Thrill U. - Dorney Park

Transcription

Thrill U.
THEPHYSICSANDMATHOFAMUSEMENTPARKRIDES
Geometry
© Copyrighted by Dr. Joseph S. Elias. This material
is based upon work supported by the National
Science Foundation under Grant No. 9986753.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Dorney Park/Kutztown University
Thrill U. - Geometry
Introduction
Welcome to Thrill U.!
This set of mathematics activities focuses on Geometry. We believe there is something for
everyone as the collection represents a breadth of adventures. Collectively, the geometry activities span
a wide range of topics found in informal geometry, plane geometry, analytic geometry and foundations
of trigonometry. Some activities require straightforward data collection and calculations, many
require collaboration and some involve interdisciplinary considerations. Several of the activities can be
easily adapted for special needs students, some can be adapted for use as follow-up activities, and still
others are appropriate for schools with intensive scheduling programs.
Each activity is preceded by an "Information Sheet" to help guide teachers in selecting
appropriate activities for their particular group(s) of students. In addition to identifying objectives
based on state and national mathematics standards, these pages provide a list of equipment needs and
suggestions that may lead to the activity's successful completion. Teachers should feel welcome to
adapt activities to the specific needs of their students. A "solution manual" may be requested by a
teacher by contacting Dr. Joseph S. Elias at [email protected].
We invite any questions you may have prior to your participation in Thrill U. as well as
constructive criticism following your adventure.
Join us in May and challenge your students to experience geometry in action!
Dr. Kathleen Dolgos
Professor Emeritus
College of Education
Kutztown University
Dr. Deborah Frantz
Department of Mathematics/CIS
College of Liberal Arts and Sciences
Kutztown University
Thrill U.
Table of Contents
Acknowledgments
Page
i
Tips for Mathematics Teachers
Page
ii
Things to Bring/Dorney Park Information
Page
iii
Algebra Activities (separate manual)
Pages 1 - 60
Geometry Activities
Specific Rides:
The Antique Carrousel
Page
61
The Ferris Wheel
Page
69
The Sea Dragon
Page
75
Steel Force
Page
81
White Water Landing – The Bridge
Page
85
A Geometric Walk
Page
91
Polygons in the Park
Page
98
Rides That are Inscribed
Page
103
The Fish Pond
Page
110
Features in the Park:
Dorney Park/Kutztown University
Acknowledgments
During the winter of 1997, area teachers of physics and mathematics,
professionals from Dorney Park, and faculty from Kutztown University gave birth to
Thrill U.: The Physics and Math of Thrill Rides. In May 1999, the Thrill U. - Physics
Planning Committee presented physics activities in the form of Thrill U.
The Thrill U. - Mathematics Planning Committee was established in the fall of
1999. The Committee began by creating algebra activities that would help teachers
achieve goals set forth by the state and national “Mathematics Standards.” As a result,
the mathematics component of Thrill U. consisted of algebra activities that were
introduced in May 2001. Geometry activities have been developed and have been a part
of the mathematics component since May 2002.
Thrill U. is the culmination of effort and time of many people. Its existence
would not have been possible without the collaborative efforts of: the professional staff
at Dorney Park and Wildwater Kingdom; the administrators and academic faculty at
Kutztown University; teachers who had taken students to the Park and provided
feedback; and (most importantly) members of the planning committees. Each planning
committee consists primarily of area high school teachers of physics or mathematics.
The leadership and creativity of these teachers resulted in impressive sets of activities.
Members of all planning committees have worn out shoes in the Park, endured days of
inclement weather, fretted over success (or failure) of their students to complete
preliminary versions of activities while in the "piloting" stages, spent many hours in
meetings, and countless hours designing and editing the activities. In short, we admire
and appreciate the efforts of all who have contributed to the success of the Thrill U.
project.
Geometry Planning Committee
Mrs. Susan Barnett
Ms. Karen Comegys
Mrs. Terri Costenbader
Dr. Kathleen Dolgos
Dr. Joseph Elias
Dr. Deborah Frantz
Mrs. Rose Gadbois
Mr. Keith Koepke
Mr. Fran McGouldrick
Mrs. Brenda Snyder
Mr. Charles Waitkus
Northwestern Lehigh High School
William Allen High School
Allentown Central Catholic High School
Kutztown University
Kutztown University
Kutztown University
Dorney Park and Wildwater Kingdom
Kutztown University
William Allen High School
i
KUTZTOWN UNIVERSITY OF PENNSYLVANIA
Tips for Mathematics Teachers
Think of this as an adventure! To help make a "stress-free" day at the Park, we have created this list
of suggestions to guide you through your planning stage of Thrill U..
•
Above and beyond all else, bring your sense of humor. Experienced teachers know that there
will be mistakes. Allow students to have fun as well as complete your selection of activities.
•
While mathematics is an "exact" discipline, applications of mathematics are much "less exact."
ALL measurements and collected data will have inherent errors.
Accept it.
•
Please do not forget copies of activity sheets, equipment and supplies. You might also consider
bringing a camcorder to record aspects of the rides for use in the classroom after Thrill U., or to
use as introductory preparation for next year.
•
If your comfort level is low with orchestrating lab-type activities, consider consulting a science
teacher for assistance with logistics.
•
Carefully peruse the complete list of activities and select those that will best fit the needs and
abilities of your students. (That is, do not expect your students to complete all of these
activities!) The difficulty levels are quite varied among the activities. Consider doing parts (but
not all) of some activities. You may modify them, or assign groups of students to them.
•
Some activities take longer than others to complete. Keep in mind that it may be necessary to
observe, ride, or take measurements several times in order to obtain good data.
•
As much as is feasible, introduce the students to the concepts to be studied during the weeks
leading up to the event. Consider planning time in class for calculations and analysis during the
days following the experience.
•
In our opinion, students who may be fearful of some rides should not be forced to ride.
•
Kutztown University students will serve as general assistants for you. They will be stationed at
designated Thrill U. rides from approximately 10:00 A.M. to 2:00 P.M. Inform your students
that they may ask the university students questions related to the activities. University students
will help students discover the "answers," but will not give them answers. Instruct your students
NOT to ask Dorney Park employees to give answers.
•
Teachers are welcome to utilize a designated grove at Dorney Park to chat with other
teachers and members of the planning committees, or to use as a place for your students to
work. Please do not leave equipment and other valuables unattended at the grove.
ii
Things to Bring
We present this list for your convenience and hope that it helps make your day at the Park
enjoyable as well as productive. You may wish to bring some or all of these items with
you to Thrill U.
•
•
•
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•
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•
•
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tickets for you, your students and your chaperones
copies of your selection of activities, enough for your group
pencils and paper
stopwatches
calculators (depends on activities: basic, basic with tangent key, or graphing)
camcorder
clipboards
inclinometers or protractors with a plumb bob
appropriate clothing and perhaps a change of clothing
sunscreen, hats, raincoats
money for food, drinks, phone
measuring tape or string
masking tape
backpacks or plastic bags to keep papers and equipment dry and together
maps of the Park (can be picked up at the entrance to the Park)
a good reserve of energy and enthusiasm for exploration
Dorney Park Information
General Information:
(800) 551-5656 or
Group Sales Information:
(610) 395-2000
(610) 395-3724
For specific questions about ticket sales for Thrill U., call Bernie Bonuccelli at (610)3953724 or e-mail him at [email protected]
Visit our website:
Dorney Park Thrill U.
iii
Information Sheet
topics:
central angle
interior and exterior angles
inscribed polygon
sector of a circle
arc length
objectives:
to use the number of sectors in various circles to determine the
measure of central angles and number of sides of inscribed
polygons.
to examine the relationship between the number of sides of the
inscribed polygons, the measure of each central angle, and the
measure of each exterior angle of these polygons.
to approximate the arc length of a sector of a circle and use this to
calculate the radius of the circle using the formula
arc length = (measure of central ÷360°)  2r
equipment:
pencil
calculator (basic four function, non-graphing type)
activity sheets
extra paper
25-foot tape measure
notes to the teacher:
Prior to your visit to Dorney Park, you may want to introduce the
Polygon Grid Activity. Introduce the vocabulary terms associated
with polygons. Please consider that all polygons in this activity are
regular polygons.
In addition to the Antique Carrousel, this activity includes the
circular rides Waveswinger, Enterprise and Apollo 2000.
61
Please notice that there are two carousels at Dorney Park. The Antique Carrousel
has 4 rows of horses mounted to the rotating platform. Be sure you are doing this
activity at the Antique Carrousel.
Before you get started on this activity, let’s define some terms.
F
Sector of a circle
E
A
CAB is a central angle.
B
C
D
CBD is an exterior angle of the polygon
inscribed in circle A. It is adjacent and
supplementary to interior CBE.
AF , AE , and arc FE bound sector FAE. A
slice of pizza is a good illustration of a sector
of a circle.
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Study questions for the Antique Carrousel:
1.
Observe the structure that encloses the Antique Carrousel and decide how many
sides are in this regular polygon. Calculate the measure of one central angle, the
measure of one interior angle and the measure of one exterior angle for this
polygon.
Number of
sides
in polygon
2.
Measure of one
central 
Measure of one
interior 
Measure of one
exterior 
Now look at the Antique Carrousel. As the ride turns, count the number of sectors
in the top of the carousel. Each sector is bounded by rows of lights.
How many sectors are there? _________________
This is one sector
of the ride.
Complete the grid:
Number of
sides
in polygon
Measure of one
central 
Measure of one
interior 
Measure of one
exterior 
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3.
Now take a look at the fence around the Antique Carrousel.
Support pillar
Measure from here
To here
Measure the distance between two adjacent fence posts. ___________________
We will use this distance as an approximation of the length of the arc for the circular area
bounded by the fence. We will assume that all fence sections are congruent and are the
same width as each of the 8 pillars, which hold up the roof over the carousel. There are
40 fence sections and 8 pillars.
Now that you know the arclength of 1/48th of this circle, we are going to find the length
of the radius of the circle.
a.
First find the measure in degrees of the central angle for this 48-gon.
Use π = 3.14
Measure of central =_____________
b.
Find the radius by solving this for r.
arclength = (measure of central 360)  2  r
Radius = _____________
c.
Area =  r²
Use
to calculate the area of the circle.
Area =_______________
d.
Use
circle.
Circumference = 2  r
to calculate the circumference of this
Circumference =______________
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4.
Now go to the Wave Swinger and complete the grid.
Use this regular polygon
Number of sides in
polygon
5.
Measure of one
central 
Measure of one
interior 
Measure of one
exterior 
Complete a grid for the Enterprise. The radius of the Enterprise is 8.5 meters.
Use this to find the arclength between each pair of consecutive radii. A car serves
as an arc.
Number of sides in
polygon
Measure of one
central 
Measure of one
exterior 
Arc length between
radii
65
6.
Complete a grid for the Apollo 2000
The connecting arms create the polygon you are
studying. Make a sketch of it in this space.
Why do the angle measures in this regular polygon
have repeating decimals?
Number of sides in
polygon
Measure of one central 
Measure of one exterior 
What makes this polygon different from the previous
rides you’ve studied?
