Intro to Points and Lines

First, state the definitions to students in student friendly language. (source: http://mathworld.wolfram.com/ )

• A point lives in the 0 dimension. It has no size and no shape. It only has a location.
  
• A line lives in 1 dimension. It has no thickness. It extends infinitely in both directions.
  
• A line segment is a finite portion of an infinite line. It has two ending points.
  
• A ray is a half-infinite or "half-line" with a starting point and no ending point.

Next, have students watch the video, The Dot and The Line: A Romance in Lower Mathematics based on the book by Norton Juster. This is an award winning cartoon from Chuck Jones which was completely hand drawn! Tell students that while watching the cartoon, they must determine if the dot is really a point and if the line is really a line. They must use the definitions just presented in class and support their reasoning.

Here's what some of my students came up with for Question 1:
1) According to the mathematical definition of a point, can the dot in the cartoon be called a point? Support your answer.

• No, the dot had a shape.
  
• No, the dot had a shape and it kept moving, not in one location.
  
• Yes, because it was on the line.
  
• Yes, it was a point because it had no dimension.
  
• No, because it had a shape.
  
• No, because it wasn't named.
  
• No, because the dot had dimensions.
  
• No, because points don't move.
  
• No, because a point has no shape or size.
  
• No, because there is no way to know how big or wide it is.
  
• Yes, because points can be anywhere.
  
• No, because it moves all the time.
  
• No, because it didn't belong to a graph.
  
• No, because it is not fixed in one spot.
  

I love this! Every student used some sort of knowledge about the point to support their reasoning. The response "Yes, because it was on a line." was very thought-provoking for me...because that is a way to describe a point. I thought it interesting that many students connected a point with being stationary because the definition includes that it describes a location. I think a follow up question to this can now be: Can a point move? Why or why not?

Question 2 asks:
Is the line depicted in the cartoon actually a line? Justify your reasoning.

Here's what I got:

• No, because it turned into other shapes.
  
• No, it wasn't always a line, sometimes it curved.
  
• No, when it was different shapes it was in different dimensions.
  
• Yes, it is just a line that formed into different shapes.
  
• Yes, even though it changed shapes it was still a line.
  
• No, because lines never end and this one did.
  
• Yes, because no matter what shape it took, it was made up of lines.
  
• Yes, because it has an infinite number of collinear points.
  
• Yes, it was almost always a line because it kept continuing.
  
• Yes, it was always a line but not always straight.
  
• No, because sometimes it stopped at the bottom.
  
• Yes, because it went on forever.
  
• No, because it turned into shapes and connected to itself so it was not infinite.
  
• No, the line because different shapes.
  
• Yes, it was always a line until it became an angle with the dot.
  
• Yes, because it never stopped.
  
• Yes, because never once did it stop. It was so long and it looked like it had two arrows.
  
• No, not always because it had an end at sometimes.
  
• Yes, because a line is an infinite number of collinear points.
  

Again, amazing ways to think about a line! Some were "textbook" answers, completely fine and described infinte number of collinear points. But I think my favorite is "Yes, it was always a line until it became an angle with the dot." I will be presenting these to the class and see what they have to say about the answers. I'm thrilled by the complexity of their reasoning. Can't wait to see what they come up with next!