Final Draft of Unit Plan

8/9/2019 Final Draft of Unit Plan

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“The Circle of Life”

A mathematical unit about circles

Karly SachsProfessor Hodgkinson

Education 132

Unit Plan assignment


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Karly Sachs

April 23, 2014

Professor Hodgkinson

Unit Plan Assignment

Unit Plan: The Circle of Life

“The Circle of Life” is a high school geometry unit on Circles. This unit should last

about four weeks and covers a wide range of topics about circles. “The Circle of Life”

covers all of the Common Core State Standards in the Circle Section of the High School

Geometry Portion of the Iowa Core. Some of the topics that will be covered in this unit

include estimating the area and perimeter of a random curved figure, deriving the

formulas for the area and circumference of a circle using the formula for the area of a

regular n-gon, calculating the sector area and arc length of different parts of a circle,

identifying and describing relationships among inscribed angles, central angles, radii,

diameters, chords, tangent lines and secant lines, and inscribing and circumscribing

geometric shapes amongst other shapes to prove different properties of angles.

The purpose of this unit is simple. “The Circle of Life” unit is suppose to teach

students all of the information in the circle section of the Common Core State

Standards so students will do well on district, state, and national assessments. Another

purpose of this unit is to get students to think critically and enhance their problem

solving skills. I wish I could distinctly say that this unit will relate to most students’ every

day lives, but in reality it will not. Besides being asked in a high school geometry class or

on a state or national assessment, students will rarely need to solve for an arc or chord

measure. One particular topic that does relate to this unit, though, is landscape design.

During this unit I hope to relate the topics we learn to real world examples such as


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landscape design as well as have students come up with ways they can use this

information. The one aspect I really enjoy about this unit is the fact that this unit does

enhance a student’s ability to think critically because students are asked to prove

different theorems, derive different formulas, and determine the relationships between

different formulas as well as different parts of a circle. Problem solving and critical

thinking skills are important to have in today’s job market and in order for students to

acquire these skills they must practice using them. This is what this unit is all about!

The learning goals and learning objectives for this unit go hand in hand with the

Common Core State Standards on circles in the high school geometry portion of the

Common Core. Students will be able to approximate the area of a curved figure, find

the area of different regular n-gons, derive the area and circumference formulas of a

circle, find the exact area and circumference of a circle using the formulas they

derived, and they will be able to prove that all circles are similar by definition. Students

will also be able to find the arc length and sector area of certain sectors of a circle,

identify and name all the different parts of a circle, inscribe and circumscribe angles

and other shapes about a circle, and prove a handful of theorems relating to circles,

chords, arcs, central angles, inscribed angles, tangent lines, and secant lines.

As one can probably see, there is a lot of information that will be covered in this

unit, meaning I must be on my game when it comes to monitoring what students

actually learn during each lesson. Just because I teach something does not mean

students will learn it right away. Sometimes I will need to reteach or review past material

to enhance student learning. Some ways I will monitor student learning are by using pre,

formative, and summative assessments. Some assessments will be informal while others

will be formal. In “The Circle of Life” unit I will have one informal pre assessment (in class


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quiz), three formal formative quizzes, three formal summative assessments (homework

quiz, study guide, unit test), and a lot of different versions of informal formative

assessments. Some examples of informal formative assessments include journal entries,

self-assessment slips, homework checks, openers, and exit slips. There are a lot more

informal formative assessment techniques I will use in this unit. They are thoroughly

described in stages two and three of the Unit Plan.

The major assessment in the “The Circle of Life” unit is the Unit Test at the end of

the twenty days. This assessment will cover everything we learned in this unit as well as

give students the opportunity to think critically, problem solve, and practice their writing

skills. The questions will range from level one questions to level three questions and the

points granted for each problem will reflect such levels. Students will be given TWO fifty-

minute class periods to complete the unit test. The first fifty-minute period will be used

for the first part of the test and the second fifty-minute class period will be used for the

second part of the test. The first part of the test will have true/false, multiple choice, and

short answer problems. The second part of the test will be a writing portion where

students will be asked to answer TWO of the following three questions. The writing

portion of the unit test is geared to assess students’ problem solving skills, critical thinking

skills, and writing skills as well as assess certain standards of the Common Core. Each

writing prompt will assess writing skills, problem-solving skills, and critical thinking skills but

they will differ in what standards they cover. The topics students can choose from are 1)

arc length and sector area 2) approximating area and 3) relationships between radii,

chords, and tangent lines. If students need more time, they can turn in what they

finished and complete the rest during lunch, before school, or after school. I will also be


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Unit Title: The Circle of Life

Conceptual Lens: Relationships

Grade level: High School Geometry

Length of unit: about 4 weeks (20 days of this Lesson)Stage 1 – Desired Results

Meaning

Enduring Understandings/Generalizations:

•

The area and perimeter formulas of non-curvedfigures can be used to estimate the area and

perimeter of curved figures.

• The area and circumference formulas for a

circle were derived from formulas of non-curved

figures.

• The length of part of a circle’s circumference

can be found by relating it to an angle in the

circle.

• The area of parts of a circle formed by the radii

and arcs of a circle can be found when the

radius and or central angle is known.

• A radius of a circle and the tangent that

intersects the endpoint of that radius on the

circle has a special relationship.• Angles formed by intersecting lines have a

special relationship to the arcs the intersecting

lines intersect. This includes (a) arcs formed by

chords that inscribe angles (b) angles and arcs

formed by lines intersecting either within or

outside of a circle and (c) intersecting chords,

intersecting secant lines, or a secant line that

intersects a tangent line

Essential Questions:

•

How can you find the area and perimeter of a curvedfigure?

• How can you use other area and perimeter formulas to

find the area and circumference of a circle?

• How can you prove relationships between angles and

arcs in a circle?

• When lines intersect a circle or within a circle, how do

you find the resulting measures of angles, arcs, and

segments?

Some of the enduring understandings and essential

questions were formed using this site:

http://images.pcmac.org/SiSFiles/Schools/CT/

MarineScienceMagnet/MSMHS/Uploads/

DocumentsCategories/Documents/Unit%2010%20

Circles.pdf


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Knowledge & Skills Acquisition

Learning Goals from the Common Core:

Understand and apply theorems about circles

• CCSS.MATH.CONTENT.HSG.C.A.1: Prove that all circles are similar.

• CCSS.MATH.CONTENT.HSG.C.A.2: Identify and describe relationships among inscribed angles, radii, and

chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a

diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the

circle.

• CCSS.MATH.CONTENT.HSG.C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove

properties of angles for a quadrilateral inscribed in a circle.• CCSS.MATH.CONTENT.HSG.C.A.4: Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

• CCSS.MATH.CONTENT.HSG.C.B.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 measure of the angle as the constant of

proportionality; derive the formula for the area of a sector.

Students will know…

• That the area is the amount of space in a 2D

shape.

• That the Circumference is the perimeter of a

circle.

•

The history of Pi• That the area of a circle = (Pi)*r^2

• That the circumference of a circle = 2*(Pi)*r

• That the area of a regular n-gon = ! * P * a

• How to approximate area and circumference.

• The definitions of apothem, circumscribe,

inscribe, arc length, sector area, central angle,

inscribed angle, circumscribed angle, arc,

chord, diameter, radius, major arc, minor arc,

Students will be able to…

Students will…

• Approximate the area of a curved figure.

• Find the area of different regular n-gons.

•

Derive the formulas for the Area and Circumferenceof a circle.

• Find the exact area and circumference of a circle

using the formulas they derived.

• Prove all circles are similar.

• Find the sector area and arc length of a circle using

the Area and Circumference formulas, proportions,

and other Theorems.

• Identify and name the parts of a circle.


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tangent and secant • Inscribe and circumscribe angles and other shapes

about a circle. They will then prove angle properties

and other theorems for said shapes and angles

• Prove certain theorems related to circles.

Resources/Materials:

• CME Project Geometry Textbook

•

Common Core State Standards

• GeoGebra (an online mathematics tool that can be used for construction)

Specific Recourses/Materials for the Students:

• Post It Notes of different colors

• Note Cards

• Compass

• Protractor

• String

• Cans

• Colored Pencils

• Dixie cups for drawing circles easily in notebooks

Specific Guided Activities, Worksheets, and Handouts:• “The Area and Perimeter of a Blob” guided activity

• “Answer Key to Homework Problems” handout

• “Investigating Pi” guided activity

• “Sector Area and Arc Length” guided activity

• “Sector Area and Arc Length: Lets think a little deeper” additional worksheet

• “Use what you know” activity

• “Jot Chart” quiz

• “Area and Perimeter Extra Credit” assignment


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• “Chords and Inscribed Angles” mini note booklet

• “Part 1 Quiz”

• “Tangents and Secants” mini note booklet

• “Review for Quiz 2” problem packet

• “Review for Quiz 2” answer key

• “Part 2 Quiz”

• “Chords, Secant Lines, and Tangent Lines Jot Chart” handout

• “GeoGebra: Chords, Arcs, Central Angles, Inscribed Angles, Secants, and Tangents Activity”

handout

• “Study Guide: Circle of Life Unit” homework assignment

•

Chapter Five Unit Test: “Circle of Life”• Chapter Five Unit Test (Writing Portion): “Circle of Life”

Premade Notecard(s) of the Day:

• “Area and Perimeter or a regular n-gon”

• “Area an Circumference of a circle”

• “Sector Area and Arc Length”


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Stage 2 – Evidence (Assessment)

Types of assessment: Selected-Response (tests, quizzes); Personal Communication (interview, oral exam, discussion);

Written Response (short constructed response questions, entrance/exit slips, essays); Performance Assessment (role-play,

Simulation, labs, dramatization)

Pre-assessment:

• I will use a pre-assessment for this unit, but it will be pretty informal. The pre-assessment activity is going to be a

short in class quiz. The in class quiz will be graded for completion only. This assessment will be given on the same

day as the previous unit’s homework quiz. This is approximately three days before the “Circle of Life” unit starts. I

will be giving it this early so I will have enough time to examine the results and figure out adjustments that need to

be made. The in class quiz will consist of twenty true/false questions dealing with new vocab and new theoremsthat will be discussed and discovered in the “Circle of Life” unit. There will be three answer choices given for the

True/False quiz though. The choices will be 1) True 2) False 3) I do not know. I will ask the students to be honest

and to not guess on the quiz. If they do not know the answer they should mark “I don’t know”. Some questions

might look like:

o 1) All circles are similar.

o 2) A central angle is half the measure of an inscribed angle.

o 3) An apothem is used to find the area of a regular n-gon.

o 4) Pi is a transcendental number that it used to find the area of a circle.

o Some vocabulary words that may be turned into true/false/I don’t know questions include:! Apothem

! Circumscribe

!

Inscribe

!

Arc length

! Central Angle

!

Sector Area

!

Inscribed angle

! Circumscribed angle

! Arc

! Chord

!

Diameter

! Radius

! Major Arc

!

Minor Arc

!

Tangent line


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!

Secant line

! Pi

! Circle

• After students take the quiz I will score them and examine the responses. I will make side notes to myself if

students know more information than anticipated. In fact, if students put “I do not know” for all of the questions, it

will be ok! If a student knows a lot more than expected, I will differentiate my lessons and activities to fit their

needs.

• Some ways I can differentiate my lessons and activities are byo

Giving advanced students an “independent study” to focus on during part of or for the whole “Circle of

Life” unit. A few examples of this might look like…

! If a student knows a lot of the theorems that will be introduced in this unit, instead of forcing the

student to sit through a lecture where they already know the information, during this time I will askthem to think more deeply on the theorems by having them try to prove them.

! If a student knows how to find the area of a sector or the length of an arc, the day this topic is

taught, I can make a different activity where the student will be asked to relate this topic to a real

life scenario.

o Depending on the student as well as the topic they already know, different independent studies or

different mini independent studies will have to be made. In order to actually know what activities I have

to make, though, I will need to score and examine all the pre assessments and go from there.

• Some other forms of pre assessment I will use during this unit are…

o Thumbs up, Thumbs down

! Who has heard of this topic before?

o Fist to Five

Formative Assessment:

• Some ways I will assess student learning along the way are by using informal as well as formal formative

assessments.

• For informal assessments I will use:

o Openers (Opening activities):

! KWL (the K and W part)

! Notecard(s) of the Day

! Sentence Starters


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! Question of the Day

o Fist to Five

o Thumbs up, thumbs down

o Think, Pair, Share

o Exit Slips (Closing Activities)

! 321 activity

! Tape sticky note to the door

! KWL (the L part)

! About—Point.

! Jot Chart

! Notecard(s) of the Day ! Sentence Starters

! Circle the Face

o In class observationso

Self assessments

! How are you feeling about the material?

! To what degree are you participating? (Taking notes? Asking Questions? Participating in small

group discussions? Participating in large group discussions? Staying on task? Responding to

thumbs up thumbs down and fist to five? Being prepared?)

! Are you keeping up with the homework?

o Basic journal entries

! Reflect on what you learned today?

! What questions do you still have about today’s lesson?

! How are you feeling about the material so far?

!

Think of a way this could be used in real life.o Homework Checks

! Homework Policy:

• Most nights, students will be asked to complete homework. Some days I will check the

homework at the door, but other times I will not. I will let the students know when homework

will be checked though.

• Because time is limited, there is not a whole lot of time to go over homework problems in

class if students have questions; this is what the classroom blog is for. If students have


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homework questions they can post them on the classroom blog and either me or other

classmates can answer the questions. If many students are having the same question,

though, I will go over that problem in class.

• Students will know if they are struggling because a) they cannot solve the problem or b)

they checked their answers on the answer key and it was wrong.

• Answer keys will be provided at the beginning of the unit!

o Little Games

! White board game

! Swat the fly

o Learning activities (graded for completion)

•

For formal assessments I will use:o Quizzes

! Jot chart quiz on the different formulas

! Two other quizzes on material

• Quiz One: Area

• Quiz Two: Chords, Tangents, Secantso

Study Guide for Unit Test: Will be graded on correctness!

• Based on the results on the informal and formal formative assessments I will adjust my lessons accordingly.

o If students seem to be grasping an idea more quickly than I anticipated, I will go over that topic a little

less by going over fewer examples.

o Conversely, if students seem to be having a lot of trouble with a topic, I will have to adjust my schedule

and allow more time for this topic than anticipated.

o If most students seem to be understanding a topic but it is clear that a few students are not I will have to

adjust my lesson as well. To better the understandings of the students who are confused, I can…• Have those students come in for help: this would be the ideal scenario

• Make a group learning activity and group the students off so a confused student is with students

who understand the topic. Maybe the students who understand the topic can explain the

material in a way the confused student will understand.

• If there is time for students to work on their homework, have the confused students get help during

that time.o

If most students seem to be having a hard time with a topic but some students seem to be


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understanding the topic I will have to adjust my lesson and allow more time for more of an explanation

on the confused topic. I do not want the students who understand the topic, though, to zone out or to

feel neglected so I will make a supplemental activity for them to work on during the “review” time.

Summative Assessment:

• I will use two different summative assessments for this unit

o Homework Quiz: The homework quiz will check for two things 1) If students did their homework and 2) if

students did their homework correctly. On the quiz, ten or so problems will be selected from the list ofhomework problems students were asked to complete during the entire unit. They will be asked either to

write down the answer to a certain problem or to write down the answer as well as their work in solving

the problem. This quiz is purely copying homework. If students did their homework, checked their

answers on the answer key, and asked questions in class and or came in for help on questions they were

confused about, this quiz should be an easy A. The goal of this quiz is to make sure students are

completing the homework assignments as well as trying their hardest on them. This quiz will be given two

days before the unit test. (This Quiz is more of a formative assessment, but since I am waiting until the

end of the unit for students to complete it, I put it in this section.)

o Unit Test: During the unit test I will have to assess student understanding of each of the State Standards

dealing with circles. On the unit test I will make a variety of questions that incorporate the main ideas of

the essential questions and the general understandings of this unit. I will also assess the skills acquired in

this unit. Since the essential questions, general understanding, and acquired skills are based off of the

Standards, if these three areas are covered, the standards will be too.

•

The Unit Test will consist of two parts; one part will be done Wednesday and one part will bedone on Thursday. The first part of the unit test will consist of true/false, multiple choice, and

short answer questions. The second part of the unit test will consist of a writing portion where

students will have to demonstrate the knowledge they acquired from the unit, critically think,

problem solve, explain their thinking, and write. Writing skills are very important to have so this is

a good way for students to practice writing in this specific content area.


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should take about 10 minutes)

2. Introduce the topic for the

day with an “I can”

statement. The “I can”

statement will be “I can use

the box method to solve for

the area and perimeter of a

random blob.” (1 Minute)

3. Pass out and explain “The

Area and Perimeter of a Blob”

guided activity. This activity

will guide the students on how

to find the area and

perimeter of a random blob. It

will teach the students the

“box” method and will show

them that as the boxes get

smaller, their estimations will

become more accurate. In

the guided activity the terms

limit, circumscribe, inscribe,

and linear approximation will

be explained. This introduces

the idea of calculus. " (25 to

30 minute activity)

4. As the students work, the

teacher will walk around the

room and observe the

students. The teacher will

chime in when needed.

5. The last ten minutes of class

will be used to discuss the

overall findings of the activity.

It will also be used to

introduce the terms limit,

inscribe, circumscribe, and

linear approximation. A

“thumbs up, thumbs down”

activity will be done to assess

2. Answer any

homework questions

that most students are

struggling with.

3. Discuss the “I can”

statement for the day.

The “I can” statement

will be “I can find the

area and perimeter of a

regular n-gon by using

the apothem and the

formula A=1/2Pa”. (1

minute)

4. Today will be a

lecture/note taking day.

The lecture will a) first

show students how to

break up a regular n-

gon to find its area b)

define the term

apothem and c) explain

how to use the formula

A=1/2Pa. (10 minutes)

5. After the lecture I will

use the gradual release

method to solve some

practice problems. I will

first show them how to

solve one, we will then

solve one together,

students will then work

together on solving one,

and then students will

solve one alone. As

students are working I

will go around and

observe/answer any

questions that students

have. (These problems

gon about a circle to

show the relationship.

(10-12 minutes)

The actual formulas:

4. Mini lecture/note

taking on the area and

circumference formulas

of a circle (10 minutes)

5. With a partner or by

yourself solve the two

area and circumference

problems that are on the

board. You have 7

minutes to complete

them and we will go

over them as a class

soon after for 5 minutes.

Do a fist to five at the

end to see the general

consensus of

understanding.

6. Notecard of the Day:

Copy down the “Area

and Circumference”

Notecard that is on the

Elmo. (7 minutes)

7. For Homework:

Students will complete a

problem set in the CME

Project Geometry

Textbook. They will be

asked to complete 1, 2,

3, 4, 5, 6, and 7 on

pages 370-372. These

problems relate to

inscribing and

the day with an “I can”


statement of the day will

be “I can describe and

derive the

transcendental number

Pi.” (2 minutes)

4. “Investigating Pi”

guided activity. This

activity will show

students how we know Pi

is 3.14 by doing a

GeoGebra activity to

discover different

patterns. Students will

then have to fill out a

table that shows these

patterns (30 minutes). If

students happen to finish

early they will be asked

to fill out another chart

to discover another

pattern. The pattern,

here, has to do with

A=1/2Cr and how area,

radius, and

circumference all relate.

5. Go over the area and

circumference formulas

of a circle again. Have

the students repeat the

formulas so they will

remember them (3

minutes).

6. Open your journals to

your KWL chart from

earlier. Fill out the L

column at this time (5

minutes)


be “Given a central

angle measure, I can

calculate the arc length

and sector area of a that

part of the circle.” (2

minutes)

4. Explain the activity for

the day. Put an

instruction sheet on the

Elmo so students know

exactly what to do. (3

minutes)

5. Do “thumbs up,

thumbs down” to make

sure everyone knows

what is expected of

them.

7. Do the “Sector Area

and Arc Length” heavily

guided activity. This

activity has students

deriving and

investigating the formulas

for Arc Length and

Sector Area. (20 minutes)

8. If students finish early

have them complete the

“Sector Area and Arc

Length: Lets Think a Litter

deeper” handout. These

questions should

enhance their learning

on a deeper level by

asking them to generalize

what they have learned

from the previous activity.

9. Large Group discussion

on the students’ findings.


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the overall knowledge for the

day.

6. For Homework: Students will

complete a problem set in

the CME Project Geometry

Textbook. They will be asked

to complete 2a, 2c, 5, 6, 7, 8,

9, and 10, and 11 on pages

362-364. (These problems all

relate to finding an

approximation of the area

and perimeter of a blob, using

the “box” method, and

inscribing and circumscribing

shapes. An answer key will be

given TODAY so students can

check their work and get

immediate feedback if they

were right or wrong.

are pretty lengthy so this

should take about 25

minutes. If students finish

early they will be asked

to answer the following

question: how can we

use the idea of inscribing

and circumscribing

regular n-gons to derive

the formulas for the area

and perimeter of a

circle?)

6. Sentence starters: For

the last 5 minutes of

class students will be

asked to fill out the

following sentence

starters to hand in

before they leave a) the

area formula or a

regular n-gon is…and b)

I am still confused

about…

7. For Homework:

Answer the following

questions a) how does

the perimeter of a

regular n-gon relate to

the perimeter of a

circle? b) How does the

area of a regular n-gon

relate to the area of a

circle? You must have

these questions

answered before you

walk into class

tomorrow!

circumscribing regular n-

gons about a circle to

find certain

measurements given the

radius of a circle.

7. For Homework: Using

the Area and

Circumference formulas

of a circle, solve for the

missing parts of each

diagram a) Find the

area and circumference

of a circle with a radius

of 2cm and b) Find the

radius and

circumference of a

circle with an area of

36*Pi. For both questions,

leave you answer in

terms of pi.

“What did you learn?

What did you notice?”

(10 minutes)

10. Notecard of the Day:

Get out a notecard and

right the following

information down on it.

Remember, sometimes

you can use notecards

on quizzes, so I suggest

you write this down. The

notecard topic for today

will be finding the Sector

Area and Arc Length of

part of the circle. Copy

the notecard I made that

is on the Elmo. When you

are done put your

notecard in your recipe

box. (5 minutes)

11. For Homework:


mini problem set in the

CME Project Geometry


asked to complete

number 2 and 3 on page

379. These problems ask

students to find the

sector area and arc

length of a circle given

the central angle

measure. It is a shorter

version on what we did in

class today.

12. For Homework:

Complete the self-


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assessment handout for

class on Monday. The

self-assessment handout

will cover a) how they

are feeling about the

material thus far? b) To

what degree are they

participating (taking

notes, asking questions,

participating in small

group discussions,

participating in large

group discussions, staying

on task, responding to

thumbs up thumbs down

and fist to fives, being

prepared, keeping up

with the notecard recipe

boxes) and c) Are they

keeping up with the

homework?

The self-evaluation form

will be neatly written out

and very easy to follow.

Using What we Know:

1. Review to Self: Look over

your “Area and Perimeter”

notecards independently.

Just in case you do not know

which ones to look over, I will

tell you. Look over your

square, rectangle,

parallelogram, triangle,

trapezoid, regular n-gon,

circle, and sector area/arc

length area and perimeter

note cards. Do this for ten

1. Study Time: Use the

first five minutes of class

to review your notecards

and class notes for the

quiz (5 minutes).

2. Jot Chart Quiz: This

quiz will be on the area

and perimeter formulas

for all the shapes we

have learned so far.

Students will be asked to

fill in the empty blanks of

the chart. Notecards will

not be used on this quiz!

1. Homework Check at

the door (2 minutes)

2. Go over the

homework problem with

your tablemates. When

the timer goes off, face

forward. (8 minutes)

3. Large group

discussion on how to

solve the proof. (10

minutes)

4. Discuss some other

facts that are related to

this Theorem such as a) if

1. Turn in Extra Credit

Assignment that was

assigned on Tuesday (3

to 5 minutes)

Applying the Theorems

to solve for parts:

2. Introduce the topic of




be “I can use the

theorems provided to

1. Study Time: Use the first

five minutes of class to

review your notecards


quiz (5 minutes).

2. The next five to ten

minutes will be used for

questions. Students will

be able to ask the

teacher questions about

the material and or the

quiz in general (5 to 10

minutes)

3. Pass out “Part 1 Quiz”


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shape and use what they

know about area and

perimeter to solve the

problems) (35 minutes).

4. As the students work, the

teacher will observe, help,

and answer any questions

students may have. The

teacher will clear up common

misconceptions as the

students work.

5. Hand in one set of

problems per group

6. 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.

7. For Homework: Study for

your “Jot Chart” quiz

tomorrow. The quiz will be on

all the area and perimeter

formulas for the shapes

mentioned in part one

(square, rectangle,

parallelogram, triangle,

trapezoid, regular n-gon,

circle, and sector area/arc

length). On this quiz you

cannot use any notecards, so

study!

and their definitions,

draw a circle and draw

and label two parts we

learned today, and write

one question you have

from today’s lesson. (10

minutes)

7. For homework: Prove

the following: two

chords are congruent if

and only if their

corresponding arcs are

congruent. Remember

that this is an “if and only

if” statement so you

have to prove it both

ways. There will be a

homework check

tomorrow!

inscribed angle is equal

to half the measure of its

intercepted arc d)

inscribed angles are

congruent if and only if

they intercept the same

arc or congruent arcs e)

any triangle inscribed in

a semicircle is a right

triangle f) for a given

point and circle, the

products of the lengths

of the two segments

from the point to the

circle is constant along

any line through the

point and circle

(a*b=c*d) g) the

opposite angles of a

quadrilateral inscribed in

a circle are

supplementary g) (As I

go over each theorem,

a thumbs up, thumbs

down activity will be

done to see if more

explanation needs to be

given.) (20 minutes)

7. Because there are so

many Theorems in this

section, a note sheet (in

the form of a mini

booklet) will be provided

with diagrams and the

theorems.

8. Sentence Starters: The

last ten or so minutes of

class students will be

asked to fill out the

Students will be able to

use ALL of their THEOREM

notecards they have

created in the past few

days on this quiz.

Notecards will be

checked before the quiz

to make sure students

are not cheating. (10

minutes)

6. The next five or so

minutes students will be

able to ask questions

about the quiz (5

minutes)

7. Tape a Sticky to the

Door: The last five

minutes of class students

will answer the following

question on their sticky

note. On their way out

they will stick their note

to the door. The question

will be a) how confident

are you about the quiz

tomorrow? Why?

(Depending on the

responses, the quiz date

may change) (5

minutes)

8. For Homework: Study

for the “Part 1 Quiz”

tomorrow!

prepared, keeping up



keeping up with the

homework?





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following sentence

starters to turn in before

they leave. They include

a) An inscribed angle

is… b) Any triangle

inscribed in a semicircle

is… and c) a central

angle is _______ an

inscribed angle.

9. For Homework: Write

down the following

Theorems we learned in

class today on your

notecards. Remember,

one theorem per card

and use pictures to help

you!

1. Go over Part 1 Quiz (15 to

20 minutes)

Tangent Lines and Secant

Lines:

2. Think, Pair, Share to the

questions “What is a secant

line? What is a tangent line?

Compare and contrast the

two.” 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

Chords, Inscribed

Angles, Secants and

Tangents

1. Answer any

homework questions

that most students are

struggling with. (5

minutes)

2. Question of the Day:

Answer the following

two questions on the

Elmo. These questions all

relate to secants and

tangents and solving for

parts. Students will have

7 to 10 minutes to solve

these problems. They

may work with their

tablemates. After 7 to 10

Review for Quiz 2



2. Go over the

homework problems

with your tablemates.

When the timer goes off,

face forward. (8

minutes)

3. There will be a quiz

tomorrow so use the

next ten minutes to go

over what is going to be

on the quiz. The quiz will

cover a) inscribed

angles b) central angles

c) chords d) secant lines

e) tangent lines f) secant

angles and g) all the

Quiz 2 Today:

1. Study Time: Use the

first five minutes of class

to review your notecards


quiz (5 minutes).

2. The next five to ten

minutes will be used for

questions. Students will

be able to ask the

teacher questions about

the material and or the

quiz in general (5 to 10

minutes)

3. Pass out “Part 2 Quiz”

4. Go over the

instructions (3 minutes)

5. The rest of the class

period will be devoted

GeoGebra Construction

Day:

1. Chords, Secant Lines,

and Tangent Lines Jot

Chart: As students walk in

the door they will be

asked to pick up and

complete the jot chart

handout on the back

table. The Jot chart will

be already made; all

students will have to do is

fill in the blanks. The jot

chart will cover all the

information that was on

Quiz Two. Students will be

asked to not use their

notecards or notes for

this informal formative


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be ready to share your pink

post it note to the class. (This

activity in general should take

about 10 to 15 minutes)

3. Introduce the topic of the

day with an “I can”


statement of the day will be “I

can use the concept of

tangents and secants to solve

for parts of a circle.” (2

minutes)

4. The rest of the class period

will be a lecture/note-taking

day. In the lecture I will draw,

identify, and define the

following terms a) secant line

b) tangent line c) secant

angle Students will draw,

label, and write as I go.

Colored pencils will be

provided. After we go over

the important terms for the

day we will discuss the

following theorems: a) If a line

intersects a circle in one point

(that is, the line is tangent to

the circle), it is perpendicular

to the radius drawn to the

point of contact b) A secant

angle with vertex inside a

circle is equal in measure to

half of the sum of the arcs in

intercepts c) A secant angle

with the vertex outside a

circle is equal in measure to

half of the difference of the

measures of the arc it

intercepts. (10 minutes)

minutes we will take the

next 5 to 7 minutes to go

over these problems as

a class. (14 to 20 minutes

in total)





be “I can use the

concept of tangents

and secants to solve for

parts of a circle.” (2

minutes)

4. Problem Solving

Activity: In your table

groups you will be asked

to solve the following

problems. Each problem

will be written on a

notecard that is taped

to each table. Since

there are six tables, you

will be asked to solve six

problems. You will have

five or so minutes to

solve each problem.

When the timer goes off

(after five minutes) each

group will stand up (yes,

you have to move) and

rotate clockwise to the

next table. Write your

answers and all of your

work on a sheet of

notebook paper. Be sure

to label what number

each problem is. You will

be turning in one set of

theorems we have

covered since the last

quiz. Students will be

able to use ALL of their

THEOREM notecards

they have created in

the past few days on this

quiz. Notecards will be

checked before the quiz

to make sure students

are not cheating. (10

minutes)

4. The next five or so

minutes students will be

able to ask questions

about the quiz (5

minutes)

5. Go over the Plan for

the day, which is a FREE

REVIEW day. Students

can a) ask the teacher

or their classmates any

questions they have

from previous homework

assignments or just

general questions b)

work on the “Review for

Quiz 2” handout

independently or with a

partner c) check the

answers to the “Review

for Quiz 2” handout with

the “Review for Quiz 2”

answer key and d) ask

any questions about the

“Review for Quiz 2”

activity. (2 minutes)

6. FREE REVIEW TIME (25

minutes)

to taking the quiz (Over

30 minutes)

6. If students finish early

they may start their

homework in class

7. For Homework:

Complete the self-

assessment handout for

class on Monday. The

self-assessment handout

will cover a) how they

are feeling about the

material thus far? b) To

what degree are they

participating (taking

notes, asking questions,

participating in small

group discussions,

participating in large

group discussions,

staying on task,

responding to thumbs

up thumbs down and fist

to fives, being prepared,

keeping up with the

notecard recipe boxes)

and c) Are they keeping

up with the homework?




assessment. They are

allowed to work with their

shoulder partners though!

(10 minutes)

2. Pass out and go over

“Part 2 Quiz”. (20

minutes)




statements of the day will

be “I can use GeoGebra

to construct chords,

tangent lines, secant

lines, central angles

inscribed angles, and

arcs. I can use

GeoGebra to determine

the relationships amongst

these parts.” (2 minutes)

4. Hand out the heavily

guided “GeoGebra:

Chords, Arcs, Central

Angles, Inscribed Angles,

Secants and Tangents

Activity” handout (2

minutes)

5. Discuss the overall goal

and the directions of this

activity. (The overall goal

is the “I can” statements)

(3 minutes)

6. Have students work on

the activity and fill out

the assignment as they

go. They may work with

their shoulder partners for

this. (15 minutes)

7. About—Point: Have


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5. After going through the

vocab words and theorems,

we will do some practice

problems that use these

theorems to solve for certain

parts. (10 minutes)

6. Basic Journal Entry: Reflect

on what you learned today.

Write this in your journal

notebook. (5 minutes)

7. For Homework: Students will

complete a mini problem set

in the CME Project Geometry

Textbook. They will be asked

to complete numbers 1, 2, 4,

5, 6, 7, and 9 on pages 410 to

411. These problems all relate

to secant lines and tangent

lines and solving for different

parts of a circle.

problems per group so

make sure everyone

writes or no credit will be

given. Work together

and communicate!

(These problems will be

asking students to solve

for different parts of a

circle using secant lines,

tangent lines, chords,

inscribed angles and arc

measures. The main goal

of this activity is to get

students practicing using

the formulas we have

learned in the past few

days. The formulas

include a*b=c*d,

m


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8. For Homework:


mini problem set in the

CME Project Geometry


asked to complete

numbers 6-8 on page

439. These problems all

relate to the problems

we did in class today.



keeping up with the

homework?




1. Thumbs up, Thumbs down:

Who is ready for the

Homework Quiz?

2. Go over any last minute

homework questions (10

minutes)

3. Go over the agenda for the

day 1) Homework Quiz 2) Pre

Assessment for the NEXT Unit

3) start “Study Guide” packet

in class (2 minutes)

4. Pass out the Homework

Quiz and go over the

directions as well as answer

any questions (5 minutes)

5. Have students complete

the Homework Quiz (15

minutes)

6. When students turn in the

Homework Quiz, make sure

they turn in their homework

along with it. This will prevent

cheating as well as gives

students an opportunity to

show the teacher what they



a) If students do not

have their homework

completed they will not

receive credit for their

homework completion

grade for the day as

well as b) will not be

able to participate in

the in class activity for

the day. Instead of

working with their

tablemates, they will be

asked to sit near my

desk and work silently on

their packets.

2. Once homework is

checked at the door the

rest of the class period

will be devoted to in the

in class review activity.

The review activity is

literally letting the

students go over each

1. Chapter Five Unit Test:

“The Circle of Life” is

Today!! When students

walk in, they are allowed

to begin the test. The

room will be rearranged

in test taking mode to

prevent cheating! (50

minutes)

2. No Homework Today!

1. Chapter Five Unit Test

(Writing Portion): “The

Circle of Life” is Today!!

When students walk in,

they are allowed to

begin the test. The room

will be rearranged in test

taking mode to prevent

cheating! (50 minutes)

2. No Homework Today!

1. Think, Pair, Share to the

questions “a) how well

do you think the test

went? b) What went

well? c) What could

have gone better? d)

What could I have done

better to be better

prepared? e) What

could Ms. Sachs have

done better to better

prepare us for this test?”

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


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know if they accidently

copied down the wrong

problem.

7. After students complete the

Homework Quiz, have

students pick up and

complete the NEXT Unit’s pre

assessment in class quiz. (10

minutes)

8. Today would be a day

where students will probably

finish up at different times. If

students happen to finish up

these two quizzes before the

bell rings, have them pick up

the homework assignment

and work on it in class.

9. If students have not already

picked up the homework

assignment, have them do so

now.

10. The last ten minutes of

class will be used to discuss

the “Study Guide: Circle of

Life Unit” assignment. Students

should know the rules of the

assignment because we do

similar assignments before

every test, but I am going to

reiterate the rules to make

sure they know I am serious

about them. The Rules of this

assignment are as follows a)

complete the “Study Guide:

Circle of Life Unit” for

HOMEWORK. If you do not

complete the assignment for

HOMEWORK, you will not

participate in the in class

problem of the Study

Guide with their

tablemates to make sure

they get every problem

correct and understand

every problem. They are

allowed to change their

answers, but they do not

have to if they believe

their answer and or

thought processes are

right. This activity relies

on student

preparedness and

student communication,

because I will only help

students if I think they

truly will not understand

the concept by the end

of the class. Students

need to ask questions

and explain their

thinking to their

classmates. Participation

will be assessed at this

time. (40 minutes)

3. As students are

working, observe what is

going on as well as

return homework (not

the homework quizzes)

to the students. They will

want their homework

back to study.

4. Hand in “Study Guide:

Circle of Life Unit”

activity

5. Remind students that

the “Circle of Life Unit

post it note to the class.

(This activity in general

should take about 10

minutes)

2. Share to the class a)

how many As b) how

many Bs c) how many

Cs, d) how many Ds and

e) how many Fs (2

minutes)

3. Share to the class a)

the class average b) the

class average compared

to their last unit test class

average and c) the

highest grade in percent

form (3 minutes)

4. Return and go over the

“Circle of Life Unit Test”.

(25 minutes)

5. Return the Homework

Quiz (2 minutes)

…

6. Introduction to the

NEW Unit


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activity TOMORROW which

will probably hinder your

grade b) remember that you

will be turning this assignment

in and it will be graded for

completion and correctness.

It WILL be graded!! Because

of this be sure to write clearly,

circle your answers, and show

all of your work (10 minutes)

11. For Homework: Complete

the “Study Guide: Circle of

Life Unit” activity

Test” is TOMORROW. It

will cover a) area and

perimeter of a blob b)

area and perimeter of a

regular n-gon c) area

and circumference of a

circle d) sector area e)

arc length f) chords g)

Inscribed angles h)

central angles I) Secant

lines and j) tangent lines.

(3 minutes)

6. Answer any questions

students have about the

test (5 minutes)

7. For Homework: Study

for the unit test

tomorrow!! Look over

your notes and your

notecards! Look over

your homework

assignments! Look over

this Study Guide!

Remember that

notecards should not be

used on Unit Tests! Good

Luck!


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Name: _______________________

Chapter Five Unit Test: “The Circle of Life”

True/False: Mark T if the answer is True and F if the answer is False. Make sure youranswer is easy to read or it will be marked wrong. (2 points each)

__________ 1. The area of a regular n-gon is equal to half of the product of its

perimeter (P) and apothem (a).

__________ 2. Pi is the numerical value of the area of a circle with radius 1.

__________ 3. The formula for the circumference of a circle is (Pi)*(r^2).

__________ 4. The measure of an inscribed angle is equal to twice the measure of

its intercepted arc.

__________ 5. The measure of a central angle is equal to the measure of its

intercepted arc.

Multiple Choice: Circle the best answer. Then write the letter on the line. Make

sure to write clear and neat or your answer will be marked wrong. (2 points

each)

__________ 6. When finding the area and perimeter of an abstract curved figure

the best method to use is

a. The area formula of a circle

b. The area formula of a regular n-gon

c. Linear approximation

d. Both a and b

__________ 7. A(n) ______________ for a circle is an angle that has its vertex at the

center of the circle.

a. Central angle

b. Inscribed angle

c. Right angle

d. Secant angle


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12. Solve for the following measures. The radius of the circle is 5m. (Do not forget

your units!)(Round your answers to 2 decimal places) (6 points)

a) Arc length of BC

b) Sector Area of DAE

Arc Length of BC = __________________

Sector Area of DAE = ________________

13. Find the area of the ENTIRE region (not just the triangle) using the given

information. AB is a semicircle. CA is a semicircle. Segment BC is 5in. Segment AB

is 5 in. (Round your answer to 2 decimal places)(Do not forget your units!) (6

points)

Area = ______________________


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14. Using the diagram and the given information find the missing measures. (Do

not forget to put your units!)(Round to the nearest tenth.)

14a. If CG=24cm, GD=72cm, and FG=106cm, find GE. (3 points)

GE = ____________________

14b. If Arc ED is 110 degrees and Arc CF is 50 degrees, what is the measure of

angle


32/43


33/43

Name: _______________________

Chapter Five Unit Test (Writing Portion): “The Circle of Life”

Pick TWO of the following problems to answer. (10 points each)

a. Essay One: Solve the problem below. After you solve it, explain in paragraph

format what steps you took and why you took them. Make sure you…

• Show all your work

• Use complete sentences

• Use correct spelling, grammar, and punctuation

• Explain what you did

• Explain why you took the steps you took

Steve wants to build a circular fence around his garden, as shown below.

Determine the total amount of fencing needed to put around the circular

garden to the nearest hundredth of a foot. AG = 5 ft.


34/43

b. Bill spilled paint and left the blob to the right. Describe how you would use the

grid method to estimate the area of Bill’s mess. Be

sure to

• Write in complete sentences

• Use correct spelling, grammar, and

punctuation•

Use the following vocabulary terms

from this chapter

o Approximation

o Limit

o Circumscribe

o Inscribe


35/43

c. PA and PB are the tangents to a circle through P. Prove PA is congruent to PB.

When writing your proof be sure to…




•

Explain what you did•

Explain why you know this is true (state a theorem)


36/43

Name: _______________________

Chapter Five Unit Test: “The Circle of Life”

True/False: Mark T if the answer is True and F if the answer is False. Make sure your

answer is easy to read or it will be marked wrong. (2 points each)

__________ 1. The area of a regular n-gon is equal to half of the product of its perimeter

(P) and apothem (a). True

__________ 2. Pi is the numerical value of the area of a circle with radius 1. True

__________ 3. The formula for the circumference of a circle is (Pi)*(r^2). False, the

formula for the circumference of a circle is (2)*(Pi)*(r). The formula for the area of a

circle is (Pi)*(r^2).

