Anthony Carruthers JOHN MUIR HIGH, PASADENA, CA
Geometry : Unit #9 - Analytic Geometry : Lesson #5

Proving the Medians in a Triangle Meet at a Point

Objective: SWBAT use analytic geometry to prove that the medians of a triangle must be concurrent.
Standards: 8.EE.C.8a HSG-CO.C.10
Subject(s): Math
60 minutes
1 Activating Prior Knowledge - 15 minutes

In this first section of the lesson, students will be working on Activating Prior Knowledge_Proving Medians of Triangle Concurrent. My purpose in this section is to make sure that my students (a) know what a median of a triangle is, (b) can find the slope a median, (c) can write the equation of a line given two points, and (d) can verify that a point is on a line given the coordinates of the point and the equation of the line.

This section also jigsaws with the next section as the equations we obtain in this section will be used in the next section as well.

I give students five minutes to work on the first problem. Then I go over the answers. As I'm going over the problems, I emphasize the importance of knowing the definition of triangle median and I strongly push students to use the point-slope formula for deriving the equation of a line. Later in the lesson when we begin to deal with abstract coordinates, students' preferred technique of calculating slope and then using y=mx+b to solve for b tends to be less practical and less efficient. 

In my modeling and explanation, I also reinforce the concept that a line is made up of all of the infinitely many ordered pairs that satisfy the equation of the line. Therefore any ordered pair that satisfies the equation of the line determines a point on the line.

Next, I give students five minutes to work on the second problem, which is slightly more challenging than the first because of fractions.

Finally, I go over the answers to the second problem again stressing the definitions and promoting point-slope formula.

 

Jigsawing problem components to reduce cognitive load
Grappling with Complexity

When I originally designed this lesson, in the next section I had students determining the location of the centroid starting with the coordinates of the three vertices. This was a very complex problem to tackle all at once. Students had to find the midpoint of each side. They had to find the slopes and the equations of the three medians. They had to set up and solve a system of equations. And they had to verify that the point of intersection was on the third median. Not to mention the fact that they needed to have the knowledge of geometry and algebra required to pull all of that off. For many students, this was a lot to be doing at once.

This happened to be a lesson for which I had a math coach observing me. Her feedback to me was, "Since you already have students practicing writing equations early in the lesson, possibly you might design the lesson so that students use the equations they write in the first part of the lesson in the second part of the lesson."

So this is how the idea came about to use the same triangle for this activating prior knowledge section and the next section. With this revised design, students already have the equations of the medians before they are asked to find the location of the centroid. In this way, they can focus more on the meaning of the centroid and the algebra required to locate it given the equations of the medians.

This seemingly small tweak significantly reduced the cognitive load on students and allowed them to access the lesson a lot better.

2 A Specific Case - 25 minutes

In this section, I give students  Specific Case_Medians Concurrent. In this exercise, students will be working with the same triangle from the previous section. In that previous section, they will have already found equations for two of the triangle's medians. Their job here will be to find the intersection of those two medians and then verify that the third median also passes through this point.

 

The exercise starts with some basic fill-in-the blank questions to make sure that students are making sense of the problem. I have my students fill these blanks in on their own first and then compare results with a partner before I reveal the answers.

 

Next, I clue students in to the fact that we are working with the exact same triangle we dealt with in the previous section and that they can carry their results forward without showing the work all over again. 

Since I explicitly give students the plan for attacking this problem, I like to give them time to work through it on their own. As students get it, they tend to help students who need help. By the time 90% of the students have finished, I've walked the room and seen who is having more significant troubles. For these students, I will try to do some one-on-one coaching to get them unstuck and I will also model the work process on the board so that students who need more direct input can access the problem.

 

Finally, when all students have had opportunity to access the problem, I call the room to order and ask students to spend 5 minutes responding to the reflection prompt.

 

3 Proving the General Case - 30 minutes

Well, there's no other way to say it. The math we're doing in this section of the lesson is tough and, for many students, will challenge their notions of what it means to do algebra. There is a lot of decision-making involved- from setting up the coordinate proof to making the most efficient algebraic moves. For this reason, I feel the need to just show students how an expert (humbly played by me) would approach the problem.

This is the idea of the worked example, and it has proven to be a useful tool, especially when it is accompanied by active thinking and metacognition by students.

