Title of Lesson: How do we use mathematical equations with circles?
UFTeach Students’ Names: Sarah Burkhardt and Julie Walthall
Teaching Date and Time: April 2nd and 3rd
Length of Lesson: 2 Days (100 minutes total)
Course / Grade / Topic: Geometry/ 10th Grade/ Equations of Circles
Source of the Lesson:
Glencoe, McGraw Hill Florida Geometry Textbook. Authors: Carter, Cuevas, Day, Malloy, and
Cummins.
http://www.mathsisfun.com/algebra/circle-equations.html
Embedding Strategies Based on Observations and CTS:
Based on the CTS and what happened in class, I am including the following teaching strategies
with these students because…
Recommended strategy based on
observations
Place students in collaborative learning
groups as pre-determined by teachers (see
Advanced Preparations) and as instructed on
exploration worksheet
Instruction by personal white boards
Recommended strategy based on CTS
Personal study of topic by teachers
Use of Formative Assessments
Reason for selecting this strategy
Students alone were disengaged from lesson and not
taking responsibility of their learning. In collaborative
learning groups, students will be responsible for their
own, as well as their classmates’, learning during
explorations and investigations.
When we observed, our mentor teacher gathered
students’ attention by telling them to take out their
white board. All students responded immediately by
taking out their boards enthusiastically. They were
focused, excited to participate and show Ms. Karis what
they knew. By doing this, the students remained
engaged (all students participated in answering
questions until class ended) throughout the entire class
as opposed to previously observed lessons.
Reason for selecting this strategy
Through the CTS, we learned that there is a strong link
between teacher expertise and student achievement.
It’s important to develop our knowledge through
studying the textbook material and strongly considering
student responses and misconceptions regarding our
topic. (Deep study of concepts after considering
“Student Responses / Possible Misconceptions” column
of this lesson plan)
Based on the CTS on Circles, I know that students are
likely to have misconceptions. Thus, including formative
assessments in the lesson will help us as teachers to
combat these. They will allow us to see what topics our
students struggle with, and provide students the chance
to show how or where they are struggling. The
formative assessments include jigsaw and whiteboards.
(The jigsaw assessment is used on day 1 exploration in
order for each student to become an expert on his/her
specific aspect of equation and then teach his/her group
mates. The whiteboards are used on day 2 after the
explanation to assess the students’ comprehension.
Students will answer questions from the board on their
own whiteboards and hold them up so the teacher can
see who has the correct answer and then the teacher
will address any common incorrect
answers/misconceptions.)
Common Core State Standards (CCSS) / Next Generation Sunshine State Standards (NGSSS):
(CCSS) / (NGSSS) with Cognitive Complexity:
Standard Number
Benchmark Description
MACC.912.GGPE.1.1
MACC.912.GMG.1.1
MACC.912.GMG.1.3
Cognitive
Complexity
Level 2
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
Use geometric shapes, their measures, and
Level 1
their properties to describe objects (e.g.,
modeling a tree trunk or an human torso as a
cylinder)
Apply geometric methods to solve design
Level 3
problems (e.g. designing an object or structure
to satisfy physical constraints or minimize cost;
working with typographic grid systems based
on ratios)
Concept Development:
Circles are everywhere is the world. Understanding circles and how they work can lead to
many discoveries. Students can apply knowledge of circles to discover facts about the natural
world in science (properties of our Earth and its spherical shape) and the arts (from art, to
architecture, to engineering systems). Without the properties and understanding of circles,
many scientists wouldn’t have made the discoveries that have made a big impact on our lives.
Not to mention the impact that circles and geometry have on the construction of our cities
and communities. In this section, we look at Equations of Circles. In order to derive the
equation of a circle, the Pythagorean Theorem and the Distance formula are applied. The
Pythagorean Theorem states that a2+b2=c2 (according to common notation). This is applied to
the equation of a circle when a triangle is graphed (on the coordinate plane for visual ease)
within a circle where the hypotenuse is the radius. Now it can be seen that the leg lengths
correspond to the difference between a point on the circle and the center point of the circle.
Thus using substitution, the Pythagorean theorem becomes (x-h)2 + (y-k) 2 = c2 where (x,y) is
the point on the circle and (h,k) is the center of the circle.
(http://www.mathsisfun.com/algebra/circle-equations.html). Now, this is the standard
equation of a circle: written as (x-h)2+(y-k)2=r2, where (h,k) is the center and r is the radius.
The center of a circle is represented by the coordinates (h,k), where h is the x coordinate of
the center and k is the y coordinate of the center. The radius tells us the distance between
the center and all the points that lie on the circle. All points on a circle are equidistant from
the center (Glencoe, Florida Geometry Textbook, pg. 744). There are multiple ways to use the
equations of circles. Given a center and radius, the equation of a circle can be determined by
plugging in the numbers into the standard form. Sometimes the diameter is given instead of
the radius. Since diameter=2(radius), the diameter can be divided by 2 to find the radius.
Then, the same process with finding the equation of a circle occurs. When a center and a
point are given, the learner must first calculate the distance between the points to find the
radius and then use it to find the equation of a circle. If the equation of a circle is given, you
can use the information to find the center and radius and then graph it onto a coordinate
plane. Once again, this can be applied to solve real world problems such as the ones listed
above under “discoveries”.
Performance Objectives
 Students will be able to identify and use the proper terminology regarding circles as
needed to solve problems (diameter, radius, center)
 Students will be able to identify properties of a circle (center and radius specifically)
given the equation or graph
 Students will be able to model and solve real world problems using geometry
(equations of circles specifically)
Materials List
 30 copies of exploration worksheet 1 per class
 30 copies of exploration worksheet 2 per class
 30 copies of formative assessment worksheet per class
 30 copies of exploration (day2) worksheet per class
 30 copies of elaboration worksheet per class
 30 copies of evaluation per class
 30 whiteboards and 30 markers per class
 1 whiteboard worksheet for each teacher (2 total)
Advance Preparations
 Advanced preparation of desks arranged in groups of 3-6




Advanced preparation of pre-sorting students into groups and written on Board at
front and names written on highlighted worksheets with group and position.
Advanced preparation of PowerPoint with directions and engagement
(Advanced preparation of video for Engagement saved to PowerPoint)
Advanced preparation of worksheets highlighted and organized by group number for
distribution
Safety
 No safety concerns for this lesson (video from Youtube but is saved to computer and
has NO advertisements and does NOT use Internet)
 (video is Flinstones and is appropriate for all audiences)
 Students will be instructed to use whiteboards and marker appropriately
5E Lesson:
DAY 1
Engagement
Time: 4 minutes
What the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible
Misconceptions
Introduce herself to the
Good afternoon everyone! My name is
Hello
students and welcome
Miss Burkhardt/ Miss Walthall and I will be
them to class.
your teacher for today and tomorrow.
Slide 1
Introduce the topic of the Today and tomorrow we are going to
Aww man not more circles
lesson
continue the study of circles.
Introduce and play video
Before we get cracking on some
I love YouTube
(see PowerPoint)
worksheets and problem solving, let’s take
Slide 2
a look at what YouTube has to offer in the
way of circles!
Segue into lesson
Okay everyone so now that we are all
I’m so ready!
primed and ready to think about circles
because of that amazing video, let’s begin!
Exploration
Time: 30 minutes
What the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible
Misconceptions
Gather sorted worksheets Today we will work together in groups on
Why are we moving
and give oral instructions a worksheet. The worksheet assigns each
groups?
for the exploration
member a job position within the group.
How do we know what
activity
For the first part, I want you to discuss
position we are?
Slide 3
your assignment with you group mates.
You will be moving groups shortly once I
call time and I will give further instructions
then.
Show students who their Here are the group numbers, members,
I don’t like my group
group mates are and what
their positions are
Slide 4/5
Pass out worksheet 1
Slide 6
and each person’s job position within the
group. Move to sit with your group at this
time.
Once you have a worksheet, make sure
So what do we do?
you have the appropriate worksheet with
your name on it, highlighted with your
group and position (1, 2, 3, 4, or 5), (A, B,
or C).
I will circulate to answer any questions you
have. You may begin!
Circulates to answer
questions and to ask
probing questions.
How did you find the radius of the circle?
[I looked at the graph
and counted the
squares]
[I knew from the
equation to take the
square root of the
number alone on the
right side of the equals
sign]
“
How did you find the center of the circle?
I guessed
[I looked at the
coordinate given on the
graph]
I guessed
“
What do notice about the number you
found for the x value of the center?
I don’t know/ I haven’t
found it yet
I didn’t notice anything
[I noticed it was the
same number as the
number after the x in
the equation]
“
What do notice about the number you
found for the y value of the center?
It’s in the equation, but
it’s negative or not quite
the same
I didn’t notice anything
[I noticed it was the
same number as the
number after the y in
the equation]
“
What do notice about the number you
found for the radius?
It’s in the equation, but
it’s negative or not quite
the same
I didn’t notice anything
I noticed it was smaller
than the number on the
right of the equals sign
in the equation
“
What can you say about the equation
given?
[I noticed it was the
square root of the
number on the right of
the equals sign in the
equation]
I don’t know
[Maybe that it relates to
the center of the circle]
[Maybe that it relates to
the radius of the circle]
“
What can you say about the numbers
given in the equation and the numbers
found for the coordinates of the center, or
the radius?
[Maybe there is a way
to graph a circle from
the equation if the
graph hadn’t been
provided]
I don’t know
I see some of the
numbers in the
equation
[I see the radius
squared in the
equation, as well as the
x and y coordinate of
the center]
Calls everyone’s
attention to provide new
I need everyone’s attention please!
At this time, I would like all of you to
I want to stay in this
group
instructions
Gives new instructions
“
“
“
Circulates to answer
questions and to ask
probing questions.
“
“
“
change groups.
I need all of the A Members to raise their
hands. Please form a new group (or two if
there are more than 6 of you A’s) in the
back left of the room.
I need all of the B Members to raise their
hands. Please form a new group (or two if
there are more than 6 of you B’s) in the
back right of the room.
I need all of the C’s remaining to form a
group (or two if there are more than 6 of
you C’s) in the front of the room.
Now that you are in new groups, I would
like you to discuss your findings. Keep in
mind that you all worked on different
problems with different numbers, but had
the same task.
So you all are the ___ (A/B/C) group. What
did each of you find individually?
Take a minute to each share what you
found regarding your group’s equation.
Now that you have shared, what
conclusions have you all drawn together
regarding the ___ (x coordinate/y
coordinate/radius)?
So each of you had different numbers, but
each of your numbers was related to the
equation given. What can you assume
I don’t want to move
I don’t want to move
I don’t want to move
There is nothing to
discuss, I’m lost!
I don’t know what we
are supposed to be
discussing!
I found a number but I
don’t know what to do
with it
[I found a number that
is also in the equation
here ___]
[In my group’s equation,
the number here ___
corresponded with the
number here ___ in the
center/radius]
I found nothing
We can’t make any
conclusions across all of
our equations
[We found that each of
our numbers for the
center/radius are in the
equation somewhere]
We don’t know what to
assume
about all equations of circles then?
[Regarding the ___ (x coordinate of the
circle’s center / y coordinate of the circle’s
center/radius)?]
We found that all of our
numbers we found for
the x/y part of the
center corresponded
with the first/second
number in the equation
but we don’t know why
We found that the
radius didn’t relate to
the equation
[We found that the
radius squared was
found in the equation]
[We can assume that
the radius squared is in
the equation every time
maybe?]
[We can assume that
the x/y coordinate of
the center can be found
by looking at the
number in parentheses
with the x/y in the
equation maybe?]
Calls everyone’s
attention to provide new
instructions
Provide Instructions
Slide 7
Circulates to answer
questions and to ask
probing questions.
Give oral instructions for
next worksheet
Slide 7
I need everyone’s attention please!
At this time, I would like all of you to
return to your original groups.
Share with your group members what you
have learned after discussing with others
of your same position (letter A, B, or C)
As a group, make conclusions regarding a
standard equation of a circle. I am passing
out a second worksheet now to guide your
discussion.
Once you have discussed, answer the
questions together with your group. The
questions ask you to write out the
conclusions you made regarding
comparing the equation to the center and
I want to stay in this
group
I learned nothing
radius.
Pass out worksheet 2
Here is a second worksheet to complete
once you have discussed with your group.
Circulates to answer
questions and to ask
probing questions.
Who has questions for me regarding
number 6?
What number in your problem
corresponds with the h?
How can we use that to draw conclusions
in general about the equation of a circle?
What number in your problem
corresponds with the k?
How can we use that to draw conclusions
in general about the equation of a circle?
What number in your problem
corresponds with the r?
How can we use that to draw conclusions
in general about the equation of a circle?
Calls attention to stop
Everyone let me have your attention. It is
working
time to stop working on the worksheet.
Student Explanation
What the Teacher Will Do Teacher Directions and Probing Questions
Transition students to
explanation
Each student has a copy
of each graph on his/her
worksheet and will be
instructed to fill in the
other parts at this time.
Formative Assessment
Each group needs to explain to the other
groups what conclusions they drew with
their own equation and graph (worksheet
1) and how that helped them answer the
questions on worksheet 2.
They then will share with the class how
this knowledge can be applied to any
equation of a circle (worksheet 2 answers).
Students not presenting should be
recording the information on their own
papers.
This discussion will continue until students
arrive at the standard equation of a circle
where students can identify the center,
and radius. (until worksheet 2 is
completed by all students)
I don’t have any
I have a lot, I don’t get it
[__ is in that spot]
I don’t know
[__ is in that spot]
I don’t know
[__ is in that spot]
I don’t know
Time: 13 minutes
Student Responses/Possible
Misconceptions
Do we have to?
Teacher response: YES
Do we have to write this
down?
Teacher response: YES
Time: 3 minutes
Teacher will call
everyone’s attention
Pass out worksheet
Great job today everyone! Before we leave Can I leave yet?
for the day, I am interested in knowing
how well this lesson helped you learn.
Is this for a grade?
Please take the remainder of the class
period to fill out this last worksheet.