Explain here:
66
Polygon Grid Activity
Please complete this activity prior to attending Dorney Park Thrill U. day.
# sides
# angles
# ’s
Sum of
measures of
interior ’s in
polygon
Measure of 1
interior  in
regular polygon
Measure of 1
exterior  in
regular
polygon
Sum of
measures of
exterior ’s in
polygon
3
5
6
7
8
9
10
12
14
16
20
Using the pattern above, fill in the rest of the table:
43
38
23
170
172.5
175
7.2
4
67
If the number of sides is expressed as N, then write a formula for the number of triangles, the
sum of the measures of the interior angles in a polygon, the measure of 1 interior angle in a
regular polygon, the measure of 1 exterior angle in a regular polygon, and the sum of the
measures of the exterior angles in a polygon.
# sides
# angles
# ’s
Sum of measures
of interior ’s in
polygon
Measure of 1
interior in regular
polygon
Measure of 1
exterior  in
regular polygon
Sum of measures
of exterior ’s in
polygon
N
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The Ferris Wheel
Information Sheet
topics:
similar triangles
circumference
velocity
perimeter and area of a polygon
angle measures
objectives:
to estimate the diameter of a circle
to determine the speed that a rider moves on the Ferris Wheel.
to calculate the perimeter, area and other measurements pertinent to
polygons.
equipment:
activity sheets
pencil
calculator
stop watch
ruler and string, or protractor with plumb and straw
Students will need to estimate the diameter of the wheel of the Ferris
Wheel. In order to determine this distance, students may use similar
triangles or trigonometry. This is an important part of this activity. The
activity cannot be completed without determining these diameter.
If using the similar triangle method on the Ferris Wheel, a 6 inch ruler
works better than a 12 inch ruler.
You may wish to direct your students to give all linear measures in feet,
and square measures in square feet.
Special note: To successfully complete this activity, students are not
required to ride the Ferris Wheel.
69
The Ferris Wheel
PART 1:
The thrill rating of this ride is __________.
1. Estimate the diameter of the wheel. This can be done using similar triangles or
trigonometry.
A. Similar Triangle Method: Stand at the curb across from the ride entrance. This
point is approximately 47 feet from the wheel. Hold your ruler up as if to
measure the diameter of the wheel. Stretch the string from the bottom of the ruler
to your eye and measure this length. Use proportional sides of similar triangles to
estimate the diameter of the wheel.
(Note that the length of side B1C1 is greater than the diameter of the wheel.)
C1
d
C
line of sight
A
string
ruler
B
h = height (ground to eye level)
h
10ft.8in.
B1
ground level
B. Trigonometry Method: Stand at the curb across from the ride entrance. This
point is approximately 47 feet from the wheel. Attach the plumb to the
protractor at the vertex mark of the protractor. Attach the straw to the straight
edge of the protractor. Use this device to determine the angle of elevation. Use
the appropriate trigonometric function to estimate the diameter of the wheel.
2. Find the circumference of the wheel.
70
The Ferris Wheel
3. Determine the average time that it takes the wheel to make one complete revolution.
Time at least three revolutions and find the average.
4. Using the results of #2 and #3, determine the velocity (speed) in feet per second that a
rider moves when the wheel is in motion.
5. If the diameter were half as large, but the time to complete one revolution remains the
same, will the rider move faster or slower on the new Ferris Wheel? How much
faster or slower?
71
The Ferris Wheel
PART 2:
1. Determine the average number of revolutions per minute (rpm) that the wheel makes.
2. Determine the angular velocity, , in radians per second.
 = rpm's x 2
60
3. Multiply the angular velocity by the radius of the wheel.
4. Compare this result with the answer to #4 of PART 1. What did you find? Why do
you think that you got your result?
72
The Ferris Wheel
PART 3:
Look at the wheel. You should see an inscribed polygon.
X
C
B
D
Z
A
O
1.
Count the number of sides.
_______________
2.
What special name applies to this polygon?
3.
Is this polygon regular?
______________
4.
Find the measure of AOB.
__________
5.
Find the measure of OAB.
__________
6.
Find the measure of ABC.
__________
7.
Radius OA = __________.
8.
Apothem OZ = __________.
9.
Find the perimeter of the polygon.
10.
Find the area of the polygon.
_______________
__________
__________
73
The Ferris Wheel
11.
DCX is called _______________.
12.
Find the measure of DCX.
13.
How does mAOB compare to mDCX?
14.
What is the sum of the measures of the angles of the polygon?
15.
What is the sum of the measures of the exterior angles (one at each vertex) of the
polygon? __________
__________
______________________________
__________
74
The Sea Dragon
Information Sheet
topics:
Pythagorean theorem
equilateral and isosceles triangles
congruent triangles
corresponding parts of triangles
right triangle trigonometry
objective:
to find the lengths of the stationary beams
to find the degree measure of all of the angles formed by the support
beams
equipment:
activity sheets
pencil
calculator (scientific or better)
The best view of the structure of Sea Dragon is at the exit side of the ride.
75
The Sea Dragon
The stationary support beams of the Sea Dragon form numerous triangles. Your
job is to calculate the measures of the unknown beams as well as all of the angles within
the stationary support structure.
Below is a drawing of the support structure as viewed from the exit side of Sea
Dragon. The solid lines represent actual beams and the dotted lines represent constructed
lines needed for relationships and calculations.
B
E
A
1.
C
D
ABC is an equilateral triangle with side 46 ft.
An equilateral triangle can also be classified as an
a.
AB =
BC =
mBAC =
º
mBCA =
triangle.
AC =
º
mABC =
º
NOTE: BD is an altitude of equilateral triangle ABC.
