We begin our discussion of the Coordinate Geometry Practice problems with Problem 4. I check the students' answers with regard to the lengths of the sides and the type of triangle that it is - I am assuming that most will agree that it is isosceles. If no one volunteers that it also looks like it could be a right triangle, I ask, "Could be a right triangle?" At this point, the students have used the Pythagorean Theorem. I explain that the Pythagorean Theorem can also be used "in reverse," to determine if a triangle is a right triangle - if the theorem works, it must be a right triangle. I ask that the students try this out on their papers.
Once we prove that it is a right triangle, I ask them to find the slopes of line segments AB and AC. I ask, "What do they notice about these slopes? What seems to be their relationship?" I explain that we will be investigating this later in the period.
For Problem 5, we take our time discussing the the relationships of the opposite sides. Not only are the opposite sides congruent but they are also parallel, and the adjacent sides are perpendicular. This provides an opportunity to investigate and discuss slopes of parallel and perpendicular lines.
The diagonals of the quadrilateral are congruent, and the students should be able to determine this. However the fact that the diagonals also bisect each other merits more discussion, as this often proves to be a difficult concept for students.
The remainder of this lesson is devoted to introducing students to dynamic geometry software and allowing them to explore their coordinate geometry concepts with this technology (MP5). See my video narrative for a little bit more explanation with regard to my anticipated use of dynamic software.
Using my interactive whiteboard, I briefly (5 minutes tops!) demonstrate and explain the geometry tools on the software that the students will be using. In particular, I make sure that everyone knows how to:
I give the students another 5 minutes or so to play and explore with the software on their own, and, while they are doing this, I hand out the Dynamic Software Practice worksheet.
For this problem set, I allow the students to choose whether they will work on their own or with a partner. Just like adults, some students are more comfortable working with technology than others!
As the students work on the problem set, I am very active, walking around the room watching in particular for technology issues that might be troublesome for the students. As always, however, I expect the students to communicate with each other and to use each other as resources for help and for sharing observations (MP3).
The material covered in the problem set is intentionally not particularly difficult. Introducing my students to the software while also solidifying their understanding of the geometry basics is my objective here. I suspect that challenging them with learning both math content and a new technology would be too much - at least I know it would be for me!
In the final problem of our problem set, the students were asked to find the slopes of three parallel and perpendicular lines. I anticipate that students will reach this point in the exercise at different times, so, as I notice students finishing up, I remind them: We spoke earlier in class about the slopes of parallel and perpendicular lines. Can you use the software to investigate this further?
When it appears that the majority of the class has had the opportunity to investigate the question, I pull the class together and ask them to present and compare their observations (MP3). I think it is easy for them to determine that slopes of parallel lines are equal; however I suspect that the relationships between the slopes of perpendicular lines (opposite reciprocals) may be more challenging, simply because the software we are using expresses slope in decimal form, rather than as a fraction. (It would be nice if the software designers would change this!) Therefore, I might, depending on the students' observations, have to do a little review of familiar fraction/decimal conversions. This is an opportunity to emphasize again the meaning of slope as a ratio, as well as to discuss precision and rounding (MP6). I might, for example, ask the students to contrast a slope of 1.33 (as reported by the software) and a slope of four-thirds. Why is it more meaningful to speak of a slope of 4/3? What does a slope of 1.33 mean? Are slopes of 1.33 and 4/3 the same thing?
I also mention to the students that we have made observations, through repeated reasoning, about the slopes of parallel and perpendicular lines, but have not yet proven these relationships, and that we will do so in our unit on transformations.
Because of our discussion of the slopes of parallel and perpendicular lines, I have the class all pulled together at this point. I ask them all to create a new window, this time without axes and a grid, and request that the students talk me through our first construction, that of bisecting a line segment. (We covered this in the fourth lesson of this unit.) At each step, I ask them to tell me what our procedure would be using a compass and straightedge, and then how we could replicate this step using the software. I complete each step on the whiteboard and then give the students time to replicate it on their own.
Next we bisect an angle using the software. I follow the same procedure, asking the students to talk me through the process, and then I allow them to do the construction on their own. To verify their construction, I ask them to measure their angles using the measuring tool on the software. This is yet another opportunity to solidify their understanding, this time with regard to the concept of an angle bisector and the relationships that are created. Additionally, it is an opportunity to revisit the vocabulary of angles (obtuse, acute, etc.)
If time permits, I ask the students to create and bisect an obtuse angle, and to verify the angle relationships, and then ask that they create an angle with a given measure (say 78 degrees), bisect it, and verify the measures.
I use the last five minutes of this class period to ask the class to summarize what they have learned with regard to the slopes of parallel and perpendicular lines, and to return to our classroom from the computer lab.