Teacher response: No, but it
will be collected and read so
that I can make changes to
tomorrow’s lesson, so do
your best.
It is important that you be honest with me I don’t want to write
about what difficulties you faced today in
comments
learning, so please be honest and write
down comments and suggestions on how
this can be improved tomorrow.
DAY 2
Introduction / Recap
Time: 5 minutes
What the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible
Misconceptions
Gather students’
Hello again!
Hello
attention
Yesterday you worked on the Explorations Are we going to be in groups
Worksheet and discussed with your peers again?
the different components of an equation
of a circle. Let’s go over what you guys
found.
Ask students what the
Who will remind the class what the
I will
formula for finding the
standard form of the equation of a circle
[(x-h)^2+(y-k)^2=r^2]
equation of a circle is.
is?
Go over what each part of What do the h and k represent in the
[The x and y coordinates of
the formula represents
formula?
the center]
Which one is the x-coordinate and which
one is the y-coordinate?
So what would the coordinates of the
center of the circle be, based on this
equation?
What if I gave you an equation that looked
like this: (x-p)2 + (y-q) 2 = r2
What would the coordinates of the center
[Where the center of the
circle is]
[The h is the x-coordinate
and the k is the ycoordinate.]
[(h,k)]
[(p,q)]
be now?
What does the r represent?
[The radius of the circle]
Exploration
8 minutes
What the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible
Misconception
There is an Exploration
We now have the equation of a circle, so
What worksheet?
worksheet that students
let’s do some practice problems!
will be doing in the same
Today, we have a worksheet that you will
Teacher response: I will pass
groups as the day before. be working on in you same groups as
them out in a minute
The teacher will give
yesterday.
instructions to students
Is it the same groups as
about the worksheet.
yesterday?
The teacher will hand out
the Exploration
Worksheet to students.
Students will then move
to their designated
groups and work on the
worksheet one problem
at a time.
After each question, the
answer will be discussed
on the Elmo. (See
Explanation below for
discussion of answers.)
The teacher will walk
around the room and ask
probing questions and
help students with any
questions they may have
“
“
“
“
Here is the worksheet for you to work on.
You may now move to your designated
groups.
In your groups, work as a team to find the
solution to the problem, and once you are
finished, sit quietly. We will review the
answers once all the groups are finished.
Teacher response: Yes, you
will be in your same groups
What if we finish early?
Teacher response: Sit quietly
with your group. You should
discuss the process with
your group to make sure
everyone understands the
problem
Raise your hand if you have any questions!
What does is mean if the center of the
circle is at the origin?
[It’s at the middle]
I don’t know
What would the x and y coordinates be if
the center is at the origin?
Given the center and diameter of the
circle, how would you write the equation?
Since the diameter is 18, what would the
radius be?
What is the distance formula?
[(0,0)]
I don’t know what to do
because we aren’t given the
radius.
[9]
Square root of 18
I don’t remember.
[d= √(x2-x1)2+(y2-y1)2]
“
[It tells us the distance
between the 2 points]
Explanation
Time: 12 minutes
What the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible
Misconceptions
The teacher gathers
Let me have your attention please!
attention of students. The
teacher will have the
Since all the groups are finished with
worksheet displayed on
question 1, let’s review how you all got
the Elmo, which will be
that answer.
projected on the board
for students to see. (One
problem will be shown at
a time).
After the students are
finished exploring a
question, the answers will
be discussed as a class.
Question 1 on Elmo
“
“
How will the distance formula help us in
this problem?
Write the equation of the circle with the
center at the origin and a radius of 5.
Since the circle has a center at the origin,
what are the coordinates of the center?
Using the coordinates (0,0) as the center,
we plug in the numbers to have the
equation… ___?
[(0,0)]
(x-0)2+(y-0)2=r2
Since we subtract 0, can we
just leave it as x2 and y2?
[yes]
“
“
“
Question 2 on Elmo
“
“
“
Whenever you have the center at the
origin, you can write it as x2+y2=r2
What would we put instead of r2?
5
[25]
[x2+y2=25]
Since the radius is 5, r2 would be equal to
25. So the final answer would be___?
Write the equation of a circle with a
center at (3,-2) and a diameter of 18.
In this case, what would our h and k values [h=3 and k=-2]
be?
What would the r be?
18
[9]
What do you notice about this problem
Nothing
that is different from the others? Hint:
What is the radius?
[They gave the diameter, not
the radius]
“
“
Go over question 3 on
Elmo
“
“
“
“
Draw graph on
Exploration Worksheet
Go over question 4 on
Elmo
“
“
“
“
“
“
Now that we know the h, k, and r values,
what is the equation of the circle?
Remember to always square your r value
in the equation!
The equation of a circle is (x-6)2+y2=25.
State the center and radius. Then graph
the equation.
Who will tell me what the center of the
circle is?
What is the radius of the circle?
[Since the diameter is 18, we
know that the radius would
be 9]
(x-3)2+(y+2)2=9
[ (x-3)2+(y+2)2=81]
(-6,0)
[(6,0)]
25
[5]
2
Since r is 25, we have to take the _____ of square
25 to get r. When we take the square root [Square root]
of 25 we get___ as the radius.
[5]
Now that we found the center and radius,
we can graph it. Take a minute to graph
what you think it looks like on your paper.
The center would be here at the
[(6, 0)]
coordinate ____ . Since the radius is 5, I
will count 5 spaces away from the center
point. If I do this in every directions and
connect the points, it will form a circle
looking like this (draws circle).
Write an equation of a circle with a center
at (2,2) and passes through the point (6,5).
This question is a little different than what I don’t know
we’ve previously done. We are now given
the center with an extra point instead of
[distance formula]
the radius. What formula would we use to
determine the radius?
I’ll give you a hint. It has to do with finding [The distance formula]
the distance between 2 points.
Who will remind us of the equation for
I will.
distance formula?
[r= √(x2-x1)2+(y2-y1)2]
We will let (x1,y1)=(2,2) and (x2,y2)=(6,5).
When we plus these numbers into the
formula we get r= √(6-2)2+(5-2)2.
So what is our radius?
[√25 or 5]
Using 5 as our radius and the center of
(2,2) that was given, we can now find our
“
equation of the circle.
Who will tell me what the equation of this
circle is?
[(x-2)2+(y-2)2=25]
Formative Assessment
Time: 15 minutes
Activity
What the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible
Misconceptions
Students will be working
Now you will be able to work on examples
on examples on their
on your own. Go ahead and take out the
personal white boards for white board under your desk. Make sure
their formative
you have a dry erase marker. Raise your
assessment. The teacher
hand if you need one.
will display a problem on
the Elmo and the students I will be displaying a problem on the Elmo.
will write their answer on Work on your white board to find this
their white board. Once
answer and hold up your white board
they find the answer, they when you are finished. Try to work alone,
will hold up their board. If but if you have a question you may ask a
their answer is incorrect,
friend near you or raise your hand.
they should consult a
peer around them about
it. Any difficult questions
will be worked out on the
Elmo.
Display question 1 on
Given the equation (x+4)2+y2=64, state the Center: (4,0)
Elmo
coordinates of the center and the measure [(-4,0)]
of the radius.
Radius: 64
[8]
Display question 2 on
Elmo
Write the equation of a circle with a
center at (3,-2) and a diameter of 24.
(x-3)2+(y+2)2=24
(x+3)2+(y-2)2=24
(x-3)2+(y+2)2=12
(x+3)2+(y-2)2=144
(x-3)2+(y+2)2=576
[(x-3)2+(y+2)2=144]
Always remember to make sure to check
whether it provides you with the radius or
diameter. Make sure it is the radius and
not the diameter that you are squaring!
Display question 3 on
Elmo.
Draw a graph of a circle with a center at
(-6,4) and a radius of 5. (Hint: Use the
The students may have
incorrect graphs drawn.
Remind students that
there is a coordinate side
on the white board that
will be useful for graphing
the circle.
Display question 4 on
Elmo. This problem is a
real world example. Tell
students to use
coordinate side of white
board to help visualize
the problem
coordinate side of the white board).
Three dancers are on a stage, which is
represented by a coordinate system.
Dancer 1 is at the coordinate (3,-3).
Dancer 2 is at the coordinate (6,-1).
Dancer 3 is at the coordinate (5,3). There
is a spotlight projected as a circle onto the
stage with the equation (x-4)2+(y+1)2=9.
Which dancers are in the spotlight?
To help visualize this problem better, use
the coordinate side of your white board.
You should begin by plotting the points of
the dancers.
Plot the points of the 3
dancers and create the
circle from the equation.
Determine which dancers
are in the spotlight.
Make sure that the circle is
drawn correctly. If some
students struggled, draw the
graph and display it on the
Elmo.
Do I need to draw it?
Teacher response: It is not
required, but drawing it will
probably help you visualize it
better.
Some students may graph
the circle incorrectly. If
students struggle, explain
how to do the problem with
help from the class.
Here are the points of the 3 dancers. Let’s [(4,-1)]
find out where the spotlight is at. Based on
our equation, who will tell me what the
coordinates of the center of the spotlight?
Someone else, tell me what the radius of
the spotlight is?
[3]
Now that we have this information, plot
the center and draw the circle
representing spotlight on your white
board.
Looking at the picture of the “dancers”
and “spotlight”, which dancers can be
found in the spotlight?