Therefore BDA  BDC because
mBDA = mBDC =
Consequently, ABD and CBD are
.
º.
triangles.
76
The Sea Dragon
b.
Since ABC is equilateral and BD is an altitude,
AD =
and CD =
.
Explain:
c.
We can now calculate the length of BD using the Pythagorean Theorem on
either ABD or CBD.
(altitude)² + (base)² = (hypotenuse)²
What information have we found that allows us to use the Pythagorean
Theorem?
Using either ABD or CBD, put the mathematical names of the sides into
the Pythagorean Theorem.
(
)² + (
)² = (
)²
Replace the names of the sides with their measures from #1a and #1b and
calculate the length of BD to two decimal places, showing all of your work.
BD =
2.
AED is a
a.
triangle.
mEDA =
º and ED = 20 ft.
From #1b we know AD =
Doing this calculation yields AE =
. How can we calculate AE ?
.
77
The Sea Dragon
b.
AED  CED because
Therefore CE =
.
because
.
c.
To calculate the measure of EAD we can use a trigonometric ratio.
Fill in the blanks for these ratios using side names from AED.
sin(mEAD) =
cos(mEAD) =
tan(mEAD) =
Replace the side names with their measures.
sin(mEAD) =
cos(mEAD) =
tan(mEAD) =
Find decimal values for the ratios above.
sin(mEAD) =
cos(mEAD) =
tan(mEAD) =
Use an inverse function on your calculator to find the angle measure.
Fill in the blanks with the appropriate decimal values from above.
Sin-1
= mEAD
Cos-1
Tan-1
= mEAD
mEAD =
º (to the nearest degree)
= mEAD
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The Sea Dragon
d.
Show two different ways that you can now calculate mECD.
mECD =
e.
Explain how you will find mAED and mCED.
mAED =
3.
º (to the nearest degree)
º
mCED =
º
AEB  CEB because
a.
Use your answer from #1c and information given in #2a to calculate BE.
(Show your work.)
BE =
b.
Based on information found in #1a and #2c show how can we find mBAE.
(Show your work.)
mBAE =
º
The degree measure of BCE =
because
79
The Sea Dragon
c.
How can we determine the measures of ABE and CBE?
Find mABE and mCBE.
mABE =
d.
º
mCBE =
º
How can we determine the measures of BEA and BEC?
Find mBEA and mBEC.
mBEA =
º
m BEC =
º
80
Steel Force
Information Sheet
topics:
angle of depression
right triangle trigonometry
slope
objective:
to approximate and compare the measures of the angles of depression and the
slopes of the first and second hills using right triangle trigonometry and the
definition of slope
equipment:
activity sheets and pencils
calculator
ruler or tape measure
Please remind students that their answers should include appropriate units of
measure.
81
Steel Force
PART 1:
Approximate the angle of depression of the first hill. The vertical drop is given. The length of
the hill can be estimated by counting the number of vertical supports on the track.
A (top of first hill)
angle of depression
Count vertical supports to
find this distance.
vertical drop
(200 feet)
B
1.
C (tunnel entrance)
Find the number of supports from the top of first hill to the point where it enters the
tunnel. (Hint: you can get a good view of these supports while standing in line at
Thunderhawk.)
The number of supports is _______________.
2.
Calculate the length of the first hill.
a.
From the answer in #1 above, subtract 2. ____________
b.
To find the approximate length of the first hill, multiply the number in #2a by 24 feet
(distance between supports). This is the approximate length of the first hill.
The length of the first hill is __________________________________.
3.
Use trigonometry to find mBAC.
mBAC is _______________.
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Steel Force
4.
Find the measure of the angle of depression.
The measure of the angle of depression of the first hill is _______________.
PART 2:
Approximate and compare the angle of depression of the second hill to that of the first hill.
D
top of the second hill
Count the number of
supports to get this length.
E
F
Pace off this distance.
1.
point where the track
crosses the water
Find the horizontal distance between the top of the second hill and the point where the
track crosses the water (EF). Stand on the midway at a point directly below the top of the
second hill. Walk on that midway in the direction of Thunder Creek Speedway. Count
the number of paces you take until you reach the point where the track crosses the water.
The number of paces is _______________.
2.
Measure the length of each pace. (Use the average of three paces to get an accurate
measurement.)
The length of one pace is _______________.
3.
Calculate the horizontal distance (EF).
The horizontal distance between the top of the hill and the point where the track crosses
the water is _______________.
83
Steel Force
4.
Count the number of supports from the highest point on the second hill to the point where
the track crosses the water.
The number of supports is _______________.
5.
Calculate the length of the second hill.
a.
From the answer in #4 above, subtract 2. ____________
b.
To find the approximate length of the second hill, multiply the number in #5a by 24 feet
(distance between supports). This is the approximate length of the second hill.
The length of the second hill is ______________.
6.
Use trigonometry to calculate mEDF.
mEDF is _______________.
7.
Find the measure of the angle of depression of the second hill.
The measure of the angle of depression of the second hill is _______________.
8.
Describe the differences between the angles of depression for the two hills on Steel
Force.
9.
Calculate the slope of each hill.
The slope of the first hill is ________________.
The slope of the second hill is ______________.
10.
Describe the differences in the slopes of the two hills.
84
Information Sheet
topics:
circle circumference as related to radius
linear and degree measure of a circle
simple right triangle trigonometry
objective:
to determine the relative location of the center of the circle
to appreciate the relationship between the circumference and radius of a circle
equipment:
activity sheets
sextant
pencil
calculator
Students must construct a 20 ft. measuring string before arriving at the
park. The string must have markings every 6 inches (½ ft.) and weighted on one
end using fishing sinkers, washers, nuts, or the like. The 6-inch markings should
be measured beginning from the outside of the weight, where the string is
attached.
Students must also construct a simple sextant in order to measure angles
vertically. Materials for the sextant: protractor with one-degree markings, a
straw, fishing line, and a weight. Tie the fishing line to the protractor at the center
of the straight edge (or through the hole if available), tape the straw to the straight
edge of the protractor, tie the weight to the other end of the line so it can swing
freely from the center of the protractor.
Please note:
Due to safety precautions, students must ride this ride to gain access to the bridge.
Remind students that all equipment must be secured when riding the ride.
85
Goal: Determine the location of the center of the circle, formed in part by the arched support of
the center section of the bridge at White Water Landing. (See photo below).
ARCHED
SUPPORT
1.
The arched support as labeled above is an arc of a circle.
a.
How many degrees are in a circle? ______
b.
If the arched support has an arc measure of 12 degrees, what fractional part of the
circle does the arched support represent? (Give your answer as a fraction and as a
repeating decimal.)
fraction______ decimal______________
2.
Find the distance (linear measure) from the top center of the arc to the water.
To calculate this distance, you will be using your sextant.
86
Point X is a point along the fence just to the right of the lamppost (with the red dot).
Point T is a point on the top of the arched support aligned with the middle of the top of the sign.
Point B is a point at the bottom of the bridge below the middle of the bottom of the sign.
Point W (marked in red) is a point on the boat track, at water level, directly below the bridge.
T
104.87
B
X
W
a.
Hold the sextant level with the top of the fence, look through the eyepiece (straw),
and locate point B. (A team member must make sure the fishing line is not
swinging and is aligned with the 90 mark on the protractor part of the sextant,
while you are looking at point B.) Raise the front of the sextant from the
horizontal to locate point T at the same position in the eye piece where point B
was. Have the team member read the degree measure on the sextant (to the
nearest 1/2-degree) when you have T in the correct position.
The measure of BXT = _______.
(Make sure the line has stopped swinging before reading mBXT.)
b.
Following the same procedure as in #2a, locate point B (recall that the line must
be still and aligned with 90), lower the front of the sextant from the horizontal to
locate point W at the same position in the eye piece where point B was. The
measure of BXW = _______.
You will be using the trigonometric function tangent (TAN) on your calculator to
find BT and BW.
side opposite the angle
Given: TAN (angle) = side adjacent to the angle
and BX = 104.87 ft., the distance from the fence to the bridge.
87
c.
Find BT using the tangent.
TAN (mBXT) =
BT
BX
TAN (mBXT) =
BT
104.87 ft.
(Use the measure of BXT from #2a to continue.)
TAN (mBXT) =
BT
104.87 ft.
BT = (104.87 ft.) TAN (mBXT)
BT = _________ft.
d.
Calculate BW using the same procedure as in #2c.
BW
104.87 ft.
(Use the measure of BXW from #2b to continue.)
TAN (mBXW) =
TAN (mBXW) =
BW
104.87 ft.
BW = (104.87 ft.) TAN (mBXW)
BW = ________ft.
Using the information from #2c and #2d, how can you determine the distance
from the top center of the arc to the
water?__________________________________________________________.
TW (arc to the water) = ___________ft.
3.
Find the circumference of the circle formed by the arc of the arched support.
a.
In #1, we determined the fractional part of the circle to be_____________ and the
decimal part of the circle that the arc represented to be________________.
88
b.
Determine the linear measure of the arc using your measuring string. Measure
along the top of the arc from the red dot to either end of the arc. (Measure to the
nearest 1/4 of a foot.)
length of ½ of the arc = _________ ft.
c.
The entire length of the arc is approximately _________ft.
d.
Based on measurements from #2d and #3c and without doing any calculations,
where would you hypothesize (educated guess) the center of the circle to be
located? (above the water, underwater but above the ground, or underground)
_______________________________
__
Explain the reasoning for your conclusion_______________________________
_________________________________________________________________
_________________________________________________________________.
e.
Using the information in #3a and #3c, the circumference of the circle is
__________ft.
Show work here:
4.
Find the radius of the circle.
a.
In #2d, we found the distance from the top center of the arc to the water line to be
_________ft. and the depth of the pool is 2.5 ft.
b.
The circumference of a circle in terms of the radius is calculated by the formula
C = ___________.
In #3e, we found the circumference to be ________ft.
How can we calculate the radius based on the circumference? _______________
_________________________________________________
___.
c.
Doing this calculation yields a radius of _________ft.
89
5.
Based on your calculation in #4c and measurements in #4a, where is the center of the
circle located? (above the water, underwater but above the ground, or underground)
_________________________________________________________________.
Explain your answer:
_________________________________________________________
_______________________________________________________________________.
a.
Does this answer match your hypothesis in #3d? _____________
b.
If not, why do you think your guess was wrong?
____________________________________
___________________________________________________________
__.
90
A Geometric Walk
Information Sheet
topics:
recognition of geometric shapes
area of square, trapezoid, triangle, circle, rectangle
circumference of a circle
volume
ratio
objectives:
to use appropriate formulas for area, volume and circumference
to identify geometric shapes in a real world situation
equipment:
activity sheets
pencil
50-foot tape measure
basic calculator
This is a good activity for working in pairs. Students need to record and measure
as a team.
When working with the outer rim of bricks, use two rows of bricks as a border. In
other words, the identical 4 geometric shapes do not overlap.
Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander for PART 2
91
A Geometric Walk
As you walk through the main gate of the Park, you will enter the plaza situated between
the gate and the Antique Carrousel.
