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Dr. Antonio Quesada – Director, Project AMP
Summary of Lesson Lab #1: CIRCLES
Authors:
Date:
Vicki Dietrich and Carrie DiLoreti
June 20, 2000
Algebra II (Advanced Algebra or 2nd semester Algebra II)
10-12
Geometry, Measurement, Problem-Solving
Circles
1) Write the equation of a circle in standard form.
2) Graph the equation of a circle using paper/pencil and a graphing
calculator.
3) Calculate area of a circle and volume of a cylinder.
4) Apply equation and area of a circle and volume of a cylinder in
problem solving situations.
5) Use and modify calculator programs for circles and cylinders.
Materials:
TI-83 graphing calculators, overhead viewing screen, and links
Activity sheet (included here)
Textbooks
Expected Time: Two 85-minute blocks of time
Subject:
Grade:
Strand:
Topic:
Objectives:
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Concept Statements:
1. The formula for calculating distance between 2 points can be used to derive the
standard form for the equation of a circle.
2. Equations of circles in standard form contain both the center and radius, which
are needed to graph circles.
3. The area of a circle is used to calculate volume of a cylinder.
Learning and Proficiency Objectives:
1. Students will use the distance formula to derive the equation of a circle. (OCBM
#11)
2. Given the center of a circle and a point on the circle or two endpoints of a
diameter, students will write the equation of the circle in standard form. (OPO
#15)
3. Given the graph of a circle, write its equation in standard form. (OPO #9)
4. Given the equation of a circle in standard or general form, find the center and
radius and graph the circle using both paper/pencil and graphing calculator
approaches. (OPO #9)
5. Apply the equation of a circle in a problem-solving situation. (OCBM #16)
6. Calculate the area of a circle when its radius is known. (OPO #10)
7. Calculate the volume of a cylinder when its radius and height are known. (OPO
#10)
8. Modify a calculator program to find the volume of a cylinder. (OCBM #15)
9. Apply the area of a circle and volume of a cylinder in a problem-solving situation.
(OPO #10, 11)
Students will write equations of circles in standard form and identify the center and
radius to graph circles with both paper/pencil and graphing calculator methods. The
students will also use a calculator program to graph circles, find areas of circles, and
find volumes of cylinders. Finally, students will extend their knowledge of circles to
solve practical problems.
Task Overview:
Each student in the Algebra II classes will work in groups of 3 or 4 to complete the
following tasks:
1. Derive the standard form of the equation of a circle using the distance formula
and given (h, k) as the center and (x, y) as a point on the circle.
2. Use paper/pencil to write equations given the center and radius, the center and a
point on the circle, or two endpoints of a diameter of a circle.
3. Use the graphing calculator and a calculator program to graph circles knowing
the center and a point on the circle.
4. Use the strategy of completing the square to convert equations of circles from
general to standard form.
5. Write equations of circles from their graphs.
6. Apply the equation of a circle in a real-world situation.
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Dr. Antonio Quesada – Director, Project AMP
7. Share results of Practice and Apply problems from the activity sheet and present
solutions to the class.
8. Use paper/pencil and a calculator program to find the area of a circular region
given its radius.
9. Use paper/pencil and a calculator program to find the volume of a cylinder given
its radius and height.
10. Apply the area and volume formulas to real world situations.
11. Present solutions of extension problems from the activity sheet and incorporate
a question and answer period.
12. Using a student survey, evaluate the lesson.
Learning Strategies:
This lesson is designed for a heterogeneous class of boys and girls in the 10th -12th
grades. The lesson lab is designed for small group interaction with student teams
randomly selected. The lesson is divided into two parts, each designed to satisfy an 85minute block of time.
Part 1: Each team of 3 - 4 students will be given an activity sheet containing a variety of
tasks involving both algebraic and graphing activities. The teacher will briefly review the
distance formula and the midpoint formula. Students will work together on practice
problems using these formulas while the teacher observes and checks team answers.
Students will then use the distance formula to generate an equation of a circle with the
distance representing the radius of the circle.
Continue with the practice questions on the activity sheet with the teacher observing
and guiding teams as needed. Provide cable links if available when students reach the
calculator program portion to save time. Students will incorporate what they have
learned to solve an application problem. Complete this half of the lab with a
demonstration of converting an equation from general form to standard form by
completing the square. Complete the Practice and Apply section of the activity sheet for
homework.
Part 2: Students will present solutions to problems from the homework. The same
teams of students from the previous day will brainstorm solutions to solve an application
problem using properties of geometric shapes, area and volume. Students will use the
calculator program on the activity sheet to confirm their findings. They will then modify
the area program to find volume and investigate the effects on area and volume when
the dimensions are changed. A student from each team will present their team's
findings from one of the sections of the application problem. The teacher will present
two extension problems on which students will individually reflect and then discuss with
their team members. Volunteers will present their solutions with time allocated for
student and teacher questions. Students will complete the survey section of the activity
sheet to reflect upon their learning.