[Dancers 1 and 2]
Elaboration
Time: 5 minutes
What the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible
Misconceptions
Urge the students to
At this time, we all agree on the standard
explore what they have
equation of a circle. But let’s explore why
learned at a deeper level
that formula is correct!
and apply it to a
challenging proof-type
problem
Pass out worksheet to
students
Reads instructions
Circulates to answer
questions and to ask
probing questions.
I am giving you all one last worksheet to
complete with your group, or individually
if you choose
Let’s read the instructions together.
Given this image, and the Pythagorean
theorem, apply your knowledge to derive
the standard equation of a circle.
Look at the image. Look at the right angle
corner. What are the coordinates of this
vertex?
How can we represent that “a” side of the
triangle using letters from the circle
equation?
Look at the image. How can we represent
that “b” side of the triangle using letters
from the circle equation?
Look at the image. How can we represent
that “hypotenuse” side of the triangle
Not another worksheet
Yes, an added challenge!
I can do this!
I don’t know
It’s got to have the same y
coordinate as the center,
and the same x coordinate
as the top point on the
circle, right?
Maybe change it to h or k?
Teacher response:
No we can’t quite change its
value or just call it
something else yet. Since we
don’t know the value of the
leg we will have to say it is
the difference between two
values that we do know.
Keep at it!
I don’t know
Maybe change it to h or k?
Teacher response:
No we can’t quite change its
value or just call it
something else yet. Since we
don’t know the value of the
leg we will have to say it is
the difference between two
values that we do know.
Keep at it!
I don’t know
using letters from the circle equation?
Maybe change it to h or k?
Teacher response:
Close! Looking at the circle,
where is the radius? Where
is the hypotenuse? What do
you notice?
Evaluation
Time: 5 minutes
What the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible
Misconceptions
If there is time, a post
Please clear your desks. I have an
evaluation will be given to assessment for you all to do to show me
the students so they can
what you have learned in the past 2 days.
display what they have
learned. Pass out post
Raise your hand when you are finished.
evaluations to students.
Once all the papers are
Thank you all for having me this semester.
collected, class should be It has been a pleasure getting to know
over.
each and every one of you.
Enjoy the rest of your day!
Group #:_____
Job Position: ______
Name: ______________________
EXPLORATION WORKSHEET
Directions:
**Each of you is part of a Team 1, 2, 3, 4, or 5.
**Within each Team, there are 3 job positions: A, B, or C.
**Follow the instructions according to your position and only complete your Problem #.
**Be prepared for further instructions by your teacher.



Job Description A: Collect and discuss information regarding the x coordinate of the
center of your circle
Job Description B: Collect and discuss information regarding the y coordinate of the
center of your circle
Job Description C: Collect and discuss information regarding the radius of the circle
Group 1 – PROBLEM 1
(x-1)2 + (y-2) 2 = 4
Center: _______
Radius: _______
Group 2 – PROBLEM 2
(x+3)2 + (y-1) 2 = 4
Center: _______
Radius: _______
Group 3 – PROBLEM 3
(x-2)2 + (y+2) 2 = 9
Center: _______
Radius: _______
Group 4 – PROBLEM 4
(x+2)2 + (y-2) 2 = 16
Center: _______
Radius: _______
Group 5 – PROBLEM 5
(x+1)2 + (y-2) 2 = 16
Center: _______
Radius: _______
Group #:_____
Job Position: ______
Name: ______________________
EXPLORATION WORKSHEET #2
Directions:
At this time, you should be in your original Groups 1, 2, 3, 4, or 5.
Share with your group what you learned from your A, B, and C group collaboration.
Once every Position has shared, as a Group, discuss the following questions and write an
answer.
Once you have finished, be prepared to share your answers with the class when called on.
1. What was your Group’s equation? Write it here: ______________________________
2. The center for you circle was: __________. The radius for you circle was: __________.
3. What connections did you make between the x coordinate of the center and the equation?
4. What connections did you make between the y coordinate of the center and the equation?
5. What connections did you make between the radius of the circle and the equation?
6. Given the standard form of the equation of a circle: (x-h) 2
What do you think the h represents?
The k ?
The r ?
+ (y-k) 2 = r2
Name: __________________________
EXPLORATION WORKSHEET
Directions: Work through each example in your small groups.
1. Write the equation of the circle with the center at the
origin and a radius of 5.
Equation of a circle
(h,k)= ( _ , _ ), r=__
Simplify
2. Write the equation of a circle with a center at (3,-2) and a
diameter of 18.
Equation of a circle
(h,k)= ( _ , _ ), r=__
Simplify
3. The equation of a circle is (x-6)2+y2=25. State the center
and radius. Then graph the equation.
4. Write an equation of a circle with a center at (2, 2) and
passes through the point (6, 5).
Step 1: Find the distance between the points to determine the radius
Distance Formula
(x1,y1)=(_,_) and (x2,y2)=(_,_)
Simplify
Step 2: Write the equation using h=2, k=2, and r=5.