PART 1:
Looking down as you walk into the plaza, you will see four identical geometric shapes
each of which look like Figure 1 below. The shaded portions indicate the location of
bricks. Do not include the geometric shapes that have a center planter.
Figure 1 - The Plaza Shape
A
B
H
G
E
F
D
C
1. Name the different geometric shapes that can be found in this section of the plaza
walkway.
2. Find the areas of the following geometric figures found on the plaza walkway. Do
not include the outer rim of bricks in your calculations.
a. Area of rectangle ABCD
b. Area of trapezoid DEFC
c. Area of square EFGH
d. Area of triangle ABD
92
A Geometric Walk
3. Knowing the areas of the geometric figures in question #2, answer the following:
a. Use the fact that the area of a geometric figure is equal to the sum of the areas of
its parts to show that the area of rectangle ABCD is equal to the sum of the areas
of its parts.
b. Remember that there are four geometric shapes on the entire plaza. What is the
total area of all the trapezoids in all four figures? Check your answer by using the
areas of the rectangle and square.
4. The rectangular shaped figures on the walkway are surrounded by and include
rectangular and square bricks.
a. Find the area of one of the rectangles that includes the outer border of bricks
(Figure 1).
b. Find the ratio of the number of rectangular bricks to the number of square bricks
surrounding and included in one of the four rectangles in the plaza.
93
A Geometric Walk
c. In #2, the areas of the rectangle ABCD and the square EFGH were found. Use
these areas to find the total area of the shaded regions in Figure 1.
d. Use the ratio found in #4b and your answer in #4c to determine the area covered
by the rectangular bricks in the shaded regions in Figure 1.
e. Use the ratio found in #4b and your answer in #4c to determine the area covered
by the square bricks in the shaded regions in Figure 1.
f. Find the area covered by the square bricks by using the measurements for the size
of the brick. Also find the area covered by the rectangular bricks using the
measurements for the size of the brick.
g. Compare the answers from parts d, e and f above. Reflect on your
answers.
94
A Geometric Walk
PART 2 (probability):
What is the percent probability of a bean bag being drop from a helicopter landing on one of the
following patterns? Hint: use the following formula P(target area) = area of target / area of total
a. The square EFGH
b. The trapezoid ABGH
c. Any trapezoid
d. The brown brick boarder of the plaza shape
PART 3:
If you walk further down the walkway past the Antique Carrousel, you will see several
rectangular planters containing trees and mulch on the main Midway. Look at the planters in line
with the fountain at the center of the Midway.
1. Find the area of the first rectangular planter that is not a square.
Use the outside edge of the concrete to measure the sides. Do not include any bricks
surrounding the planter.
95
A Geometric Walk
2. Find the area of the section of the rectangular bed that contains mulch. Explain the
procedure to find the area covered by only mulch.
(Assume that the area to be covered is clear of all electrical boxes, trees, etc.)
3. Given that mulch can be purchased by the cubic yard, estimate the number of cubic
yards of mulch that would be needed to fill one rectangular bed. (One cubic yard of
mulch covers approximately 80 to 100 square feet if spread 3 inches deep.)
Extension for Experts:
4. If the mulch can only be ordered by the bag, how many bags would be
needed? One bag of mulch contains 3 cubic feet of mulch.
96
A Geometric Walk
PART 4:
In the middle of the main Midway, you will find a circular fountain filled with water. The
fountain’s structure contains a middle circular concrete island. The six outer segments are flower
beds.
Figure 2 - The Fountain
1. Find the area and circumference of the circle formed by the outer edge of the fountain
given that the radius of the circle is 97 inches.
Area:
_______ sq. inches
Circumference:
_______ inches
2. Find the area (i.e. the surface area) of the part of the fountain that holds water. The
middle circular island has a radius of 31 inches. The distance from the edge of the
circular island to the interior wall of the fountain is 54 inches. This geometric shape
looks like a doughnut.
3. The usual depth of the water in the fountain is 8.5 inches. Find the volume of
water needed to fill the fountain to this depth.
97
Information Sheet
topics:
classification of a wide variety of polygons
identification of convex, concave, regular and non-regular polygons
objectives:
to recognize and name polygons inherent in the Park
to accurately classify triangles and quadrilaterals
to recognize characteristics of polygons (convex, concave, regular, etc.)
to order the measures of areas of triangles and quadrilaterals
to sketch polygons
equipment:
activity sheets and pencils
Please keep in mind that this activity is not an activity on measurement. Students
will be unable to obtain measurements of angles and lengths of sides for most of
the polygons. Their answers for columns 3, 4, and 5 on the “Answer Grid” will
be based on observations and reasoning skills.
For questions with multiple answers (i.e. #4 and #13), please allow for different
arrangements of the answers.
For responses about the polygon in PART 1, #11, students should respond as
though the hole were not there.
98
PART 1:
The goal of this activity is to be able to accurately describe many different polygons that can be
found in Dorney Park. (Please read through PART 2 before you start PART 1.)
Place all of your answers on the grid that is provided at the end of this activity.
For each of the following,
a.
identify (by its mathematical name) the polygon that is being described,
(If you see a polygonal region, identify the polygon that describes its boundary.)
b.
draw a sketch of what you see;
c.
identify it as regular or not regular; and
d.
identify it as convex or concave.
Your answers will be terms such as: pentagon, hexagon, decagon, dodecagon, icosagon, 36-gon,
triangle, quadrilateral, etc. If you find a triangle or a quadrilateral, list as many names for it as
you can: isosceles, equilateral, right, acute, obtuse, rectangle, square, parallelogram, trapezoid,
or rhombus.
1.
The base (bottom) of a car on the Ferris Wheel has this polygonal shape.
