As the students enter the classroom, small whiteboards with the coordinate plane on them, markers, and erasers are on their desks. A "Do Now" problem is on the board, asking the students to plot two points to create a line segment, and to find the midpoint, length, and slope of this line segment. Students compare their answers in their groups. If students seem to struggle with any of this, I repeat this process with new pairs of points, until everyone is comfortable with the concepts.
Next I present the students with a series of new tasks. I give them two points to plot that create a horizontal line segment and ask them to calculate the slope, and then repeat this process with two points that create a vertical segment. In a previous lesson entitled Basics of Coordinate Geometry, we discussed the difference between no slope and zero slope, but I think it is helpful to revisit this concept from time to time.
I then give the students two more points to plot, and ask that the students create a line segment that is parallel to it. We discuss the relationship between the slopes of the line segments. I then ask that the students create a new line segment that is neither vertical or horizontal, and use coordinate geometry to draw a line segment that is perpendicular to it. I ask that the students discuss their approach to this problem in their groups, and I move from group to group to ensure that all understand the relationships that exist between slopes of perpendicular lines.
When all seem to understand this, I conclude by asking the students to create a rectangle using our coordinate geometry concepts ( specify that no sides may be horizontal or vertical), and to find the perimeter and area of their rectangle. This requires that students pay close attention to the structure of the figure they are creating, and connect that structure to their coordinate geometry concepts of slope and distance. (MP 7) Students share their results in their groups.
Now it's time for the students to put all of their new knowledge to work! I hand out new sheets of graph paper and the problem set entitled Second Coordinate Geometry Practice.
Before the students begin, I ask them to listen carefully to my directions: Not only would I like them to solve these problems, but I would like them to focus on:
1) showing their work clearly, using good geometric notation
2) justifying their answers completely and concisely, using good geometric vocabulary (MP3)
This is really their first attempt at justifying their answers, and we will take a good amount of time to analyze and evaluate their answers. The students work in their groups.
I ask them to complete the first four problems. When it appears that all of the groups have completed them, the class comes together to compare answers. I take my time with this section - no rushing - because I think this exercise is really beneficial for the students. I also use this opportunity to stress the importance of respecting each other's efforts; I communicate clearly that even if someone arrives at an incorrect answer, there will still be much that can be learned from that person's work. As we discuss the problems and their solutions, I stress that their answers need to make sense and that their answers need to be complete, with appropriate justifications.
Problem 1 is straightforward.
Starting with problem 2, I ask students to volunteer to share their solutions using a document camera. In problem 2, I expect that some students will have found the midpoint of B and M, rather than the endpoint A, and we discuss this. Additionally, we look at how students justified their answers. How did they support their argument that their coordinates for A are correct? There are a number of ways to approach this, from using their coordinates for A and the coordinates of B to find the midpoint, to looking at the points on a graph and discussing slope, to finding the distance from A to M and from M to B to ensure that the distances are equal. We discuss all of these options and any others that students might have tried.
Problem 3 contains concepts very similar to problem 2, and again I ask students to display and explain their approaches to this problem.
In problem 4, we look at the different ways in which the students calculated the lengths of the sides of the triangle, and different approaches to finding the area of the triangle, if students used methods other than using the area formula. We also discuss the different forms in which students give the answer for perimeter, as an exact value or as an approximation. (MP6)
I then ask that the students work in their groups on problem 5. The notion of diagonals bisecting each other tends to be a tough one for the students to wrap their heads around, so as I walk around the room, I discuss what this means (in terms of the diagonals having the same midpoint) and we discuss it again as a class when all have completed the problem.
One of my emphases is problem 5 is vocabulary. What does it mean when the question asks if the diagonals are congruent? The phrase we have used in the past is "Congruent segments are equal in length" and I refer to this.
In problem 6, precision and vocabulary are again key components. (The vocabulary and circumference of circles are 7th grade concepts, and so should be familiar to the students.) As with the previous problems, students display and explain their approaches.
I assign problem 7 for homework.
I hand out the Check for Understanding to the students. This activity will, I hope, serve two purposes - to help me understand those concepts that the students might still be shaky on so that I can address these topics before the unit test, and to give each student feedback on his or her understanding of the concepts.
I also announce that the unit test will be given the next day, and that we will spend time reviewing and answering questions before the test.