Lesson: Deriving the Equation of a Circle
Goals and Objectives:
- Students will derive the equation of a circle using the distance formula and/or Pythagorean Theorem.
Standards:
- MCC9-12.G.GPE.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.- MCC9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. (Restrict to context of circles and parabolas)
Previous Knowledge: distance formula, Pythagorean Theorem, triangles, circles, radius
Student Grouping: For this activity, students will be working in pairs assigned by the teacher.
Materials: Geometer’s Sketchpad file, Math Warehouse Circle Applet, Math Open Reference Circle Applet, What’s in a Circle worksheets, How Many Triangles worksheet, Equations of Circles in Real Life worksheet, From Standard to Vertex Form worksheet, pencil
Vocabulary: circle, triangle, distance formula, Pythagorean Theorem, radius, hypotenuse, leg
Lesson Outline:
- First assign students to watch the Learnzillion video on circles as a review of what they already know in preparation for the circle standard. The video can be found at the link below:
- Students will derive the equation of a circle using the distance formula and/or Pythagorean Theorem.
Standards:
- MCC9-12.G.GPE.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.- MCC9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. (Restrict to context of circles and parabolas)
Previous Knowledge: distance formula, Pythagorean Theorem, triangles, circles, radius
Student Grouping: For this activity, students will be working in pairs assigned by the teacher.
Materials: Geometer’s Sketchpad file, Math Warehouse Circle Applet, Math Open Reference Circle Applet, What’s in a Circle worksheets, How Many Triangles worksheet, Equations of Circles in Real Life worksheet, From Standard to Vertex Form worksheet, pencil
Vocabulary: circle, triangle, distance formula, Pythagorean Theorem, radius, hypotenuse, leg
Lesson Outline:
- First assign students to watch the Learnzillion video on circles as a review of what they already know in preparation for the circle standard. The video can be found at the link below:
- Show students the first circle applet to explore. The link can be found below:
Allow students to spend time in pairs exploring the applet on their own in search of the underlying mathematics. Then give them the worksheet below to complete as they continue exploring:
- Show students how to use the second applet, specifically how the center can be moved and the radius length changed. A link to the applet is found below. The same pairs should use the applet to complete the worksheet that follows:
- If there is time, or as an assignment for homework, students should watch the equations of circles Learnzillion video to make sure they are aware of the key concepts present in the applet activities. The link is below:
- To make sure students' understanding of the points on a circle corresponds with its derivation from the Pythagorean Theorem, use the following worksheet and GSP file:
![](http://www.weebly.com/weebly/images/file_icons/file.png)
equation_of_circle.gsp | |
File Size: | 3 kb |
File Type: | gsp |
- Students should be made aware of the real world applications involving these equations of circles with the following worksheet, which should be worked on in small groups:
- Instead of simply telling students how to convert from standard to vertex form of a circle, students should try to struggle on a problem similar to the one given below:
Hint about completing the square to see if students can make the connection. If a student does, allow them to show the class. Then, give a short explanation of how this process is done.
- As a means of assessment for this concept, give the following worksheet:
- As a means of assessment for this concept, give the following worksheet:
Teacher Resources:
- Below are the keys for all of the worksheets: