SoundOffProblem.pdf

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UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science Partnership Grant Program NCLB Title II Part B Revised 12/13/12 1

Mathematics Senior Level Capstone Course

Unit Overview

Title of Unit: Sound Off! Unit Designers:

Diana Bowen Albemarle Lisa Haney Albemarle Erik Nylander Staunton Jane Thomas Charlottesville Edited by Diane Leighty, UVA-SCPS Office of Mathematics Outreach

Context:

Summary of the issue, challenge, investigation, or problem.

Students will apply the reflective properties of conic sections to design and build a structure (or model) that has special acoustic properties.

Students will explore, create, and build structures with unique acoustic properties.

Number of Class Hours:

28 - 34 hours Unit

Design: X_ Problem Based Other Subject

Areas/Disciplines Addressed:

Architecture, Physics, Engineering, Construction, Design, Acoustics

Driving Question: Can you design and build a structure with a unique acoustic property? Mathematics Content

Addressed:

Properties of conic sections

Applications of conic sections to the real-world Scale drawings for design

Calculation of cost of design materials

MPE Addressed: 1, 3, 6, 7, 12, 29 Problem Solving, Decision Making, &Integration; Procedure & Calculations; Applying Functions Assumption of Prior

Knowledge:

General understanding of quadratic functions (to be reviewed) Basic arithmetic (for calculating cost)

Understanding of use of scale (ratios and proportions)

College and Career Readiness/21st Century Skills to be taught (T) during this unit or

Collaboration:

Students work in teams to create possible designs for the structure.

Students work in teams to create their

T/E/A Research:

Students research real-world conic sections and acoustic structures using the internet and possible field trip.

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expectation (E) for student use during this unit and assessed (A): BIE Page 35-37

structure.

Communication (Oral and/or Written): Students create scale drawings of their structure.

They share their structures with the community (paper, presentation, video)

E/A Technology:

Use of the internet to research acoustics and conic sections.

Use of the internet to explore properties of an ellipse

Use of spreadsheets to determine costs for materials.

T/E/A

Critical Thinking/Decision Making: Students create a design of a structure that has special acoustic properties based on conic sections.

T/E/A Other:

Major Products and/or Performances:

Group:

Design and creation of acoustic structure.

Presentation of structure with discussion of mathematics involved. Presentation may take the form of video or demonstration of the structure’s special properties.

Presentation Audience:

X Class

X School

Individual: X Expert

X Community

Other: Launch: Event or

experience used to engage the students interest and inquiry:

Teacher Notes: These videos are from youtube.com and they feature Barossa Reservoir in Southern Australia. Teachers may want to consider a field trip to visit an example of a structure with a unique acoustic property. Videos demonstrating the power of mathematics and acoustics:

http://www.youtube.com/watch?v=ygTTJ_f8zwM http://www.youtube.com/watch?v=PYYKeLdJmvc

Possible field trips: Statuary Hall in the U.S. Capitol in Washington D.C, The Hume Fountain at the University of Virginia in Charlottesville, The Whispering Wall at the College of William and Mary in Williamsburg.

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Evaluation: Formative Assessments (During the Unit)

Interview X Practice Presentations X

Mathematicians Journal X Notes X

Preliminary

Plans/Outlines/Prototypes

X Checklists

Rough Drafts Concept maps

Field Tests Other:

Summative Assessment (End of Project)

Written Products, with a rubric X Peer Evaluation, with a rubric

X Oral Presentation with a rubric X Self Evaluation, with a

rubric Other Product(s) or

Performance(s), with a rubric Structure

X Other:

Blue prints of structure X

Resources Needed: On-site people, facilities: Permission and site for construction or site to display models Equipment/Technology: Excel, Internet Access, Tools, Work Space

Materials: Building materials, Blueprint papers Community Resources: Civil engineer, Sound engineer, Architect

Reflection Methods: Individual, Group, and/or Whole Class

Mathematicians Journal X Small/Focus Groups X

Whole Class Discussions X Fishbowl Discussions

Survey Other:

Material Adapted From: http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=65; and from notes, assessments, and activities provided by Charlottesville High School and Western Albemarle High School. Teacher Note: If you do not have access to explorelearning.com you can use explorelearning.com free for up to five minutes per gizmo. Other sites that may be substituted are http://www.intmath.com/plane-analytic-geometry/ellipse-interactive.php or www.geogebra.org or http://mathworld.wolfram.com/ .

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U N I T C A L E N D A R

page 1

TITLE: Sound OFF! Time Frame: 3 – 4 weeks

M O N D A Y T U E S D A Y W E D N E S D A Y T H U R S D A Y F R I D A Y

P R O J E C T W E E K O N E

Notes: Only use notes when necessary depending on the level of your students and prior knowledge. This calendar can be shifted to remove all of the notes days and straight to the real-world examples and partner assessment of knowledge

Engage (Show YouTube Videos)

Blog Assignment

Field Trip and/or Talk with an Expert

String Properties and Explore Learning Gizmo

Quadratics Review

Go Over Quadratics Review

Notes (Intro to Conic Sections) Notes (Parabolas)

P R O J E C T W E E K T W O

Notes: Several websites provide an interactive experience, including Explore Learning, Wolfram Math World, and even Geogebra. Notes (Circles and Ellipses) Notes (Hyperbolas) Notes (Real-world Examples)

Real-World Problems

Partner Assessment on Quadratics

Research and Design Brainstorm (Design Phase)

P R O J E C T W E E K T H R E E

Notes: Depending on the structure, more time may be needed for the building phase or if the groups are only building models this calendar is appropriate

Design Rough Draft (Design Phase)

Model Building/Drawing

Design Presentations and Evaluations (Design Phase)

Building Phase Building Phase Building Phase and Final

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Virginia’s Senior Level Capstone Course

Instructional Plan

Unit Title:

Sound Off!

Driving Question:

Can you design and build a structure with a unique acoustic property?

Task/Project/Problem:

Students will build and present a structure with a unique acoustic property.

ENGAGE

How will student’s interested be peaked so they want to engage in the inquiry in this unit?

Number of

hours: 1

Teacher Notes:

As individuals and as a class, students will brainstorm

real-world applications of conic sections. Students may come up with

famous structures such as The Gateway Arch in St. Louis or

everyday structures such as the satellite dish. Next, show videos

demonstrating the power of mathematics and acoustics from the

launch materials above.

Videos demonstrating the power of mathematics and acoustics: http://www.youtube.com/watch?v=ygTTJ_f8zwM

http://www.youtube.com/watch?v=PYYKeLdJmvc

Present the Blog Assignment (HO #2). Students may know

famous structures such as the St. Louis Arch or every day conic

sections such as satellite dishes. Teachers do not need to limit

the discussion to conic sections with acoustic properties.

Teachers should set up a secure blog in advance with a class

username and password. Teachers should post an example to

guide the blog. Teachers can use the blog to help students sort

the applications by type of conic section, structure or

non-structure, or other classifications. Teachers can also ask

students to comment on the blog post they find most interesting.

An example blog post and structures with acoustic properties

can be found below.

Possible Blog Example: The Gateway Arch in St. Louis is an example of a parabola. It is also an example of American expansionism and it links the eastern and western United States. I could use the height of the arch and the length between the bases to find an equation for the

parabola. The equation would represent a cross-section of the arch.

Examples of Conic Sections with Acoustic Properties: Anechoic Chamber, Whispering Wall, Whispering Gallery, Echo Chamber,

Mathematician

Journal

Prompts:

Where do we

see and how

do we use

conic sections

in our

everyday

lives?

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Amphitheater, Noise Barriers, Sound Tube in Melbourne, Australia, Noise Abatement Wall, Parabolic Microphone (non-structure)

Materials and/or Resources Needed:

HO #2

Products:

Interactive Blog

EXPLORE

Teacher provides

guidance for the explorations to prepare students with the

knowledge and skills to engage in the task. Students will self-assess on the prior knowledge and skills assumed for the unit

Number of

hours: 4 – 10

depending on

prior

knowledge of

conic sections

Teacher Notes: Students will activate prior knowledge of

quadratic functions to complete a GNAW on Quadratics Activity

(HO #3) in groups. Students should be familiar with vocabulary.

Students may have difficulty determining the exact values of the

maximum (vertex), x-intercepts, and y-intercepts. Graphing

calculators may be used to supplement analytical methods.

Teacher Notes: Students will explore properties of parabolas,

circles, and ellipses using gizmos from explore learning and

construction by-hand methods and String Properties of Ellipses

and Parabolas (HO #4). Students should be able to label the

focus and directrix of a parabola in the construction given

directions and recognize that every point on the parabola is

equal distance from the focus and directrix. Students should be

able to construct a circle using prior knowledge. Students

should be able to construct an ellipse and be able to label the

foci in the construction. If teachers do not have access to

Explore Learning, they may do all of the constructions by-hand.

Teachers may also want to show the gizmo to the students

before letting them work independently. Teachers should

practice the constructions ahead of time. [You can get a

temporary account with Explore Learning for free.]

Screenshot 1: Example of the String Property of a Parabola with a Horizontal Directrix

Mathematician

Journal

Prompts:

What are the

“important”

points on a

quadratic

relation?

How can we

use math to

solve

real-world

problems

involving

quadratic

relations?

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Screenshot 2: Example of the String Property of an Ellipse

Teacher Notes: Students must review/learn properties of conic

sections before beginning the project. Attached are basic notes

on conic sections. Teachers may need additional notes to

support student learning. The first set of notes is labeled

Introduction to Conic Sections – Class Notes (HO #5) and it

reviews quadratic functions, introduces students to the four

quadratic relations, and provides examples of parabolas and

circles. Students will be able to graph the parabolas and they

can use a graphing calculator to support their work. Students

will struggle with the derivation of the equation of a circle using

the distance formula. Teachers may take example 4 out of the

notes if they do not want to show this relationship. Showing this

relationship is highly recommended because it relates directly to

the construction of a circle using a string.

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exploration. The second set of notes is labeled Parabola

Properties – Class Notes (HO #6)

Teacher Notes: Students must learn additional properties of an

ellipse including the geometric definition and the relationship to

the foci. For many students, this may be new material. Students

will have limited exposure to these concepts from the

exploration. The third set of notes is labeled Ellipse Properties –

Class Notes (HO #7)

Teacher Notes: Students must learn additional properties of a

hyperbola including the geometric definition and the

relationship to the foci. For many students, this may be new

material. The fourth set of notes is labeled Hyperbola Properties

– Class Notes (HO #8). Students should read examples in

groups and complete the Your Turn portion of the notes.

Teachers should support learning but students will be filling in

these notes primarily unassisted in groups of 2 – 4.

Teacher Notes: Students should complete the Assessment on the

Properties of Conic Sections (HO #9) with a partner, a graphing

calculator, and notes. Students do not need to memorize the

properties and equations for conic sections but they should be

able to recognize, use, write and apply the equations using

notes.

Materials/Equipment/Resources Needed:

HO #5 – 9

Pencils

String

EXPLAIN

Teacher introduces the main task of the unit and

prepares

students to work in small group independent work...

Number of

Hours: 4 - 8

Teacher Notes: Students need to be introduced to real-world

applications of conic sections. If available, there is a flashlight

example file on Geometer’s Sketchpad that demonstrates how

adjusting a foci affects the parabolic lens and resulting light

beam. If this example is not available teachers may search for

similar examples on the Internet. Teachers will present

Real-World Applications of Conic Sections – Class Notes (HO #11)

and students will solve problems in groups of 3 – 4 to show an

understanding of these applications by completing Real-World

Applications of Conic Sections – Class Activity (HO #12). This

will introduce students to real-world applications including

applications with acoustic properties. Teachers can also present

students Real-World Applications of Conic Sections – Class

Notes Filled In (HO #13) or use these notes for their own

guidance. Students should be encouraged to use pictures and

Mathematician

Journal

Prompts

How can we

use math to

solve

real-world

problems

involving

quadratic

relations?

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diagrams when completing the class activity. If these notes are

not appropriate for your class-level, please go to Kuta Software

and use the free worksheets available through their website.

Teacher Notes: Teachers should introduce the project and

students or teachers should design the class into teams of 3 – 5

students. Teachers should allow students to use the Internet, the

library, and experts to conduct research on conic sections with

acoustic properties. Students should complete Research and

Design Brainstorm (HO #14) to determine the direction of their

design. During this time teachers should observe student

progress and help students determine if designs are using

cross-sections, 3-dimensional structures, or 2-dimensional conic

sections. Teachers will need to approve of each basic sketch and

description before students design the structure. Once students

complete HO #14 and you have approved the basic design then

students should begin work on a design with measurements,

equations, sketches, and possible materials and estimated costs.

Students will organize this information on the Design Rough

Draft Worksheet (HO #15) and they will use this information to

formulate a 5-minute presentation to the class about their

design. This 5-minute presentation will help the class decide

which design is most feasible for the class. Students will also

complete peer evaluations during the 5 minute presentation.

The Peer Evaluation for Designs (HO #16) will be used to select

the design. Teachers and students may also decide to use a

combination of two designs based on results from HO #16.

Here is a student example of a scaled drawing:

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Materials/Equipment/Resources Needed:

HO #11 – 16

Pencils

String

ELABORATE

The student groups are working independently with teacher consultations.

Number of

Hours: 10 – 14

Suggested Teams for Design Phase:

Teams of 3 – 5 students design a real-world conic section with a unique acoustic property to be built as a model or as a structure in the school community or the local community.

Encourage each team to pick a

different structure.

Teacher Notes: During this phase of the project, students design

and build a real-world or model of a quadratic relation to be

displayed in the community. It is highly suggested students build

a whispering gallery or whispering wall but teachers can adjust

this project as needed. After the design phase is completed,

students vote on the top design. Once the top design is selected

students break down into teams for the building phase of the

project. Each structure should be unique to the classroom of

students and the teacher.

Teacher Notes: There are no handouts for the next phase of the

project. Each team should create products based on the

guidelines below. Teachers may modify team products based on

classroom size or group size. Teachers must be able to review

designs, scales, materials lists, cost models, and community

letters before any part of the design is finalized. Students will

struggle with the open-endedness of this part of the project.

However, each group has specific tasks to support the building

process. Once the materials are purchased and the build site is

determined, all students should participate in the building of the

project. Teachers and experts need to train students on proper

construction and usage of tools. It is the responsibility of the

leadership team to create materials to self-evaluate peers

throughout the implementation phase. The leadership team is

also responsible for making sure the teams stay on task during

the building phase. Teachers may need an engineer or architect

to determine if the structure meets local safety and workplace

guidelines.

Mathematician

Journal

Prompts

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Suggested Teams for Building Phase:

Teams of 3 – 5 students are placed into task-based groups.

Teachers may adjust group size or group roles to meet individual

class needs.

Leadership Team

(oversees project, provides additional

research, supports each of the other four teams)

Construction Team

(provides designs through each phase)

Materials Team

(determines all materials, quantities, and costs

and displays information in an excel spreadsheet)

Implementation Team

(purchases materials and decide the order

in which to process and build the structure)

Community Correspondence and Model Team

(researches

possible locations and makes community contacts for materials,

site location and they design the model to display our final goal,

decides the direction of the presentation as well)

Formative Assessments:

Peer Evaluations for Designs (HO #16)

Student Designed Peer Assessments (Created by Leadership

Team with the aid of the teacher)

EVALUATE

Working groups submit products or make

presentations

Number of

Hours: 1 – 3

Teacher Notes: Students test the model or structure during this

phase of the project. Each team also submits a one-page

summary of the group’s contribution. Guidelines for this

summary are found on Group Summary Guidelines (HO #17)

and each group’s contribution and summary will differ.

Students use this information and multimedia to field test the

structure and present the project to the selected community. The

final product is a one-page summary and a short multimedia

presentation to highlight each individual group’s

accomplishments and the class accomplishments.

Teacher Evaluation of Summary and Presentation and Group

Work (HO #18)

Mathematician

Journal

Prompts

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Map the Unit

What do students need to know and be able to do to complete the task/project/problem

successfully? How and when will they assess their own necessary knowledge and skills?

How will they remediate their own gaps or weaknesses in knowledge and skills? Look at

each major task for the unit and analyze the tasks necessary to produce a high-quality

product.

Project: Can you design and build a structure with a unique acoustic property?

KNOWLEDGE AND SKILLS

NEEDED

Assumed

already

learned

Students

will

self-assess

Will be

taught

during the

unit

1. Scale drawing

X

2. Properties of conic sections

X

3. Research into acoustic walls

X (research) X

X

4. Create an Excel spreadsheet to

calculate costs (Only some groups)

X

5. Ability to build wall

X

6. Ability to solve problems involving

properties of conic section (Notes and

problems included for students without

prior knowledge)

X

7. Ability to solve problems involving

real-world applications of conic sections

X

8. Create designs and rough drafts of

possible structures

X

9. Create a multimedia presentation

X

What project tools will student’s use?



Know/need to know lists



Daily goal sheet

X Mathematician’s Journals

X Briefs/Memos

X Task lists

X Planning Calendar

□ ________________________________

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HO #1 Sound Off! Field Trip Letter

Thursday, January 12, 2012

Dear Parent/Guardian:

We are beginning a new unit on Conic Sections and we plan to explore many of the real-world

applications throughout the unit. To explore the unique acoustic properties associated with conic sections we will be visiting one of the following places in Virginia:

- Statuary Hall in the United States Capitol in Washington D.C. - The Hume Fountain at the University of Virginia in Charlottesville

- The Whispering Wall at the College of William and Mary in Williamsburg

This will help your student better understand the driving question for our next project: Can we design a structure with a unique acoustic property? Students will have the opportunity to test the acoustic

properties on site and find out the history of each structure. Please fill out the following permission form allowing your child to participate in this field trip.

Sincerely,

Your Name Your School

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HO #2 Name:

Sound Off! Date:

Brainstorm and Blog on Conic Sections Password:

I. Launch: Brainstorm a list of conic sections in the real-world.

II. Blog Assignment: You will explore the real-world applications of conic sections by posting on the Conic Sections Blog. Be creative and do not repeat previous posts! Look for an example on the blog. Your post needs to include the following information:

1. A real-life application of a conic section 2. The name of the conic section

3. Explain the application 4. Post a picture or sketch

Blog Website: ____________________________________________________________

User Name: _______________

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HO #3 Name:

Sound Off! Date:

GNAW on Quadratic Functions Period:

A GNAW is when you examine a mathematical problem using graphs, numbers, analytical work, and words

Directions: Fill out the GNAW starting with the numerical section in groups of 2 or 3.

Quadratic Functions: A rocket is fired from a platform into the air. The following quadratic represents the height (in feet) of the rocket over time (in sec).

2. Graphical:

1. Label the axes, create a scale, and graph the function.

2. Draw an axis of symmetry and label any intercepts and the vertex.

1. Numerical

:

1. What is the real-world domain of the

function? Why?

2. Fill in the table.

3. Analytical:

1. How high was the platform the rocket was fired from?

2. How high does the rocket get in the air?

3. How long does it take for the rocket to reach its maximum height?

4. When does the rocket hit the ground?

4. Words:

1. What is the real-world range of the

function? Why?

2. Explain the real-world meaning of the

x- and y-intercepts.

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HO #3b Solution Key

Sound Off!

GNAW on Quadratic Functions

A GNAW is when you examine a mathematical problem using graphs, numbers, analytical work, and words

Directions: Fill out the GNAW starting with the numerical section in groups of 2 or 3.

Quadratic Functions: A rocket is fired from a platform into the air. The following quadratic represents the height (in feet) of the rocket over time (in sec).

2. Graphical:

1. Label the axes, create a scale, and graph the function.

2. Draw an axis of symmetry and label any intercepts and the vertex.

1. Numerical

:

1. What is the real-world domain of the

function? Why?

0 < t < 4.12

2. Fill in the table.

t h(t)

0 12 0.5 39.5

1 59 1.5 70.5

2 74 2.5 69.5

3 57 3.5 36.5

4 8 4.12 0

3. Analytical:

1. How high was the platform the rocket was fired from?

12 feet high

2. How high does the rocket get in the air? Approx. 74 feet or 74.02 feet

3. How long does it take for the rocket to reach its maximum height?

Almost 2 seconds or 1.97 seconds

4. When does the rocket hit the ground? After 4.12 seconds

4. Words:

1. What is the real-world range of the

function? Why?

0 < h(t) < 74.02

2. Explain the real-world meaning of the

x- and y-intercepts.

x-intercept: when object hits ground

y-intercept: height of object at time zero

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HO #4 Name:

Sound Off! Date:

String Properties of Ellipses and Parabolas Period:

To understand some of the unique properties of ellipses, parabolas, and circles we are going to use internet-based activities and handmade constructions to determine characteristics unique to each conic section.

Do Now: Using your knowledge of circles and the pencil and string on your desk, complete the “do now”. 1. Explain to someone how you could construct a circle using a string tied to a pencil.

2. Demonstrate your explanation in the space below using the materials provided.

Exploration 1: Go to http://www.explorelearning.com and enter the username and password given you, or sign up for a free trial. Search for the gizmo “Parabola with Horizontal Directrix” and launch the gizmo. Turn on “show polynomial form” and “show string property” and adjust the “p” value to explore then complete the following questions. [Another option is http://www.intmath.com/plane-analytic-geometry/ellipse-interactive.php]

1. What happens to the length of the “string” as you move away from the vertex?

2. One end of the “string” is attached to the focus of the parabola. The focus always lies on this unique line associated with parabolas. ____________________

3. The other end of the string moves along a line called the directrix of the parabola. What happens to the directrix as you increase the “p” value? What happens to the directrix as you decrease the “p” value?

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Exploration 2: Search for the gizmo “Ellipse Activity B” and launch the gizmo.

1. Adjust the “a” value to various values greater than 4. What happens to the ellipse as you increase and decrease the “a” value?

2. Where is the “a” value located in the equation of an ellipse? Where is the “b” value located in the equation of an ellipse?

3. The foci (plural of focus, red dots on the gizmo) of an ellipse lie on the major axis. How can you determine which axis is the major axis? How could you determine which axis is the major axis if no foci were present on the graph?

4. The other axis is referred to as the minor axis. Set the “b” value to 6 and the “a” value to 3. Which axis is the major axis? How do you know from the equation?

Exploration 3:

1. Create three different ellipses using the string method demonstrated in the gizmo and label

the major axis, minor axis, and foci on a plain sheet of paper. Be neat!

2. Use sketches and words to explain what happens to an ellipse as the foci get closer

together and what happens when the foci get farther apart.

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HO #5 Name:

Sound Off! Date:

Introduction to Conic Sections – Class Notes Period

Do Now: Graph. You may use a calculator to support your work.

1. 1



3



2 1

2

y  x  2. 2

2 12 17

y x  x

Vertex: ____________

Axis of symmetry: _____________

Vertex: ___________

Axis of symmetry: ____________

Conic Sections

 Relation specified by an equation of the form: 2  2   

0

Ax Bxy Cy Dx Ey F

 Basic quadratic relations are _____________, _______________, and ________________ with _____________ being a subset of _______________.

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Example 1: Accurately graph 2

xy Vertex: ________________

____________ 2

xy Axis of Symmetry: ______________

Example 2: Accurately graph x



y1



22 ______________ x



y1



22

Vertex: ______________ Axis of symmetry: _______________

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Parabolas:



Y-parabolas ya x



h



2k

Vertex: _____________ Axis of symmetry: ____________________________



X-parabolas xa y



k



2h

Vertex: _____________ Axis of symmetry: ____________________________ Example 3: Accurately graph

x





2 y

2



10 y



7

Vertex: __________________

Axis of symmetry: _______________

X Y X Y

-2 -1

0

1 2

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Example 4: The Distance Formula and Midpoint Formula

Example 5: Find an equation of a circle having center (4, 5) and radius 6.

Example 6: Find the center and the radius and then graph the circle

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HO #6 Name:

Sound Off! Date:

Parabola Properties – Class Notes Period:

Parabola: The set of all points in a plane that are the same distance from a fixed line and a fixed point not on the line. The fixed point is called the ___________ . The fixed line is called the ____________. Example 1: Using the Definition of a Parabola

Write an equation for graph that is the set of all points in the plane that are equidistant from the point )

3 , 0 (

F and the liney3.

Example 2: Writing the Equation of a Parabola

Write an equation of a parabola with a vertex at the origin and a focus at (-5, 0).

Example 3: Identifying Focus and Directrix

Identify the focus and the directrix of the graph of the equation 2

16

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HO #7 Name:

Sound Off! Date:

Ellipse Properties – Class Notes Period:

Example 1: Solve the following for y and graph on your graphing calculator.

2 2 1 25 16

x y

 

Example 2:

Solve the following for y and graph on your graphing calculator.



 



2 2

2 3

1

25 16

x y 

 

Ellipses Summary:



 

2



2

2 2 1

x h y k

a b

 

 



 



2 2

2 2 1

x h y k

b a

 

 

 ___________ square terms; sum  Vertex: ______________

(25)

 Minor axis = ___________

Geometric Definition:

Reflexive Property:

 To find foci: __________________

Example 3: Sketch and identify all significant points.



 

2



2

1 2

1

9 34

x y 

 

Example 4: Sketch and identify all significant points.

2 2

(26)

Example 5: Write the equation of an ellipse with…

Endpoints of the major axis at (2, 12) and (2, -4)

(27)

HO #8 Name:

Sound Off! Date:

Properties of Hyperbolas – Class Notes Period:

 Formed when a double cone is cut through perpendicular to the base.

 Two squared terms; difference

 Vertex form:



 



1 

 



1

2 2 2 2 2 2 2 2









b

h

x

a

k

y

or

b

k

y

a

h

x

 Major axis



2 a

; always the first axis in the hyperbola; also called the transverse axis; vertices are at the end of the major axis; difference of the distances to the foci

 Minor axis



2 b

 Center:

 

h,

k

 Asymptote lines: lines formed by the diagonal of a rectangle

2 a

by

2 b

 To find foci:

a

2



b

2



c

2

 Geometric Definition: set of points such that the difference of the distances from the foci to a point on the hyperbola is a constant.

 Reflective Property: energy emitting from a focus bounces off the other branch of the hyperbola as if it originally came from the other focus.

Example 1: Solve for y and graph on your calculator

1

16

25

2 2





y

x

Center (0, 0)

Out 5 on x, out 4 on y; draw rectangle

2 2 2

c

b

a





41

16

25 





c

Foci:

(

41 ,

0 )

,

(

41 ,

0 )

(28)

Your Turn:

1

25

16

2 2





x

y

Example 3:

1

25

16

2 2





y

x

Center (0, 0)

Out 4 on x, out 5 on y; draw rectangle

2 2 2

c

b

a





41

25

16 





c

Foci:

(

41 ,

0 )

,

(

41 ,

0 )

Your Turn: Graph the following hyperbola.



 



1

16

5

9

2

_



2

_



y

(29)

Example 5: Put the following hyperbola in correct form and identify the center, vertices and slope of the asymptote lines.

y

2



4 x

2



6 y



16 x



11 

0

(30)

HO #9 Name:

Sound Off! Date:

Assessment on Properties Conic Sections Period:

Directions: Complete this assessment with a partner. You may use any notes or practice problems to support your work. You may use a graphing calculator.

I. True or False. Classify each statement as true or false. 1. A parabola with a

y

2 term opens up or down.

2. The vertex of y 2



x1



23 is

 

1,3

.

3. The major axis in

2 2

1

16

25 x

_

y

_

, is vertical and 10 units long.

4. A circle has an eccentricity of 1.

5. An ellipse is formed by slicing a cone parallel to the base of the cone.

II. Multiple Choice. Choose the best answer from the given choices and fill the corresponding bubble on your answer sheet.

6. What is the graph of (x2)2(y3)2 4?

A. An ellipse B. A circle

C. A hyperbola D. A parabola

7. When graphed, which of the follow equations would produce a parabola?

A.

4

1 )

4 )(

2 (

y



x





B.

(

2 )

2

4

1 )

4

(31)

C.

(

2 )

4

1 )

4 (

y





x



D. 2

(

2 )

2

4

1 )

4 (

y





x



8. When graphed, which of the following equations would produce a hyperbola?

A.

x

2



y

2



8

B.

x



y



8

C. x2  y2 8 D. x y28

9. Find the center and radius of the given circle:



x4

 

2 y1



2 18.

A.

center

:

(



4 ,

1 )

radius:18 B.

center

:

(

4 ,



1 )

radius

:

18

C. center:(4,1) radius:9 D. center:(4,1) radius:9

For 10 – 12, use parabola

x



2 y

2



2

10. Which below could be the graph of the parabola?

(32)

C. D.

11. The focus and directrix of the parabola are…

A. directrix:

2

1

8 x

 

; focus:

1 , 0

7

8 

_











B. directrix:

1

8 y

 

; focus:

2,

1

8 











C. directrix:

1

8 x

 

; focus:

1 , 2

8 











D. directrix:

1

2

8 y

 

; focus: 0, 17 8

 _ 

 

 

12. What is the axis of symmetry for the parabola?

A. y2 B. _y0 C. x 2 D. x 0

Use the following graph for problems 13 – 15: 13. What is the equation of the conic?

A.



 



2 2

3

2

1

100

196 x



y



(33)

B.



 



2 2

3

2

1

25

49 x



y





C.



 



2 2

3

2

1

100

196 x



y





D.



 



2 2

3

2

1

25

49 x



y







14. How long is the major axis?

A. 5 B. 10 C. 7 D. 14

15. What is the distance from the center to the foci?

A.

296

B.

74

C.

24

D.

96

Match each description below with the most appropriate conic it describes below.

A. Parabola B. Circle C. Ellipse D. Hyperbola

16. The foci are the same as the center.

17. The equation used to find the foci is a2b2 c2.

18. The set of points such that the difference of the distances to the foci is the major axis. 19.

_x

2



6 x



7 

y

2

20. These are found in flashlights and car headlights.

21. Formed by slicing a cone parallel to the base of the cone. 22.

2 x

2



32 

2 y

2

(34)

HO #10 (answer key) Name:

Sound Off! Date:

Assessment on Properties Conic Sections Period:

Directions: Complete this assessment with a partner. You may use any notes or practice problems to support your work. You may use a graphing calculator.

I. True or False. Classify each statement as true or false. 1. A parabola with a

y

2 term opens up or down. false

2. The vertex of y 2



x1



23 is

 

1,3

. true

3. The major axis in

2 2

1

16

25 x

y





, is vertical and 10 units long. true

4. A circle has an eccentricity of 1. false

5. An ellipse is formed by slicing a cone parallel to the base of the cone. false

II. Multiple Choice. Choose the best answer from the given choices and fill the corresponding bubble on your answer sheet. Answers on right in bold.

6. What is the graph of (x2)2(y3)2 4?

A. An ellipse B. A circle

C. A hyperbola D. A parabola B. A circle

7. When graphed, which of the follow equations would produce a parabola?

A. 4 1 ) 4 )( 2

(y x  B. ( 2)2

4 1 ) 4

(y  x

C. ( 2)

4 1 ) 4

(y  x D. 2 ( 2)2

4 1 ) 4

(y  x B. ( 2)2

4 1 ) 4

(35)

8. When graphed, which of the following equations would produce a hyperbola?

A. x2 y2 8 B. xy8

C. x2 y28 D. x y28 C. x2 y28

9. Find the center and radius of the given circle:



x4

 

2 y1



2 18.

A. center:(4,1)

radius

:

18

B. center:(4,1)

radius

:

18

C. center:(4,1)

radius

:

9

D. center:(4,1)

radius

:

9

B. center:(4,1)

radius

:

18

For 10 – 12, use parabola

x



2 y

2



2

11. Which below could be the graph of the parabola?

(36)

C. D.

C. 11. The focus and directrix of the parabola are…

A. directrix: 21 8

x  ; focus: 1 , 07 8

_ 

 

  B. directrix:

1 8

y  ; focus: 2,1 8

 

 

 

C. directrix: 1 8

x  ; focus: 1, 2 8

 

 

  D. directrix:

1 2

8

y  ; focus: 0, 17 8

 _ 

 

 

A 12. What is the axis of symmetry for the parabola?

A. y2 B. _y0 C. x 2 D. x 0

B. _y0

Use the following graph for problems 13 – 15: 16. What is the equation of the conic?

A.



 



2 2

3

2

1

100

196 x



y



(37)

B.



 



2 2

3

2

1

25

49 x



y





C.



 



2 2

3

2

1

100

196 x



y





D.



 



2 2

3

2

1

25

49 x



y







B 17. How long is the major axis?

A. 5 B. 10 C. 7 D. 14

D. 14 18. What is the distance from the center to the foci?

A.

296

B.

74

C.

24

D.

96

C.

24

Match each description below with the most appropriate conic it describes below.

A. Parabola B. Circle C. Ellipse D. Hyperbola

26. The foci are the same as the center. B. Circle

27. The equation used to find the foci is a2b2 c2. D. Hyperbola

28. The set of points such that the difference of the distances to the foci is the major axis. C. Hyperbola 29.

_x

2



6 x



7 

y

2D. Hyperbola

30. These are found in flashlights and car headlights. A. Parabola 31. Formed by slicing a cone parallel to the base of the cone. B. Circle 32. 2x2 322y2B. Circle

(38)

HO #11 Name:

Sound Off! Date:

Real-World Applications of Conic Sections – Class Notes Period: I. Applications of Parabolas

Example 1: A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from the satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. If the dish is 10 feet across at its opening and is 4 feet deep at its center, at what position should the receiver be located?

Example 2: The cables of a suspension bridge are in the shape of a parabola, as shown in the figure. The towers supporting the cable are 600 feet apart and 80 feet high. If the cables touch the surface of the road midway between the towers, what is the height of the cable at a point 150 feet from the center of the bridge?

II. Applications of Hyperbolas

(39)

HO #12 Answer Key to HO #11

Sound Off!

Real-World Applications of Conic Sections – Class Notes Filled In I. Applications of Parabolas

Example 1: A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from the satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. If the dish is 10 feet across at its opening and is 4 feet deep at its center, at what position should the receiver be located?

Solution: The receiver should be located at the focus.

Draw a picture, Place the vertex of the dish at (0, 0) (-5, 4) (5, 4) Label ordered pairs using the dimensions above

Find a by substituting (5, 4) into

y



ax

2 4a52 so… a =

25

4 _{Use the relationship}

c a

4 1

 to find c the

distance from the focus to the vertex.

c 4 1 25 4 _ 16 25  c

so the receiver should be placed at (0, 25/16) to be at the focus.

Example 2: The cables of a suspension bridge are in the shape of a parabola, as shown in the figure. The towers supporting the cable are 600 feet apart and 80 feet high. If the cables touch the surface of the road midway between the towers, what is the height of the cable at a point 150 feet from the center of the bridge?

Solution: You need to determine an ordered pair on the bridge with an x-coordinate of 150 ft.

Draw a picture using the information above. The bridge is 80 feet high and the towers are 600 ft. apart Find a by substituting (300, 80) into

y



ax

2 (-300, 80) (300, 80)

2

300

80a so a =

1125 1 2 1125 1 x

y and (150, y)

Find the height by substituting in 150 for x

2 ) 150 ( 1125 1 

(40)

II. Applications of Hyperbolas

Example 3: Two signaling stations are 200 miles apart. A ship at sea receives a signal from one station 0.00038 seconds after receiving the signal from the other station. [Note: The speed of each radio signal is 186,000 miles per second.]

Write the equation of the hyperbola representing the ships position. Solution: Draw a picture. Place each station on a foci 200 miles apart. The difference in time between the stations is equal to the major axis This means using the definition of a hyperbola we know

68 . 70 000 , 186 * 00038 . 0

2a  miles this means a = 35.34

Since 2c = 200 then c = 100. Find b using a2b2c2

10000 9156

.

1248 b2

b28751.08

The equation of the hyperbola is ₁

08 . 8751 92

. 1248

2 2



 y

(41)

HO #13 Name:

Sound Off! Date:

Real-World Applications of Conic Sections – Class Activity Period:

Directions: Complete the following problems in groups. Draw a picture. Show all work. Real-World Problem Solving

Adapted from Precalculus by Michael Sullivan, 6th ed’n

1. A hall 100 feet in length is designed as a whispering gallery with an elliptical ceiling. If the foci are located 25 feet from the center, what is the height of the ceiling at its center? What is the equation of the ellipse used describe this gallery? You may assume it is centered at the origin.

2. An elliptical wall, 320 feet long and 150 feet wide is designed to be a whispering wall. How far would the listener have to be from the source of the sound in order to hear it?

(42)

4. The cables of a suspension bridge are in the shape of a parabola. The towers supporting the cable are 500 feet apart and 200 feet high. If the cables just touch the road surface midway between the towers, what is the height of the cable 75 feet from the tower?

5. A ship at sea receives a signal from one station 0.000625 seconds after it receives signal from another station. Write the equation of the hyperbola representing the ships possible location if the stations are 130 miles apart and the signals travel at 186,000 miles per second.

6. A racetrack is in the shape of an ellipse 80 feet long and 40 feet wide. What is the width 10 feet from the side?

80 ft

40 ft

10 ft ?

(43)

HO #13 b Solutions Sound Off!

Real-World Applications of Conic Sections – Class Activity

Directions: Complete the following problems in groups. Draw a picture. Show all work. Real-World Problem Solving

Adapted from Precalculus by Michael Sullivan, 6th ed’n

1. A hall 100 feet in length is designed as a whispering gallery with an elliptical ceiling. If the foci are located 25 feet from the center, what is the height of the ceiling at its center? What is the equation of the ellipse used describe this gallery? You may assume it is centered at the origin.

and 502 – b2 = 252 so b = 43.3 Therefore, the height of the ceiling is 43.3 feet.

2. An elliptical wall, 320 feet long and 150 feet wide is designed to be a whispering wall. How far would the listener have to be from the source of the sound in order to hear it?

so, 1602-1502 = c2 c = 55.68 so the distance the listener must be from the source of

the sound is 2 x 55.68 or 111.36 feet away.

3. A hall 64 feet in length is designed as a whispering gallery. If the ceiling reaches a height of 30 feet above the 5 foot vertical walls at the highest point, how far from the nearest walls should two people stand to be able to whisper to one another?

and 322 - 302 = 124 so the solution is 11.14 feet from the center of the room, or 32-11.14 = 20.86 feet from the wall.

4. The cables of a suspension bridge are in the shape of a parabola. The towers supporting the cable are 500 feet apart and 200 feet high. If the cables just touch the road surface midway between the towers, what is the height of the cable 75 feet from the tower?

Vertex is (0.0) and the parabola contains the points (-250, 200) and (250, 200).

(44)

HO #14 Name:

Sound Off! Date:

Research and Design Brainstorm

Period: Driving Question: Can you build a structure with a unique acoustic property?

Directions: To determine which structure interests your design group you should research examples and designs for structures with acoustic properties. After researching structures with acoustic properties your group should answer the questions below.

Here are some websites to support your research:

- http://en.wikipedia.org/wiki/Architectural_acoustics (Architectural acoustics)

- http://www.architechweb.com/ArticleDetails/tabid/254/ArticleID/3496/Default.aspx (More about architectural acoustics)

- http://en.wikipedia.org/wiki/Whispering_gallery (a collection of whispering walls and their locations)

Which structure interests your group?

Why does this structure interest your group?

What shape (conic section or 3-dimensional conic section) is used to create this structure?

What equations do you need to support your research?

What materials would you build your structure out of?

(45)

HO #15 Name:

Sound Off! Date:

Design Rough Draft Worksheet

Period:

Directions: Complete each of the following. Teacher approval should be obtained after each step.

Sketch of Design with Measurements:

Equations with Analytical Support: