Capstone

I Will Derive
Spin off of Cake’s song “I will survive”

I will derive.
As long as I know how to problem solve
I know I'll be alive.
I've got all my life to live.
I've got all my love to give.
I will derive.
I will derive.

I Will Derive is a collection of lesson plans that
incorporate coming up with or “deriving” equations
in each mathematical task. Each lesson involves
guided discovery, collaborative learning, and
differentiated instruction.
Lesson One: Deriving the formula for sector area and arc length
Lesson Two: Deriving the formula for a trapezoid
Lesson Three: Deriving the formulas for polar equations

…

Lesson Number One:

Secondary Lesson: Arc Length and Sector Area (Procedures with
Connections: Guided Discovery)
Name of class: Geometry

Length of class: 90 minutes

LEARNING GOALS to be addressed in this lesson (What standards or umbrella
learning goals will I address?):
MA2013 (9-12) Geometry 28. Derive, using similarity, the fact that the length
of the arc intercepted by an angle is proportional to the radius, and define
the radian measure of the angle as the constant of proportionality; derive
the formula for the area of a sector. [G-C5]
LEARNING OBJECTIVES (in ABCD format using verbs from Bloom’s Taxonomy):
Students will find the area of a sector given its arc length and radius
Students will find the arc length of a sector given its area and radius
Students will find the area and or arc length of a circle given the measure of
the central angle and radius.
Objectives that will be hit without knowing:
Students will review computing the area and circumference of circles.
Students will review constructing and measuring angles.
Students will use inductive reasoning to compare their measurements and
calculations and to generalize their results.
CONTENT (What specific
concepts, facts, or vocabulary
words will I be teaching in this
lesson?):

SKILLS: (What skills will students acquire or
practice?)

Arc Length
Sector Area
Radius
Circumference
Diameter

Students should know how to find the
circumference of a circle

Prerequisites:

Students should know how to find the
area of a circle

Area
Central Angle

Given the area and or circumference of
a circle, students should know how to
solve for the diameter or radius of that
circle
Skills acquired during the lesson:
Students will find arc length
Students will find Sector Area
Students will enhance their inductive
reasoning skills

RESOURCES/MATERIALS NEEDED (What materials and resources will I need?):
Student Materials
String
Protractors
Rulers
Compasses
Calculators
Measuring Tape
Cans
Teacher Materials:
Elmo
Colored Pencils
Handouts
Sticky Notes (yellow and pink)
Instructions
Guided Activity
1 Example
Ticket Out
LEARNING PLAN (How will you organize student learning in this lesson?

ACTIVATE (How will I pre-assess my students’ understanding, activate their prior
knowledge, or get them excited about my lesson?)
I will activate my lesson with a Think, Pair, Share. I will have my students…
Think, Pair, Share to the question “What do you remember about finding the area
and circumference of a circle?” First think to yourself and write down some ideas on
the yellow post it note in front of you. After two minutes or so, turn to your shoulder
partner and discuss your ideas. Together, write down some ideas on the pink post it
note in front of you. After three minutes or so, face front and be ready to share your
pink post it note to the class. (This activity in general should take about 10 minutes)
ACQUIRE & APPLY (What instructional strategies will I choose to help my students
acquire and apply the knowledge, skills, attitudes, and behaviors outlined above?)
The main strategy I am going to use for this lesson is guided discovery. Students will
be given a heavily guided activity to complete. By the end of the activity, students
should be able to come up with the formulas for the area of a sector of a circle as
well as arc length.
This activity will involve cooperative learning
This lesson will include small group and large group discussions
ASSESSMENT (How will you asses student understanding?):
At the end of the lesson students will fill out a “ticket out” that I will then collect as
they walk out the door. The ticket out will informally asses what they learned. The
ticket out will consist of two parts…
Part One: Circle the Face—
Circle the face that best resembles how you are feeling after this activity. Put your
name on it and turn it in on your way out.
Part Two: Answer the following Problem—
Students will answer the following problem. It has to do with arc length and sector
area. It is a computational problem that will let me know if students understand the
general process of finding arc length and sector area.

LESSON PLAN SEQUENCE & PACING (How will I organize this lesson? How
much time will each part of the lesson take?)
1. Students will complete the Launch Activity (10 minutes)
2. Introduce the topic of the day (Arc Length and Sector Area) and go over
the instructions for the activity. Answer student questions during this time. (5
minutes)

3. Pass out materials and have students work on the guided activity.
Students will form groups of three. While students are working, walk around
the room during this time and pose questions to guide students thinking if
needed. (45 minutes)
The big ideas of the lesson include…
Problems 1-4 involve activating students’ prior knowledge and allow students
to get familiar with the materials at hand.
Problems 5-7 involve deriving the formula for arc length
Problem 8 involves comparing and contrasting arc length to the total
circumference of a circle as well as coming up with a general understanding
of arc length.
Problems 9-11 involve deriving the formula for sector area as well as coming
to a general understanding of sector area in relation to the total area of a
circle.
Problems 12-13 involve coming up with a general pattern for both arc length
and sector area as well as developing formulas for both arc length and
sector area.
The “Think About It” questions all involve higher ordered thinking. In these
questions students will get at the concept that the length of the arc
intercepted by an angle is proportional to the radius of that circle.
4. If students finish early have them write out formal formulas for the area of a
sector as well as a formal formula for arc length. Have them compare and
contrast the two formulas as well as discuss how these formulas relate to
similar figures and proportionality.
5. Join 2 sets of small groups to discuss their findings (10 minutes)
6. Class discussion of findings (10 minutes).
7. After students finish the activity and the class discussion is complete they
will have to complete one example problem. (3 minutes)
9. Go over example problem as a class (3 minutes)

12. Ticket Out (4 minutes)

Instructions (This will be put on the Elmo)
1. In your groups, complete the guided activity on how to find
arc length and sector area of circles. Everyone must write. If you
have any questions raise your hand. (45 minutes)
2. Hint: To find the circumference of your can, use string and a
ruler
3. If you finish before the timer goes off, raise you hand.
4. When the timer goes off, you will be asked to combine with
another group. Get into these groups as quickly and as
efficiently as possible.
5. Discuss your findings. What did you observe? Answer the
following questions during the discussion. Jot down the answers.
(10 minutes)
a. How did your group arrive at the answers to each question?
Not only discuss what the answers are, but how you got there.
b. What mistakes did your group make before arriving at the
answers?
c. Is there an easier method you could have used to solve each
problem?
d. What information helped you find the answers?
6. When the timer goes off, it is time to discuss as a class. Face
forward and be ready to listen  (10 minutes)

Arc Length and Sector Area (Answer Key)
Our Goal: The goal of this activity is to derive (or come up with) the formulas for arc
length and sector area using the questions as a guide.
Deriving the formula for Arc Length:
1. Which can did your group receive? EXAMPLE: Can A
2. What is the circumference of your can in centimeters (round to the nearest millimeter)?
20.73 cm
3. Using your measurement for the circumference, find the length of the diameter and
radius using algebra. Be sure to show your work. (Remember to write the units!)
Circumference = 2 * Pi * radius
20.73 = 2 * (3.14) * (radius)
10.365 = (3.14) *(radius)
3.30 = radius
Diameter = 2 * (radius)
Diameter = 2 * (3.30)
Diameter = 6.60 cm
OR…
Circumference = Diameter * Pi
20.73 = Diameter * 3.14
Diameter = 6.60 cm
Radius = Diameter / 2
Radius = 6.60/2
Radius = 3.30 cm

Diameter: 6.60 cm
Radius: 3.30 cm
Check your answers for the diameter and radius by measuring the length across the can
with a ruler. What did you get? Were you close?

4. In the space provided, use the compass to draw a circle with the radius and diameter
you computed in #3 above. Then, use radii to divide the circle into four equal parts. (This
is not drawn to scale: just for seeing purposes)

5. What is the measure of each central angle in the circle you constructed in #4? Show
your work algebraically.
Central Angle = Total Angle Measure of the Circle / Number of Parts
Degrees:
Central Angle = 360 / 4
Central Angle = 90 degrees
Radians:
Central Angle = 2*Pi / 4 = Pi / 2
Central Angle: (Pi) / 2 radians
Each Angle is 90 degrees.
Each Angle is (Pi/2) radians.
6. Write a fraction that compares the measure of the central angle to the total number
of degrees at the center of the circle. This fraction is the constant of proportionality. Be
sure to simplify your fraction.
The Fraction in degrees is: 90 / 360 = 1 / 4
The Fraction in radians is: (Pi / 2) / (2 * Pi) = 1 / 4
7. Keeping in mind your answer to #2, above, what would be the length of the arc
formed by one of the central angles mentioned in #5?
Circumference = 20.73

Radius = 3.30

Diameter = 6.60

Thoughts: Circumference relates to arc length. If we know the total circumference
is 20.73, the arc length will be smaller than the total circumference. We want the
arc length of (1/4) of the circle…so (1/4) of 20.73 cm. So divide 20.73 by 4. This is
5.18 cm.
OR…
(Arc Length)/(Circumference) = (Angle Measure) / 360
Cross Multiply and Solve
(Arc Length) / 20.73 = 90 / 360

Arc Length = 5.18 cm
OR…
Arc Length = 20.73 * (90/360)
Arc Length = 5.18 cm
The arc length is:

5.18 cm

(remember the units!)

Describe how you found this arc length. I found the arc length by multiplying the
circumference by ¼. I found the arc length by setting up proportions. I knew the angle
measure was a part of the whole circle. I also knew the arc length is a part of the whole
circumference. Because of these relationships, I can use the idea of similar triangles to
set up a proportion, cross multiply, and solve. I found the arc length by multiplying the
circumference by the fraction of the circle we want.
8. Compare the arc length to the overall circumference of the circle. What do you
notice?
Arc Length / Circumference
5.18 / 20.73
1/4

Arc Length / Circumference = 5.18/20.73
Simplify: 1/4
I notice that: I notice that (arc length / circumference) is the same as (central angle /
360 degrees). I also noticed that (arc length / circumference) is the same as (central
angle / 2*Pi radians). These all equal ¼.

Deriving the formula for Sector Area:
9. What is the area of the circle you drew in #4?
Area = Pi * (radius)^2
Area = (3.14)*(3.30)^2
Area = 34.19 cm squared
The Area of the circle is: 34.19 cm squared (remember the units!)
10. What is the area of one of the four sectors formed in #4?
Area / 4
34.19 / 4
8.54 cm squared
The area of each sector is: 8.54 cm squared (remember the units!)
11. Compare the area of the sector to the total area of the circle. What do you notice?
Sector / Total Area
8.54 / 20.73
1/4

Sector Area / Total Area = 8.54 / 20.73
Simplify: 1/4
I notice that: I notice that (sector area / total area) is the same as (central angle / 360
degrees). I also noticed that (sector area / total area) is the same as (central angle / 2*Pi
radians). I also noticed that these all equal (arc length / circumference) which all equal
¼.
12. What pattern do you see in questions #6, 8, and 11? Why does this pattern
occur? The pattern I see is that you multiply everything by ¼. The ¼ comes from

the fraction of the circle we are trying to evaluate. If a person wants to find the
length of an arc, they should multiply the total circumference by ¼. If one wants
to find the area of a sector, they should multiply the total area by ¼. Finding arc
length and sector area is just finding “parts” of the circumference and area of a
circle.
13. What formulas can you write to express the pattern you see? (Write your
answer in terms of Radians)

Formula for Arc Length: (Central Angle / 2*Pi) * Circumference
Formula for Sector Area: (Central Angle / 2*Pi) * Total Area
Think About It:
1. Can you write the formula for arc length another way?
Yes. I can write the formula as a set of proportions and solve for the missing
value. The proportion set up would be…
(Arc Length / Circumference) = (Central Angle in Radians / 2 * Pi)
2. Can you write the formula for sector area another way?
Yes. I can write the formula as a set of proportions and solve for the missing
value. The proportion set up would be…
(Sector Area / Total Area) = (Central Angle in Radians / 2 * Pi)
3. Is it possible to find the area of a sector if you are just given the arc length and
radius? If so, how? Give an example.
Yes. If you are given the radius of the circle, you can divide the circumference
of the circle as well as the area. Then you can set up a proportion using the arc
length, circumference, sector area, and area. It would look something like this.

(Arc Length) / (Circumference) = (Sector Area) / (Total Area)
Now all you have to do is plug values in, cross multiply, and solve for the sector
area.
4. Is it possible to find the arc length of a piece of a circle if you are only given
the area of a sector and the diameter? If so, how? Give an example.
Yes. If you are given the diameter of a circle, you can easily find the radius.
Once you find the radius, you can find the circumference and total area of the
circle. You can then use the formula from number five, plug in numbers, cross
multiply, and solve for arc length.
5. Can you use the central angle of a circle to find arc length and sector area?
Why?
Yes. In order for one to do this though, you will need to be given more
information, for example, a circumference, an area, a diameter, or a radius. If
you are given one or more of these pieces of information, you can use the
central angle over 360 to find arc length and sector area.
Sector Area = (central angle / 2 * Pi) * Area of Circle
Arc Length = (central angle / 2 * Pi) * Circumference
Using Algebra, we can also come up with the idea that the arc length is
proportional to the radius as well as that sector area = angle in radians *
diameter

In Class Example Problem: (Projected on the Elmo)
Find the Arc Length and Sector Area of the Shaded Region: (You
may do this in your notes or on the back of the guided activity)

Name:________________________
Date:_________________________

Arc Length and Sector Area
Our Goal: The goal of this activity is to derive (or come up with) the formulas for arc
length and sector area using the questions as a guide.
Deriving the formula for Arc Length:
1. Which can did your group receive? ________________________
2. What is the circumference of your can in centimeters (round to the nearest millimeter)?
______________________________________________________________________________________
3. Using your measurement for the circumference, find the length of the diameter and
radius using algebra. Be sure to show your work. (Remember to write the units!)

Diameter: _________________________
Radius: ___________________________
Check your answers for the diameter and radius by measuring the length across the can
with a ruler. What did you get? Were you close?
4. In the space provided, use the compass to draw a circle with the radius and diameter
you computed in #3 above. Then, use radii to divide the circle into four equal parts.

5. What is the measure of each central angle in the circle you constructed in #4? Show
your work algebraically.

Each Angle is _______________________ degrees.
Each Angle is _______________________ radians.
6. Write a fraction that compares the measure of the central angle to the total number
of degrees at the center of the circle. This fraction is the constant of proportionality. Be
sure to simplify your fraction.
The Fraction in degrees is: _____________________________
The Fraction in radians is: ______________________________
7. Keeping in mind your answer to #2, above, what would be the length of the arc
formed by one of the central angles mentioned in #5?

The arc length is: ____________________ (remember the units!)

Describe how you found this arc length. _______________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
______________________________________________________________________________________
8. Compare the arc length to the overall circumference of the circle. What do you
notice?
Arc Length / Circumference = _____________________________
Simplify: ___________________________________________________
I notice that:
______________________________________________________________________________________
______________________________________________________________________________________
Deriving the formula for Sector Area:
9. What is the area of the circle you drew in #4?

The Area of the circle is: ___________________ (remember the units!)
10. What is the area of one of the four sectors formed in #4?

The area of each sector is: __________________ (remember the units!)

11. Compare the area of the sector to the total area of the circle. What do you notice?

Sector Area / Total Area = ________________________
Simplify: _________________________________________
I notice that:
______________________________________________________________________________________
______________________________________________________________________________________
12. What pattern do you see in questions #6, 8, and 11? Why does this pattern
occur?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________

13. What formulas can you write to express the patterns you see? (Write your
answer in terms of Radians)

Formula for Arc Length: ____________________________________
Formula for Sector Area: ____________________________________

Think About It:
1. Can you write the formula for arc length another way?

2. Can you write the formula for sector area another way?

3. Is it possible to find the area of a sector if you are just given the arc length and
radius? If so, how? Give an example.

4. Is it possible to find the arc length of a piece of a circle if you are only given
the area of a sector and the diameter? If so, how? Give an example.

5. Can you use the central angle of a circle to find arc length and sector area?
Why?

Name: _____________________
Date: ______________________
Ticket Out—Circle the Face: How do you feel about Arc Length and Sector
Area?

