“They never meet,” we reply. “They go on and on and never cross. They are always the same distance apart.”
“How sure are you?”
“Very sure,” we state with conviction. [insert pic of train tracks. possibly better if the pic is inside a thought bubble, maybe even a cartoon head/torso & head]
“Consider the following puzzle."
Molly leaves her house and walks 1 km south. She then walks 1 km west and encounters a bear. Startled, Molly runs 1 km north and comes back to her house, safe. How is this possible? Oh . . . and what colour was the bear?”
We look around at each other confused. We draw diagrams on chart paper with smelly markers to no avail.
“Here. Catch!” An object is casually tossed to each of us. “Look closely.”
“Oh, my gosh! Everything we knew is wrong!” we exclaim. Then, laughing, “But now we know what colour the bear was!”
Have you ever experienced a pleasurable surprise? A great twisting ending of a movie, a delicious plot twist in a novel, a sudden reveal in a documentary. These surprises are why we love watching movies or reading great stories. We experienced a mathematical surprise that was so profound, it rose to the level of an epiphany. How once, our mathematical worlds were organized, they were now completely different: they were bigger . . . and better. Once we had had this surprise, we were determined to share it with our students and their parents.
The Surprising Mathematics
For such an important relationship, the concept of parallelism is never developed in either the elementary or secondary curricula in Ontario. The first time parallel lines are mentioned is in grades 3 and 4 where it is assumed that students already know what it is so they can identify rhombi and parallelograms as having parallel sides. In grade 7, students are constructing parallel lines, and grade 8s are looking at the properties of transversals to parallel lines. The only real mention of what parallel lines are is in the glossary where it states that parallel lines are, “lines in the same plane that do not intersect.” Except that this is not the whole truth and nothing but the truth. A better definition might be that parallel lines are straight lines that go in the same direction.
In high school, you would think that parallelism might be taught a bit more in depth, but sadly this is not the case. In grade 9, students are to investigate the slopes of parallel lines (Um, they’re the same. That’s it.), and in grade 12 Calculus and Vectors, the students investigate parallel planes. That is all. The sum total of all we are supposed to teach our students about parallel lines. No wonder we all thought that parallel lines never ever meet.
To paraphrase Euclid’s parallel postulate, if any of the angles a or b in the diagram to the right are less than 90°, then the two lines will meet. In other words, if they are both 90°, the lines do not meet, i.e., they are parallel. This works if and only if the lines are all on the same plane, in 2-dimensional space. Once you get into three dimensions, the postulate does not on a 2-dimensional plane.
Take a look at what objects were tossed at us at the beginning.
Watch mathematician Megumi Harada explain how parallel lines can meet.
Surprising the Students
The students were able to experience the mathematics of parallel lines in almost exactly the same way that we the teachers did. Teaching these concepts to the students took on a form almost like that of a comedy skit. A story is presented, interaction with the audience encourages everyone to look stage left, and when the performers are sure the audience is ready, the reveal or punchline comes out of stage right.
When you are surprised, your mind opens up. It is open to new possibilities that you might not have been open to before. It is this state that we wished to get the students into so that we could introduce them to even more geometric shenanigans that happen when you only think in three rather than two dimensions. The students watched a quick demonstration about string and straight lines. A length of string was loosely held between two people.
“Is it straight?”
“No. Not at all.”
“How could we make it straight?”
“Pull it tight.”
So we did. Nice, straight line. Shortest distance between two points. Look stage left, please.
The students were asked to draw a line on a map they thought would be the straightest, shortest line between Toronto and Tokyo. Next they were given a piece of string and asked to trace their paths on the globes.
“Make sure you tighten the string once you have traced your path!”
“Hey it slipped up and over the North Pole!”
“So that’s the shortest path?” we asked.
“It must be.”
“So the shortest line on a globe must be . . .”
“And a straight line along the surface of a sphere must be . . .”
“If you were flying from Toronto to Tokyo, which route would use less fuel?”
“Cool. You get to fly over the north pole.”
One student in particular, one who did not consider himself good at math, held up a globe and stated that we were all just looking at it wrong. “Look,” he said, “If you want to understand why the Toronto to Tokyo line goes over the north pole, stop looking at it from the side (the equator nearest, poles at top and bottom). All you have to do is turn it (he turned the globe to let us look straight down at the north pole) and it’s obvious.” Now a whole new world of mathematics opened up to him in which he was good, in which he was able to notice things that even some adults can’t see easily. This experience led him to become a confident math student who knew that even when something is difficult, you just have to patiently search for a different perspective to understand it. Turn the globe on its side.
The students, of course, wanted to surprise their parents with this puzzle, too, so they carefully practiced their new math with each other to “learn their lines.” Off they went home, stories and globes in hand.
The parents had their surprise, too. We asked them for responses about what they had learned with their children about parallel lines:
“Yes. I know that parallel lines never meet each other. But what a wonderful truth, I got from this math activity. That only in spheres, parallel lines meet together.”
“Parallel lines do meet, but only on round surfaces. To get to a place the shortest path will not always be along the latitude line. Only the longitude lines will meet.”
“We might think the answer is easy, but we must always take a minute to reflect and make sure.”
“I learned how things that seem obvious are not necessarily so. Parallel lines on a square or other right angled object will intersect but a sphere has different properties that change the outcome.”
“Things weren't as straight forward as we expected.”
“I was pleasantly surprised to learn that parallel lines do meet, I had never thought about longitude/latitude.”
Involving parents with the learning of their children is always a good idea. Moving beyond practice type homework to activities where the children can teach their parents and share a pleasurable surprise is even better. The challenge is to continue teaching the math that is worth talking about.
Implications for Practice
Teaching mathematics in this way affords teachers a whole new way to differentiate their lessons. This activity started off with a low floor: straight lines, parallel lines, things we already think we know. Every student was able to get on and understand everything at the onset. Next, inserting a mathematical surprise allows everyone to notice that there is a high ceiling, and that we can all reach up to touch it in our own ways: parallel lines meet at the poles, the shortest distance on a sphere is a curve, the best way to get to Tokyo is over the north pole, great circle routes are best for conserving fuel on airplanes . . . oh, and the bear? It’s white. All the students felt immeasurably smarter after these experiences. This especially included those exceptional students that we differentiate for, ESL students, IEP students. We need to consider all students as exceptional. We need to change our idea of an exceptional student from a deficit model to a surplus model.
Children are much more capable of sophisticated mathematical thought than any of our curricula currently give them credit for. Bringing in complex mathematics that starts at their level allows a high portion of our students to excel and reach beyond the artificial barriers of the grade level curriculum - all while easily meeting grade level expectations. All students become aware that mathematics is big, wonderful, surprising, and of course, fun. Mathematics may be big, but their minds are bigger!