Ideas in Geometry/Constructible Numbers

This lesson was based around Group Project 8 that we did in class on November 8th.

The packet began by introducing the idea of constructible numbers, it said

“Given a unit of measure (like inches or meters or miles), what distances can we precisely construct using a compass and straightedge? These distances are what we call constructible numbers.”

Then the packet had us create a number line when we were given the distance between 0 and 1.

To do this you should set your compass so that one leg of the compass was touching point 0 and the other leg was touching point 1. This was the distance. Then you should place the non-pencil leg on point 1 and swing the pencil side around to mark the point that was the same distance from 0 to 1 on the right of point 1. This created point 2. This is what the number line looks like now:

You could use this to create the next consecutive point for infinity.

This technique can also be used to create negative points on the number line. If the same distance measure was kept on the compass, the non-pencil leg could be placed on 0 and the pencil leg could be swung to the left to find the point -1. This is what the number line looks like now:

This could also be continued to negative infinity.

Next you are given two segments (a and b) and are asked to have them add together and subtract.

To add segments a and b together you should first draw a straight line using your straightedge. Mark an arbitrary point on the lawn you have drawn. This will be the left end point of the new segment you are creating. Then set the compass to length a. Place the non-pencil leg on the point on the line and swing the pencil leg to find the other endpoint of segment a.

Then set your compass to the length of b. Place the non-pencil leg on the right endpoint of segment a on the line you have drawn and swing the pencil side the mark the other endpoint of segment b. You have now created segment a+b.

To create a-b you must once again draw a line with your straightedge and mark a point to be the left end of segment a. Then set your compass to distance a. Place the non-pencil side on the point you have marked and swing it so that the pencil leg marks the other end of a.

Then set your compass to segment b. Place the non-pencil leg of the compass on the right endpoint of a and swing it to mark the left endpoint of b. The part of a that is not also encompassed in b is a-b.

Next you learn to create fractions by using the concept of similar triangles. Similar triangles are triangles that have the same angle measures and whose sides are proportional. This can be done using the following example:

You are given segments a and b, how can you construct a line segment of length a/b. This can be done be creating a right triangle where a is the vertical leg and b is the horizontal leg.

Now add a point that is 1 away from the right angle and create a triangle so that the vertical side is of length a/b. It is of side length a/b because the side lengths are proportional.

The main goal of this lesson was to introduce the idea of constructible numbers and how they can be created.