Circles and Triangles
Sophomore/ Junior
Geometry
Circles, Sectors, Arc Lengths
Kristen Appleby
9
14 Days
Mount Blue High School
In this unit, students will understand the applications of circles in everyday life. As well as how to
solve problems in everyday life using circles, and circles and triangles together. They will know
how to find the area of a sector by comparing it's angle to that of a circle. They will know how
circles and triangles can be interrelated when solving problems. Students will also learn about arc
lengths and how to find them on a circle. There will be pre-tests (ungraded), formative (some
graded) and summative (all graded) assessments throughout the unit to help guide the learning in
the unit. Students will be given formulas and expected to memorize some of them for any
summative test. Students will also be given the opportunity to practice the problems and learn how
to apply them to their lives.
Common Core Standards
Content: Geometry
Grade Level: High School
Domain: Circles
Cluster: Find arc lengths and areas of sectors of circles
Standards: G-C 5: Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius, and define the radian formula for the area of the sector.
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Circles and triangles can be interrelated when solving problems.
You can find the area of a sector by comparing it's angle to that of a circle.
Circles are a part of everyday life.
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
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How can you solve problems by using circles and triangles together?
Why can you find the area of a sector by using a circle?
How is a circle a part of everyday life?
 formulas such as: the area of a circle, how to find the radius of a
circle, how to find sectors, angles, inscribed angles, the central angle,
the triangle chord angle, the diameter of a circle, chord of a circle,
circumference of a circle, arc of a circle, and the sector area of a
circle.
 vocabulary such as: angle, arc length, arc, biject, center, circle,
circumference, circumscribed, concentric circle, converse, diameter,
equidistant, inscribed angle, major arc, minor arc, perpendicular,
radius, secant, sector, semicircle, tangent, and triangle.
 critical details such as: how to find the area of a circle, how to find
the area of a sector, and how circles and triangles can be used to
solve problems together.
 demonstrate that you can find the area of
a sector by using a circle.
 evaluate that circles are in everyday life.
 solve problems using circles and triangles.
 analyze the area of a sector.
 consider that circles and triangles can be
used together to solve problems.
 recognize that circles are in everyday life.
Goal: Use Comic Life to create a presentation about how circles are important using examples given
in class.
Role: Create a comic for Marvel Comics to explain the importance of circles in real life.
Audience: Marvel Comics
Situation: You’re on a quest to solve circle equations and find their usefulness
Product/Presentation: Create a comic life to show your understanding of circles by animating the
above story.
Standards (Criteria from both rubrics - product and presentation): Preparedness 25%, Stays on
Topic 20%, Enthusiasm 5%, Vocabulary 20%, Speaks Clearly 5%, Comprehension 25%,
Mathematical Concepts 30%, Explanation 10%, Diagrams and Sketches 5%, Mathematical
Terminology and Notation 25%, Completion 15%, Neatness and Organization 15%. Requirements
5%
•Educreation: create a presentation about the steps on how to find the area of a sector by using a circle,
how to solve, and examples.
•Podcast: discuss the applications of circles in everyday life and explain how to use them (like a lesson plan).
•Digital Story: Come up with examples on how to use circles and triangles.
•Google docs and ibook Author: write a textbook section about the area of a sector, use google docs to help
with discussion and ibook Author to publish.
•Adobe illustrator: locate circles and triangles in artwork.
•Comic Life: students will work individually or in pairs to create story to explain about circles in everyday life
•Pre-assessment: Survey: I would ask open ended questions about if they know what circles
and triangles are, if they know what angles are, arc lengths and sectors.
•Checking for Understanding: Fist to five, circle triangle square, entrance/exit tickets, pop it,
take and pass, and word sort.
•Self-assessment: Student, peer, teacher, timely feedback.
•Circles and triangles can be interrelated when solving
problems.
•You can find the area of a sector by comparing it's angle
to that of a circle.
•Circles are a part of everyday life.
•Circles: Derive using similarity the fact that
the length of the arc intercepted by an angle is
proportional to the radius, and define the
radian formula for the area of a sector.
•Geometry
•Circles
Marvel Comics is creating a line of comics based in education and they have asked you to create a
comic showing the importance of circles. You need to come up with a storyline explaining the steps,
formulas, theorems, graphs etc. related to at least two types of circles. The comic story line can be
using anything from superheroes to a how to comic, as long as it is child appropriate. Bring in your
expanse knowledge of geometry and circles. Marvel wants you to help students understand the
importance of circles in everyday life, what arc lengths, and sectors are. Use Comic Life to bring
your story alive and to show your understanding of circles. Once you have created your comic submit
it to Marvel. If your comic is selected you will be chosen to do a line of educational math comics for
them.
•Comic Life
•Presentation
By what criteria will student products/performances be evaluated?
•Mathematical Concepts 30%
•Explanation 10%
•Diagrams and Sketches 5%
•Mathematical Terminology and Notation 25%
•Completion 15%
•Neatness and Organization 15%
•Preparedness 25%
•Stays on Topic 20%
•Enthusiasm 5%
•Vocabulary 20%
•Speaks Clearly 5%
•Comprehension 25%
Lesson 1
Consider the W.H.E.R.E.T.O. elements. (L)
(W) 1.1 Students will understand that circles are a part of everyday life. (Where) Reallife: Circles are
everywhere in real life, over the course of this lesson, students will come up with examples, and I will open
with a couple of examples as well. (Why) Derive using similarity the fact that the length of the arc
intercepted by an angle is proportional to the radius, and define the radian formula for the area of a
sector.(What)
(H) 1.2 I will find a short video about circles in the real world, preferable a music video.
(E) 1.3 Students will know formulas such as: the area of a circle, hot to find the radius of a circle, the
diameter of a circle, and circumference of a circle. Students will know vocabulary such as: circle, triangle,
arc, arc length, radius, angle, circumference, diameter, center, circumscribed, semicircle, perpendicular,
biject, and equidistant. Students will know critical details such as how to find the area of a circle.(Equip)
The graphic organizer, spider map, will be used to help students organize and write down the types of
circles they know of in every day life. The cooperative learning model, give one, get one, will help students
share their knowledge of circles with each other and help them delve deeper into their knowledge of circles.
(Explore) Students will work individually or in pairs to create a comic life presentation about circles in
everyday life. (Experiences)
(R) 1.4 Fist to five consists of the students holding the number of fingers up to show their understanding of
the question posed by the teacher. This will help the teacher to know that the students understand about
circles in everyday life. (Rethink) Students will assess themselves and their peers using a checklist.
(Rethink/Revise) The teacher uses a rubric to assess students on their product, Comic Life. (Revise/Refine)
(E) 1.5 Formative Assessment - Pre-Assessment: The teacher would ask open ended questions about if they
know what circles and triangles are, if they know what angles, arc lengths and sectors are. Checking for
Understanding: Fist to five. Timely Feedback: peer, checklist; teacher, detailed rubric. (Evaluate)
(T) 1.6 Tailors
Verbal: Have students get into groups and discuss the types of circles after the sticky notes part. The
students will then say and/or write out their own definitions of one or more of the circle types.
Logic: Have students answer questions such as how do these circles apply to the real world? How would
you use the circles? Analyze the graphs of circles shown in class.
Visual: During the lesson, the students as a well as the teacher on the board, will have many chances to
draw graphs and examples of circles.
Musical: Either find a song that involves circles in the real world, or create a song about circles in the real
life. Find a way rhythmically to remember. circles in the real world.
Kinthestic: At the beginning of the discussion about the types of circles, have each student in each group
write two or more ideas on sticky notes and then go up to the board and have them organize them by type,
order and so on.
Intrapersnal: In the discussion the students will be reflecting on the answer to where circles are in real life
before they go into the groups. The fist to Five exercise will also help the student reflect on what they know
about circles.
Interpersonal: The students will be getting into groups to discuss circles in the real world.
Naturalist: Have the students think of examples of circles in nature. How do circles apply to nature? What
types of circles do you use when you do things outside in the nature setting?
(O) 1.7 Students will be able to recognize that circles are in everyday life (Self-Knowledge) Product:
Comic Life. Number of days: 2 (Organize)
Lesson 2
Consider the W.H.E.R.E.T.O. elements. (L)
(W) 2.1 Students will understand that circles are a part of everyday life. (Where) Reallife: Circles can be
used to solve everyday problems; students will be able to evaluate that circles are in everyday life.
(Why) Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to
the radius, and define the radian formula for the area of a sector.(What)
(H) 2.2 I will have students watch a short clip of a teacher who can draw a large circle perfectly because
he used to be part of a competition of people drawing perfect circles.
(E) 2.3 Students will know vocabulary such as: circle, triangle, arc, arc length, radius, angle,
circumference, diameter, center, circumscribed, semicircle, perpendicular, biject, and equidistant. Students
will know critical details such as how to find the area of a circle. (Equip) The graphic organizer, echart,
will be used to help students organize their thoughts on circles in real life. The cooperative learning model
Paired Heads Together, will help students share their knowledge on circles as well as expand it by sharing
their answers to discussions questions and recording them to help guide them in their podcast. (Explore)
Students will work individually or in pairs to create a podcast discussing the applications of circles in
everyday life and explain how to use them (like a lesson plan). (Experiences)
(R) 2.4 Word Sort consists of the students receiving the vocabulary words from lesson 1 and 2, and sorting
them into categories that they believe they should be sorted into. This will help the teacher to know that the
students can evaluate circles in everyday life. (Rethink) Students will assess themselves and their peers
using a checklist. Students will assess themselves based on how they thin they did personally, and peers will
give positive specific feedback to the student so the student will know what they did well. (Rethink/Revise)
The teacher will use a rubric for the podcasts, to grade the student't use of circles, the correctness of the
circles chosen and their formulas, as well as how well they podcast is done.
(Revise/Refine)
(E) 2.5 Formative Assessment - Pre-Assessment: The teacher would ask open ended questions about if they
know what circles are used in everyday life. Checking for Understanding: Word Sort. Timely Feedback:
self, peer, checklist; teacher, detailed rubric. (Evaluate)
(T) 2.6 Tailors
Verbal: The Podcast, and the discussion groups will be opportunity for verbal learners.
Logic: Have students ask and answer questions during the discussion.
Visual: The video clip in the hook, as well as the notes in class and graphs shown, and the graphic
organizer are all ways to include the visual learner.
Musical: Find a way to musically remember formulas for circles.
Intrapersonal: In the discussion the students will be reflecting on the answer to where circles are in real
life before they go into the groups. The word sort exercise will also help the student reflect on what they
know about circles, and how to solve problems using them.
Interpersonal: The discussion will be getting into groups and discussing how to use the circles and solving
problems; the podcast as well can be done in pairs.
Naturalist: Have the students work out problems related to circles in nature.
(O) 2.7 Students will be able to evaluate that circles are in everyday life.(Interpretation) Product:
Podcast. Number of days: 2 (Organize)
Lesson 3
Consider the W.H.E.R.E.T.O. elements. (L)
(W) 3.1 Students will understand that circles and triangles can be interrelated when solving problems. (Where)
Reallife: Circles and triangles are used together to find angles; there are many situations in life where
(Why) Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the
radius, and define the radian formula for the area of a sector.(What)
(H) 3.2 I will use the paper activity handout I printed out to give students a look into circles and triangles; this
is a hands-on activity.
(E) 3.3 Students will know formulas such as: the area of a circle, how to find the radius of a circle,how to find
angles, inscribed angles, the central angle, the triangle chord angle, and the arc of a circle. Students will know
vocabulary such as: circle, triangle, arc, arc length, radius, angle, circumference, diameter, tangent, converse,
secant, center, concentric circle, circumscribed, minor arc, major arc, semicircle, inscribed angle,
perpendicular, biject, and equidistant.Students will know critical details such as how to find the area of a
circle, and how circles and triangle can be used to solve problems together. (Equip) The graphic organizer,
flow chart, will be used to help the students organize their thoughts on circles and triangles being used
together. The cooperative learning exercise, Mix-Pair-Share, will be used to help the students find fresh ideas
on circles and triangles being used together. (Explore) Students will come up with examples on how to use
circles and triangles, they will then create a digital story, such as a video or comic, to show the examples and
work out the examples. (Experience)
(R) 3.4 Take an pass consists of the students writing a problem on a paper, they will pass it to the right and so
on, until they get their paper back, this way students will be able to see the ways that other students solve these
types of problems. (Rethink) Peers will use a checklist to make sure that the student met all the criteria on
fining the circles in artwork. (Rethink/Revise) The teacher will use a detailed rubric to record the number of
circles and sectors found, the legitimacy of the circles and the appropriateness of the artwork. (Revise/Refine)
(E) 3.5 Formative Assessment - Pre-Assessment: The teacher would ask open ended questions about if they
know what circles and triangles are, if they know what angles, arc lengths are. Checking for Understanding:
Take and Pass. Timely Feedback: peer, checklist; teacher, detailed rubric. (Evaluate)
(T) 3.6 Tailors
Verbal: The discussion will allow students to work through their issues in solving angles verbally.
Visual: The adobe illustrator project will give visual learners the chance to thrive, as well as notes, graphs,
and the graphic organizers.
Musical: While using the flow chart students can use it to help them remember the steps rhythmically.
Kinthestic: The paper activity will allow students to get involved in the learning physically.
Intrapersonal: The flow chart organizer will allow students to reflect on the process of working on solving
angles.
Interpersonal: The mix freeze pair exercise will allow students to discuss the material in pairs.
(O) 3.7 Students will be able to consider that circles and triangles can be used together to solve problems.
(Empathy) Product: Adobe Illustrator. Number of days: 2-3 (Organize)
Lesson 4
Consider the W.H.E.R.E.T.O. elements. (L)
(W) 4.1 Students will understand that circles and triangles can be interrelated when solving problems.
(Where) Real life: Circles and triangles can be used to solve problems such as finding the angle of a
shadow of a building so that you know where to put a picnic table. (Why) Derive using similarity the fact
that the length of the arc intercepted by an angle is proportional to the radius, and define the radian formula
for the area of a sector.(What)
(H) 4.2 I will use students to create a human graph to solve problems using circles and triangles.
(E) 4.3 Students will know formulas such as: the area of a circle, how to find the radius of a circle, angles,
inscribed angles, the central angle, the triangle chord angle, the diameter of a circle, chord of a circle,
circumference of a circle, and the arc of a circle. Students will know vocabulary such as: circle, triangle,
arc, arc length, radius, angle, circumference, diameter, tangent, converse, secant, center, concentric circle,
circumscribed, minor arc, major arc, semicircle, inscribed angle, perpendicular, biject, and equidistant.
Students will know critical details such as how to find the area of a circle, and how circles and triangle can
be used to solve problems together.(Equip) Students will use the graphic organizer, step by step chart, to
help the students organize the steps of using circles and triangles, and write a description of how to do it.
Students will use the cooperative learning model, Rally Coach, to help students help each other work on the
problems given in class for students to work on. (Explore) Students will write a textbook section about the
area of a sector, use Google docs to help with discussion and iBook Author to publish. (Experience)
(R) 4.4 Entrance tickets will be used to ask the students a couple of major questions that we have been
talking about to make sure all students understand where we are at at this point in the lesson. (Rethink)
Students will use checklists to make sure they as well as themselves met criteria such as solving problems
using circles and triangles, the number of problems shown etc. (Rethink/Revise) The teacher will use a
detailed rubric to make sure the student met the criteria such as the students did, but also items such as
accuracy. (Revise/Rethink)
(E) 4.5 Formative Assessment - Pre-Assessment: The teacher would ask open ended questions about if they
know what circles and triangles are, if they know what angles, arc lengths are. Checking for
Understanding: Entrance/Exit tickets. Timely Feedback: self, peer, checklist; teacher, detailed rubric.
(Evaluate)
(T) 4.6 Tailors
Verbal: The discussion is a chance for the student to learn verbally.
Visual: The graphic organizers, notes, graphs, and human graph are all chances for the visual learner.
Kinthestic: The human graph is a chance for students to learn whole moving.
Intrapersonal: The entrance/Exit tickets, step by step chart, and rally coach are all chances for the students
to reflect on their knowledge.
Interpersonal: The human graph, and discussion are chances for the student to learn while around others.
Naturalist: The student can ask questions like how can you slove problems like this in the real world. The
human graph is also a way for students to learn outside.
(O) 4.7 Students will be able to solve problems using circles and triangles. (Application) Product: Digital
Story. Number of days: 3-4 (Organize)
Lesson 5
Consider the W.H.E.R.E.T.O. elements. (L)
(W) 5.1 Students will understand that you can find the area of a sector by comparing it's angle to that of a
circle. (Where) Reallife: Analyzing the area of a circle is used in life to find answers to questions such as
how much room with this rug take up? (Why) Derive using similarity the fact that the length of the arc
intercepted by an angle is proportional to the radius, and define the radian formula for the area of a
sector.(What)
(H) 5.2 I will allow students to explore the
website http://www.emaths.co.uk/tutorials/geometrytoolbox.swf for a few minutes and then they will be
allowed to use this website if desired to help aid finding the area of a sector.
(E) 5.3 Students will know formulas such as: the area of a circle, how to find the radius of a circle, how to
find sectors, angles, inscribed angles, the central angle, the triangle chord angle, the diameter of a circle,
chord of a circle, circumference of a circle, arc of a circle, and the sector area of a circle. Students will
know vocabulary such as: circle, triangle, arc, arc length, radius, angle, sector, circumference, diameter,
tangent, converse, secant, center, concentric circle, circumscribed, minor arc, major arc, semicircle,
inscribed angle, perpendicular, biject, and equidistant. Students will know critical details such as how to
find the area of a circle, how to find the area of a sector, and how circles and triangle can be used to solve
problems together. (Equip) Students will use the graphic organizer, tic tac toe, to help students organize
their thoughts for the product; they will also use it for organizing the steps of finding the area of a sector.
(Explore) Students will use Adobe Illustrator to identify circles, as well as sectors in artwork.
(Experience)
(R) 5.4 Circle, triangle, square will be used to help understand what the students know, don't know or are
unsure of, so we can help them move ahead. (Rethink) The student will assess themselves using a checklist
to make sure that they did everything they should have in their google docs char. (Rethink/Revise) The
teacher will use a detailed rubric to assess the student's chapter section to make sure they did it all, that
they included the items that were supposed to be in their section, and how well they did. (Revise/Rethink)
(E) 5.5 Formative Assessment - Pre-Assessment: The teacher would ask open ended questions about if
they know what circles and triangles are, if they know what angles, arc lengths are. Checking for
Understanding: Circle, Triangle, Square. Timely Feedback: self, checklist; teacher, detailed rubric.
(Evaluate)
(T) 5.6 Tailors
Verbal: The discussion, tool box exploration, and general class notes , are an opportunity for students to
learn verbally.
Logic:The discussion is a chance for logical learners to ask questions such as how do we use sectors in
real life, how do we use arc lengths in real life?
Visual: The graphic organizer, notes and graphs are all opprotunites for the students to learn visually.
Kinthestic: The tool box website is a chance for the studemt to learn hands on.
Intrapersonal: The toolbox website exploration, the tic tac toe organizer, and the pair discussion are
opportunities for the student to reflect on their learning.
Interpersonal: Pair discussion is an oppritunity for the student to learn with as pairs.
(O) 5.7 Students will be able to analyze the area of a sector. (Perspective) Product: Google Docs. Number
of days: 3-4 (Organize)
Lesson 6
Consider the W.H.E.R.E.T.O. elements. (L)
(W) 6.1 Students will understand that you can find the area of a sector by comparing it's angle to that of a
circle. (Where) Reallife: One way that finding the area of a sector will help you in real life is by finding the
area that an irrigation pipe can irrigate a feild, so that you know how many you need for your field.
(Why) Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the
radius, and define the radian formula for the area of a sector.(What)
(H) 6.2 Students will explore http://www.sketchometry.com/ for a few minutes, and they will be able to use
this as a tool to help students in solving sector problems.
(E) 6.3 Students will know formulas such as: the area of a circle, hot to find the radius of a circle, how to find
sectors, angles, inscribed angles, the central angle, the triangle chord angle, the diameter of a circle, chord
of a circle, circumference of a circle, arc of a circle, and the sector area of a circle. Students will know
vocabulary such as: circle, triangle, arc, arc length, radius, angle, sector, circumference, diameter, tangent,
converse, secant, center, concentric circle, circumscribed, minor arc, major arc, semicircle, inscribed angle,
perpendicular, biject, and equidistant. Students will know critical details such as how to find the area of a
circle, how to find the area of a sector, and how circles and triangle can be used to solve problems together.
(Equip) Students will use the graphic organizer, Sequence Chart, to help them organize the steps of solving
the area of a circle, they could also use it to help them with their product. (Explore) Students will use
Educreation app to tell a story about the steps on how to find the area of a sector by using a circle, how to
solve, and examples. (Experience)
(R) 6.4 Students will use pop-it to track their progress in the class. (Rethink) Students will use checklists to
make sure they did everything they were supposed to in their Educreation product, such items such as,
checking to make sure all problems and examples are correct and look professional. (Rethink/Revise) The
teacher will use a rubric to grade the final project, making sure all items are correct and professional, and
that the number of problems is sufficient. (Revise/Refine)
(E) 6.5 Formative Assessment - Pre-Assessment: The teacher would ask open ended questions about if they
know what circles and triangles are, if they know what angles, arc lengths and sectors are. Checking for
Understanding: Pop it. Timely Feedback: self, checklist; teacher, detailed rubric. (Evaluate)
(T) 6.6 Tailors
Verbal: Circle the sage is a chance for verbal learners to learn.
Visual: The sequence chart, and the sketchometer website are both chances for visual students to learn.
Musical: Students can use the sequence chart to help them learn the steps musically.
Kinthestic: The pop it exercise and the sketchometer website are both chances for students to learn handson, as well as circles the sage.
Intrapersonal: The sketchometer exercise is a chance for the students to reflect on their knowledge and to
expand it.
Interpersonal: Circle the sage is a chance for students to help each other learn in a discussion/group
atmosphere.
(O) 6.7 Students will be able to demonstrate that you can find the area of a sector by using a circle.
(Explanation) Product: Educreation. Number of days: 3-5 (Organize)