Classroom: Teacher’s Guide
High School Math
Classroom: Teacher’s Guide
High School Math
The Port of Long Beach
925 Harbor Plaza
Long Beach, CA 90802
(562) 590-4121
www.polb.com
© 2008 Port of Long Beach
Classroom: Teacher’s Guide
High School Math
Contents
Preface
About Classroom
Introduction
Lesson One
v
Lesson Four
vii
Linear Depreciation
ix
Terminal Objective
Lesson
Keywords
Closure
Worksheet
Worksheet Answers
1
Data Analysis
Terminal Objective
Lesson
Keywords
Closure
1
2
2
8
Lesson Two
9
Percentages
Lesson Five
49
Terminal Objective
Lesson
Closure
Handout
Handout Answers
49
50
56
59
61
63
9
10
14
Lesson Three
15
Lesson Six
15
16
27
Terminal Objective
Lesson
Closure
Handout
Handout Answers
Appendix
Terminal Objective
Lesson
Closure
29
30
30, 35, 41
44
45
47
Clearance
Terminal Objective
Lesson
Closure
Distance Calculations
29
Building a Bridge
63
64
69
71
73
75
contents
| iii
Preface
The Port of Long Beach is an industry-leading, environmentally
friendly, global seaport. Every year about $100 billion worth of
cargo passes across the Port of Long Beach’s docks. Imported cargo
arrives at the Port bound for store shelves, factories and other
destinations locally and across the United States; and exports
leave, bound for foreign ports and international consumers. These
goods include everything from electronics and machinery to food,
cars and petroleum products.
As a key international trade hub, the Port of Long Beach supports
nearly 1.5 million jobs across Southern California and the nation,
and these jobs – as engineers, environmental scientists, freight
forwarders, crane operators and logistics specialists, just to name
a few – require a highly skilled workforce.
Port of Long Beach: Classroom aims to make students aware of the Port
of Long Beach and to prepare them for port career opportunities.
These lessons combine real-world Port of Long Beach situations
with content from the California state-approved curriculum.
The result is an engaging and interactive series of lessons that
fully conform to the state content standards while getting
students excited about the major global seaport right in their own
backyards.
If you want more information about any of the information in
these lessons, please visit our Web site at www.polb.com.
Let’s get started!
preface
|v
About Port of Long Beach: Classroom
High School Math Teacher’s Guide
These math lessons were developed in partnership with the Long
Beach Unified School District. The teacher’s guide should be used
in conjunction with the Powerpoint presentations to provide
an interactive and visual representation of the content. These
lessons are fully aligned to the California State Curriculum and are
intended to complement the teacher’s own lesson plans.
The lessons are divided into several parts, which should be
completed in order.
Terminal Objective: The overall lesson objective
Content Standard Reference: The California Content
Standard to which the lesson teaches
Materials: Materials required for the lesson
Time Required: The estimated number of 1-hour classes
needed to complete the lesson
Introduction of Lesson: Pre-lesson preparation
Anticipatory Set: Information or activities that prepare
students for the upcoming lesson
Student Objective: What the students will get out of the
lesson
about CLASSROOM
| vii
Purpose: Description of real-life applications of the content
and why the skills are important for students
Procedure: The body of the lesson
Input: Information needed to understand the lesson, such
as definitions or formulas
Modeling: A description of how to use the formulas or
complete the problems
Check for Understanding: Practice problems and activities
to check for student learning
Guided Practice: An opportunity for students to apply their
skills to new problems
Closure: A summary of the lesson, which may include
homework or additional activities
viii | about CLASSROOM
Introduction:
Port of Long Beach
In order to understand the math lessons that follow, you will
need to become familiar with the Port of Long Beach, particularly
with some of the special terminology used in goods movement
and international trade. This section and the accompanying
Powerpoint presentation (Port of Long Beach Basics.ppt) provide
background information to facilitate student learning.
About the Port of Long Beach
The Port of Long Beach is the second busiest seaport in the United
States and a major gateway for U.S.- Asian trade. Every year
about $100 billion worth of cargo passes across the Port of Long
Beach’s docks. Imported cargo arrives at the Port bound for store
shelves, factories and other destinations locally and across the
United States; and exports leave, bound for foreign ports and
international consumers.
The Port of Long Beach is a full-service seaport. Everything from
electronics and machinery to food, cars and petroleum products
are shipped through the Port. The Port generates roughly 30,000
jobs in Long Beach, or about 1 in 8 jobs.
Types of Cargo
Cargo coming through the Port of Long Beach is divided into four
categories:
Introduction
| ix
1. Containerized cargo
2. Dry bulk cargo
3. Liquid bulk cargo
4. Break bulk cargo and roll-on, roll-off cargo
Containerized cargo is cargo that comes in containers. These
containers hold just about anything…iPods, tennis shoes,
furniture, you name it.
Containers come in two sizes. The smaller ones are twenty-foot
containers.
The larger containers are forty-foot containers. Most of the
containers you see on the road are 40-foot containers.
A 20-foot container can hold 320 19-inch LCD televisions.
8.6 ft
holds 320
LC D T V s
20 ft
x | introduction
8 ft
A 40-foot container holds 640 of these LCD televisions.
40 ft
8 ft
holds 640 L C D T V s
8.6 f t
The number of containers a ship can hold is measured in TEUs, or
a 20-foot-equivalent. One TEU is one 20-foot container. A fortyfoot container is 2 TEUs. Large ships carry about 8,000 TEUs.
1 TEU = 1 Twenty Foot Equivalent Unit = 1 20-foot container
2 TEUs = 2 Twenty Foot Equivalent Units = 1 40-foot container
8.6 ft
8 ft
20 ft
40 ft
8 ft
8.6 f t
Tell your neighbor:
1. What does TEU stand for? Answer: Twenty-Foot Equivalent Unit
2. What size are the containers you normally see on the back of
semi trucks? Answer: 2 TEUs. They are 40 feet long.
Introduction
| xi
Dry bulk cargo is dry stuff that does not come in containers. It is
measured by weight or volume. Some examples are salt, cement,
gravel, sand and grain.
Tell your neighbor:
1. What is dry bulk? Answer: Dry stuff that does not come in
containers.
2. Give an example. Answer: salt, cement, gravel, sand, etc.
Liquid bulk cargo is wet stuff that does not come in containers.
It is measured by weight or volume. Some examples are oil,
gasoline, and chemicals.
Tell your neighbor:
3. What is liquid bulk? Answer: Wet stuff that does not come in
containers.
4. Give an example. Answer: oil, gasoline, chemicals.
Break bulk cargo is comprised of large or heavy items moved on
pallets, bundles or rolls. Some examples are steel, lumber, paper
on rolls, machinery and food products.
Roll On-Roll Off cargo is comprised of items that are driven
on and off the ship. Examples are cars, trucks, buses and
construction vehicles. Roll on-roll off cargo is often called ro-ro
cargo, pronounced just as it looks – row, row.
Tell your neighbor:
1. What is break bulk? Answer: large or heavy items moved on
pallets, bundles or rolls
2. Give an example of ro-ro cargo. Answer: cars, trucks, buses and
construction vehicles
xii | introduction
Create this table in your notes and fill it in to summarize the
lesson:
Type of Cargo
Description
Example
Containerized
In containers
Shoes, computers, etc.
Dry bulk
Dry material
Salt, etc.
Liquid bulk
Liquid material
Oil, etc.
Break bulk, ro-ro
Large or rolling
Wood, etc.
Introduction
| xiii
Data Analysis Lesson 1
Data Analysis
Lesson one
Terminal Objective
Students will identify claims based on
Content Standard
Reference:
Beach and then analyze and evaluate
Grade 6 Number Sense
6.2.5: Identify claims based on
statistical data and, in simple
cases, evaluate the validity of
the claims.
statistical data involving the Port of Long
the validity of the claims by answering
multiple choice questions.
Materials
Data Analysis PowerPoint
Time Required
1 class
Data Analysis
|1
Data Analysis Lesson 1
Introduction of Lesson
Anticipatory Set
In an effort to help reduce pollution, the Port of
Long Beach collects data on the environment at
the Port. These statistics are used to analyze and
prepare for future anti-pollution efforts.
Student Objective
Students will learn how to answer questions about
bar, circle, and line graphs using Port of Long
Beach graphs.
Purpose
The ability to interpret graphs is critical to
understanding the Port of Long Beach’s economic
impacts and environmental efforts.
Lesson
Keyword
1. Statistics - the organization
and analysis of data
Input
Statistics is the organization and analysis of data.
The data can be stored in charts and displayed in
different ways. These charts include bar graphs,
circle graphs and line graphs.
ba r g raph :
90
80
70
60
50
East
40
West
30
North
20
10
0
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
circle g raph:
O the r
R aw
12%
Ma teria ls
20%
O verhead
26%
2 | Data Analysis
Sa la ries
42%
B u s iness
Ex p ens e
s
Data Analysis Lesson 1
Lesson cont’d
line g raph:
35
30
25
20
15
10
5
1 9 98
1 9 99
2 0 00
2 0 01
2 0 02
2 0 03
2 0 04
2 0 05
2 0 06
Ye a r s
Modeling
Studying graphs like these can be useful in
estimating the Port of Long Beach’s economic
impacts. One of the ways the Port benefits the
economy for the city of Long Beach is by providing
jobs, both full- and part-time employment. The
Port supports nearly 30,000 jobs, or roughly one
out of eight in the city.
According to the circle graph shown…
Retail
Sales
Others
3,079
3, 036
Port user
Port
Industries
5,262
18,407
…approximately how many jobs in the Port are
due to retail sales and port industries?
A 3,036 B 18,407
C 21,443
D 29,784
Input
Steps to analyze graphs:
• Look over the graph
• Carefully read the question
• Read each choice
• Eliminate possible wrong answers
• Decide on a strategy
Data Analysis
|3
Data Analysis Lesson 1
Lesson cont’d
Modeling
Approximately how many jobs in the Port are due
to retail sales and port industries?
A 3,036 B 18,407
C 21,443
D 29,784
Eliminate 3,036 and 18,407.
Add retail sales and port industries together.
3,036 + 18,407 = 21,443
Answer: C
Check for Understanding
Have students explain to their neighbors:
Why is C the best answer for this question?
Guided Practice
According to the circle graph below, which of the
following conclusions is true?
Others
$1.54 billion total
wages/salaries at the
Port of Long Beach
$143.4
million
Retail
Sales
$69.2
million
Port user
$396.4
million
Port
Industries
$934.1
million
A P
ort industries created less than half of the
wages/salaries at the Port.
B Port users created more wages/salaries than the
retail sales and others combined.
4 | Data Analysis
Data Analysis Lesson 1
Lesson cont’d
C M
ore wages/salaries came from others than Port
users.
D The Port’s wages/salaries totaled in the millions
of dollars.
(Correct answer: B)
Modeling
The Port’s pollution control measures have been
instrumental in reducing emissions from cargohandling equipment.
Based on the bar graph above, which of the
following conclusions is true?
A Particulate matter pollution decreased by less
than 50 tons from 2002 to 2005.
B In 2002, 140 tons of particulate matter pollution
were emitted by cargo handling equipment.
C Particulate matter pollution decreased by more
than 50% from 2002 to 2005.
D Th
e region’s particulate matter pollution is
increasing.
(Correct answer: C)
Data Analysis
|5
Data Analysis Lesson 1
Lesson cont’d
Check for Understanding
Is the following statement true or false?
When answering multiple choice questions about
a graph, one needs to eliminate answers before
thoroughly reading the question.
