Geometric constructions continued

The second day of constructions started with students practicing duplicating an angle. This embedded constructing segments of the same length, so it was good practice for both constructions introduced on the first day.

Some students were still struggling a bit with the congruent angle construction. As I worked with these students, I could see the light bulb going off as to what they were doing and why they were doing it.

I then presented the next challenge. I drew a line segment on the board and labeled it as segment AB. I then placed a point C on the board such that C was not on the segment. I told students their task was to construct a line parallel to segment AB that passed through point C.

I let students play around with this for about 5 minutes. Most were stymied. I told them to think about what they knew about parallel lines and angle relationships. This got a few students moving. I told students to look in their notes to see what the relationships were. Some said they had no notes on this topic. As I looked through their notes, I pointed out where this information was recorded.

More and more students started to realize they needed to make use of corresponding angles, yet were reluctant to draw a new line (the transversal) to help. I pushed them to think about how they would duplicate the angle without adding any new lines. Most came to realize that they indeed had to draw a transversal.

With the transversal drawn, many students still struggled with how to proceed. I encouraged them to think about the process they used for duplicating angles. Tentatively, students started to create the congruent angle they needed. I noticed many students wanted to fall back on using a protractor or making use of markings on the compass as to the width of the compass opening. I had to reinforce that the compass was the measuring device and that any markings on the device was irrelevant.

As I walked around, I noticed one girl had done something entirely different. I asked her to explain her method to me. She said she had used the compass to measure the separation between points A and C. She then went to point B and drew an arc the same distance. Next she measured the distance between points A and B. Placing the compass on point C, see drew an arc that intersected with the first arc she had drawn and labeled the intersection point D.

She said that point D was the same distance away from B as C was from A. She drew the segment CD and said this was parallel because C and D were the same distance away from the segment and so every point in between will also be the same distance. She had, in effect, constructed a parallelogram. I congratulated her on her thinking. She said it just made more sense to her based on one of the definitions of parallel lines that we had examined.

I had a student present the construction of the parallel line by using corresponding angles. We discussed this construction and students asked questions for clarification. I could tell that many still felt shaky on this construction. I then had the one girl share her parallel points construction. The reaction from the class was this was so much simpler and easier to understand. I had to agree.

I spent the last few minutes of the class going over a couple of problems from a quiz I gave last week. I was disappointed in students not thinking about the information given to them. I discussed two problems in particular. One, the problem stated a ray was an angle bisector and students had to justify the equation they set up to solve the problem. The second involved midpoints; many students wrote on the quiz that they didn't remember the midpoint formula.

This disturbed me because I hadn't asked them to remember formulas but, rather, to use reasoning and common sense to find solutions. These problems are perfect examples of why students do not perform well on state and national assessments. They have been hammered with memorizing formulas that have no meaning to them versus understanding the facts of the situation and using some basic knowledge to construct a solution. Hopefully the class will become more comfortable with this approach as we move along.

The geometry team has a common assessment scheduled for next week that covers the first unit. I will spend the two class days before working with students to get better at identifying the relationships they are seeing. Their algebra skills seem adequate to solve the equations that result in the problems, it's the recognition of the relationships that appears to be the problem.

I was to introduce algebraic proofs as part of this unit. There is no way that I can do this justice prior to the assessment, so I'll hold off assessing this piece until later.