Introducing Geometric Constructions

After devoting a class to just practicing work with angle relationships and midpoints, I needed to move on to geometric constructions. I wanted to take a more inquiry-based approach to this topic.

My first challenge was thinking about what exposure students may have had to compasses. I know that I have seen a lot of movies with sailing ships and the chart scenes normally included the use of a compass. I decided to look up a clips that showed the use of a compass in ship navigation.

To start class, I held up a compass and asked students if they had ever seen this before. When I asked where, a couple of students said it was something pirates used. Perfect! I asked the class what pirates would use the compass for. Most shrugged their shoulders. A few replied that they could draw circles with them.

I then showed the first clip I found. It's short and has no sound but shows someone using a compass with a chart. I asked the class what the person was doing with the compass. The class responded it appeared the person was using it to measure distance. I re-emphasized the idea of using the compass to measure distance.

It was now time to show the second clip. This clip shows how the compass is used to measure distance and make markings with arcs.

I then drew a line segment on the board and labeled it as segment AB. I marked a third point C on the board. I told the class the challenge was to make an exact copy of AB so that AB = CD by using the compass as a distance measuring device and a ruler solely to draw straight lines.

I then let students struggle through the challenge. Some students were done quickly. Checking their work, I asked how they copied the line. These students said they used the ruler. I told them that wasn't allowed. The only device they had for measuring distance was the compass. I told them to think about what they saw on the video.

Slowly students started to get the idea. I did have to walk around a lot and talk through how the compass could be used to measure distance with quite a few groups. After everyone had the general idea, I asked them to draw another segment and then make a copy that was congruent. This time students seemed to get what they needed to do.

I next drew an angle on the board and labeled the vertex A. I drew a second point B and told them the challenge was to make an exact copy of angle A so that the measures of both angles were the same. Again, they were to use only the compass and straight edge.

Students worked on this for 15 plus minutes. I walked around and checked on their work. Many students had drawn two angles with both angles having side segments that were the same length. I asked how they knew the angles were the same measure. They were stumped by this. Others had measured the separation of the rays but hadn't considered that they weren't measuring the width of the angles from equivalent points.

After about 15 minutes, several students were honing in on some productive ideas. One student in particular said he though he had it. He went through his process and it was exactly what I would have shown if I were giving step-by-step instructions. I had him share his method with his group before letting him share it with the class.

As a class, we discussed why this process duplicated the angle and related it back to the video and distance measurement done in the clip. I asked students to try using this to duplicate their angle.

By this time, almost the entire class was comfortable with duplicating the length of a line segment. They weren't as comfortable with the ideas of using the different lengths to ensure the angles were congruent.

Next class, we'll work on duplicating another angle and then I'll turn them loose on trying to construct a parallel line through a point not on the given line.

Overall, I was pleased with the outcome of this class. Students gained a better understanding of how to use a compass to measure distance, how to use arcs as markings, and how to construct congruent line segments. They also were exposed to how to put these ideas together to construct congruent angles.