__________ 4. The measure of an inscribed angle is equal to twice the measure of its

intercepted arc. False, the measure of an inscribed angle is equal to HALF the

measure if its intercepted arc.

__________ 5. The measure of a central angle is equal to the measure of its intercepted

arc. True

Multiple Choice: Circle the best answer. Then write the letter on the line. Make sure to

write clear and neat or your answer will be marked wrong. (2 points each)

__________ 6. When finding the area and perimeter of an abstract curved figure thebest method to use is

a. The area formula of a circle

b. The area formula of a regular n-gon

c. Linear approximation

d. Both a and b

__________ 7. A(n) ______________ for a circle is an angle that has its vertex at the

center of the circle.

a. Central angle

b. Inscribed angle

c. Right angle

d. Secant angle


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__________ 8. A(n) ____________ for a circle connects two points on a circle. The center

point does not need to be one of these points.

a. Arc

b. Chord

c. Diameter

d. Radius

_________ 9. A(n) __________ is an angle that has its vertex in the circle and has sides

that are chords of the circle.

a. Central angle

b. Inscribed angle

c. Right angle

d. Secant angle

_______ 10. When a line has only one intersection point with a circle, it is said to be a…

a. Secant Line

b. Parallel line

c. Perpendicular Line

d. Tangent Line

Short Answer: Solve each problem. Show all your work or no credit will be given.

11. Find the area and perimeter of a regular hexagon with a side length of 12 cm.

Draw a picture to help you. (Don’t forget your units in your answer!)(Round your

answer to 2 decimal places.) (5 points)

Perimeter = 12 (6) = 72cm

Area = (1/2)*P*a

Area = (1/2)*72*a

Base of Triangle: 6cm

Hypotenuse of Triangle: 12cm

(a)^2+(6)^2=(12)^2

(a)^2+36=144(a)^2=108

a=10.39

So…Area=(1/2)*P*a

Area = (1/2)*72*(10.39) = 374.04 cm^2

Area = 374.04cm^2

Perimeter = 72cm


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12. Solve for the following measures. The radius of the circle is 5m. (Do not forget your

units!)(Round your answers to 2 decimal places) (6 points)

a) Arc length of BC

b) Sector Area of DAE

Arc Length of BC = 11.08m

Sector Area of DAE = 13.52m^2

13. Find the area of the ENTIRE region (not just the triangle) using the given information.

AB is a semicircle. CA is a semicircle. Segment BC is 5in. Segment AB is 5 in. (Round

your answer to 2 decimal places)(Do not forget your units!) (6 points)

Area of the Whole Shape = 41.94 in^2

a) C=(2)(Pi)(r)

C=(2)(Pi)(5)

C=10(Pi)

C= 31.4m

Arc length = C*(angle measure/360)

Arc length =(31.4)*(127/360)

Arc Length = 11.08m

b) A=(Pi)(r)(r) Sector Area = A*(angle

measure/360)

A= (Pi)(25) Sector Area = (78.5)*(62/360)

A=78.5 Sector Area = 13.52m^2

Area of Circle with diameter AB = (3.14)*(2.5)^2

= 19.63

Area of half of this circle = 9.81

Area of Circle with diameter AC = (3.14)*(3.53)^2

= 39.25

Area of half of this circle =19.63

Area of the Triangle = (b)(h)(1/2)

= (5)(5)(1/2)

= 12.5

Area of the whole shape = 12.5+19.63+9.81= 41.94


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14. Using the diagram and the given information find the missing measures. (Do not

forget to put your units!)(Round to the nearest tenth.)

14a. If CG=24cm, GD=72cm, and FG=106cm, find GE. (3 points)

GE = 16.3cm

14b. If Arc ED is 110 degrees and Arc CF is 50 degrees, what is the measure of angle


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15. Find the measure of A, B, C, J, K, and angle


41/43

Name: _______________________

Chapter Five Unit Test (Writing Portion): “The Circle of Life”

Pick TWO of the following problems to answer. (10 points each)

a. Essay One: Solve the problem below. After you solve it, explain in paragraph format

what steps you took and why you took them. Make sure you…




• Explain what you did

•

Explain why you took the steps you took

Steve wants to build a circular fence around his garden, as shown below. Determine

the total amount of fencing needed to put around the circular garden to the nearest

hundredth of a foot. AG = 5 ft.

The amount of fencing needed for the garden is 23.55 feet.

For the rest of the problem, the students’ answers may vary. There are a few different

ways to go about solving this problem so as long as students get the right answer,

show their work, and thoroughly explain what they did, they should receive full credit

on this problem.


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The purpose of this problem is to get students thinking about how this unit relates to the

real world. Some other purposes of this particular problem are to get students

practicing their writing skills, problem solving skills, and critical thinking skills as well as

demonstrate what they know about Arc Length.

b. Bill spilled paint and left the blob at the right. Describe how you would use the gridmethod to estimate the area of Bill’s mess. Be sure to

•

Write in complete sentences

• Use correct spelling, grammar, and

punctuation

• Use the following vocabulary terms

from this chapter

o Approximation

o Limit

o Circumscribe

o Inscribe

The students’ answers will vary for this problem as well. As long as students write in

complete sentences, use correct spelling, grammar, and punctuation, use all the

vocabulary words listed above, and seem to have the right idea they will receive full

credit.

The purpose of this problem is to get students thinking about how this unit relates to the

real world. Some other purposes of this particular problem are to get students

practicing their writing skills, problem solving skills, and critical thinking skills as well asdemonstrate what they know about approximation and estimation.

c. PA and PB are the tangents to a circle through P. Prove PA is congruent to PB. When

writing your proof be sure to




•

Explain what you did

• Explain why you know this is true (state a theorem)


43/43

The students’ answers will vary for this problem as well. As long as students write in

complete sentences, use correct spelling, grammar, and punctuation, use proper

vocabulary, and seem to have the right idea they will receive full credit.

The main purpose of this particular problem is to get students practicing the idea of

the proof. Some other purposes of this particular problem are to get students

practicing their writing skills, problem solving skills, and critical thinking skills as well asdemonstrate what they know about tangent lines, radii, and triangles.

Date post:	01-Jun-2018
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