AmazingAmusement.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 21st Century Teaching and Learning: Project-based Unit

Snapshot of Unit Content and Student Expectations

Title of Unit: The Amazing Amusement Unit Designers:

Lynn Shelton (Arlington Public Schools) Mirela Geagla (Arlington Public Schools) Context of the Project: Amusement Parks

Number of Class Hours:

16-17 hrs. Unit

Design: Project-based Unit Other Subject

Areas/Disciplines Addressed in the Unit:

Physics Statistics Economics

Driving Question: Can you design the ultimate ride for a new amusement park?

Mathematics Content Addressed:

Modeling of motion using linear, quadratic and/or sine waves. Data collection and analysis, including normal distribution. Use of geometric scale modeling

Mathematical Process Goals Addressed

_X_ Problem Solving _X_ Communication _X_Reasoning _X_Connections _X_Representations

Assumption of Prior Knowledge:

Linear functions and modeling Quadratic Functions

Courses for Which the Unit is Appropriate

AFDA, Algebra 2 or above

College and Career Readiness/21st Century Skills

BIE Page 35-37

T for skills to be taught and expected to use during the unit,

E for skills student are expected to know and be able to use during this unit

A for skills that will be assessed during this unit

_E, A_Collaboration _E_Research _E,A_Communication (Oral and/or Written)

_T,A_Technology _E,A_Critical Thinking/Decision Making

_A_Other: (Describe) Creativity

Major Student Products and/or Performances:

Group

Oral presentation Design of the ride

Presentation Audience:

x Class

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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 Individual Written reflection Expert Community Other:

Engage the students interest and inquiry:

Class discussion about the “best ride” at area amusement parks

Evaluation: Formative

Assessments (During the Unit)

Interview x Practice Presentations

Mathematicians Journal x Notes

Preliminary

Plans/Outlines/Prototypes

x Checklists

Rough Drafts Concept maps

Field Tests Other: quiz x

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

x

Other Product(s) or

Performance(s), with a rubric

x Other:

Resources Needed: On-site people, facilities:

Physics teacher Engineering teacher

Equipment/Technol ogy:

Computers, graphing calculators

Materials: Poster board paper, rulers, writing utensils

Community Resources:

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

Mathematicians Journal x Small/Focus Groups x

Whole Class Discussions x Fishbowl Discussions

Survey Other: written product x

Material Adapted From:

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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

Quick Snapshot for the Sequence of Unit Activities

UNIT TITLE: The Amazing Amusements MINUTES PER CLASS 90 minutes

D a y 1 D a y 2 D a y 3 D a y 4 D a y 5

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

Notes

Whole class discussion about amusement rides and introduce project: HO#1

Assign groups and roles

Group task on linear modeling and quadratic functions: HO#2

Math journal entry

Quadratic modeling of pumpkins: HO#3

Project brainstorming

Individual modeling assessment HO#4

Project work time

Project update

Peer evaluation rubric HO#5

Statistics activity (letters in

name) HO#6

Project work time

Project workday

Give rubric HO#7

Group interviews

D a y 6 D a y 7 D a y 8 D a y 9 D a y 1 0

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

Notes

Individual statistics assessment HO#8

Ferris wheel activity HO#9

Project work time

Preliminary design due for peer evaluation (rubric): HO#7

Finalize design and draft advertisement for their ride

Complete advertisement

Plan oral presentation

Begin work on individual paper

Captain meeting

Finalize individual papers

Practice for oral presentation

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What do students need to know and be able to do to complete the unit successfully?

Project: The Amazing Amusement

Analyze each major task and the final project for the unit for the knowledge and skills necessary to produce a high-quality product.

How and when will students assess their own necessary prior knowledge and skills?

How will they remediate their own gaps or weaknesses in expected prior knowledge and skills?

KNOWLEDGE AND SKILLS NEEDED Assumed

already learned Students will self-assess Will be learned and assessed during the unit 1. Linear functions and modeling x

2. Quadratic functions and modeling x

3. Measures of central tendency x x

4. Normal distribution and z-scores x

5. Periodic data modeling x

6. Scale drawing x

7.

8.

What project tools will students use to monitor their progress through the unit and especially the project?

X Student developed Know/need to know lists

X Student Mathematician’s Journals X Student developed Task lists X Teacher developed Rubrics

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Unit Title: The Amazing Amusement

Driving Question: Can you design the ultimate ride for a new amusement park?

Project Description: The design and advertisement of a new amusement park ride, including a written product and oral presentation

ENGAGE

How will

students’ interest be piqued so they want to engage in the inquiry in this project-based unit?

Number of min _30 __

1. Teacher leads class discussion about popular amusement park

rides.

“What is your favorite amusement park ride?” “Why?” “What is the maximum drop, the maximum velocity? “ “How long is the ride?”

“What amusement park has your favorite ride and what is the entry fee for that park?”

HO#1: Discussion questions with photos of rides

2. Assign groups (3 – 4 students) and roles including a captain

Mathematician Journal Prompts

EXPLORE

Teacher provides guidance for the explorations to prepare students with the

knowledge and skills to engage in the project-based unit.

Students will self-assess on the prior knowledge and skills assumed for the unit

Number of hours_4.5__

Group tasks involving linear and quadratic modeling

Activity 1: HO#2: Linear and Quadratic Functions Review Students will work together to complete a review of linear and quadratic functions and modeling(HO#2). This activity reviews using graphs to model linear and quadratic functions.

Materials/Equipment/Resources Needed: HO#2, HO#2A

Activity 2: HO#3: Pumpkin Chucking

Students are expected to graph data by hand and using vertex form find a quadratic equation of best fit. Students use the graphing calculator to graph another set of data and using standard form find the equation of best fit. Students use their equations and graphs to make predictions.

(HO#3: Pumpkin Chucking) Answer sheet for teachers HO#3A

Students need Graphing calculators

List of support linear websites

http://home.windstream.net/okrebs/page4.html Mathematician Journal Prompts What additional information do you need to complete this project?

21 Century Teaching and Learning Inquiry Learning Project-based Learning Unit

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/Notes/1a%20-%20linear%20functions.htm

Activity 3: Project brainstorming

Students are in groups to begin working on their project. Possible teacher posed questions to assist student groups?

What type of park do they want to design?

Is there a theme?

What will make the owner pick their design? Costs?

Activity 4: HO#6: What is in a Name?

Students are expected to collect data and calculate the measures of central tendency graph the data, sketch the normal curve for the data and calculate probabilities of the normal distribution.

Answer sheet for teachers HO#6A Students need graphing calculators

Activity 5: HO#9 Ferris Wheel

Students are expected to model the motion of a ferris wheel using a cosine function with transformation and make predictions.

Answer sheet for teachers HO#9A Students need graphing calculators.

EXPLAIN

Teacher introduces the project and prepares students to work independently in small groups Number of Hours_0.5_

The teacher launches the project and explains the following:

1. The final project is to design a ride and to create an

advertisement that will convince an interested buyer to purchase your idea.

2. The expectations are as follows: to write a letter describing

the product, the mathematics that was used, and the reasons that support those ideas.

3. Present your project to the class.

HO#7: Assessment Rubrics

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ELABORATE

The student groups are working independently with teacher consultations.

Number of Hours__9___

Student groups will develop the advertisement, create the visual and write the paper for their new ride.

Teacher interviews student groups regarding project progress to date.

Teacher asks students to peer assess preliminary designs of other groups (HO#7)

Teacher meets with group captains to determine project completion.

EVALUATE

Working groups submit products or make

presentations

Number of Hours_1.5__

Teacher collects peer evaluation rubrics (HO#5)

Teacher gives two assessments . ( HO#4 and HO#8)

Teacher grades the project using the rubrics (HO#7)

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Amusement Park Rides

What makes an amusement park ride thrilling or popular?

Check out some interesting facts about the Anaconda Ride at Kings Dominion:

Maximum speed 50mph

Sends visitors into a 360 ° vertical loop after exiting the tunnel tower

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HO#2

Linear and Quadratic Functions Review The Dance Class

A dance class meets once a week for 15 weeks in a studio.