Solve the Problem: Find the Arc Length and Sector Area of the shaded region:

Lesson Number Two:
Secondary Lesson Plan: Transformations and Area
Name of class: 8th Grade Geometry

Length of class: 50 minutes

LEARNING GOALS to be addressed in this lesson (What standards or umbrella
learning goals will I address?):
The goal of the task is to come up with as many ways to transform a trapezoid
into a rectangle as possible. In the close of the lesson, students will see that
there are many ways to transform this shape into a rectangle and that there is
NOT only ONE right method to solve this problem. The goal of the lesson is to
get students problem solving as well as documenting how they are going
about solving a problem.
LEARNING OBJECTIVES (in ABCD format using verbs from Bloom’s Taxonomy):
Given the area formula for a rectangle and properties of a trapezoid, students
will derive the area formula for a trapezoid by transforming the trapezoid into a
rectangle. Students will draw upon their knowledge of rotations, translations,
reflections, and polygon properties to complete the transformation and make
connections with algebraic formulas.
Given the formula of a rectangle as well as specific properties of a trapezoid,
students will derive the formula of said shape using transformations.
Standards:
CCSS.MATH.CONTENT.8.G.A.1. Verify experimentally the properties of rotations,
reflections, and translations [8. G. 1]
CCSS.MATH.CONTENT.8.G.A.2. Understand that a two-dimensional figure is
congruent to another if the second can be obtained from the first by a
sequence of rotations, reflections, and translations; given two congruent
figures, describe a sequence that exhibits the congruence between them.
[8.G.2.]
CCSS.MATH.CONTENT.8.G.A.5. Use informal arguments to establish facts about
the angle sum and exterior angle of triangles, about the angles created when
parallel lines are cut by a transversal, and the angle-angle criterion for similarity
of triangles. For example, arrange three copies of the same triangle so that the
sum of the three angles appears to form a line, and give an argument in terms

of transversals why this is so. [8.6.5]
Objectives that will be hit without knowing:
Students will review computing the area of a rectangle as well as a trapezoid.
Students will review the concepts of rotations, reflections, and translations.
Students will review the basic properties of the shape they are given.
CONTENT (What specific
concepts, facts, or
vocabulary words will I
be teaching in this
lesson?):
Shapes:
• Trapezoid
• Rectangle
• Parallelogram
• Triangle
• Angles
Concepts:
• Transformation
• Rotation
• Reflection
• Translation
• Area
• Angle
Relationships
• Slope

SKILLS: (What skills will students acquire or practice?)
Prerequisites:
Students should know how to find the area of a
rectangle, parallelogram, and triangle.
Students should know the basic ideas of
transformations.
• Reflection—Flip
• Translation—Slide
• Rotation—Turn
Students should know how to transform a
parallelogram, triangle, and kite (optional) into a
rectangle.
• Students should know how to transform a
triangle into a rectangle.
• Students should know how to transform a
parallelogram into a rectangle.
• Students should know how to transform a kite
into a rectangle (optional).
Students should be familiar with deriving the
formulas for a parallelogram and triangle using
transformations and the area formula of a
rectangle.
Skills acquired during the lesson:
Students will transform a trapezoid into a rectangle.
While transforming a trapezoid into a rectangle,

students will describe the transformations used as
well as the reasoning behind each step.
Students will derive the formula for the area of a
trapezoid given the area of a rectangle.
Students will build on the understanding that the
area of a two-dimensional figure is preserved
through transformations.
RESOURCES/MATERIALS NEEDED (What materials and resources will I need?):
Handouts:
“Venn Diagram” handout
“Instruction Sheet” handout for the Elmo
“Transformation Task” handout
“Paper Trapezoid” handouts
“Ticket Out—321” handout for the Elmo
Materials:
Paper
Red colored pencil
Blue Colored pencil
Colored Pencils
Scissors
Graph Paper
Rulers
Sticky Notes
LEARNING PLAN (How will you organize student learning in this lesson?
ACTIVATE (How will I pre-assess my students’ understanding, activate their prior
knowledge, or get them excited about my lesson?)
The teacher will activate the lesson with a Think, Pair, Share. He/she will have
their students…
Think, Pair, Share to the question “What are the properties of a trapezoid? What
are the properties of a rectangle? Write these properties in the Venn Diagram
handed to you.” First think to yourself and write down some ideas with the blue
colored pencil in front of you. After two minutes or so, turn to your shoulder
partner and discuss your ideas. Together, write down some ideas with the red
colored pencil in front of you. After three minutes or so, face the front of the
room and be ready to share your thoughts to the class. (This activity in general

should take about 10 minutes)
ACQUIRE & APPLY (What instructional strategies will I choose to help my
students acquire and apply the knowledge, skills, attitudes, and behaviors
outlined above?
The instructional strategy the teacher will be using in this lesson will be a “Doing
Math” Mathematics Task. The task will ask students to transform a trapezoid into
a rectangle in order to derive the formula for a trapezoid using the formula for
a rectangle. They will be asked to work together with their tablemates.
While students are working, the teacher will walk around and monitor student
learning. The teacher will write down misconceptions the students are having
as well as write down how students are solving this problem. The teacher will
also assist the students by asking guiding questions when needed.
After about twenty minutes or so, the teacher and students will come together
as a class and discuss their findings. The teacher will have already “ranked” the
order in which to call on his or her students based on the level of mathematics
preformed in each technique. When students are called on, they will be asked
to share their way of thinking.
For this task, students will be grouped by achievement level so all students will
have the opportunity to contribute to solving the problem. If there is a group of
struggling learners in the classroom, which in most cases there will be, the
teacher can assist them by asking guiding questions when needed. Some
guiding questions the teacher can ask include:
• How can we cut up this trapezoid and arrange its pieces to make a
rectangle?
• When we cut the trapezoid this way, what happens to the base? What
happens to the height?
• Think back to when we did this activity with the parallelogram or triangle.
Are there similar methods you can use with this activity?
The task is designed so students can use whatever method they choose to
come up with a solution to the problem. They may use manipulates (like
colored pencils, paper, graph paper, and scissors) if they so choose.
If students finish before time is up, I will either ask them to:
1) Find another way to transform the trapezoid into a rectangle
• Can you think of another way to transform the trapezoid into a
rectangle?

Could you double the area of the trapezoid (use two trapezoids) to
transform your figure? How does this method differ from the method you
exhibited?
2) Write out their thinking in a more formal manner or
• I see that you cut your trapezoid up into many pieces and placed them
in certain spots. Can you describe in words what you did?
o What transformation did you use to move this piece? Why can you
do that?
3) Transform a regular trapezoid into a rectangle and then compare and
contrast the transformations they used for each shape.
•

This task includes:
1) Cooperative learning
2) Small and large group discussions
3) Critical thinking
ASSESSMENT (How will you asses student understanding?):
At the end of the lesson students will fill out a “ticket out” that I will then collect
as they walk out the door. The ticket out will informally asses what they learned.
The ticket out will be a 321. They students will be familiar with this form of a
ticket out. They will be asked to write down…
3: Three properties of a trapezoid
2: Two facts about transformations and area computations
1: One aspect of the lesson they are still confused about
They may write their answers on a half sheet of paper or on the sticky note
provided. They will place this assessment on the front table before they leave.
LESSON PLAN SEQUENCE & PACING (How will I organize this lesson? How much
time will each part of the lesson take?)
1. Students will complete the Launch Activity. (10 minutes)
The Launch will activate students prior knowledge
2. Introduce the topic of the day (transforming a trapezoid into a rectangle)
and go over the instructions for the activity. Answer student questions during
this time. (3 minutes)
3. Pass out the task and have students work on it. They will work with their
tablemates. They will be in groups of four. (20 Minutes)

4. If students finish before the twenty minutes is up, have them either 1)
transform the trapezoid into a rectangle another way 2) write out their thinking
in a more formal manner or 3) ask them to transform a regular trapezoid into a
rectangle and then compare and contrast the transformations they used for
each shape. If struggling learners are having a difficult time, encourage them
to use the manipulates on the back table as well as ask them guiding questions
to further their thinking.
5. After the twenty minutes is up, come back together as a class and have a
teacher led discussion on the different techniques the students used to
transform the figure. During this time, the teacher will call on students to talk (in
a particular order) as well as write down what the students say. Students will be
asked to record what their classmates say as well. (15 minutes)
6. Ticket out (2 minutes)

Instructions for the Day
1. With your assigned groups transform the
trapezoid into a rectangle. Make sure to
label the height of your trapezoid before
you start!
You will be assigned a group. Everyone
must write. If you have any questions raise
your hand. (20 minutes)
2. If you finish before the timer goes off,
raise you hand.
3. When the timer goes off, we will begin
our class discussion. (15 minutes)
Please face the front of the room and be
ready to take some notes! 

