When the students enter the room, they find Right Triangles Diagram on the board. I begin by reviewing old vocabulary:
This is all vocabulary that is very important to this unit.
I then ask how many triangles there are in the diagram. Many students will only see two triangles at first, but usually their classmates are quick to point out their mistake! At this point, the students should be ready to delve into the meat of the lesson.
Next I hand out the Diagrams Handout.
Diagram 1 Exploration
Using Diagram 1 (triangles like those we just discussed) of the hand out, I ask the students to use scissors to cut out the two smaller triangles at the bottom of the page. I ask the students to stack the triangles, with the large triangle ABC on the bottom and the small triangle on the top, and with angle A from each triangle lined up, as shown in the diagram.
This diagram is reminiscent of the similar triangles we worked on the previous unit, in which a line parallel to one side of a triangle created similar triangles and proportional segments. I ask the students:
What could be the relationship between these three triangles? If the answer of similar is not forthcoming, I ask if the triangles are congruent. Why or why not? If not congruent, then what are they?
I then have the students label all three of the triangles’ angle A as x (front and back on the two smaller triangles), draw the right angle symbols in all the right angles (front and back of the smaller triangles), and then ask them to determine an expression for the remaining angles, hoping that they will determine that the third angle must be 90 – x. (Some students may need some assistance with 180 – (90 + x), if they choose that route, in order to arrive at 90 – x. This provides a good opportunity to review some algebra concepts.) I ask them to label all of the remaining angles of the triangles 90 – x (front and back on the smaller triangles).
This is a chance to review old concepts:
I have the students place the triangles side by side and then ask:
How do you know these triangles are similar?
Here I am hoping that someone will offer up the Angle-Angle Similarity Postulate. Having labeled the angles x, 90 – x, and drawing in the right angles on their triangles, the angle congruences should be clear to the students.
I ask the students to place the two smaller triangles on top of triangle ABC, so that their triangles look like the diagram on the board. Working with a partner, I ask the students to explore the following:
What transformation or composition of transformations would map triangle ADB onto triangle BDC? triangle BDC onto triangle ABC? triangle ADB onto triangle ABC?
When the students have arrived at answers, the class is given the opportunity to share and compare their transformations. All of the offered transformations should include a dilation, and this exercise will help not only to familiarize the students with relationships between these three triangles, but will help to drive home the connection between dilations and similarity.
Diagram 2 Exploration
Moving on to Diagram 2 on the handout, I ask the students to outline the three triangles in different colors. Then I ask them to label the sides. Beginning with the small triangle, we label the short leg (SL), long leg (LL), and hypotenuse (H). We repeat this process with the middle-sized triangle and then the big triangle (which always leads me to make dumb references to Goldilocks and the Three Bears. I don’t know if many students even know this story anymore, though!). The results are shown in the Similar Right Triangles snapshot.
At this point, we take a step back and look at the diagram as a whole. What do they notice about , for example? While is the short leg in the green triangle, what is it in the red triangle? I ask students for their observations about the diagram in an effort to continue to familiarize them with the diagram and all of the different angle and segment relationships that result from the similar triangles.
Still looking at the colored triangles and the sides labeled in the diagram, I ask the students to tell me what proportions are possible. We arrive at the proportions indicated in the snapshot. We also discuss variations; is it okay, for example, for you to use H/SL instead of SL/H? As a result of the previous chapter on similar polygons, students don’t seem to have much problem with this.
Having established an understanding of all of the possible proportions, students are now ready to find the length of a segment. I ask the students to look at Diagram 3 of the handout, on which there is a numerical problem. Once again, I ask the students to outline the triangles in color, and then focus on the small triangle. What two sides are labeled in the small triangle? What ratio, therefore, can we use? In this case, it involves the short leg and the long leg, and I ask them to write that ratio down, next to the problem, so that it is there as a reminder when they set up their proportion. I then ask which of the other two triangles is associated with x in this problem. Since neither the short nor the long leg of the large triangle are known or labeled, we must therefore focus on the middle-sized triangle; once all seem convinced, I ask them to set up a proportion and solve for x.
At the bottom of the page, I have written, “Does your answer for x make sense? Why or why not?” These are questions that I ask constantly, hoping that the students will model this behavior and ask these questions of themselves when they problem-solve.
I hand out Similar Right Triangles Problems. These problems can be pretty challenging for students. First of all, in a diagram with three triangles, they need to establish which two triangles are used in the problem, and then they need to figure out an appropriate ratio. I begin this lesson slowly and methodically, asking the students to stay on pace with me for the first couple problems.
I have found that it helps to emphasize to the students that they should begin by:
This will establish the ratio that they use. I ask them to write that ratio down next to the diagram. Then I ask them to find another triangle in which those same two sides have been indicated, and to outline and label that triangle.
Students are now ready to set up and solve their proportion, and this is where the second challenge comes in for many students. The rigor of the algebra throughout this unit is (or can be) pretty high. I will be asking my students to simplify radicals, to factor quadratics, to multiply and square binomials, and to solve both first and second degree equations. This is another reason that I take this problem set pretty slowly and methodically, and circulate through the room, with an eye toward those students who are struggling with algebra.
In the third problem, the students will need to add 2 and 4 in order to find the length of the hypotenuse. This is intuitive and is usually not a problem; however, in the final four problems in the problem set, they need to add a number and a variable, and this usually causes more difficulties. It is a concept, however, that we worked on in the Similar Polygons unit, where the students did error analysis, and I am hopeful that the work that we did in that unit will help to make this unit progress smoothly.
After the class has completed and discussed the first three problems, I ask them to take a step back and look at their three proportions. Do they see any patterns? If their answers are no, then I continue to ask this question as they complete each problem. Eventually, students begin to observe that their proportions always have one value diagonally across from itself and we take a moment to discuss this as a class. I have found that this observation helps students; it’s a way of checking themselves – if something is not diagonally across from itself, they can be positive that they have made a mistake, and know that they need to reexamine their steps in setting up the problem.
By the fourth or fifth problem, I have found that the students begin to take off on their own, comparing answers with their neighbors and gaining in confidence. At this point, I walk around the room, watching for struggling students and for any concepts that seem to be particularly troubling.
I hand out the Ticket Out The Door. In it, I look to see if the students can see and appropriately name the similar triangles in the diagram, indicating to me whether or not they have come away with an understanding of the triangle relationships.
I also ask for feedback on their understanding. Are there any concepts in which they are having trouble? Do they feel confident in what they are doing, or are they finding it difficult? If so, what aspects are causing them difficulty?