I am a firm believer in handing tests back during the following lesson. Sometimes it is definitely painful to get tests corrected so quickly, but I think it's important to provide prompt feedback to the students.
At the start of today's class I hand the Unit 1 Test back and provide students time to talk within their Groups about their answers (MP3). I then ask if there are any problems on which anyone would like more explanation. If so, I explain these on the whiteboard.
In the previous unit we used constructions to find a midpoint. What about finding midpoints on the coordinate plane?
According to Math Open Reference:
If you know the coordinates of a group of points you can:
In this lesson we will explore a number of these topics, including midpoint, distance, slope, and the slopes of parallel and perpendicular lines. This in turn will set us up for future lessons on finding area and perimeter, studying equations of lines, parabolas, and circles, and exploring transformations.
To begin today's foray into Coordinate Geometry, I have on hand lots of graph paper and I hand out one sheet to each student, along with a straight edge. I have the students fold the graph paper into quarters (as they would say, "Like a hot dog roll and then like a hamburger roll") and draw a set of axes on the top left quadrant of their paper.
I am not particularly picky about having the students fold their papers, label their axes, number the axes from -10 to 10 and so on; in fact, I know my kids have been coached in middle school, for developmentally sound reasons, to do all these things, and usually at least one student will ask about having to do them. I always respond, "These are your notes and you can choose how and what you write on your notes." I stress this because I want wean them away from relying on their teacher to tell them exactly what to do and how to do it; instead I'd like to help them move toward taking responsibility for their own learning.
My students (the honors or accelerated students) are pretty solid on which axis is which, the direction the numbers go, and so on, and therefore I don't feel the need to emphasize these concepts; I realize, however, that groups of students who are more diverse in ability and experience may, in fact, need a lot more practice on these skills.
I have the students plot the points (6,8) and (8,10) and draw a connecting line segment, and then I ask: What is the midpoint of this line segment? My students are usually able to find it by inspection.
We repeat this process with (1,2) and (5,0), then with (-2,5) and (-6,1), continuing with more if necessary. At this point, I ask a series of questions:
The most common error students make with regard to midpoints is confusing the Midpoint Formula with slope or distance (i.e., "Should I add the coordinates or subtract?") For this reason, I really hammer home the notion that finding the coordinates of midpoints is nothing more than averaging the x-coordinates and averaging the y-coordinates, because they all know how to average two numbers.
I give the students several more pairs of coordinates and ask them to find the midpoints without plotting the points, and, as we go over these, I constantly pose the questions:
Finally, only after all this, do I ask what the midpoint formula might be. Most of the time, the students are easily able to volunteer this, particularly after the discussion relating midpoints to averages. I find it is helpful at this point to take the time to discuss and contrast subscripts and exponents, as there are some students who confuse them. And someone usually asks at this point, "Do we have to use the formula?" My answer to this question is, "No."
Using the top right-hand quadrant of a sheet of graph paper, I have the students plot the points A(2,2) and B(5,6), and I ask them to figure out the length of line segment AB. I give them time to confer with their groups and, if they seem to be struggling, I remind them of the work that we did in the opening lesson on the isosceles right triangles.
How did we figure out the length of the hypotenuse?
Eventually, perhaps with some further suggestions from me if needed, the students should draw a right triangle with line segment AB as its hypotenuse and use the Pythagorean Theorem to calculate the length of the line segment.
This section of the lesson is a good opportunity to frequently remind students of the different notations used for the length of a segment and the name of the segment. As I write the length of segment AB on the board, I ask various questions about the notation I should use and, as I walk around the room observing the students' work, I watch for their use of notation.
Now, on the bottom left-hand quadrant of the graph paper, I have the students repeat the same process with C(-1,-2) and D(2,4), asking them to find the exact length of line segment CD. When students have successfully completed this, we discuss the triangle that they used to find this length.
I'd like my students to think about their answers whenever they find length or distance, and to always ask themselves, "Does this answer make sense?" (MP1, MP6).