Transform the Trapezoid (Answer Key)
Instructions: Draw in the height of the trapezoid. Label it h, for height. After you
label the height, transform the following trapezoid into a rectangle. Describe the
steps you took to transform your shape in the space provided.
After you transform the trapezoid into a rectangle, find an area formula for a
trapezoid using 1) the transformations you made, and 2) the area formula of a
rectangle using the variables: B1, B2, and h.

Goals:
1. While preserving the area of a given shape, I can transform a trapezoid into a
rectangle.
2. I can find an area formula for a trapezoid.

Part One: Transform the trapezoid into a rectangle. Hint: Remember that the
area of a two-dimensional figure is equivalent to another if the second can be
obtained from the first.
First transform the trapezoid into a rectangle by actually cutting up the shape
and rearranging the pieces into a rectangle.
Describe and sketch the transformations you preformed in the space below.
Sketches:
Parallelogram Method:

Description: We know that b1 is parallel to b2. We also know the trapezoid AECD
is a quadrilateral. Since b1 is parallel to b2, construct a segment congruent to b2
attached to b1. Since b1 is parallel to b2, construct a segment b1 attached to
b2. Since (b1+ b2) is parallel to (b1 + b2), translate AD over. What actually
happened here was that we doubled the area of the trapezoid because we
reflected and translated the trapezoid over and added it to its original position.
We know line AD is parallel to line D1A1. Now we have a parallelogram that is
double the area of the trapezoid. We can then transform this parallelogram into

a rectangle. To do this we drop a perpendicular from A to b2. This creates a
right triangle. We can then translate the triangle over to the other side.
Algebraic Method:

Description: Construct a median through the trapezoid. By the definition of a
median we know b1, b2, and the median are all parallel lines. We also know
segment congruencies. Construct a perpendicular line from the median to b2
on both ends. Then rotate the triangles upward.
**Some other methods that can be done involve cutting the trapezoid into
triangles or making the trapezoid look like stair steps. Just see what the students
come up with and go from there!

Part Two: Now it is time to derive your area formula. Remember to rename
segment AE and call it B1 and rename segment DC and call it B2. Show your
work in the space provided.
Sketch Diagrams:
Parallelogram Method:

The area of this shape would be (b1+ b2) *h. The area of the figure currently,
though, is double the area of the original trapezoid because of how we
manipulated it. Because this area is double the area, all we need to do is divide
it by two and presto!
The Area Formula for a Trapezoid is: (1/2)*(b1+ b2)*(h)
The algebraic method is more difficult, but it can be done. If there is time left at
the end of the period, we will discuss this method in more detail.
Depending on how students transformed their figure, their explanations will be
different. They should all come to the same conclusion though!

The Area Formula for a Trapezoid is: (1/2) * (b1 + b2) * h

321
Write Down…
3: Three properties of a trapezoid
2: Two facts about transformations
and area computations
1: One aspect of the lesson they are
still confused about
**Turn this in at the front table before you
leave class

Name: _____________________________
Date: ______________________________

Opener for the Day: The Transformation of a Trapezoid
Question: What are the properties of a trapezoid? What are the
properties of a rectangle? Write these properties down in the Venn
diagram below.
Trapezoid

Both:

Rectangle:

Name: ______________________________
Date: _______________________________

Transform the Trapezoid
Instructions: Draw in the height of the trapezoid. Label it h, for height. After you
label the height, transform the following trapezoid into a rectangle. Describe the
steps you took to transform your shape in the space provided.
After you transform the trapezoid into a rectangle, find an area formula for a
trapezoid using: 1) the transformations you made, and 2) the area formula of a
rectangle using the variables: B1, B2, and h.

Goals:
1. While preserving the area of a given shape, I can transform a trapezoid into a
rectangle.
2. I can find an area formula for a trapezoid.

Part One: Transform the trapezoid into a rectangle. Hint: Remember that the
area of a two-dimensional figure is equivalent to another if the second can be
obtained from the first.
First transform the trapezoid into a rectangle by actually cutting up the shape
and rearranging the pieces into a rectangle.
Describe and Sketch the transformations you preformed in the space below.
Sketches:

Description:
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________

Part Two: Now it is time to derive your area formula. Remember to rename
segment AE and call it B1 and rename segment DC and call it B2. Show your
work in the space provided.
Sketch Diagrams:

The Area Formula for a Trapezoid is: ________________________________

Need some extra trapezoids?

Lesson Number Three:
Secondary Lesson Plan Template: Polar Graphs (“Doing Math” Lesson)
Name of class: Pre Calculus

Length of class: 90 minutes

LEARNING GOALS to be addressed in this lesson (What standards or umbrella
learning goals will I address?):
Represent complex numbers on the complex plane in rectangular and polar
form (including real and imaginary numbers), [and explain why the rectangular
and polar forms of a given complex number represent the same number]. (NCN.4.)
**The part in brackets is not addressed in this lesson but will be addressed in the
lesson the day before.
LEARNING OBJECTIVES (in ABCD format using verbs from Bloom’s Taxonomy):
Students will identify the different types of polar curves and parametric curves.
Students will compare and contrast the difference between polar curves and
parametric curves.
Students will analyze the different polar graphs by finding maximum and
minimum R-values as well as the line of symmetry.
By using the equation given, students will predict the graph of the polar function
or vise versa.
I can statements for the students:
1. I can identify the different types of polar curves and parametric curves.
2. I can compare and contrast the difference between polar curves and
parametric curves.
3. I can analyze the different polar graphs by finding maximum and minimum Rvalues as well as the line of symmetry.
4. By using the equation given, I can predict the graph of the polar function or
vise versa.

CONTENT (What specific
concepts, facts, or vocabulary
words will I be teaching in this
lesson?):
•
•
•
•

•
•
•

Polar Curve
Parametric Curve
Rose Curve
Limacon Curve
o Limacon with an
inner loop
o Cardioid
o Dimpled Limacon
o Convex Limacon
Lemniscate Curve
R-Value
Theta

SKILLS: (What skills will students acquire or
practice?)
Prerequisites:
Students will be taught the section on Polar
Coordinates the day prior to this lesson.
Students will know how to graph equations in
the rectangular plain.
Skills acquired in this lesson:
Define polar curve (x,y plane) and
parametric curve
Identification of polar graph types given an
equation (rose, limacon, lemniscate)
Identification of polar graph types given a
graph
Determine r values given an equation or
graph (maximum)
Determine symmetry types (if any)
Determine other characteristics of polar
curves
o Domain
o Range
o Bounded or Unbounded
o Continuous

RESOURCES/MATERIALS NEEDED (What materials and resources will I need?):
Launch:
Yellow Post-it Notes
Pink Post-it Notes
Lesson:
“Instructions” handout
Timer

“Mini Guided Instruction” handouts for each person in the “discovery” group
• Rose Curve Guided Discovery
• Limacon Guided Discovery
• Lemniscate Guided Discovery
Colored Cards (to form groups)
Graphing Utility
Graph paper
Colored Pencils
“Notes Outline” handouts for each person in the “home” group
Assessment:
“Ticket Out” handout
LEARNING PLAN (How will you organize student learning in this lesson?
ACTIVATE (How will I pre-assess my students’ understanding, activate their prior
knowledge, or get them excited about my lesson?)
The teacher will activate the lesson with a Think, Pair, Share. He/she will have
their students…
Think, Pair, Share to the question “What are the important characteristics to look
for while graphing an equation? Make a list of these characteristics.” First think to
yourself and write down some ideas on the yellow post it note in front of you.
After two minutes or so, turn to your shoulder partner and discuss your ideas.
Together, write down some ideas on the pink post it note in front of you. After
three minutes or so, face front and be ready to share your pink post it note to the
class. (This activity in general should take about 10 minutes)
ACQUIRE & APPLY (What instructional strategies will I choose to help my students
acquire and apply the knowledge, skills, attitudes, and behaviors outlined
above?
The activity the teacher will use to teach the lesson will be a JIGSAW Activity. First
we will divide the class into six different groups (there will be two groups working
on each topic). Each group will be given one of three different topics to
discover. They will become the “experts” on this topic. The three topics include:
• Rose Curve
• Limacon
• Lemniscate
Each group will be assigned a topic strategically. Once the topics are assigned
and the groups are formed, students will use the “Important Characteristics of a
Graph” list from the launch to investigate their equation. They will be asked to
find the general forms of the equation as well as diagrams and different rules.

