OBJECTIVE - G.GPE.A.1

 

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

 

INTERPRETATION OF OBJECTIVE - G.GPE.A.1

 

This objective connects the familiar shape of a circle to the coordinate grid. We use the definition of a circle and the distance formula to derive the equation of a circle. We also learn how to complete the square so that we can manipulate the equation into the vertex form.

 
     

 

 

SKILLS

(1) The student will be able to derive the equation of a circle.

 

(2) The student will be able to determine the center and radius of a circle when given an equation of a circle.

 

(3) The student will be able to complete the square to transform the equation into vertex form.

  

 

THE BIG IDEA

The study of conic sections provide us a nice connection from the world of geometry to the world of algebra. When we move our geometric shapes onto the xy coordinate plane we are able to describe them as equations and then study their behavior. Circles are the easiest of the conic sections. A good place to start!!

 
     
 

 

TRAPS & PITFALLS

The biggest trap is that the equation of a circle has a constant value that represents the square of the radius... not the radius.

Lots of students assume that

has a radius of 4.

 

Another area of difficulty is completing the square. Students with weak algebra skills struggle to see what the purpose of the steps are and they often forget to balance the equation as they add new values to the equation.

  

 

PAST CONNECTIONS

The students connect to some algebra skills here - the distance formula and completing the square. This is a very nice objective to introduce algebraic relationships in geometry. The previous concepts aren't very difficult and yet they provide a nice bridge to the new concepts.

 

FUTURE CONNECTIONS

We are preparing students to study all kinds of relationships through coordinate analysis. While we cover circles here we will look at parabolas next and then later they will look at the rest of the conic sections.

 

 
     
   

 

MY REFLECTIONS (over line l)

The equation of a circle is a very nice connection between the two worlds... geometry and algebra. Quickly we see how by moving from the generic plane to the coordinate grid that we are able to analyze things much more thoroughly. We are able to derive equations, plot circles, and determine facts about the circles.

 

I teach this unit (unit #4) after unit #5 because I like to do all of the circle properties and relationships first and then move to the coordinate grid. This makes an easy transition from unit to unit as well.

 

Teaching completing the square is also review for them but many struggle with it so I work backwards using binomial multiplication first.