What's the Point?

Task

The students in Ms. Sun's class were drawing geometric figures. First she asked them to draw some points, and then she asked them to draw all the line segments they could that join two of their points.

1. Joni drew 4 points and then drew 4 line segments between them:

   

   Are there other line segments that Joni could have drawn?

2. Tony drew 3 points and then drew 3 line segments between them:

   

   Are there other line segments that Tony could have drawn?

3. Here are 5 points. Draw all the line segments you can connecting pairs of them.

   

4. Starting with just two points, how many line segments can you draw between them?

5. Tony decided that he could actually draw two line segments between two points, and maybe even more. This is what he drew:

   

   What do you think of Tony's idea? Discuss it with a partner.

IM Commentary

The purpose of this task is to use what students intuitively understand about connecting points or “dots” with lines to generate a discussion about what points are and how they should be represented. It is important to note that there is a sense in which Tony is correct: there is more than one line segment joining points in the circles that represent the points he drew. The idea of a point is that it has a location but no length or width. Of course, we can't literally draw an object with no length or width, so any representation of a point must "take up space." So Tony's idea stems from an artifact of the way points are necessarily represented. Adding to this potential confusion is that we sometimes represent points with a relatively large dot or with shapes other than a dot in order to bring attention to them, which can cause even more confusion.

This task is intended to lead into a class discussion about how we think about points vs. how we represent points. The summary conversation led by the teacher will determine the value of this instructional task. A similar discussion about lines and line segments, which have length but no width, will help students understand the difference between the idea of a line and a representation of a line.

This task includes an experimental GeoGebra worksheet, with the intent that instructors might use it to more interactively demonstrate the relevant content material. The file should be considered a draft version, and feedback on it in the comment section is highly encouraged, both in terms of suggestions for improvement and for ideas on using it effectively. The file can be run via the free online application GeoGebra, or run locally if GeoGebra has been installed.

Attached Resources

Solution

1. There are two more segments that Joni could draw:

   

2. Tony has drawn all the segments that are possible to draw between the three points.

3. It is possible to draw 10 different line segments between these five points:

   

4. There is only one line segment joining any two points.

5. Tony is right that there is more than one line segment joining the circles he has drawn to represent the points. A point has a location but no length or width, and there is only one straight path (aka a line segment) starting at the location indicated by one point and ending at the location indicated by a second point. However, there are lots of curved paths between two points; for example: