Proving the Medians in a Triangle Meet at a Point
Lesson 5 of 7
Objective: SWBAT use analytic geometry to prove that the medians of a triangle must be concurrent.
Activating Prior Knowledge
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.
A Specific Case
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.
Proving the General Case
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:
- I give them time to analyze the step and think privately about their justification/explanation.
- A or B partner (alternating) shares his or her justification/explanation.
- B or A partner (alternating) agrees or disagrees with what has been said.
- 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.
- 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.
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.
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.