Next I ask the class about the legs of their right triangle. Why does one leg have length 3? Where does this 3 come from? Why does the other leg have length 6? In their answers to these questions, I'm expecting to hear that these values are the differences in the x and y coordinates, but, even better, I'd love to hear that the lengths of the legs represent the distance between the x-coordinates and the distance between the y-coordinates. I have found that finding the lengths of the legs of a triangle makes a lot more sense to some students when I say, "How far is it from -1 to 2 on the number line? How far is it from -2 to 4 on the number line?", particularly for those students who struggle with operations with signed numbers. I will model this line of thought often when doing problems involving distance.
Lastly, I ask the students what the formula for distance would be. I help set the stage for this by reminding them: What process did we use to find distance or length? How did we find the lengths of the legs of the triangle? I give them time in their groups to work on this, while I sketch on the board a line segment with endpoints (x1,y1) and (x2,y2). This helps to suggest notation for those who are struggling with it.
Once we have arrived at the Distance Formula and tried it out on sample coordinates, I take the time to remind the students again that they can simply ask themselves "How far is it? How far is it from one x coordinate to the other? How far is it from one y coordinate to the other?" This helps those students who struggle with subtraction, and helps those students who use subtraction make certain that their calculations make sense.
My students have encountered slope in Algebra and in 8th grade, but, invariably, when I ask them what slope is, they respond mechanically, offering up either "delta y over delta x" or "y equals mx plus b." (Maybe this will change as students are exposed to the Common Core!) When I ask what these phrases mean, I'm almost always met with silence. So we go to the bottom right-hand quadrant on our graph paper and proceed similarly to my midpoint process.
Once we begin to make some progress, I ask the students to plot (3,-5), (6,-5), and draw a line. I'll ask, "What is the slope of this line?"
When we think rise over run, the change in the y-coordinates is 0 and the change in x-coordinates could be any number (other than zero), depending on what points they choose to use. What is 0 divided by a number equal to? We do the same thing with (4,2) and (4,-3). Here the change in the y-coordinates could be any number, but the change in x-coordinates is 0. What is a number divided by 0 equal to?
I ask the students why 0 divided by 2 is 0, while 2 divided by 0 is undefined. My experience has been that they have no idea, so I quickly run through a division by zero conversation. I think this is an important conversation to have with them in terms of number sense, and is also key in helping them to understand why horizontal lines have zero slope and vertical lines have no slope. Once they own this knowledge, they won't have to rely on memorization but will know conceptually what the slope of a horizontal or vertical line is (MP 2).
All of this leads us to the slope formula. Once again, I ask the students to figure out and tell me what the slope formula is. We practice using it on some points that I write on the board, and we revisit the notion that the difference in the x values and the difference in the y values is really just the distance between these values on the number line (MP8).
Lastly, I bring up the fact that someone had mentioned "y equals mx plus b" when I asked about slope. What is y = mx + b? At this point, they realize that the m represents the slope of a line and most recall that b is the y-intercept of a line. I explain that we will be working with equations of lines later in the course.
I hand out the Coordinate Geometry Practice problems. I also provide graph paper and small white boards and markers that have the coordinate grid drawn on them. Students can choose which of these tools (MP5) they would like to use.
In this set of problems, the students first practice the concepts of midpoint, distance, and slope. Then they apply their knowledge to two problems that require them to determine which of these concepts they need to use. These problems also require that the students begin to justify their answers.
The students work in their groups, discussing approaches and comparing answers. (MP3) Any problems that are not completed during the class period will be assigned for homework.
With 5 minutes left in the class, I ask the students to complete the Coordinate Geometry Practice problems for homework.
I then hand out the Ticket Out the Door. In this short assignment, I am hoping the students will tie together one of the geometric definitions they have learned previously with the coordinate geometry that they learned in this lesson. It also asks them to justify a congruency relationship using a definition, which is a skill leading directly to the geometric proofs in a future unit.