What is a Line?

On Monday I was tutoring a girl in year nine, and teaching her about linear graphs, and the whole y = mx + c thing. She asked what linear meant, and I told her that it comes from the Latin word linearis, which means like a line. She was momentarily puzzled: "Aren't all graphs lines?", she asked. I drew a vaguely-quadratic looking graph on the mini whiteboard and said, "This one's not". To which she replied, "Er, yeah it is".

This made me question my concept image of a line. Can a line be curved? If no, then why do we bother saying "straight line" when this would apparently be tautologous. If yes, then how can we use the word "linear" to describe relationships that correspond to strictly straight line graphs?

I decided to do some research. I asked a few members of my department how they would define a line. Here are some of the answers I got, along with an evaluation of each.

Something that joins together two points
The person who gave me this definition said that a curve would count as a line by this definition, as a curve can join two points. However, this would mean that the line joining two points would not be unique, and that makes me feel uneasy. Also, this definition seems to be more for a line segment, as opposed to a line: I have always been taught that lines are infinitely long in two directions.

The shortest path between two points
A little bit nicer, as it now has uniqueness (if by "shortest" we're referring to the usual metric). However again this is just a line segment, not a line.

The locus of points where...
This definition was never finished, because he couldn't think what the locus would be. I decided to still include this definition because I think he's on to something.

A sequence of points
Take any sequence of numbers and plot them on cartesian axes. Then join these up, and what you have is a line. I would refine this by saying "linear sequence", but that would be a bit of a circular definition, because I would define a linear sequence as one that would make a straight line when drawn on a graph. This definition isn't very satisfying because it only makes sense on a set of axes, whereas obviously lines occur elsewhere. In geometry, for example, lines can exist with absolutely no plane of reference.

I suppose the definition I would go with would be the second one, except I'd alter it to make it infinitely long:
"A line is the infinitely long path that goes between two points, such that the path from one point to the other is the shortest possible" Arggh this doesn't work! The line could be squiggly outside of the two points and this could still hold! How do I say "it's straight" when I'm trying to define straight in the first place?

I'm afraid I'll have to do what I always end up doing when it comes to maths debates:

Wikipedia to the rescue!

"Line (geometry), an infinitely-extending one-dimensional figure that has no curvature:

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects.

Thus, until seventeenth century, lines were defined like this: "The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [...] will leave from its imaginary moving some vestige in length, exempt of any width. [...] The straight line is that which is equally extended between its points"

In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.

Is it just me or is that a really hand-wavey definition? And it's still not clear whether a line has to be straight: the pre-seventeenth century definition allows for curvature, but the analytic geometry definition does not. And what does "equally extended" mean anyway?

You might wonder why on earth I care about these things. I do too sometimes. But when I asked people in my department, we got a good discussion going, and that for me is an amazing thing: people arguing about maths. I love it! I remember year 11 RE lessons when I was at school: we used to argue all the time (and every comment would start with "Surely..." - Do any of my ex-classmates remember this?) but we never argued in maths. I love having a good debate (pro tip: never use the expression "maths debate" in a lesson. Trust me) and it is something we should be doing more of in maths. Maths is not about blindly accepting rules and definitions. We should question them, challenge them! Ask a child this and watch their mind explode: "How did the first ruler making factory make the ruler perfectly straight?"

As you can probably tell from the length of this post, I have a lot of work to be doing which I am trying to avoid. There is something quite stressful happening next week in my department, and my stress reduction technique is to immerse myself in mathematical pedantries.

Bye for now,

Emma x x x

I thought you might be interested to read the hilarious text message conversation that took place between me and my dad as I was researching this post:

Me: how would you define a line?
Dad: a set of points determined by a linear function.
Me: define "linear".
Dad: an equation that forms a line.
Me: that's a circular definition.
Dad: define "circular".
Me: a shape that contains no lines.
Dad: define "line".
Me: our conversation has come full circle!
Dad: define "circle".

My dad never should have been given a Blackberry.

Emma x x x

A Grouping Activity

In today's blog post, I'm going to be sharing with you one of the first classroom activities I was ever taught. As a would-be teacher I mean, not as a student.

It was the second day of my PGCE, the first day that we were separated into subject groups, although a lot of us maths lot managed to find each other on the first day anyway. I suppose geeks, like insects, give off pheromones that signal to others of the same species.

We were each given a number. For the sake of simplicity, let's say we were given the numbers 1 to 36. As memory serves, there were actually over forty of us. But let's not let accuracy get in the way of a good story.

On each of the six tables there was a piece of paper saying one of the following:
Prime numbers
Square numbers
Triangle numbers
Numbers greater than 20
Even Numbers
Numbers with more than four factors.

We had to join a group corresponding to our number. As I recall my number was 3. I could have joined either the prime group or the triangle group. Some numbers could have belonged to three different groups. The number 5, for example, had no choice in the matter. The catch was this: every group had to have exactly six members.

I could tell you the solution to this problem, but I'll leave that for you to work out. Feel free to email me (nqtpi@gmail.com) if you get stuck. The one hint I will give you is that a girl named Lydia worked out pretty quickly that because both of our numbers were triangular but not square, we would have to be in the triangle group. And that is the story of how how me and Lydia became friends* and how Team Hopper (previously known as Team Triangle Numbers) was formed.

*OK not really, but I repeat: let's not let accuracy get in the way of a good story.

I think this activity would be a cool way of getting a class of pupils into groups. It's pretty challenging, so maybe best saved for A level classes? Obviously it will need adapting, as if you have 36 students in your A level maths class, you are clearly working at the wrong school.


Emma x x x

PS My blogiversary is coming up, so expect some interesting posts in the near future.