Equation of a circle
(h,k)=(_,_), r=__
Simplify
WHITE BOARD EXAMPLES
Directions:
(One problem will be displayed at a time on the Elmo)
*Write answers on your own white board.
*Hold up answers when you are finished.
*If wrong, ask a neighbor for help
1. Given the equation (x+4)2+y2=64, state the coordinates of
the center and the measure of the radius.
2. Write the equation of a circle with a center at (3,-2) and a
diameter of 24.
3. Draw a graph of a circle with a center at (-6,4) and a
radius of 5. (Hint: Use the coordinate side of the white
board).
4. Three dancers are on a stage, which is represented by a
coordinate system. Dancer 1 is at the coordinate (3,-3).
Dancer 2 is at the coordinate (6,-1). Dancer 3 is at the
coordinate (5,3). There is a spotlight projected as a circle
onto the stage with the equation (x-4)2+(y+1)2=9. Which
dancers are in the spotlight? (Hint: It may help to draw it.)
Name: _________________________
POST ASSESSMENT
1. Find the equation of a circle with a center at (-10,5) and a diameter
of 20.
2. The equation of a circle is (x-13)2+(y+11)2=81. Find the center and
radius of this circle.
3. Three dancers are on a stage, which is represented by a coordinate
system. Dancer 1 is at the coordinate (2,-2). Dancer 2 is at the
coordinate (0,-1). Dancer 3 is at the coordinate (5, 3). There is a
spotlight projected as a circle onto the stage with the equation (x4)2+(y+1)2=9. Which dancers are in the spotlight? (Hint: It may help
to draw it.)
ELABORATION WORKSHEET
Directions: Using the image provided and the Pythagorean Theorem, derive the equation
of a circle.
Pythagorean Theorem: a2 + b2 = c2
Where a and b are lengths of the legs, and c is the length of the hypotenuse of a right
triangle.
In the image, C represents the center of the circle, (x,y) is a point on the circle and a
coordinate, (h,k) is the center of the circle and also a coordinate.
Note: r is the radius AND the hypotenuse.
ELABORATION WORKSHEET KEY
Directions: Using the image provided and the Pythagorean Theorem, derive the equation
of a circle.
Pythagorean Theorem: a2 + b2 = c2
Where a and b are lengths of the legs, and c is the length of the hypotenuse of a right
triangle.
In the image, C represents the center of the circle, (x,y) is a point on the circle and a
coordinate, (h,k) is the center of the circle and also a coordinate.
Note: r is the radius AND the hypotenuse.
Use the Pythagorean Theorem: a2+b2=c2 (according to common notation).
This is applied to the equation of a circle when a triangle is graphed (on the coordinate plane
for visual ease) within a circle where the hypotenuse is the radius.
Now it can be seen that the leg lengths correspond to the difference between a point on the
circle and the center point of the circle.
Thus using substitution, the Pythagorean theorem becomes (x-h)2 + (y-k) 2 = c2 where (x,y) is
the point on the circle and (h,k) is the center of the circle.
Now, this is the standard equation of a circle: written as (x-h)2+(y-k)2=r2, where (h,k) is the
center and r is the radius.
ANSWER
Group #:_____
Job Position: ______
Name: ______________________
EXPLORATION WORKSHEET KEY
Directions:
**Each of you is part of a Team 1, 2, 3, 4, or 5.
**Within each Team, there are 3 job positions: A, B, or C.
**Follow the instructions according to your position and only complete your Problem #.
**Be prepared for further instructions by your teacher.



Job Description A: Collect and discuss information regarding the x coordinate of the
center of your circle
Job Description B: Collect and discuss information regarding the y coordinate of the
center of your circle
Job Description C: Collect and discuss information regarding the radius of the circle
Group 1 – PROBLEM 1
(x-1)2 + (y-2) 2 = 4
Center: (1,2)
Radius: 2
Group 2 – PROBLEM 2
(x+3)2 + (y-1) 2 = 4
Center: (-3,1)
Radius: 2
Group 3 – PROBLEM 3
(x-2)2 + (y+2) 2 = 9
Center: (2,-2)
Radius: 3
Group 4 – PROBLEM 4
(x+2)2 + (y-2) 2 = 16
Center: (-2,2)
Radius: 4
Group 5 – PROBLEM 5
(x+1)2 + (y-2) 2 = 16
Center: (-1,2)
Radius: 4
Project Based Interactions: 2-Day Lesson Plan
Group #:_____
Job Position: ______
Name: ______________________
EXPLORATION WORKSHEET #2 KEY
Directions:
At this time, you should be in your original Groups 1, 2, 3, 4, or 5.
Share with your group what you learned from your A, B, and C group collaboration.
Once every Position has shared, as a Group, discuss the following questions and write an answer.