We will guide them to come up with the following facts:
• General form of the equation
o How did you find this?
o What does each “part” of the general form tell you?
• Sketch(s) of the graph
• Line(s) of symmetry (if any)
o How do you know this?
o Are there different formulas you can use to check?
• Continuous?
o Why?
• Bounded or unbounded?
o Why?
• Is there a max?
o Why?
• Is there a min?
o Why?
• What is the domain?
o How did you find this?
• What is the range?
o How did you find this?
• Compare the graph using polar coordinates and rectangular coordinates.
Sketch them both. How are they the same? How are they different?
o Domain?
o Range?
o Max?
o Min?
o Bounded?
o Continuous?
**Depending on the needs of each group, they will be guided with different
questions. This is where the differentiation aspect of the lesson comes into play.
While each group collaborates and learns about their topic, the teacher will put
colored cards on each student desk. These cards will determine what “home
group” students are in. After forty-five minutes or so, students will move groups.
They will change from their “discovery group” to their “home group”. In each
home group, there should be 4 to 6 members (depending on the classroom size);
each member should be an “expert” on a different topic. For the next twenty
minutes or so, students will teach their group members about their topics.
After the twenty minutes, students will be asked to compare and contrast each
type of graph. We will do this as a class.
• How do the general forms of the equations differ?

•

•

How do the graphs differ?
o How do the three graphs differ? (Limacon, Rose, Lemniscate)
o How does the graph of the rose curve in the polar plane differ from
the graph of the rose curve in the rectangular plane?
o How does the graph of the Limacon curve in the polar plane differ
from the graph of the Limacon curve in the rectangular plane?
o How does the graph of the Lemniscate curve in the polar plane
differ from the graph of the Lemniscate curve in the rectangular
plane?
Students will address what they think is important to address at this time as
well.

ASSESSMENT (How will asses student understanding?):
To assess student understanding, a ticket out will be given at the end of class.
The ticket out will ask students to match different formulas with the different
graphs. They will then be asked to sketch out their “favorite” graph of the day
and write three facts about it.
LESSON PLAN SEQUENCE & PACING (How will I organize this lesson? How much
time will each part of the lesson take?)
1. Launch: Think, Pair, Share (10 minutes)
2. Discovery Group: Part One of the JIGSAW (45 minutes)
3. Home Group: Part Two of the JIGSAW (20 minutes)
4. Come together as a class and compare and contrast the polar graphs. We
will discuss general findings during this time as well. (15 minutes)
5. Ticket Out (5 minutes)

Goals for the Lesson:
1. I can identify the different types of polar curves
and parametric curves.
2. I can compare and contrast the difference
between polar curves and parametric curves.
3. I can analyze the different polar graphs by finding
maximum and minimum R-values as well as the line of
symmetry.
4. By using the equation given, I can predict the
graph of the polar function or vise versa.

Instructions for the Day
1. In your “discovery” groups, complete the guided activity
on either the…
a) Rose Curve
b) Limacon Curve
c) Lemniscate Curve
You will be assigned a group. Everyone must write. If you
have any questions raise your hand. (45 minutes)
2. If you finish before the timer goes off, raise you hand.
3. When the timer goes off, you will be asked to get in your
“home” groups. Get into these groups as quickly and as
efficiently as possible. They will be assigned as well. In these
groups… (17 minutes)
a) Each member of the group with teach the rest of the
group their topic.
b) Take notes while classmates are talking
c) Do NOT just copy each other’s packets and call it a
day
4. When the timer goes off again, it is time to discuss as a
class. Face forward and be ready to listen  (15 minutes)

Name: ___________________________
Date: ____________________________

Polar Graph Exploration
What are some important characteristics of graphs? List as many as
you can.
Zeroes/Solutions/Intercepts
Y-intercept
Maximum (absolute/local)
Minimum (absolute/local)
Domain
Range
Symmetry
Altitude
Period

The Rose Curve:
Use the following website to explore the different properties of a Rose Curve.
The website is https://www.desmos.com/calculator
Type in the following functions:

a. r = 2 sin(4θ)
b. r = 4 cos (3θ)
c. r = 4 sin (3θ)

1. Notice how each equation begins with “r” instead of “y”. Does this make a
difference on how the graph looks? Sketch the graphs to draw a conclusion.
r = 2 sin(4θ)

y = 2 sin (4x)

Do the graphs differ? Answer with yes or no. Yes
**As you probably have guessed, for the Rose Curve, the equation has to be
equal to “r” in order for it to be a true Rose Graph.! The “r” term as well as the “θ”
are actually indicators that you are graphing in the Polar Plane versus the
rectangular plane, which is what we are used to.

2. What are some characteristics of a Rose curve? Sketch it below.
Sketch of the Rose Curve:

General Characteristics of a Rose
Curve:
1. Looks like a flower
2. Looks symmetrical
3. Contains both sine or cosine
4. Has Petals

3. The following functions are also Rose Curves. By looking at the functions and
graphing the equations, can you come up with a general form for a rose curve?
Write the general form of the equation below. How did you come up with this?
(Hint: the general form of a function is an equation with variables)
a. r = 5 sin (2 θ)
b. r = 7 sin (5 θ)
c. r = 8 sin (6 θ)

d. r = 5 cos (2 θ)
e. r = 7 cos (5 θ)
f. r = 8 cos (6 θ)

The general form(s) of the function are:
1. r = a sin (n θ)
2. r = a cos (n θ)
I came to this general conclusion because: I came to this general conclusion by
looking at the patterns of the equations above.
There are two forms of the equation because: Of the Sine and Cosine

4. Number of Petals:
How can one tell how many petals a rose curve contains by just looking at an
equation? If n is an odd number, the number of petals is n. If n is an even
number, the number of petals is 2n.
Can one tell if the graph has an odd or even number of petals just by looking at
the equation? If so, how? Yes. Look at the value of n.
How can one tell how long a petal is by just looking at the equation? Look at the
value of a.
5. Sin(θ) and Cos(θ) and Tan(θ) oh my!
If one replaces Sin(θ) with Cos(θ), how does the graph change? Pick one of the
graphs from number three. Sketch the graph using both the “Sin (θ)” version and
the “Cos(θ)” version of that graph. How does the graph change?
Sine Graph

r = 5 sin (2 θ)

Cosine Graph

r = 5 cos (2 θ)

Describe the transformation occurring between the sine graph to the cosine
graph. The cosine graph is the sine graph rotated 90 degrees clockwise.
If one were to replace Sin(θ) or Cos(θ) with Tan(θ), does the graph still look like a
Rose curve? Why are why not? It looks kind of similar but really elongated…the
petals never end though.

6. Symmetry
Is the Rose Curve symmetrical about the x-axis, y-axis, and origin? Use the space
below to show your work algebraically as well as graphically. (Hint: When
answering these questions pay attention to when the number of petals is even
or odd. Also pay attention to when the function contains sine or cosine).

a. The y-axis? If so, when? 1) When n is even and 2) When n is odd and r = a cos
(n θ)
b. The x-axis? If so, when? 1) When n is even and 2) when n is odd and r = a sin
(n θ)
c. The origin? If so, when? When n is even
7. Compare and Contrast the Graphs in DIFFERENT planes
Come up with an example of an equation of a Rose Curve.
•

•

•

Graph this equation in the polar plane. (Hint: Remember when we are
graphing in the Polar Plane, we are using “r” instead of “y”. We are also
using “θ” instead of “x”)
Graph the same equation in the rectangular plane. (Hint: Remember
when we are graphing in the Rectangular Plane, we are using “y” instead
of “r” and “x” instead of “theta”)
Sketch both graphs in the table below.

My equation in the Polar Plane is: r = 4 cos (3θ)
My equation in the Rectangular Plane is: y = 4 cos (3x)
Polar Plane:

Rectangular Plane:

Analyze the graph in the rectangular plane. Find the…
a. Maximum and Minimum: absolute value of a
b. Domain: All real numbers
c. Range: [- |a|, |a|]
d. Altitude: a
e. Period: Not pertinent information

Do these characteristics relate to any of the characteristics of the graph in the
polar plane? If so, which ones? How do they relate? Explain your answer.
The Maximum R Value and Altitude
Relate
The are both continuous
They are both bounded
There are no asymptotes

Name: ___________________________
Date: ____________________________

Polar Graph Exploration
What are some important characteristics of graphs? List as many as
you can.
Zeroes/Solutions/Intercepts
Y-intercept
Maximum (absolute/local)
Minimum (absolute/local)
Domain
Range
Symmetry
Altitude
Period
The Lemniscate Curve:
Use the following website to explore the different properties of a Lemniscate
Curve.
The website is https://www.desmos.com/calculator
Type in the following functions:

a. ! ! = 3! sin(2θ)

d. ! ! = 3! cos(2θ)

b. ! ! = 5! sin (2θ)

e. ! ! = 5! cos (2θ)

c. ! ! = 4 sin (2θ)

f. ! ! = 4 cos (2θ)

(You may need to solve for “r” in order to graph the equation using the website.
To solve for “r”, all you have to do is square root both sides of the equation  )

1. Notice how each equation begins with ! ! instead of ! ! or y. Does this make a
difference on how the graph looks? Sketch the graphs to draw a conclusion.
! ! = 3! sin(2θ)

! ! = 3! sin(2θ)

y= 3! sin(2θ)

Does ! ! and ! ! differ? Answer with a yes or no. Yes
Does ! ! and y differ? Answer with a yes or no. Yes
**As you probably have guessed, for the Lemniscate Curve, the equation has to
be equal to ! ! in order for it to be a true Lemniscate Graph.! The “r” term as well
as the θ are actually indicators that you are graphing in the Polar Plane versus
the rectangular plane, which is what we are used to.
2. What are some characteristics of a Lemniscate curve? Sketch it below.
Sketch of the Lemniscate Curve:

General Characteristics of a
Lemniscate Curve:
1. Looks like an infinity sign
2. Symmetric about the origin
3. Seems like the general form always
contains (2θ)

4. Can have a sine or cosine

3. The following functions are also Lemniscate Curves. By looking at the functions
and graphing the equations, can you come up with a general form for a
Lemniscate curve? Write the general form of the function below. How did you
come up with this? (Hint: the general form of a function is an equation with
variables)

a. ! ! = 9! cos (2θ)
b.!! ! = 4 sin (2θ)
c. ! ! = cos (2θ)
**Hint: In order to graph these equations on the graphing website provided, you
will need to solve for r.
The general form(s) of the function are:
1. ! !

= ! ! sin(2θ)

2. ! !

= ! ! cos(2θ)

I came to this general conclusion because: I came to this general conclusion by
looking at the patterns from above. I noticed that there was always a ! ! , 2θ, and
a sine or cosine.
There are two forms of the equation because: Of the Sine and Cosine

4. Sin(θ) and Cos(θ) and Tan(θ) oh my!
If one replaces Sin(θ) with Cos(θ), how does the graph change? Pick one of the
graphs from number three. Sketch the graph using both the “Sin (θ)” version and
the “Cos(θ)” version of that graph. How does the graph change?
Sine Graph
!

!

! = 3 sin(2θ)

Cosine Graph
!

!

! = 3 cos(2θ)

Describe the transformation occurring between the sine graph to the cosine
graph.
The graph seems to rotate 90 degrees clockwise
If one were to replace Sin(θ) or Cos(θ) with Tan(θ), does the graph still look like a
Lemniscate curve? Why are why not?
Not one bit. It actually looks like a wired x shape graph. Weird.

5. Symmetry
Is the Lemniscate Curve Symmetrical about the x-axis, y-axis, and origin? Use the
space below to show your work algebraically as well as graphically. (Hint: Pay
attention to when the function contains sine or cosine).

a. The y-axis? If so, when? The Cosine graph is
b. The x-axis? If so, when? The Cosine graph is
c. The origin? If so, when? The Sine and Cosine graph are

6. Compare and Contrast the Graphs in DIFFERENT planes
Come up with an example of an equation of a Lemniscate Curve.
•

•

•

Graph this equation in the polar plane. (Hint: Remember when we are
graphing in the Polar Plane, we are using “r” or in this case, “! ! ”. We also
use “θ” instead of “x”)
Graph the same equation in the rectangular plane. (Hint: Remember
when we are graphing in the Rectangular Plane, we are using “y”. We are
also using “x”.)
Sketch both graphs in the table below.

My equation in the Polar Plane is: ! ! = 4 sin (2θ)
My equation in the Rectangular Plane is: y = 4 sin (2x)
Polar Plane:

Rectangular Plane:

Analyze the graph in the rectangular plane. Find the…
!

!!

!

!

a. Maximum and Minimum: ( , 4) is one of the maximum and (
the minimums…The value of a determines the Range
b. Domain: All real numbers
c. Range: [-4, 4]
d. Altitude: 4

, -4) is one of

e. Period: Pi
Do these characteristics relate to any of the characteristics of the graph in the
polar plane? If so, which ones? How do they relate? Explain your answer.
Both graphs go through the origin
The length of a petal is half the Altitude

Name: ___________________________
Date: ____________________________

Polar Graph Exploration
What are some important characteristics of graphs? List as many as
you can.
Zeroes/Solutions/Intercepts
Y-intercept
Maximum (absolute/local)
Minimum (absolute/local)
Domain
Range
Symmetry
Altitude
Period
The Limacon Curve:
Use the following website to explore the different properties of a Limacon Curve.
The website is https://www.desmos.com/calculator
Type in the following functions:
a. r = 3+2sin(θ)
b. r = 6+4sin(θ)
c. r = 4+4sin(θ)

d. r = 3+2 cos (θ)
e. r = 6+4 cos (θ)
f. r = 4+4 cos (θ)

g. r = 3 - 2sin(θ)
h. r = 6 - 4 cos (θ)
i. r = 4 - 4sin(θ)

(You do not need to type all of these in…just a few)

1. What are some general characteristics of a Limacon Curve? Sketch it below.
Sketch of the Lemniscate Curve:

General Characteristics of a
Lemniscate Curve:
1. Looks Roundish

2. Can contain a plus sign

3. Can contain a minus sign

4. Can contain sine or cosine

2. Notice how each equation begins with “r” instead of “y”. Does this make a
difference on how the graph looks? Sketch the graphs to draw a conclusion.
r = 3+2sin(θ)

Do the graphs differ? Answer with yes or no. Yes

y = 3+2sin(x)

**As you probably have guessed, for the Limacon Curve, the equation has to be
equal to “r” in order for it to be a true Limacon Graph.! The “r” term as well as
the “θ” are actually indicators that you are graphing in the Polar Plane versus
the rectangular plane, which is what we are used to.
3. The Limacon Curve can be divided up into four different graph types.
a. Limacon with an inner loop-- When (
b. Cardioid-- When (

!
!

)=1

c. Dimpled Limacon-- When 1 < (
d. Convex Limacon-- When (

!
!

!
!

!
!

)<1

)<2

) ≥!!2

Sketch the graphs below and label them with the proper term above. (Hint: in
the first example, a=5 and b=4)
a. r = 5 + 4 sin (θ)
Dimpled Limacon

b. r = 7 + 7 sin (θ)
Cardioid

c. r = 8 + 2 sin (θ)
Convex

d. r = 2 + 3 sin (θ)
Inner Loop

By looking at the functions above and graphing the equations, can you come
up with a general form for a Limacon curve? Write the general form of the
equation below. How did you come up with this? (Hint: the general form of a
function is an equation with variables)
The general form(s) of the function are:
1. r = a ± b sin (θ)
2. r = a ± b cos (θ)
I came to this general conclusion because: I came to this general conclusion by
looking at the general patterns of the equations above.
There are two forms of the equation because: Of the Sine and Cosine

4. Sin(θ) and Cos(θ) and Tan(θ) oh my!
If one replaces Sin(θ) with Cos(θ), how does the graph change? Pick one of the
graphs from number three. Sketch the graph using both the “Sin (θ)” version and
the “Cos(θ)” version of that graph. How does the graph change?
Sine Graph

r = 3+2 sin (θ)

Cosine Graph

r = 3+2 cos (θ)

Describe the transformation occurring between the sine graph to the cosine
graph.
The graph seems to rotate 90 degrees clockwise.
If one were to replace Sin(θ) or Cos(θ) with Tan(θ), does the graph still look like a
Limacon curve? Why are why not?
Not one bit! I kind of looks like two weird cucumbers.

5. Minus or Plus
Pick one of the graphs from number three. Sketch the graph using first a plus sign
and then a minus sign. How does the graph change? Does it change at all?
Using a Plus Sign (+)

r = 7 + 7 sin (θ)

Using a Minus Sign (-)

r = 7 - 7 sin (θ)

How Does the Graph Change?
The Graph Reflects over the X-axis
6. Symmetry
Is the Limacon Curve Symmetrical about the x-axis, y-axis, and origin? Use the
space below to show your work algebraically as well as graphically. (Hint: Pay
attention to when the function contains sin or cos.)

a. The y-axis? If so, when? r = a ± b sin (θ)
b. The x-axis? If so, when? r = a ± b cos (θ)
c. The origin? If so, when? Never
7. Compare and Contrast the Graphs in DIFFERENT planes
Come up with an example of an equation of a Limacon Curve.
•

•

•

Graph this equation in the polar plane. (Hint: Remember when we are
graphing in the Polar Plane, we are using “r” instead of “y”. We are also
using “θ” instead of “x”)
Graph the same equation in the rectangular plane. (Hint: Remember
when we are graphing in the Rectangular Plane, we are using “y” instead
of “r” and “x” instead of “theta”)
Sketch both graphs in the table below.

My equation in the Polar Plane is: r = 2 + 3 sin (θ)
My equation in the Rectangular Plane is: y = 2 + 3 sin (x)
Polar Plane:

Rectangular Plane:

Analyze the graph in the rectangular plane. Find the…
a. Maximum and Minimum: 5 or in general form a + b
b. Domain: All real Numbers
c. Range: [a-b, a+b]

d. Altitude: 5 or a+b
e. Period: Irrelevant to the problem
Do these characteristics relate to any of the characteristics of the graph in the
polar plane? If so, which ones? How do they relate? Explain your answer.
The Maximum R Value and
Altitude Relate
The are both continuous
They are both bounded
There are no asymptotes

Name: ___________________________
Date: ____________________________

Polar Graph Exploration
What are some important characteristics of graphs? List as many as
you can.

The Rose Curve:
Use the following website to explore the different properties of a Rose Curve.
The website is https://www.desmos.com/calculator
Type in the following functions:

a. r = 2 sin(4θ)
b. r = 4 cos(3θ)
c. r = 4 sin(3θ)
1. Notice how each equation begins with “r” instead of “y”. Does this make a
difference on how the graph looks? Sketch the graphs to draw a conclusion.
r = 2 sin(4θ)

y = 2 sin(4x)

Do the graphs differ? Answer with yes or no. ___________________________________
**As you probably have guessed, for the Rose Curve, the equation has to be
equal to “r” in order for it to be a true Rose Graph.! The “r” term as well as the “θ”
are actually indicators that you are graphing in the Polar Plane versus the
rectangular plane.
2. What are some characteristics of a Rose curve? Sketch it below.
Sketch of the Rose Curve:

General Characteristics of a Rose
Curve:
1.

2.

3.

4.

3. The following functions are also Rose Curves. By looking at the functions and
graphing the equations, can you find two general forms for a rose curve? Write
the general forms of the equation below.
(Hint: The general form of a function is an equation with coefficients and
parameters written as variables.)
a. r = 5 sin (2 θ)
b. r = 7 sin (5 θ)
c. r = 8 sin (6 θ)

d. r = 5 cos (2 θ)
e. r = 7 cos (5 θ)
f. r = 8 cos (6 θ)

The general forms of the function are:
1. ___________________________________________________________________________
2. ___________________________________________________________________________

I found the general forms by: _________________________________________________
______________________________________________________________________________
______________________________________________________________________________
There are two forms of the equation because:
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
4. Number of Petals:
How can one tell how many petals a rose curve contains by just looking at an
equation?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Can one tell if the graph has an odd or even number of petals just by looking at
the equation? If so, how?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
How can one tell how long a petal is by just looking at the equation?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________

5. Sin(θ) and Cos(θ) and Tan(θ) oh my!
If one replaces Sin(θ) with Cos(θ), how does the graph change? Pick one of the
graphs from number three. Sketch the graph using both the “Sin (θ)” version and
the “Cos(θ)” version of that graph. How does the graph change?
Sine Graph

Cosine Graph

Describe the transformation occurring between the sine graph to the cosine
graph.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
If one were to replace Sin(θ) or Cos(θ) with Tan(θ), does the graph still look like a
Rose curve? Why are why not?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________

6. Symmetry
Is the Rose Curve symmetric about the x-axis, y-axis, and origin? Use the space
below to show your work algebraically as well as graphically. (Hint: When
answering these questions pay attention to when the number of petals is even
or odd. Also pay attention to when the function contains sine or cosine).

a. Symmetric about the y-axis? If so, when?
______________________________________________________________________________
______________________________________________________________________________
b. Symmetric about the x-axis? If so, when?
______________________________________________________________________________
______________________________________________________________________________
c. Symmetric about the origin? If so, when?
______________________________________________________________________________
______________________________________________________________________________

7. Compare and Contrast the Graphs in DIFFERENT planes
Provide an example of an equation of a Rose Curve.
•

•

•

Graph this equation in the polar plane. (Hint: Remember when we are
graphing in the Polar Plane, we are using “r” instead of “y”. We are also
using “θ” instead of “x”)
Graph the same equation in the rectangular plane. (Hint: Remember
when we are graphing in the Rectangular Plane, we are using “y” instead
of “r” and “x” instead of “theta”)
Sketch both graphs in the table below.

My equation in the Polar Plane is: _____________________________________________
My equation in the Rectangular Plane is: ______________________________________
Polar Plane:

Rectangular Plane:

Analyze the graph in the rectangular plane. Find the…
a. Maximum and Minimum: __________________________________________________
b. Domain: __________________________________________________________________
c. Range: ____________________________________________________________________
d. Altitude: __________________________________________________________________
e. Period: ____________________________________________________________________

Do these characteristics relate to any of the characteristics of the graph in the
polar plane? If so, which ones? How do they relate? Explain your answer.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________

Name: ___________________________
Date: ____________________________

Polar Graph Exploration
What are some important characteristics of graphs? List as many as
you can.

The Lemniscate Curve:
Use the following website to explore the different properties of a Lemniscate
Curve.
The website is https://www.desmos.com/calculator
Type in the following functions:

a. ! ! = 3! sin(2θ)

d. ! ! = 3! cos(2θ)

b. ! ! = 5! sin (2θ)

e. ! ! = 5! cos (2θ)

c. ! ! = 4 sin (2θ)

f. ! ! = 4 cos (2θ)

(You may need to solve for “r” in order to graph the equation using the website.
To solve for “r”, take the square root of both sides of the equation  )

1. Notice how each equation begins with ! ! instead of ! ! or y. Does this make a
difference on how the graph looks? Sketch the graphs to draw a conclusion.
! ! = 3! sin(2θ)

! ! = 3! sin(2θ)

y= 3! sin(2θ)

Does ! ! and ! ! differ? Answer with a yes or no. ________________________________
Does ! ! and y differ? Answer with a yes or no. _________________________________
**As you probably have guessed, for the Lemniscate Curve, the equation has to
be equal to ! ! in order for it to be a true Lemniscate Graph.! The “r” term as well
as the θ are actually indicators that you are graphing in the Polar Plane versus
the rectangular plane.
2. What are some characteristics of a Lemniscate curve? Sketch it below.
Sketch of the Lemniscate Curve:

General Characteristics of a
Lemniscate Curve:
1.

