Post #2

 Welcome back! We're talking about Geometry today!

While I'm not teaching Geometry now, last Spring I put together a geometry lesson involving constructions to bring to job interviews. At the time that I was putting it together, I was taking a geometry course where I found a lot of enjoyment in geometric constructions with my straight edge and compass. I was inspired to explore technology that could help me make even more precise constructions. I found that Geogebra offered lots of easy to use tools that captured my attention and made geometric constructions fun and simple to explore.

In the larger lesson that this Geogebra activity is a part of, the objective is that students will be able to draw a circle through three given points. This construction uses the construction of perpendicular bisectors, so students will also learn that. This relates to New York State Next Generation Mathematics Learning Standard GEO-G.CO 12: Make, Justify, and Apply Formal Geometric Constructions.

My goal in exploring this software and creating this lesson is three-fold:

1. I want to find engaging technology to teach constructions to students

2. I want to provide students with strategies to check their work and feel confident in their constructions. I want them to appreciate precision and see that they're able to make something cool!

3. I want students to explore geometric constructions and see that constructions work no matter the placement of the points or line segments. I want them to see that the purpose of constructions is to make powerful, general statements about mathematics.

If I taught this lesson, I would model how to use Geogebra by demonstrating the construction of perpendicular bisectors myself. This would give students a tour of the software and a tutorial on how to use it. It would also scaffold their construction of a circle through three points later in the lesson.

"I do" Demonstration: Perpendicular Bisector Construction

1. Students would receive this handout of steps so that they could follow along. (From mathopenref.com)

2. I create two points and a line segment on Geogebra using the point and line segment tools.

3. I use the compass tool to set a radius length and make two circles, one with center A and the other with center B

4. I use the intersect tool to make points where the two circles intersect each other. I use the line tool to connect those points.

5. The line that I have created is the perpendicular bisector of segment AB!

6. I can check my work by using the perpendicular bisector construction tool on Geogebra, and it turns out that my line (h) is exactly the same line that Geogebra gives (i) when I use their construction tool.

"You do" Activity: Circle through three points construction:

1. Students receive this handout of steps to follow (from mathopenref.com)

2. Students use the point tool to make three random points on the graph (it doesn't matter where they are!)

3. Students use the line segment tool to make two line segments using those points.

4. Using what they learned from the "I do," the students create perpendicular bisectors for those line segments. They use the intersect tool to create the point where the perpendicular bisectors intersect each other.

5. They use the compass tool to make a circle with radius LA (or LB or LC) and center L. They see that this circle passes through all of their original random points perfectly!

6. Then, to check their work, students can use the circle through 3 points tool to see that that circle is precisely the same as the one they constructed!

7. I would ask students to see what happens to the circle when the points are closer or further away from each other. What happens if they are right next to each other? Students should see that these steps always work (when the points are collinear, the radius of the circle is infinity!), and they've worked out a very powerful geometric construction that can be applied to any three points.

Reflection:

Mathematics is a beautiful academic discipline which is centered around precision and logic. I think this Geogebra lesson introduces those values of mathematics in a very accessible way. The various tools on the software allow students to break down what the computer does with the one-step construction specific tools, and see just how precise their own work can and should be. Additionally, the technology allows students to explore different cases easily. They can try putting three points in different formations and positions and see that they can always create a circle that goes through all three of them with the same process. This construction is so powerful that it applies to all cases. I think that these values of mathematics can benefit students in other fields and across many of their interests such as computer science, writing, and debating.