Once you have finished, be prepared to share your answers with the class when called on.
1. What was your Group’s equation? Write it here: ______________________________
2. The center for you circle was: __________. The radius for you circle was: __________.
3. What connections did you make between the x coordinate of the center and the equation?
The x coordinate corresponded to the number being subtracted from x in the equation
4. What connections did you make between the y coordinate of the center and the equation?
The y coordinate corresponded to the number being subtracted from y in the equation
5. What connections did you make between the radius of the circle and the equation?
The radius was the square root of the number on the right side of the equals sign
6. Given the standard form of the equation of a circle: (x-h) 2
What do you think the h represents?
The x coordinate of the center
The k ?
The y coordinate of the center
The r ?
The radius of the circle
+ (y-k) 2 = r2
Project Based Interactions: 2-Day Lesson Plan
Name: __________________________
EXPLORATION WORKSHEET KEY
Directions: Follow along with the examples as we go over them on the Elmo
1. Write the equation of the circle with the center at the origin and a
radius of 5.
x2+y2=25
Equation of a circle
(h,k)= (0,0), r=5
Simplify
2. Write the equation of a circle with a center at (3,-2) and a diameter of
18.
(x-3)2+(y+2)2=81
Equation of a circle
(h,k)= (3,-2), r=18/2=9
Simplify
Project Based Interactions: 2-Day Lesson Plan
3. The equation of a circle is (x-6)2+y2=25. State the center and radius.
Then graph the equation.
Center: (6,0)
Radius: 5
4. Write an equation of a circle with a center at (2,2) and passes through
the point (6,5).
Step 1: Find the distance between the points to determine the radius
r=(6-2)2+(5-2)2
25= 5
Distance Formula
(x1,y1)=(2,2) and (x2,y2)=(6,5)
Simplify
Step 2: Write the equation using h=2, k=2, and r=5.
(x-2)2+(y-2)2=25
Equation of a circle
(h,k)=(2,2), r=5
Simplify
Project Based Interactions: 2-Day Lesson Plan
WHITE BOARD EXAMPLES KEY
Directions:
(One problem will be displayed at a time on the Elmo)
*Write answers on your own white board.
*Hold up answers when you are finished.
*If wrong, ask a neighbor for help
1. Given the equation (x+4)2+y2=64, state the coordinates of the center
and the measure of the radius.
Center: (-4,0)
Radius: 8
2. Write the equation of a circle with a center at (3,-2) and a diameter of
24.
(x-3)2+(y+2)2=144
3. Draw a graph of a circle with a center at (-6,4) and a radius of 5.
(Hint: Use the coordinate side of the white board).
(graph will shown on Elmo)
Project Based Interactions: 2-Day Lesson Plan
4. Three dancers are on a stage, which is represented by a coordinate
system. Dancer 1 is at the coordinate (3,-3). Dancer 2 is at the
coordinate (6,-1). Dancer 3 is at the coordinate (5,3). There is a
spotlight projected as a circle onto the stage with the equation (x4)2+(y+1)2=9. Which dancers are in the spotlight? (Hint: It may help
to draw it.)
Dancers 1 and 2
Project Based Interactions: 2-Day Lesson Plan
Name: _________________________
POST ASSESSMENT KEY
 Find the equation of a circle with a center at (-10,5) and a diameter of 20.
(x+10)2+(y-5)2=100
 The equation of a circle is (x-13)2+(y+11)2=81. Find the center and radius of this circle.
Center: (13,-11)
Radius: 9
Three dancers are on a stage, which is represented by a coordinate system. Dancer 1 is at
the coordinate (2,-2). Dancer 2 is at the coordinate (0,-1). Dancer 3 is at the coordinate (5,
3). There is a spotlight projected as a circle onto the stage with the equation (x4)2+(y+1)2=9. Which dancers are in the spotlight? (Hint: It may help to draw it.)
Dancer 1 only
Project Based Interactions: 2-Day Lesson Plan
Name: _________________________
Formative Assessment END OF DAY 1
1. Write the equation of a circle with a radius of 4 and a center at (0, 2)
2. What did you find most difficult about today’s lesson?
3. What improvements can the teacher make for tomorrow’s lesson?
Name: _________________________
Formative Assessment END OF DAY 1
1. Write the equation of a circle with a radius of 4 and a center at (0, 2)
2. What did you find most difficult about today’s lesson?
3.
What improvements can the teacher make for tomorrow’s lesson?
Project Based Interactions: 2-Day Lesson Plan
Name: _________________________
Formative Assessment END OF DAY 1 KEY
Write the equation of a circle with a radius of 4 and a center at (0, 2)
Radius: 4, so r2 =16
(x-0)2+(y-2)2=16  (x)2+(y-2)2=16
What did you find most difficult about today’s lesson?
What improvements can the teacher make for tomorrow’s lesson?