Suppose that:

- it costs m dollars to rent the studio for 15 weeks.

- the number of students is n.

- each student pays f dollars per course

- the teacher makes a profit of p dollars at the end of the course.

1. Given that m = $500 and f = $70, write an equation to show how the profit, p depends on n,

the number of students.

2. Graph the equation and explain the significance of the point where the graph crosses the

y-axis.

3. What is the minimum number of students required in order for the teacher to make some

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The Sidewalk Stones

In Prague some sidewalks are made of small square blocks of stone.

The blocks are in different shades to make patterns that are in various sizes.

Pattern #1:

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Pattern #3:

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2. Which pattern has a total of 841 gray blocks?

3. How many white blocks has that pattern?

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HO#2A

Linear and Quadratic Functions Review - Answer Key

The Dance class

A dance teacher runs a dance class for 15 weeks. The class meets once a week in a studio. Suppose that:

- it costs the teacher m dollars to rent the studio for 15 weeks.

- the number of students is n.

- each student pays a single fee of f dollars.

- the teacher makes a profit of p dollars at the end of the course.

1. Given that m = 500 and f = 70, write an equation to show how the profit, p depends on n, the number

of students.

p  70n500

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14 .

7

70

500

0

500

70 







n

therefore, at least 8 students.

The Sidewalk Stones

In Prague some sidewalks are made of small square blocks of stone.

The blocks are in different shades to make patters that are in various sizes.

1. How many blocks of each kind will pattern # n need?

Pattern #1:

White stones:

1 

3 

4

Gray stones: 12432

Pattern #2:

White stones:

2 

5 

4

Gray stones: 22452

Pattern #3:

White stones:

3 

7 

4

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Pattern #n:

White stones:

n



(

2 n



1 )



4 

8 n

2



4 n

Gray stones:

n

2



4 

(

2 n



1 )

2



8 n

2



4 n



1

2. Which pattern has a total of 841 gray blocks?

Students should write and solve the following quadratic equation:

2

21 ,

10

841

1

4

8

2







n

In conclusion pattern #10 has 841 gray blocks.

3. How many white blocks has that pattern?

There are 840 white blocks in that pattern.

4. Explain your work and show your calculations.

Students can use multiple ways to find the solutions; these include:

- solving a quadratic equation using factoring

- solving a quadratic equation by graphing

- solving a quadratic equation using the quadratic formula

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Pumpkin Chucking

You are attending the Fall Festival. There is a pumpkin tossing contest where you can win a prize for guessing the distance a pumpkin will be tossed or for figuring out how to get the longest distance. You have been watching the pumpkins in action and have collected the following data.

A) Sketch a scatter plot of the data. B) What do you notice about the data?

C) Find an equation that fits the data.

D) Using your model, predict the distance for a 33◦ angle toss.

E) Using your model, how would you achieve the longest distance?

Angle Distance

20 272

30 362

40 409

50 401

60 337

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The following table shows the population of a town from 1996 to 2004. Assume that t is the number of years since 1996 and P is measured in thousands of people.

Year,t 0 1 2 3 4 5 6 7 8

Population, P 22.8 25 26.5 27.1 27.8 28.1 27.9 26.9 26.1

a) Put the data on the graphing calculator.

 Press STAT, ENTER

 Put year data into L1 and population data into L2

b) Turn on the scatterplot for this data.

 Press 2nd Y= ENTER

 Turn on plot 1 for your data

 Create a “good” window for your data and press GRAPH

c) What do you notice about this data?

d) Use a graphing calculator to find the best fitting model for the data.

 Press STAT, right arrow(calc), 4 (line) or 5 (quad)& press ENTER

 Round numbers to two decimal places if necessary

e) Using your model, what is the population in 2007?

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Pumpkin Chucking

You are attending the Fall Festival. There is a pumpkin tossing contest where you can win a prize for guessing the distance a pumpkin will be tossed or for figuring out how to get the longest distance. You have been watching the pumpkins in action and have collected the following data.

B) What do you notice about the data?

Non linear, goes up and then down

C) Find an equation that fits the data.

y= -.3(x – 45)2 + 415

D) Using your model, predict the distance for a 33◦ angle toss.

x = 33 y = 372

E) Using your model, how would you achieve the longest distance?

415

The following table shows the population of a town from 1996 to 2004. Assume that t is the number of years since 1996 and P is measured in thousands of people.

Year,t 0 1 2 3 4 5 6 7 8

Population, P

22.8 25 26.5 27.1 27.8 28.1 27.9 26.9 26.1

Angle Distance

20 272

30 362

40 409

50 401

60 337

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a) Put the data on the graphing calculator.

a. Press STAT, ENTER

b. Put year data into L1 and population data into L2

b) Turn on the scatterplot for this data.

a. Press 2nd Y= ENTER

b. Turn on plot 1 for your data

c. Create a “good” window for your data and press GRAPH

c) What do you notice about this data?

Non linear, looks quadratic

d) Use a graphing calculator to find the best fitting model for the data.

a. Press STAT, right arrow, 4 (line) or 5 (quad)& press ENTER

b. Round numbers to two decimal places if necessary

y= -.21x2 + 2.08x + 22.96

e) Using your model, what is the population in 2007?

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Individual Modeling assessment

A bottle is filled with water. The water is allowed to drain from a hole made near the bottom of the bottle. The table shows the level of the water y measured in centimeters from the bottom of the bottle after x seconds.

Time (s) 0 20 40 60 80 100 120 140 160 180 200 220

Water level (cm)

42.6 40.7 38.9 37.2 35.8 34.3 33.3 32.3 31.5 30.8 30.4 30.1

Find and graph a linear regression equation and a quadratic regression equation.

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Individual Modeling assessment

A bottle is filled with water. The water is allowed to drain from a hole made near the bottom of the bottle. The table shows the level of the water y measured in centimeters from the bottom of the bottle after x seconds.

Time (s) 0 20 40 60 80 100 120 140 160 180 200 220

Water level (cm)

42.6 40.7 38.9 37.2 35.8 34.3 33.3 32.3 31.5 30.8 30.4 30.1

Find and graph a linear regression equation and a quadratic regression equation.

Determine which equation is a better fit for the data.

Linear model:

Quadratic model:

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HO#5

Collaborative work skills Peer evaluation rubric

Category 4 Leader 3 Assistant 2 Novice 1 Needs to

improve

Working with others

Almost always listens to, shares with and supports the efforts of others;

keeps the group members working

well together

Usually listens to, shares with, and supports the efforts

of others; does not cause “waves” in

the group.

Often listens to, shares with, and supports the efforts of others, but sometimes is not a good team

member.

Rarely listens to, shares with, and

supports the efforts of others.

Often is not a good team player.

Focus on the task

Consistently stays focused on the task

and what needs to be done; he/she is very self-directed.

Focuses on the task and what needs to be done most of the

time. Other group members can count

on this person.

Focuses on the task and what needs to be done some of the time.

Other team members must sometimes nag, prod, and remind to keep this person

on-task.

Rarely focuses on the task and what needs to be done. Lets others do the

work

Quality of Work

Provides work of the highest quality.

Provides good quality work.

Provides work that occasionally

needs to be checked/redone by

other group members of ensure quality.

Provides work that needs to be checked/redone by

others to ensure quality.

Attitude

Is never publicly critical of the project

or the work of others. Always has

a positive attitude about the task(s).

Rarely is publicly critical of the project or the work of others. Often has a positive attitude

about the task(s).

Occasionally is publicly critical of

the project or the work of other members of the group. Usually has

a positive attitude about the task(s).

Often is publicly critical of the project or the work of other members of the group. Often has a

negative attitude about the task(s).

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“What is in a Name?”

A) Class Data collection: Each student will count the number of letters in their first

name and record their data in the table below.

B) Calculate the mean for this data.

C) Calculate the median for this data.

D) Calculate the mode for this data.

E) How do these three measures of central tendency compare?

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G) If we sampled the whole school, how do you think these measures would change?

H) Which of these measures do you think most accurately represents the typical length

of a student’s first name? Why?

I) Calculate the standard deviation for this data.

J) Draw a bar graph for this data

K) Do you notice any patterns with this graph?

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N) What’s the probability that someone’s first name is between 5 and 10 letters?

O) Suppose there are 1500 students in this school, about how many students would you

expect to have a first name less than 4 letters long?

P) Another class collects last name data. They find that the length of a last name is also

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HO#6A

“What is in a Name?”

A) Class Data collection: Each student will count the number of letters in their first name and

record their data in the table below.

5 6 5 5 4

7 6 3 5 5

6 6 6 7 5

6 4 9 7 3

7 3 4 5 6

B) Calculate the mean for this data.

5.4

C) Calculate the median for this data.

5

D) Calculate the mode for this data.

5 and 6

E) How do these three measures of central tendency compare?

All are pretty close to 5

F) Do we appear to have any outliers?

Maybe the 9

G) If we sampled the whole school, how do you think these measures would change?

Answers will vary

H) Which of these measures do you think most accurately represents the typical length of a

student’s first name? Why?

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1.4

J) Draw a bar graph for this data

K) Do you notice any patterns with this graph?

Bell curve

L) Is this data “normal”? Why or why not?

Pretty normal due to bell curve shape and mean=median=mode

M) Draw the normal curve for this data

N) What’s the probability that someone’s first name is between 5 and 10 letters?

61%

O) Suppose there are 1500 students in this school, about how many students would you expect to

have a first name less than 4 letters long?

240 students

P) Another class collects last name data. They find that the length of a last name is also

normally distributed and has a mean of 8 and a standard deviation of 1.8. Is it more unusual to have a first name of 9 letters or a last name of 12 letters?

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HO #7

RUBRICS

Oral Presentation Rubric

Outstanding 5

 Exceptional knowledge of subject

 Effective integration of visual aids

 Elaborates on topic enthusiastically and in detail

 Well organized with effective transitions

 Polished confident delivery: appropriate volume, clear

articulation, and solid eye contact

4 Satisfactory

3

 Adequate knowledge of subject

 Used visual aids

 Communicates topic clearly

 Logically organized

 Satisfactory delivery

2

Needs Improvement 1

 Minimum knowledge of subject

 Inadequate use of visual aids

 Unclear communication of topic

 Poorly organized

 Unsatisfactory delivery

Overall Oral Presentation rating:

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Advertisement (Visual) Rubric

Outstanding 5

 Information displayed is result of thorough, insightful research

 Mastery of all artistic principles: balance, rhythm, unity, contrast,

and emphasis

 A very creative, professional product with audience appeal

 No errors in spelling, grammar, or punctuation.

4 Satisfactory

3

 Information displayed is result of sufficient research

 Some evidence of artistic principles: balance, rhythm, unity,

contrast, and emphasis

 A neat and functional product

 Few errors in spelling, grammar, or punctuation.

2

 Information displayed is result of insufficient research

 Little evidence of artistic principles: balance, rhythm, unity,

contrast, and emphasis

 A product that lacks visual appeal; not carefully prepared

 Too many errors in spelling, grammar, or punctuation.

Overall Advertisement (Visual) rating:

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Essay/Paper Rubric

Outstanding 5

 Exceptional knowledge of topic demonstrated

 Catchy, interesting and clear introduction

 Carefully organized

 Works cited page included

 Few, if any, errors in grammar, spelling, usage and

mechanics

4 Satisfactory

3

 Adequate knowledge of topic demonstrated

 Clear introduction

 Somewhat organized

 Works cited page incomplete

 Some errors in grammar, spelling, usage and mechanics

2

 Minimal knowledge of topic demonstrated

 Ineffective or unclear introduction

 Poorly organized

 Works cited page missing

 Too many errors in grammar, spelling, usage and mechanics

Overall Essay/Paper rating:

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Individual Statistics Assessment

Refrigerator temperature in degrees Fahrenheit

36 38 35 33 33

35 32 34 37 37

36 35 36 34 35

34 35 33 36 34

37 36 32 35

1) Calculate the

a) mean b) median

c) mode d) standard deviation

2) Draw the bar graph for the above data.

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4) Draw the normal curve for this data including the z-scores.

5) What is the z-score for 34◦ ?

6) What percent of the data is between 33◦and 37◦ ?

7) What would be the temperature of the coldest 10% of the refrigerators?

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Individual Statistics Assessment

Refrigerator temperature in degrees Fahrenheit

36 38 35 33 33

35 32 34 37 37

36 35 36 34 35

34 35 33 36 34

37 36 32 35

1) Calculate the

a) mean b) median

34.916 35

c) mode d) standard deviation

35 1.613

2) Draw the bar graph for the above data.

3) Does this data look normal? Why or Why not?

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4) Draw the normal curve for this data including the z-scores

5) What is the z-score for 34◦ ?

-.625

6) What percent of the data is between 33◦and 37◦ ?

79%

7) What would be the temperature of the coldest 10% of the refrigerators?

Below 33

8) If you tested 300 refrigerators, how many would you expect to have a temperature above 36◦ ?

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Ferris’s Wheel

When you ride a Ferris Wheel, your distance (y) in feet from the ground, varies with time (x) in minutes since you got on the wheel. Suppose that the Ferris Wheel at the County Fair has a diameter of 40 feet and that its axle is 25 feet above the ground. This super-fast wheel makes 3 revolutions per minute.

1) Draw a picture of this Ferris Wheel and label given information.

2) At x=o seconds, you get on the Ferris Wheel. How long does it take to get to the top of the

wheel? How long until you are back to the starting point?

3) Fill in the following data chart.

Time in seconds (x) Distance in feet (y)

0

5

10

15

20

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Because the distance varies periodically with the time, we need a periodic function to fit this graph. We will be using a cosine curve.

y=Acos(B(x - C)) + D

We will need to determine the values of A, B, D and C. These values are transformations of the parent function y=cos(x).

5) D = mid-line of the wave (which is the vertical shift). Find D for our graph.

6) A = the amplitude of the wave (which is how high/low the waves goes from the mid-line). Find

A for our graph.

7) C = the time of the first maximum of the wave (which is the horizontal shift). Find C for our

graph.

8) B= the number of cycles in 360˚ = 360 ÷ period (time for one cycle). Find B for our graph.

9) Write the cosine equation for our graph.

10)Graph this equation on your calculator with an appropriate window. How well does it match your

sketch in #4?

11)What is your distance after riding the Ferris Wheel for 17.5 minutes?

12) How long have you been on the Ferris Wheel when your height reaches 35 feet for the second

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Ferris’s Wheel

When you ride a Ferris Wheel, your distance (y) in feet from the ground, varies with time (x) in minutes since you got on the wheel. Suppose that the Ferris Wheel at the County Fair has a diameter of 40 feet and that its axle is 25 feet above the ground. This super-fast wheel makes 3 revolutions per minute.

1) Draw a picture of this Ferris Wheel and label given information.

5) At x=o seconds, you get on the Ferris Wheel. How long does it take to get to the top of the wheel? How long until you are back to the starting point?

10 seconds, 20 seconds

6) Fill in the following data chart.

Time in seconds (x) Distance in feet (y)

0 5

5 25

10 45

15 25

20 5

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Because the distance varies periodically with the time, we need a periodic function to fit this graph. We will be using a cosine curve.

y=Acos(B(x - C)) + D

We will need to determine the values of A, B, D and C. These values are transformations of the parent function y=cos(x).

5) D = mid-line of the wave (which is the vertical shift). Find D for our graph. 25

2) A = the amplitude of the wave (which is how high/low the waves goes from the mid-line). Find A for our graph.

20

3) C = the time of the first maximum of the wave (which is the horizontal shift). Find C for our graph.

10

4) B= the number of cycles in 360˚ = 360 ÷ period (time for one cycle). Find B for our graph.

18

5) Write the cosine equation for our graph.

y=20cos(18(x – 10)) + 25

6) Graph this equation on your calculator with an appropriate window. How well does it match your sketch in #4?

Very well

7) What is your distance after riding the Ferris Wheel for 17.5 minutes?

45 ft

8) How long have you been on the Ferris Wheel when your height reaches 35 feet for the second time?