Classroom/Information Management:
Teams of students will be formed as explained in the Learning Strategies. Each team
member will have his/her own TI-83 graphing calculator. Each team will be provided a
cable link for transfer of programs. The links will be returned to the teacher at the end
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of the first day. Each team member is responsible for completing his/her own activity
sheet to be submitted by the end of the second day.
Assessment:
To assess students' understanding of the concepts associated with circles, the teacher
will utilize a number of strategies. These include observation of student interaction in
teams, completion of the guided practice and homework problems on the activity sheet,
student presentations of problem solutions, peer and teacher questioning, and student
surveys and written reflections.
Tools and Resources:
1. TI-83 graphing calculators, overhead viewing screen, and links
2. Activity sheet
3. Textbooks
Results Sharing:
Through teacher and peer conferencing, team presentations of problem solutions, and
question and answer sessions, students' understanding of the concepts can be
demonstrated. Students may also orally reflect upon their learning, including positive
features of the lesson lab as well as suggestions for improvement.
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Dr. Antonio Quesada – Director, Project AMP
LESSON LAB
ACTIVITY SHEET
CIRCLES
NAME___________________
DATE___________________
BLOCK__________________
Use this sheet for notes and class activities.
PART 1
Distance Formula
Midpoint Formula
Guided Practice
1. Find the distance between P(-7, 12) and Q(6, 5). Give an exact answer
and an approximate answer to the nearest hundredth.
2. Find the coordinates of the midpoint of the line segment whose
endpoints are P(-3, 10) and Q(6, -12).
3. The endpoints of the diameter of a circle are at A(-5, 2) and
B(3, 4). Find the center and radius of the circle.
Standard Equation of a Circle
Given (h, k) as the center of a circle, (x, y) as a point on the circle, and
using the distance formula to find the radius, write the equation for a circle.
(Eliminate the radical.)
Guided Practice (algebraic)
Write the standard equation of a circle with the given conditions:
1. center (0, 0); radius = 4
2. center (2, 3); radius = 4
3. center (-2, -2); radius = 1/3
4. center (7, -4); radius = 5
5. The circle passes through (7, -1) and has center at (-2, 4).
6. The endpoints of a diameter are at (-2, -3) and (4, 5).
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Explore Translations of Circles with Paper/Pencil and Graphing Calculator
Examples: To graph x2 + y2 = 9 with paper and pencil, simply plot (0, 0) as
the center and use radius = 3 to move out 3 places in each direction from
the center.
To graph x2 + y2 = 9 with a calculator, solve for y and graph the two
resulting equations together. (See fig. below)
Guided Practice (graphic)
1. Use a graphing calculator to graph the circle defined by x2 + y2 = 25.
Sketch the graph.
2. Graph the circle defined by (x+3)2 + (y-2)2 = 25. Sketch the graph.
3. Describe the transformation of the graph in #1 to the graph in #2.
4. Determine the center and radius of the circle defined by
(x+2)2 + (y+1)2 = 16 and graph the circle to support your answer.
Note: Another way to graph a circle in addition to solving for y is to
use the following key strokes: 2nd PRGM 9 A, B, R ENTER where
A & B are coordinates of the center and R is the radius. Verify this
with x2 + y2 = 9.
Use the following program to verify your answers to #1, 2, and 4 above.
Application
Vibrations from a certain earthquake were noticeable up to 120 miles away
from the earthquake’s epicenter. The center of the closest city to the
earthquake’s epicenter is 85 miles east and 90 miles south of the epicenter.
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Dr. Antonio Quesada – Director, Project AMP
If Alexandra was 15 miles west of this city, could she have noticed the
vibrations at the time of the earthquake? Justify your answer.
General Form of the Equation of a Circle: x2 + y2 + Dx + Ey + F=0 where D,
E, and F are constants
Example: Write the standard equation for the circle given by
x2 + y2 - 8x + 7 = 0. State the coordinates of the center, give the radius, and
sketch the graph.
You try: Write the standard equation for the circle given by
x2 + 6x + y2 - 4y - 3 = 0. Give the center, radius, and sketch the graph.
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Dr. Antonio Quesada – Director, Project AMP
Practice and Apply – Homework
I.
Write the standard equation for each circle graph.
II.
Write the standard equation for each circle. Find the center and radius. Then
sketch the graph.
a. x2 + y2 = 100
b. 3x2 + 3y2 = 36
c. (x-1)2 + (y-7)2 = 81
d. x2 + y2 - 10x + 8y + 5 = 0
e. x2 + y2 + 8y + 4x – 5 = 0
III.
Write the equation of the circle that meets the conditions:
a. Circle passes through origin and has center at (-3, 4).
b. Endpoints of a diameter are at (-3, 4) and (2, 1).
IV.
A radio program is broadcast from a van that is located 40 miles east and 30
miles north of a radio tower. The van sends the radio signal to the tower, which
then transmits the signal a maximum distance of 70 miles. If Jessica is 30 miles
east of the van, is she able to receive the radio program? Justify your response.