2.
The metal bars that join the supports between the swings on Charlie Brown's Swing form
this polygonal shape.
3.
At Musik Express, each letter of the sentence "MIT MUSIK GEHT ALLES
BESSER" is written on one of these.
4.
The center structure on the wheel of Musik Express is comprised of these three
polygonal regions.
5.
The base of the center support of the Scrambler has this shape.
6.
The roofs on the cars on the Enterprise, laid end to end, form this polygon.
7.
At the top of the Dominator, in a plane parallel to the ground, there is a polygon that
"connects" the three vertical supports. Name the polygon.
8.
The roof of Patio Pizza is decorated with black and white polygons of this type.
9.
This polygon can be viewed on the ground (made from red/brown bricks) and is the
smallest polygon that surrounds the Antique Carrousel.
99
10.
Above the windows at the SUBWAY sandwich shop, there are polygonal signs that say
"PAY HERE" or "ORDER HERE." What is the shape of each sign?
11.
On the center support of the Scrambler, this polygonal region has a circular hole in it.
(When working with this polygon, ignore the hole.)
12.
At the Funzone Arcade, there are two different shapes that make up the three-piece
emblem above the neon "FUNZONE" sign. Identify both polygons.
13.
The arms that hold the cars on Apollo 2000 are connected to each other at the top of the
ride. In a plane parallel to the ground, name the polygon that is formed by the
connections of all the arms of the cars.
14.
The perimeter of the pavilion in Camp Snoopy has this shape.
PART 2:
1.
Arrange the areas of all of the regions enclosed by triangles that were found in PART 1
from smallest to largest. Do this by listing the numbers of the objects. (Do you really
need any formulas or computations of areas?)
2.
Arrange the areas of all of the regions enclosed by quadrilaterals that were found in
PART 1 from smallest to largest. Do this by listing the numbers of the objects. (Do you
really need any formulas or computations of areas?)
100
Answer Grid for Polygons in the Park
Problem number and
Attraction
1. Ferris Wheel
Names for the
polygons
Sketch
Regular or
not regular?
Concave or
convex?
2. Charlie Brown’s
Swing
3. Musik Express
(letters)
4a. Musik Express
(center)
4b. Musik Express
(center)
4c. Musik Express
(center)
5. Scrambler (base)
6. Enterprise
7. Dominator
8. Patio Pizza
101
Problem number and
Attraction
9. Antique Carrousel
Names for the
polygons
Sketch
Regular or
not regular?
Concave or
convex?
10. Subway
11. Scrambler (hole)
12a. Funzone Arcade
12b. Funzone Arcade
13. Apollo 2000
14. Pavilion at Camp
Snoopy
102
Information Sheet
topics:
properties of regular polygons (interior & exterior angles)
area of regular polygon
central angles
area of circle
area of sectors
objectives:
to use properties of regular polygons
to use formulas for determining measures of interior and exterior angles
to calculate lengths and areas associated with polygons inscribed in circles
equipment:
activity sheets
pencil
basic calculator
notes to teacher:
This activity contains three parts.
PART 1 – Chart form investigation of interior and exterior angles.
PART 2 – Question and answer investigation of regular polygons.
PART 3 – Area investigation of inscribed polygons.
PART 1: Regular Polygons – Interior & Exterior Angles Chart
Attendance at the Park is needed to complete the Name of Ride column of the chart.
If time is a factor, the other columns may be completed/calculated at a later time.
http://mathforum.org/dr.math/ is a good source for the polygon names for 11 and
up, if you are not going to use 11-gon, 12-gon, etc.
PART 3: #1, 2, & 3
The circle provided has divisions of 5.
103
PART 1: Regular Polygons – Interior & Exterior Angles Chart
Walk around the park. Look carefully at each ride. You should start to notice that most of the rides
listed are circular in nature. An actual or visualized polygon is inscribed within the circle. For example
an actual polygon exists on the Ferris Wheel. The vertices of the polygon are located at the ends of the
spokes. The Tilt-A-Whirl’s polygon is visualized by joining segments from each seat’s pivot point.
Examples of how the rides may appear (bird’s eye view) as inscribed polygons:
KEY:
1.
Actual structure
Visualized structure
Why do you think most of the rides have polygons with an even number of sides?
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Instructions for Regular Polygons – Interior & Exterior Angles chart.
a.
Determine where the 10 rides listed below fit into the chart.
Ride/Structure list: Antique Carrousel
Apollo 2000
Dominator
Ferris Wheel
Monster
Musik Express
Pavilion near Fossil Find and Snoopy Bounce
Scrambler (configuration of seating area)
Tilt-A-Whirl
Wave Swinger.
b.
c.
d.
Fill in the Name of Polygon column. You may need your math book for this.
Place the # of Vertices in that column.
Complete the Measure of each Exterior  column, using your information from # of
Vertices and Sum of the measures of Exterior ’s columns.
Use the information in your chart to complete the last two columns.
e.
104
Regular Polygons – Interior & Exterior Angles
Sum of the
measures
of
Exterior
’s
3
360
Sum of the
measures
of
Interior 
+
Exterior 
180
4
360
180
5
360
180
6
360
180
7
360
180
8
360
180
*
9
360
180
*
10
360
180
*
11
360
180
12
360
180
13
360
180
14
360
180
15
360
180
16
360
180
17
360
180
18
360
180
19
360
180
20
360
180
Name of Ride
*
*
*
*
*
#
Sides
Name
of Polygon
# of
Vertices
Measure of
each
Exterior 
Measure of
each
Interior 
Sum of the
measures
of
Interior
’s
* See if you can find a ride or structure to match the number of sides.
