In this section of the lesson, my goal is for students to understand what the Triangle Midsegment Theorem says and to understand what we will need to do in order to prove it (MP1).
To achieve this goal, I guide students through the first page of Part 1_Proving Triangle Midsegment Theorem. We start with the generalized statement of the theorem. Then I introduce a diagram of a triangle. Next I ask students to restate the theorem in the context of the triangle in the diagram. I provide a sentence frame for this restatement and students must fill in the blanks to complete it. I give the students 1 or 2 minutes to fill in the blanks on their own before consulting with a partner. After that, I reveal the correct re-statement while carefully explaining each part of the statement and how it follows from the original statement of the theorem.
Next students translate the restated theorem into precise mathematical notation (MP6). In order to do this, they will need to introduce the midpoints of the sides of the triangle and name them. I give students time to figure this out as this has been one of my overarching goals for the course: getting students to realize that we introduce and name figures in diagrams so that we can introduce them into language using mathematical notation.
When students have had time to write their statements, I have them compare with a neighbor to do some initial quality control. Then I show the correct statement under the document camera.
So at this point, students have basically understood what the theorem says and what they will need to prove for this particular triangle. Now they are ready to start proving.
This section is called 'School of Hard Knocks' because, in it, students will have to pay their dues by grappling with some messy abstract coordinates. This is all because of my intentionally horrible choice of position/orientation for the triangle on the coordinate plane. SEE THE REFLECTION IN THIS SECTION FOR MY THINKING ON DESIGNING THE LESSON THIS WAY.
On page 2 of Part 1_Proving Triangle Midsegment Theorem Students have three tasks:
1. Use the midpoint formula to find the coordinates of the midpoints.
2. Use the slope formula to show that the midsegment and third side are parallel.
3. Use the distance formula to show that the length of the midsegment is half that of the third side.
Recognizing that students can easily get stuck or off track I structure this portion of the lesson pretty tightly so that students get immediate feedback after they've taken a risk to attempt a problem.
First, I give them 2 minutes to find the coordinates of midpoints D and E. Then I model the process. I show, for example, how I like to leave leading terms positive when possible (e.g. b-a as opposed to -a+b) SEE REFLECTION IN THIS SECTION FOR OTHER OPPORTUNITIES TO MODEL SEEING STRUCTURE IN EXPRESSIONS.
Next we move on to #2, which asks students to show that the midsegment is parallel to the third side of the triangle. I begin with a pair-share, asking students how we might use coordinate (analytic) geometry to show that two segments are parallel. After that, students try their hand at finding the slopes of the two segments. I caution my students to be careful with signs, use parentheses where appropriate, and remember to "distribute the negative". When students have had enough time, I show the process for obtaining the answers. Again, there is lots to model with regard to seeing structure and performing mindful algebraic manipulations so I definitely take advantage of these at this point in the lesson.
Finally, we go through a similar process with #3 as we did with #2. There are good opportunities here to model mindful algebraic manipulation as well. For example factoring the 4 out of the radicand to reveal that the third side is twice as long as the midsegment.
At this point, we've proven the midsegment theorem. However, since the activity was scaffolded, it's possible that some students have just been going through the motions without understanding what happened. For that reason, I have students take turns explaining what happened in #2 and #3. The student explaining will explain in their own words the work that was done, why it was done, and what it establishes. Then the non-explaining partner will re-voice what they have heard. Then the roles reverse.
After the last section, this section should be a welcome relief for students. I give each student Part 2_Proving Triangle Midsegment Theorem. As they read, they learn (if they didn't realize already) that we made our lives more difficult than we needed to in Part 1. Now they have a chance to make a better choice.
On the first page of the handout, students need to re-sketch the triangle from Part 1 with position and orientation that will facilitate the proof writing process. Then they need to explain how their choice will make things more convenient. This is an independent process because I want each student to think on their own and make their own choices. They have experience with this type of thinking from a previous lesson on proving the medians of a triangle are concurrent.
After students have completed their sketches and responded to the prompt at the bottom of the first page, we move on the next page to see how the triangle should be positioned and oriented.
Then students will be left alone to prove the Triangle Midsegment Theorem using this diagram. Although easier than the Part 1 task, it is a good way for me to know if students have understood the work from part 1. One thing I tend to see is that students use the distance formula to find the length of horizontal segments. This is a good opportunity to talk about staying present current scenario and not just repeating the exact same procedures from Part 1. It's also a good time to reinforce (MP8)...when we want the length of horizontal segments, we simply subtract their x coordinates and take the absolute value.
When students have had enough time to finish working, I have them get together with their A-B partners and take turns rehearsing what they would say if they were called up to present their proof.
When they have had time to do this, I call on two or three randomly chosen non-volunteers to come to the front of the class and present their proofs.
In this final section, students work on the last page of Part 2_Proving Triangle Midsegment Theorem. The first item gets at the idea that we do not need to prove the theorem for all three midsegments because we can perform rigid transformations (preserving all lengths and angles/slopes) such that a any vertex can be brought to the origin and any other vertex brought to the x-axis.
Items 2 and 3 require students to apply their knowledge of the Triangle Midsegment Theorem in order to classify quadrilaterals.
Finally, items 4 and 5 deal with the fact that the midsegment triangle is similar to the original triangle with scale factor 1:2.
When students are finished, I collect the papers and post model responses on the internet for students to see and learn from.
In this take-home assignment for the lesson, students will need to transfer what they have learned to the related, but somewhat novel, task of proving the relationships between the bases and median of a trapezoid.
I give each student a copy of Independent Task_Proving Triangle Midsegment Theorem to take home. I explain that the successful completion of this task will depend on their ability to apply what they have learned in this lesson to a new context. It will not be exactly the same, but will require the same type of thinking and procedures.
Then all that's left is to collect the papers at the next class meeting and see how well students have done.