Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 61395
Circle Reasoning
Students use the Pythagorean Theorem (Distance Formula) to derive the Standard Equation of a Circle; then move between descriptions and
equations of a circle.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Computer for Presenter,
Basic Calculators, LCD Projector, Adobe Acrobat Reader
Instructional Time: 2 Hour(s)
Resource supports reading in content area: Yes
Freely Available: Yes
Keywords: center, circle, diameter, distance, Pythagorean, radius
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
Circle Reasoning - Activity 1.pdf
Circle Reasoning - Activity 2.pdf
Circle Reasoning - Activity 3.pdf
Circle Reasoning - Closure.pdf
Circle Reasoning - Extra Practice.pdf
Circle Reasoning - Summative Assessment.pdf
Circle Reasoning - Visual Aid.pdf
Circle Reasoning - PowerPoint.pptx
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will be able to:
derive the equation of a circle, with given center and radius, using either the Pythagorean Theorem or the Distance Formula
write the equation of a circle, with given center and radius or some other descriptive information like ends of diameter, length of diameter, or center and known
point on the circle, using the Standard Equation of a Circle
describe similarities among the Pythagorean Theorem, the Distance Formula, and the Standard Equation of a Circle
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should have prior knowledge of and be able to:
draw a circle with a tool, such as a compass
plot a point in any quadrant of the rectangular coordinate plane
accurately draw a circle on the rectangular coordinate plane either free-hand or with an appropriate tool
correctly recall and use the Pythagorean Theorem
correctly recall and use the Distance Formula
page 1 of 3 compute a square root
factor trinomials
Guiding Questions: What are the guiding questions for this lesson?
1. What are the similarities and differences among the Pythagorean Theorem, the Distance Formula and the Standard Equation of a Circle? (The Pythagorean
Theorem, the Distance Formula, and the Standard Equation of a Circle are all analytically equivalent and represent the same relationship among points within a
rectangular coordinate plane. There are slight differences in the naming of the points dependent upon which of the three situations is of interest.)
2. What relationship is there between the radius of a circle with regard to Standard Equation of a Circle and the hypotenuse of a right triangle with regard to the
Pythagorean Theorem? (The radius of the circle is the same as the hypotenuse of the right triangle.)
Teaching Phase: How will the teacher present the concept or skill to students?
The teacher will distribute and direct students to follow along as the class collectively completes Circle Reasoning - Activity 1. The Circle Reasoning - Visual Aid may be
reproduced and distributed, or projected, for view and reference while the activity, explanation, and discussion occur.
Possible questions and expected student responses are included here for the teacher to use throughout the lesson to ensure understanding by students. These can be
asked at any time to help students with the concepts being learned. The teacher will use student responses as formative assessment to shape the scope and pace of
the lesson.
1. What is the definition of a circle? (A circle is the set of all points in a plane equidistant from a fixed point called the center.)
2. How do we draw circles on the rectangular coordinate grid? (To draw a circle on the coordinate grid, locate the center and use the length of the radius to identify
points on the circumference - either free-hand or with a compass.)
3. What is the Pythagorean Theorem and how is it used? (The Pythagorean Theorem is a relationship among the three sides of a right triangle expressed as
; it is used to calculate unknown lengths in right triangles and, if all lengths are known, verify a triangle is a right triangle.)
4. What do a, b, and c represent in the Pythagorean Theorem? (In the Pythagorean Theorem, c represents the length of the hypotenuse, and a and b represent the
length of the other two sides, sometimes called legs.)
5. What is the Distance Formula and how is it used? (The Distance Formula is expressed as
, and used to calculate the distance between two
points on the rectangular coordinate plane.)
6. What do
,
and d represent in the Distance Formula? (Within the Distance Formula the first point is
and the second point is
, the distance
between the two points is represented by d.)
7. What is the Standard Equation of a Circle and how is it used? (The Standard Equation of a Circle is
where (h, k) is the center and r is the
radius.This equation is used to easily identify the center and radius of the circle as well as facilitate graphing the circle on a rectangular coordinate plane.)
8. What do x, y, h, k, and r represent in the Standard Equation of a Circle? (The center of the circle is (h, k), the radius is r, and all points on the circumference of the
circle are written in the form (x, y).)
