Where we've been: In the past two lessons, students have dealt with segments lengths abstractly (segments expressed as algebraic expressions) and concretely (using rulers to measure).
Where we're going: In this lesson, students will determine segment lengths and midpoints using coordinate formulas.
So in this section of the lesson, we want to make sure that students have a command of the notation involved in measuring segment lengths, and that they have a conceptual understanding of segment length and midpoint of a segment.
Materials: Ruler with centimeter markings; Calculator (Optional); Paper
1. Create segment AB with length 13 cm. Be precise.
2. Find the exact midpoint of segment AB and label it S.
3. Measure to verify that S is the midpoint of segment AB. If it is, mark the diagram appropriately to communicate this.
4. Write a concise paragraph, using geometric notation where appropriate, explaining how you verified that S is the midpoint of segment AB.
See a Sample Paragraph here.
In this lesson there are some essential understandings that we want students to develop. First is the idea that coordinate planes do not exist in and of themselves; we define them in order to suit our purposes. Next is the idea that in a two-dimensional coordinate system, horizontal and vertical distances are simpler to determine than diagonal distances. Finally, the idea that diagonal distances can be analyzed in terms of their horizontal and vertical components is important.
We’ll also be reinforcing some basic notation that students have been learning.
So to begin this section of the lesson, I give each student a sheet of grid paper. Working from the document camera, I give the directions contained in the resource, Distance Formula Concept Development: Student Instructions. [I intentionally do not give students a copy of the directions, nor do I show all of the directions at once. When students receive a list of instructions all at once, they tend to approach it as a checklist rather than a learning activity.]
After #14 on Distance Formula Concept Development Student Instructions, we pause for a mini-lesson on Deriving the Distance Formula.
Once students have a good feel for the distance formula, and a good set of worked examples for reference, it's time for them to work on some examples on their own.
The goal is for them to develop fluency and automaticity with the formula. Ideally, they will not have to draw out a right triangle every time they get a distance problem.
Instead, they should start to see the coordinates and have a good feel for the value of (x2 - x1) and (y2 - y1) just by asking themselves how far apart x1 and x2 are on the number line. Same for y1 and y2. And they should quickly be able to find the square root of the sum of the squares of these two quantities.
Which reminds me, make sure that students are skilled at squaring numbers (as opposed to multiplying them by two), that they know their basic multiplication facts, and that they can estimate irrational roots so that they know when their answers are unreasonable.
Finally, this is definitely a lesson suited for using calculators. As much as students need to learn math facts and number sense, they also need to know how to use basic technology and this skill needs to be developed too.
To check for understanding, let's see if students can transfer what they've learned about the distance formula to a 'not so novel' context: Perimeter and Area of Triangles, Rectangles and Squares. At this point, students may only be comfortable using the distance formula when instructed to do so. The goal of this section is to have students practice selecting the formula as a tool when appropriate. Having this type of command of the formula will be useful, for example, later when students will have to derive equations of parabolas and circles based on their geometric definitions (CCS HSG-GPE)
The Area and Perimeter in the Coordinate Plane resource is intended to be used as part of an in-class activity.
At this point in the lesson, I am intentional about fostering student independence. Students are all too willing to hand over the reins of their cognitive architecture to the teacher. For example, as they look at (as opposed to read) the directions, students will say things like “What is it you want me to do here?” I clarify that it is not ME that wants them to do anything. The DIRECTIONS are calling for them to do something and it’s their job to figure out what that is. So, the bottom line: we have to get students to read and interpret the directions.
The next type of question I get: “So should I use the distance formula to find the perimeter (or area)?” I recognize this, again, as an attempt to put the onus on the teacher. So I redirect with questions like:
Why does it make sense to you to use the distance formula?
What does perimeter mean?
What information would you need in order to calculate its perimeter?
What is the distance formula for?
What specifically could the distance formula tell you about the figure?
Another issue that arises is that students are unsure which coordinates to “plug into” the distance formula when presented with the multiple ordered pairs that form the vertices of a figure. Students will say things like, “Which points do I use?” The gut feeling seems to be that they should just throw all of the points into the formula and out would come the perimeter. Directing their attention to the graph, I ask, “What are you trying to find?” Some might say "The Perimeter"...."Ok then, if you had a ruler, what would you do first?" Then, based on their response I might say “Oh, if you want to find the length of segment AB, then which points would be appropriate to use in the formula?”
So the basic idea while enacting this section of the lesson is to place the onus for learning with the student and force them to grapple with question: When do I apply the distance formula?