105
PART 2:
Answer the following questions about polygons:
1.
What makes a polygon regular?
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
2.
Use the figures below. Draw the diagonals from one vertex to help complete the
following chart:
Number of sides
3
4
5
6
7
8
9
n
Number of diagonals
from one vertex
Number of triangles
created by diagonals
Sum of the m’s in a
Sum of the measures of
the interior’s in the
polygon
3.
Write a formula for the sum of the measures of the interior angles of a convex polygon with n
sides.
________________________________________________________________________
4.
Write a formula for the measure of each interior angle of a regular polygon with n sides.
________________________________________________________________________
5.
What is the sum of the measures of the exterior angles of a convex polygon (one angle at
each vertex)?
________________________________________________________________________
6.
Write a formula for the measure of each exterior angle of a regular polygon with n sides.
________________________________________________________________________
106
Select True or False. Support your decision with a brief explanation.
7.
The sum of the measures of the interior angles of a polygon is always a multiple of 180.
True or False Explanation: _____________________________________________
_____________________________________________________________________
_____________________________________________________________________
8.
There is a regular polygon in which the sum of the measures of the interior angles is 310.
_____________________________________________________________________
_____________________________________________________________________
9.
As the number of sides of a regular polygon increase, the measure of each interior angle
will decrease.
_____________________________________________________________________
_____________________________________________________________________
10.
As the number of sides of a regular polygon increase, the sum of its exterior angle
measures will also increase.
_____________________________________________________________________
_____________________________________________________________________
11.
Each exterior angle of a regular hexagon is acute.
_____________________________________________________________________
_____________________________________________________________________
107
PART 3:
In the circles provided, accurately inscribe the regular polygon represented by each of the
following rides: Dominator (completed for you), Scrambler (seat configuration), and Monster.
Draw at least one central angle and one apothem. For each, calculate the length of the apothem,
perimeter of the polygon, measure of one central angle, area of one triangle, and area of the
polygon.
1. Dominator: Draw regular polygon ABC inscribed in circle O.
Given: AB = 40 feet
Find the following:
A. apothem a of polygon ABC = __________
C
B. perimeter of polygon ABC = __________
C. measure of central AOB = __________
D. area of AOB = _______
O
E. area of polygon ABC = __________
A
a
B
EXTENSION:
F. area of sector AOB = __________
G. area of region bounded by
AB and arc AB = __________
H. difference between the area of
polygon ABC and circle O = __________
2. Scrambler (seat configuration): Draw regular polygon ABCD inscribed in circle O.
Given: diameter = 260 inches
Find the following:
A.
apothem of polygon ABCD = __________
B. perimeter of polygon ABCD = __________
D. area of AOB = __________
E. area of polygon ABCD = __________
EXTENSION:
AB and arc AB = __________
polygon ABCD and circle O = __________
108
3. Monster: Draw regular polygon ABCDEF inscribed in circle O.
Given: diameter = 42 feet
Find the following:
A. apothem of polygon ABCDEF = __________
B. perimeter of polygon ABCDEF = __________
D. area of AOB = __________
E. area of polygon ABCDEF = __________
EXTENSION:
AB and arc AB = __________
polygon ABCDEF and circle O = __________
109
The Fish Pond
Information Sheet
Topics:
Circumference
Area
Volume
Trigonometry
Objectives:
To use the circumference of a circle to determine its radius.
To calculate the area and volume of a cylinder given its dimensions.
To calculate the height of an object using trigonometric ratios.
Equipment:
Pencil
Calculator
Activity sheets
Tape measure
String
Inclinometer
KUTZTOWN UNIVERSITY OF PENNSYLANIA
Special thanks to Sara Bechtel, Ken Eicheldinger and Angela Xander
110
The Fish Pond
The historic fish pond has been here since the 1920s. It is located near the entrance to Gasoline
Alley.
Part I:
1. Find the circumference of the pond.
a. Walk around the fish pond to find the gate. It is near the hedge in the direction of
Steel Force.
b. Measure around the inside of the cement base of the pond from middle of post to
middle of post on the gate. Use a string to do so, then measure length with a
measuring tape.
c. Measure one section on either side of the gate. Both are congruent, so we only
need to measure once. Again, use the string and go from post to post around the
inside of the cement base.
d. The remaining sections of fence are congruent. Take a moment to walk around
the pond and count the number of remaining fence sections.
e. Measure one of these sections and record your results here.
f. Calculate the circumference of the fish pond by adding all of your measurements.
Remember to include all sections of fence and the gate.
111
The Fish Pond
2. Calculate the area and volume of the pond.
a. Recall that circumference of a circle is found using the formula C = 2  r. Using
the circumference, find the radius of the pond.
b. In order to calculate the area of the fish pond, we must take into consideration the
fact that there is an island in the center. Given that the radius of the island is 10
feet, calculate the area of the pond. Recall that A =  r2.
c. Given that the depth of the pond measures 4 feet, calculate the volume of the
pond. This will tell us the maximum amount of water that it could be contained
within the pond. Don’t forget about the island in the center!
3. Find the height of the center tower.
a. Using the top railing of the fence as your starting point, steady the inclinometer.
Tilt it upward until the top of the tower is in your sight. Have a teammate read
the angle of inclination.
112
The Fish Pond
b. Using the angle of inclination and the radius of the pond, calculate the height of
the tower from the railing upward. Use the diagram as a reference.
Height
Angle
Radius
c. Because we started at the railing, instead of the ground, this is an incomplete
height. You must now find the height from the ground to the top of the
inclinometer as it rests on the top railing.
d. Adding these two lengths together will give you the height of the tower.
113

Thrill U. - Dorney Park

Transcription

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