The teacher may wish to show the attached presentation titled Circle Reasoning - PowerPoint, a slide show to provide a general overview of the concept.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
The teacher will distribute Circle Reasoning - Activity 2. Students will be directed to complete this activity either individually, with a partner, in triads, or via small
cooperative learning groups - at the teacher's discretion. The teacher will circulate around the room and provide guidance, clarification, assistance, feedback, and
praise as appropriate.Discussion, debriefing, and consensus should occur. Misconceptions should be clarified. The teacher should monitor for use of radius, r, rather
than diameter, d; attention to the difference between r and
; use of correct signs when writing equations; and knowledge that (x + h) is the same as (x - (-h)).
Connections among graphic, verbal, and algebraic representations of circles should be emphasized.
If students need additional - practice with the process of completing the square, use Circle Reasoning - Extra Practice.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
The teacher will distribute Circle Reasoning - Activity 3 (best printed in landscape format). Students will be given directions to complete this activity either individually,
with a partner, in triads, or via small cooperative groups. The teacher will circulate around the room and provide guidance, clarification, assistance, feedback, and
praise as appropriate. Discussion, debriefing, and consensus should occur. Use of proper algebraic notation is of utmost importance.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
The teacher may pose a question where students are asked to compare and contrast diagrams, either verbally or in writing, that relate to the Pythagorean Theorem,
the Distance Formula, and the Standard Equation of a Circle. See attached file titled Circle Reasoning - Closure for suggested prompt.
Summative Assessment
The teacher will distribute Circle Reasoning - Summative Assessment. Students will complete the three multi-part exercises and submit their justified solutions.
Formative Assessment
As the lesson is taught and students are lead through Activities 1, 2, and 3, the teacher should:
circulate around the classroom
interact with all students
ask probing/guiding questions (some suggested questions and possible answers are listed in the Teaching Phase)
provide positive and constructive feedback in both verbal and written form
correct misconceptions and calculation errors
adjust scope, sequence, and pacing of teaching to facilitate learning of all skills necessary to demonstrate mastery of the Standard
ensure the process of completing the square is understood prior to Activity 3, and provide remediation as necessary
Feedback to Students
Students will receive feedback via internal reflection, oral discussion, written responses, and verbal praise throughout the lesson from themselves, the teacher, and
classmates. The lesson is structured such that active participation in and mastery of Activity 1, should establish a strong basis for success with Activity 2, followed by
Activity 3. Student understanding should be checked for (individually and collectively) via informal means such as questioning, dialogue, and discussion prior to the
undertaking of subsequent activities, and formal avenues like spot checking written work and/or collecting and grading worksheets.
page 2 of 3 ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Provide students or groups with rectangular graph paper or gridded dry-erase boards (and markers) so they may plot points and draw circles as needed throughout
the lesson
Provide the attached Circle Reasoning - Visual Aid either in hard-copy form or project on screen
Circle Reasoning - Activity 1: provide list for students to choose from for fill-in responses [center, circle, equidistant, radius, square, h, k, r, x, y]
If students need more practice with the process of completing the square, provide Circle Reasoning - Extra Practice
Provide students with, or encourage or require students to have, a traditional geometric construction tool - the compass, so circles are easily and accurately drawn
Extensions:
The teacher may want to have students recreate the derivation process for the Standard Equation of a Circle as outlined in Activity 1. Students should use their own
words and format, yet employ proper algebraic notation to move from the Pythagorean Theorem to the Distance Formula to the Standard Equation of a Circle. Specific
directions are left to the teacher's discretion. A suggested prompt is: "In your own words write at least three sentences to summarize the relationship between the
Pythagorean Theorem and the Standard Equation of a Circle."
or
Have students investigate the relationship between the three-dimensional Distance Formula,
and the Standard Equation of a Sphere,
.
Suggested Technology: Computer for Presenter, Basic Calculators, LCD Projector, Adobe Acrobat Reader
Special Materials Needed:
Suggested: rectangular graph paper, construction compasses, gridded dry-erase boards (and markers)
Further Recommendations:
The teacher should print, photocopy, and be familiar with all Circle Reasoning activities (answer keys provided) before the lesson is taught.
Consider two-sided copies.
Suggested timing, without remediation, is as follows:
Activity 1 = 10 minutes
Activity 2 = 45 minutes
Activity 3 = 35 minutes
Closure = 10 minutes
Summative Assessment = 20 minutes
Additional Information/Instructions
By Author/Submitter
This lesson supports the following Standard for Mathematical Practice: MAFS.K12.MP.5.1 - Use appropriate tools strategically.
SOURCE AND ACCESS INFORMATION
Contributed by: Dana Yancoskie
Name of Author/Source: Dana Yancoskie
District/Organization of Contributor(s): Miami-Dade
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-GPE.1.1:
Description
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.
page 3 of 3