The Circle (Day 1 of 2)
Lesson 1 of 13
Objective: SWBAT meaningfully develop the definition of a circle.
As usual students receive the Learning Targets for the unit. This allows students to focus on what is being taught.
Today we start our study of the definition of a circle. We use the definition to determine the equation of a circle. As they arrive in class, students pick up the supplies needed to do Activity 1 of Wax Paper Conics. This activity requires students to read and follow directions as well as analyze the construction that they are making with wax paper. As the students are working, I move round the room clarifying the task and answering questions.
Today's work begins a 4-day activity with one conic section construction each day. I push the students to get started so that we can use class time efficiently and have time to make generalizations that build on the construction activities. I continually prompt the students as I move around the room. I say, "read the directions, explain the directions to yourself, and then do what the instructions say to do."
The biggest challenge students face is completing Step 2. Many have difficulty with the meaning of the instruction, "the center point lies on the circle." I think some confusion results from a misunderstanding of the definition of a circle. Many students have the misconception that the circle is "the area bounded by the circle, rather than the locus of points equidistant from the center point." If necessary, I will demonstrate this step to the class. If I do, I will also ask students where the circle is located so that we address the misconception directly. I might say something like, "you created the points on the circle, so put the center point on one of those points, then crease the wax paper."
After about 10 minutes I stop the students to see if they can see what shape is being made by the lines. I expect that some students will not be finished, but the construction is a starting point. I may ask the students to finish the activity at home or later in the class, depending on how the lesson proceeds.
Definition of a circle
I am now ready to introduce the definition of a circle and then use the definition to find the equation of a circle. I first have the students answer the questions on the Activity Sheet.
- What shape seems to be made by following these instruction?
- Why does it make this shape?
In a typical lesson, at least one student will say that the points where the lines intersect are all the same distance from the center point of the circle. Other students are not sure what to make of this observation. I have found that this is a good time to introduce a formal definition of a circle. I project a definition on the board (see Circle presentation page 1). Then, we read the definition as a class. While students consider this definition, I ask the following questions:
- What is meant by all points?
- If they are all the same distance from the center, how do the points appear?
Next, I will draw a circle on the board. I will ask students to identify points on the board that fit the illustrate the definition of a circle. Then, I will put a point inside the circle and ask if the point fits the definition. In order to help students realize why they might have arrived in class today with a different definition of a circle in mind, I like to ask students how learned about circles in elementary school. I put a colored circle up similar to the ones that are often used in K-5 mathematics. Of course, students recognize this as a circle. I then explain that the coloring represents the area inside the circle or bounded by the circle, not the geometric circle.
I have a joke that I always use at this point. "How many sides does a circle have?" Students say no sides or an infinite number of sides. I explain that the idea of an infinite number of sides helps develop the formula for the area of a circle which is a topic discussed in calculus. I now give the punch line "A circle has 2 sides." They look at me like I am crazy and I then respond "an inside and an outside." Of course I hear a lot of moans but students do use this to help them remember the circle is not the area bounded by the circle.
Once we have discussed the definition of a circle, I transition to using the definition to find the standard equation for a circle. I move to Page 2 of the Circle presentation. I explain that it is standard notation in algebra to represent the coordinates of the center point as (h, k). Then, I locate a random point (x,y) on the graph. I ask, "Using this diagram, how can we describe the other points on the circle?"
After letting students make some suggestions, I draw a segment from (h, k) to (x, y) and label it as r for radius. I ask, "How can we determine the length of r if we know the center and the point (x,y)?" I will wait until students nominate the Distance Formula as the appropriate tool to use. If they need a hint, I will draw a right triangle so that they can think about using the Pythagorean Theorem to write the equation (see Developing Formula).
Once we complete the work of writing a general equation for the circle, I discuss standard form for the equation of a circle. Following our practice from the Trig Identities unit, I have students begin to create a reference sheet for conic section. I ask students to divide the sheet into 4 sections. The first section is labeled "Circle" and they write the standard equation and identify (h,k) as the center along with r as the radius. We will talk about the general form of conic later so we will add this information later.
Practice and Closure
Since we are getting close to the end of the class I have the students work Practice problems. The first problem should take about a minute to complete. It just involves replacing (h,k) and r with their known values.
The second problem will take some problem solving since the students will need to determine the radius from the given information. Students work on the problem for a few minutes and then we share the work out to the class. If we run out of time before the students have really worked on the problem, I will assign the problem to be finished at the beginning of the next class.