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Dr. Antonio Quesada – Director, Project AMP
PART 2
1. Ashley built a new house and had just planted grass for a new lawn. She wanted to
be able to put her dog on a leash in the yard, but she didn’t want the dog to tear up
her young lawn. Her plan was to anchor a pin in the ground up against the side of
the house to which she could attach the end of the leash. She decided to put gravel
down on all of the ground that the dog could reach while on the leash. Note: The
length of the leash cannot be greater than the distance from the pin to the corner of
the house.
a. Draw a diagram of the problem. What geometric shape represents the part of the
lawn that would need gravel?
b. If the leash is 10 feet long, how many square feet could the dog reach while on
the leash?
c. If she wanted the gravel to be 4 inches deep, how many cubic feet of gravel
would she have to buy?
d. If one bag contains one cubic foot of gravel and costs $0.79 per bag, how much
would she have to spend on gravel?
2. Here is a program for the TI-83 Plus to find the area of a circle. Explain the purpose
of each step and enter it into your calculator.
:ClrHome
:Input “RADIUS?”, R
:R2A
:Disp “AREA =”,A
a. Run the program using R = 10 to check your calculation of the area of the lawn
that the gravel would need to cover in Ashley’s yard. What effect does doubling
the radius have on the area of the circle?
b. Modify the program for finding the area of a circle to give you the volume of a
cylinder if you are given the radius and the height.
3. Run the program using R = 10 and H = 1/3 to check your calculation of the volume of
gravel that Ashley would need to purchase.
a. If she used an 8 foot chain, how much gravel would she need?
b. If she used a 10 foot chain, but wanted the gravel to only be 3 inches deep, how
many cubic feet of gravel would she need?
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Dr. Antonio Quesada – Director, Project AMP
EXTENSION:
1. Referring to the original dimensions of the chain and the depth of the gravel, if the
pin that anchors the leash was moved out 2 feet from the house instead of being
pinned against the house, how could you determine the amount of gravel needed?
2. Instead of attaching the leash to a pin, Ashley considered attaching the chain to an 8
foot run. Modify the program on the calculator to find the area of the yard that the
dog could cover and the volume of gravel that she would have to buy. How much
would she have to spend on gravel if the leash is 6 feet long and the gravel is 4
inches deep?
Survey:
On a scale of 1 to 5 (1 = bad and 5 = excellent) how would you rate this lesson?
What did you like most about this lesson?
What did you like least?
What did you learn?
What questions do you still have about the material covered in the lesson?
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Solutions to Activity Sheet
Part 1
1. 218, 14.76
2. (3/2, -1)
3. C(-1, 3) r = 17
1.
2.
3.
4.
5.
6.
x2 + y2 = 16
(x – 2)2 + (y – 3)2 = 16
(x + 2)2 + (y + 2)2 = 1/9
(x – 7)2 + (y + 4)2 = 5
(x + 2)2 + (y – 4)2 = 106
(x – 1)2 + (y – 1)2 = 25
1.
2.
3.
4.
C(0, 0), r = 5
C(-3, 2), r = 5
Center moved 3 places left and 2 up from the origin.
C(-2, -1), r = 4
Application: Yes. Alexandra’s position at coordinates (70, 90) falls on the
equation of the circle x2 + y2 = 14400 found using the center as the origin
and the radius = 120.
Example: (x – 4)2 + y2 = 9, C(4, 0), r = 3
You Try : (x + 3)2 + (y – 2)2 = 16, C(-3, 2), r = 4
Practice and Apply – Homework
I. x2 + y2 = 16
(x – 4)2 + (y + 3)2 = 9
(x + 5)2 + (y + 3)2 = 25
(x + 3)2 + (y – 4)2 = 16
II.
a. x2 + y2 = 100, C(0, 0), r = 10
b. x2 + y2 = 12, C(0, 0), r = 23
c. (x – 1)2 + (y – 7)2 = 81, C(1, 7), r = 9
d. (x – 5)2 + (y + 4)2 = 36, C(5, -4), r = 6
e. (x + 2)2 + (y + 4)2 = 25, C(-2, -4), r = 5
III.
a. (x + 3)2 + (y – 4)2 = 25
b. (x + .5)2 + (y – 2.5)2 = 8.5
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V.
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No; x2 + y2 = 2500 is the equation of the circle; Jessica’s position at
coordinates (70, 30) falls outside the circle.
Part 2
1. a. semicircle
b. 50  157.08 ft2
c. (50/3)  52.36
d. $41.36
2. ClrHome :
Input”Radius?”, R:
Clears the screen
Asks user to type in length of the radius and
assigns the value calculated to the variable A
Disp”AREA =”, A:
Displays the words “AREA =” on the screen
then displays the value of A
a. Quadruples the area
b. Add to the circle area program:
Input”H?”,H
AHV
Disp”VOLUME =”,V
3. a. 33.51 ft3
b. 39.27 ft3
Extension:
1. Add the area of the circular segment from the area of the entire
circle. (Area of a circular segment = (m/360)r2)
2. $ 55.04