reasonable operation to look into–composition.

Do we have division in the world of matrices or the world of functions? If you multiply a matrix A by a matrix B, is there another matrix to multiply by that will get you back to A? What about something that undoes composition by $f$ or multiplication by $f$ in the same way that subtracting $f$ undoes adding $f$?

## What is an Algebra? (Part II)

(This is Part II of a multi-part series. See Part I)

Consider the following statements about integer arithmetic.

even + even = even
even + odd = odd
odd + even = odd
odd + odd = even

even * even = even
even * odd = even
odd * even = even
odd * odd = odd

Speaking loosely, we can say that we have an Algebra consisting of the elements “even” and “odd.” We call this Algebra the Integers modulo 2. We write 0 as shorthand for even and 1 as shorthand for odd, and use the following notation to distinguish this modular arithmetic from our standard arithmetic.

0 + 0 = 0 (mod 2)
0 + 1 = 1 (mod 2)
1 + 0 = 1 (mod 2)
1 + 1 = 0 (mod 2)

0 * 0 =  0 (mod 2)
0 * 1 = 0 (mod 2)
1 * 0 = 0 (mod 2)
1 * 1 = 1 (mod 2)

A note on notation: think of the (mod 2) as describing the statement as a whole, not the element on the right-hand-side of the equals sign. What we are really saying is that in the Algebra of the Integers modulo 2, we have 1 + 1 = 0.

——————————-

The following statements are a bit more of a mouthful, but are analogous to the example above.

[A Multiple of 3] + [A Multiple of 3] = [A Multiple of 3]
[A Multiple of 3] + [A number that is one more than a multiple of 3] = [A number that is one more than a multiple of 3]

Some simple notation is necessary. We use (mod 3rather than (mod 2) here to indicate that we are thinking about multiples of 3 rather than multiples of 2. We use to mean [A Multiple of 3], to mean [A number one more than a multiple of 3], and to mean [A number that is two more than a multiple of 3].

Then, we can write the statements above as
0 + 0 = 0 (mod 3)
0 + 1 = 1 (mod 3)

We leave it to you to write out the remaining 7 statements of addition and the 9 statements of multiplication. These statements describe the Algebra of the Integers Modulo 3.

So far, we have seen the Integers Modulo 2 and the Integers Modulo 3. It’s not hard to imagine the notation or interpretation for the Integers Modulo 4 or the Integers Modulo 37.

So if you are asked to find 2+2, you better know what Algebra you are working in! In the Algebra of the Reals or the Integers, 2+2=4. But in the Integers Modulo 3, we have 2+2=1(mod 3).

Given that our motivation is to understand what “An Algebra” is rather than to understand the intricacies of any particular Algebra, we’ll close our discussion of this family of Algebras here. The curious reader may want to explore whether division makes sense in this Algebra.

## What is an Algebra? (Part I)

We’ll answer this question in this series of posts by giving a hands-on introduction to a variety of different Algebras (yes, there are lots!). Loosely speaking, an Algebra is a system that consists of elements and one or more operations. For now, you can think of the elements as numbers and the operations as the basic arithmetic you’re familiar with: addition or multiplication, for example.

So, our first example of an Algebra is the one we generally study in high school: the Real Numbers. And our operations are the basic four we know from elementary school.

Our second example of an Algebra is the set of integers. Here, our Algebra does not include all of our basic operations: division is problematic in that the quotient of two integers is not necessarily another integer. So, the Algebra of the integers includes addition, subtraction, and multiplication; but not division.

We’ll close this post with one more Algebra that is familiar from high school: the Algebra of the Complex Numbers. Addition, subtraction, and multiplication all behave nicely. Division is less clear to the novice eye: Given two complex numbers $z=a+bi$ and $y=c+di$, does the notion of $\frac{z}{y}$ make sense? Is it still a complex number? If the answers to these questions are yes, the Algebra of the Complex Numbers seems to be more like the Reals. And if not, the Algebra of the Complex Numbers seems to be more like the Integers. We’ll leave these questions about complex division for you to consider.

In the next post, we’ll consider an extremely useful family of Algebras: the Integers modulo $n$.

## A Novel Take on Graphing

Here’s the pitch:

1. Draw a horizontal axis and label it with the numbers $1,10,100,...$ (What numbers should go to the left of 1?)
2. Draw a vertical axis, labelled similarly.
3. Draw a downward sloping line at a 45 degree angle through the “origin” (which seems to now be the point $(1,1)$. The equation $y=-x$ no longer seems appropriate. Can you write an equation that fits this graph?
4. Try going in the other direction. If you write down the equation that would normally give a circle and try to graph it on these axes, what do you get?
5. Try lines, parabolas, circles, exponential functions, logarithmic functions, rational functions, etc.
6. Try other ways of labelling your coordinate axes.

Note on Notation: My students and I reserve $x$ and $y$ for the standard axes labelled in the standard way. We generally use $a$ to represent the strange values along the horizontal axis and $b$ to represent the strange values along the vertical axis.

## A Fractal Inducing Problem about Polygons

Here’s a question I like: take an n by n grid (start small), and draw straight lines connecting dots. If you end where you started and don’t cross lines, you get a polygon. What’s the most number of sides the polygon can have?

I received those words in an e-mail from Dan Finkel in April of 2014. I read the e-mail on the way from my front door to my car, and I visualized the 3×3 case for most of my commute to Tacoma.

Convinced that the problem would be perfect for my younger of two groups of elementary age homeschoolers that day, I launched into it–still with no clear ideas about the solutions. By the end of the hour, I still hadn’t had a chance to pick up a pencil, and I was skeptical that the this group of children had managed to produce the best solution–it’s hard to be satisfied by an asymmetrical solution to a problem like this. I pitched the problem to two groups that day and spent multiple hours after class exploring the different sizes–the 4×4, 8×8, 16×16 are particularly interesting to me for a particularly elegant solution. This simple problem has captivated me more than any other I’ve considered since, and I’m looking forward to hearing what you think.