 

So the way I approach this is to have the students read a chunk of text to themselves and then have them respond (first privately, then in conversation with a partner) to the "discuss", "think", etc. prompts that are posed in the handout.

I also err on the side of caution with respect to explaining things that are mentioned in the text. If I sense that students may not understand, I will clarify, elaborate, re-phrase and ask questions to check for understanding. I also share my own responses to the "discuss", "think", etc. prompts so that students can, again, hear the perspective of a relative expert.

When we get to page three, students have to do some heavier lifting to find the slopes and equations of lines. There are three rows in the table that students need to fill in. I give them 5 minutes or so to complete the first row and then I show my process for getting the answer. I use point-slope to find the equation and definitely share with students why I choose to use the point (0,0) instead of (a+b,c), which would be valid, but not expedient. Again, a lot of why I chose to present the lesson this way is so that I would be able to convey all of the decision-making that goes into completing this type of task.

Next, I have students try their hands at completing the second and third rows of the table. When they have had ample time, I model the process for filling in these rows as well.

When we get to pages 4 and 5, students are presented with a fully worked example. All of the algebra has been completed for them. Their job is to understand the procedures that have taken place and why they are valid. For each step, we go through the following protocol:

  1. I give them time to analyze the step and think privately about their justification/explanation.
  2. A or B partner (alternating) shares his or her justification/explanation.
  3. B or A partner (alternating) agrees or disagrees with what has been said.
  4. I call on non-volunteers to share their justification/explanation with the class. We have dialogue until we reach a workable justification/explanation if necessary.
  5. I provide extra commentary as needed. This could mean cleaning up a student explanation or it could mean explaining how the given step was one of many steps that could have been taken and explaining why, among those, it made the most sense in my estimation. #4 on page 4 is a good example of this. See the reflection in this section for more on how getting students to recognize structure allows them to be more adept at using algebra.

 

After all of this explanation and modeling, I give students time to express synthesize the whole experience from their perspective. As I read these, it will give me a good idea of how well students understood the lesson. I tell students that this is also a good way for them to really lock the learning into their long term memory by making sense of it as a coherent process. I remind them that they will have to repeat the process on their own independently (without aids) and advise them to take this opportunity to make sure that they really know their stuff. 

Although we start these reflections in class, I allow students to take them home so that they can continue to rehearse and study in preparation for the independent task (i.e., quiz) they will take on the next class day.

 

Knowing when to use the worked example
Modeling

With the advent of the Common Core, I think one misconception is that everything should be student-centered and constructivist. Math is still an esoteric discipline and in many cases students are uninitiated to the techniques that they will need to complete a given task. 

In these instances, I have found that it is much more efficient and more effective to use worked examples. So what led me to decide that this would proof would best be taught through a worked example? For one, the sheer number of variables involved in the problems makes the algebra very intimidating for students who are not accustomed to dealing with these kinds of problems. There will come a time when they will actually see these problems as simpler problems since they require no computation, but at this point they are just plain intimidating. Secondly, there are algebraic moves that streamline the proof and I suspect that the overwhelming majority of students would not ever discover these moves on their own. Using the worked example allows me to give students experience with these types of problems so that they will not see them as so intimidating, and it also allows me to model and explain the expert algebraic moves that were made in the proof so that they understand why they were made and why they were better than other moves that could have been made.

Basically, it was clear to me that the expertise required to write this proof was not something that students would magically discover or arrive at. I needed to take them through an apprenticeship in order for them to begin acquiring this expertise. Hence my decision to employ the worked example. Then in the next section, students get to put what they have seen in the worked example into practice.

 

4 Independent Task - 30 minutes

At least one day will have passed before students are required to complete the Independent Task_Prove Medians Concurrent. They also will have had time to study the exercises, notes, and worked examples.

So when it's time for them to start this task, they will be alone with the handout, a blank sheet of paper, and their thoughts. I give them 30 minutes to complete the task, and if it seems they need more time, I give more time. For students who tend to finish more quickly, I'm sure to have some Bonus Problems handy that relate (at least tangentially) to the lesson. 

5 Extension Task - 20 minutes

For use after the next lesson on partitioning line segments, I have the Extension Task_Centroid as Partitioner ready.

 

After the lesson on partitioning line segments, it will be easier to see that the centroid partitions each median in the ratio 2:1.

It's a neat connection that reinforces the learning from both lessons. This extension task appears again in the next lesson.