Whiteboarding in Geometry: Points, Lines, and Planes

We did whiteboarding in Geometry today, and it was amazing! On Friday, we did a mini lesson on points, lines, and planes, basically just definitions. Their homework involved answering questions about a diagram that showed two parallel planes intersected by a line. I felt like they might not have gotten enough practice with these concepts, and we hadn’t done any drawing of points, lines, and planes, so today I planned to have them do more practice. Out come the white boards! It was their first time using them. Following is the list of statements I used. They had to draw what each statement described:

• Points X and Y lie on line CD.
• Points A and B are collinear.
• Two planes do not intersect.
• Two planes intersect.
• Line LM and line NP are coplanar but do not intersect.
• Line m intersects plane R at a single point.
• Lines s and t intersect, and line v does not intersect either one.

What I liked about this experience of whiteboarding is that the problems weren’t drill problems, like I had done in the past. These required some real thinking. The diagrams of points, lines, and planes that students drew were entirely new to them – they hadn’t seen anything like it. And many of them didn't think to represent some of the diagrams in 3-dimensional space as opposed to 2D. So for example, when students were asked to draw a line m that intersects plane R at exactly one point, they came up with things like:

and

 when they really needed something like:

We got to have some good conversations about why some of these diagrams didn't work. For example, in the second picture, you can see the student was thinking about the line intersecting at only one point on the plane. But planes go on forever. Even though the diagram makes it look like the plane is contained, it really goes on infinitely in all directions (in 2D). So if you extended it, it would look like the first picture, where the line is on the plane, rather than intersecting it at a single point. 

It was fun to watch students think about how to draw these things. The ones who struggle to pay attention and take notes were into it, and so were the ones who get bored easily by class. So I'd say it was a success!



Students also had to work on an Always, Sometimes, Never activity for points, lines, and planes. I like the level of challenge of some of these questions. I stole these from someone, but I can't remember who, so thanks to that anonymous person! It came from here.

Always True, Sometimes True, or Never True?

Points and Lines

1. Given one point, only one distinct line can be drawn through it.

2. Given two points, only one distinct line can be drawn through both of them.

3. Given three points, only one distinct line can be drawn through all of them.

4. Given four points, only one distinct line can be drawn through all of them.

Points and Planes

5. Given one point, only one distinct plane can be drawn through it.

6. Given two points, only one distinct plane can be drawn through both of them.

7. Given three points, only one distinct plane can be drawn through all of them.

8. Given four points, only one distinct plane can be drawn through all of them.

Lines and Planes

9. A given line is contained in a plane.

10. A given line is contained in two planes.

11. A given line is contained in three planes.

12. Two lines are contained in the same plane.

13. Three lines are contained in the same plane.