2.

3.

4.

3. The following functions are also Lemniscate Curves. By looking at the functions
and graphing the equations, can you find two general forms for a Lemniscate
curve? Write the general forms of the function below. (Hint: The general form of
a function is an equation with coefficients and parameters written as variables.)

a. ! ! = 9! cos (2θ)
b.!! ! = 4 sin (2θ)
c. ! ! = cos (2θ)
**Hint: In order to graph these equations on the graphing website provided, you
will need to solve for r.
The general forms of the function are:
1. ___________________________________________________________________________
2. ___________________________________________________________________________

I found the general forms by : _________________________________________________
______________________________________________________________________________
______________________________________________________________________________

There are two forms of the equation because:
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________

4. Sin(θ) and Cos(θ) and Tan(θ) oh my!
If one replaces Sin(θ) with Cos(θ), how does the graph change? Pick one of the
graphs from number three. Sketch the graph using both the “Sin (θ)” version and
the “Cos(θ)” version of that graph. How does the graph change?
Sine Graph

Cosine Graph

Describe the transformation occurring between the sine graph to the cosine
graph.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
If one were to replace Sin(θ) or Cos(θ) with Tan(θ), does the graph still look like a
Lemniscate curve? Why are why not?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________

5. Symmetry
Is the Lemniscate Curve Symmetrical about the x-axis, y-axis, and origin? Use the
space below to show your work algebraically as well as graphically. (Hint: Pay
attention to when the function contains sine or cosine).

a. Symmetric about the y-axis? If so, when?
______________________________________________________________________________
______________________________________________________________________________
b. Symmetric about the x-axis? If so, when?
______________________________________________________________________________
______________________________________________________________________________
c. Symmetric about the origin? If so, when?
______________________________________________________________________________
______________________________________________________________________________

6. Compare and Contrast the Graphs in DIFFERENT planes
Provide an example of an equation of a Lemniscate Curve.
•

•

•

Graph this equation in the polar plane. (Hint: Remember when we are
graphing in the Polar Plane, we are using “r” or in this case, “! ! ”. We also
use “θ” instead of “x”)
Graph the same equation in the rectangular plane. (Hint: Remember
when we are graphing in the Rectangular Plane, we are using “y”. We are
also using “x”.)
Sketch both graphs in the table below.

My equation in the Polar Plane is: _____________________________________________
My equation in the Rectangular Plane is: ______________________________________
Polar Plane:

Rectangular Plane:

Analyze the graph in the rectangular plane. Find the…
a. Maximum and Minimum: __________________________________________________
b. Domain: __________________________________________________________________
c. Range: ____________________________________________________________________
d. Altitude: __________________________________________________________________

e. Period: ____________________________________________________________________
Do these characteristics relate to any of the characteristics of the graph in the
polar plane? If so, which ones? How do they relate? Explain your answer.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________

Name: ___________________________
Date: ____________________________

Polar Graph Exploration
What are some important characteristics of graphs? List as many as
you can.

The Limacon Curve:
Use the following website to explore the different properties of a Limacon Curve.
The website is https://www.desmos.com/calculator
Type in the following functions:
a. r = 3+2sin(θ)
b. r = 6+4sin(θ)
c. r = 4+4sin(θ)

d. r = 3+2 cos (θ)
e. r = 6+4 cos (θ)
f. r = 4+4 cos (θ)

g. r = 3 - 2sin(θ)
h. r = 6 - 4 cos (θ)
i. r = 4 - 4sin(θ)

(You do not need to type all of these in…just a few)

1. What are some general characteristics of a Limacon Curve? Sketch one
below.
Sketch of the Lemniscate Curve:
General Characteristics of a
Lemniscate Curve:
1.

2.

3.

4.

2. Notice how each equation begins with “r” instead of “y”. Does this make a
difference on how the graph looks? Sketch the graphs to draw a conclusion.
r = 3+2sin(θ)

y = 3+2sin(x)

Do the graphs differ? Answer with yes or no. ___________________________________
**As you probably have guessed, for the Limacon Curve, the equation has to be
equal to “r” in order for it to be a true Limacon Graph.! The “r” term as well as
the “θ” are actually indicators that you are graphing in the Polar Plane versus
the rectangular plane.

3. The Limacon Curve can be divided up into four different graph types.
a. Limacon with an inner loop-- When (
b. Cardioid-- When (

!
!

)=1

c. Dimpled Limacon-- When 1 < (
d. Convex Limacon-- When (

!
!

!
!

!
!

)<1

)<2

) ≥!!2

Sketch the graphs below and label them with the proper term above. (Hint: In
the first example, a=5 and b=4)
a. r = 5 + 4 sin (θ)

b. r = 7 + 7 sin (θ)

c. r = 8 + 2 sin (θ)

d. r = 2 + 3 sin (θ)

By looking at the functions above and graphing the equations, can you find a
general form for a Limacon curve? Write the general forms of the equation
below. (Hint: The general form of a function is an equation with coefficients and
parameters written as variables.)
The general forms of the function are:
1. ___________________________________________________________________________
2. ___________________________________________________________________________
I found the general forms by: _________________________________________________
______________________________________________________________________________
______________________________________________________________________________
There are two forms of the equation because:
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
4. Sin(θ) and Cos(θ) and Tan(θ) oh my!
If one replaces Sin(θ) with Cos(θ), how does the graph change? Pick one of the
graphs from number three. Sketch the graph using both the “Sin (θ)” version and
the “Cos(θ)” version of that graph. How does the graph change?
Sine Graph

Cosine Graph

Describe the transformation occurring between the sine graph to the cosine
graph.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
If one were to replace Sin(θ) or Cos(θ) with Tan(θ), does the graph still look like a
Limacon curve? Why are why not?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
5. Minus or Plus
Pick one of the graphs from number three. Sketch the graph using first a plus sign
and then a minus sign. How does the graph change? Does it change at all?
Using a Plus Sign (+)

Using a Minus Sign (-)

How Does the Graph Change?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
6. Symmetry
Is the Limacon Curve Symmetrical about the x-axis, y-axis, and origin? Use the
space below to show your work algebraically as well as graphically. (Hint: Pay
attention to when the function contains sin or cos.)

a. Symmetric about the y-axis? If so, when?
______________________________________________________________________________
______________________________________________________________________________
b. Symmetric about the x-axis? If so, when?
______________________________________________________________________________
______________________________________________________________________________
c. Symmetric about the origin? If so, when?
______________________________________________________________________________
______________________________________________________________________________

7. Compare and Contrast the Graphs in DIFFERENT planes
Provide an example of an equation of a Limacon Curve.
•

•

•

Graph this equation in the polar plane. (Hint: Remember when we are
graphing in the Polar Plane, we are using “r” instead of “y”. We are also
using “θ” instead of “x”)
Graph the same equation in the rectangular plane. (Hint: Remember
when we are graphing in the Rectangular Plane, we are using “y” instead
of “r” and “x” instead of “theta”)
Sketch both graphs in the table below.

My equation in the Polar Plane is: _____________________________________________
My equation in the Rectangular Plane is: ______________________________________
Polar Plane:

Rectangular Plane:

Analyze the graph in the rectangular plane. Find the…
a. Maximum and Minimum: __________________________________________________
b. Domain: __________________________________________________________________
c. Range: ____________________________________________________________________
d. Altitude: __________________________________________________________________
e. Period: ____________________________________________________________________

Do these characteristics relate to any of the characteristics of the graph in the
polar plane? If so, which ones? How do they relate? Explain your answer.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________

Ticket Out
Two-way match up: Match the graph to its general form. Write the name of the
curve on the line.

r = a cos (n θ)
r = a sin (n θ)
Where n > 1
r = a ± b sin(θ)
r = a ± b cos (θ)
Where a > 0 and b > 0

! ! = a sin (2θ)
! ! = a cos (2θ)

What is your favorite graph of the day (Rose, Lemniscate, Limacon)? Sketch it.
Write three characteristics about it.
My favorite graph is _______________________________________________

1.
2.
3.