False – you should start by reading the question/
statement
Guided Practice
For questions 1 – 4:
• Answer each of the following questions in your
notes.
• When prompted, check your answers with your
neighbor.
• Be prepared to explain your answers to the class.
1) Which of the following statements are true?
(there may be more than one)
A
issolved oxygen concentration decreased
D
between 1976 and 1980.
B Harbor water quality dropped below the state
minimum in 2003.
C The line graph indicates water quality being
consistently above state minimum standards.
D 1974 and 2005 had approximately the same
oxygen concentration.
(Correct answers – C & D)
6 | Data Analysis
Data Analysis Lesson 1
Lesson cont’d
2) Which time period had the greatest reduction of
airborne dust?
A Jan 00 to Dec 00
C Jan 04 to Sep 04
(Correct answer – A)
B
D
Dec 01 to Mar 02
Sept 02 to Dec 02
3) The Port monitors protected species of birds,
including peregrine falcons, least terns, and blackcrowned night herons, that could be affected by
their projects.
True or False: The average number of birds at the
harbor has more than doubled since the 1970s.
(Correct answer – True)
4) In an effort to restore aquatic ecosystems, the Port
monitors the number and diversity of fish species
in the Long Beach harbor. Using the graph below,
which statement is true?
Data Analysis
|7
Data Analysis Lesson 1
Lesson cont’d
A
Fish diversity has generally decreased from
1997 to 2004.
B The number of fish species increased from 1980
to 1985.
C There were 3 fish species observed per 5 minutes
sampling time in 1993.
D There were 5 fish species observed per 5
minutes sampling time in 2004.
(Correct answer – D)
Closure
Tell students:
In your notes, write a summary of how to read
a graph and answer questions about it.
Exchange papers with your neighbor.
Add any statement to your summary that will
help you to remember today’s lesson.
8 | Data Analysis
Percentages Lesson 2
Percentages
Lesson two
Terminal Objective
Students will solve percent of increase
Content Standard
Reference:
Port of Long Beach by subtracting the two
Grade 7 Numbers and Sense
7.1.6: Calculate the percentage
of increases and decreases of a
quantity.
or decrease word problems involving the
numbers, making a fraction comparing
the difference to the original, and
changing the fraction to a percent.
Materials
Percentages PowerPoint
Time Required
1 class
Percentages
|9
Percentages Lesson 2
Introduction of Lesson
Anticipatory Set:
Every year thousands of cars are brought into the
Port of Long Beach from overseas for sale in the
U.S. They are literally “parked” in the ship for
the ride here. In 2005 approximately 328,000 cars
were brought into the Port of Long Beach. In 2006
approximately 426,000 were brought in. What was
the percent of increase in the amount of cars from
2005 to 2006?
Student Objective:
Students will learn how to solve this question and
other problems involving percent of increase and
decrease.
Purpose:
Port employees need to understand how to
calculate increases and decreases in percentages.
Students will learn about post secondary career
opportunities at the Port of Long Beach.
Lesson
Input
A fraction can be divided to obtain the equivalent
decimal which can be changed to a percent.
Modeling
Use a picture of containers being unloaded from a
ship as an example.
1
1. 1 out of 5 containers = 5 = 20%
2. 2 out of 5 containers= 2 = 40%
5
Input
Find the percent of increase to represent a
situation.
10 | Percentages
Percentages Lesson 2
Lesson cont’d
1. Calculate the increase.
2. Rewrite the problem: increase is what % of the
original number.
3. Write a fraction representing the situation.
4. D
ivide the denominator into the numerator to
three decimal places.
5. Write the percent.
Modeling
In 2005, approximately 328,000 cars were
brought into the Port of Long Beach. In 2006,
approximately 426,000 were brought in. What
was the percent of increase in the amount of cars
from 2005 to 2006?
1. 426,000 – 328,000 = 98,000
2. 98,000 is what % of 328,000?
3. 98,000
328,000
4. 328,000
98,000 = 0.298
5. 30%
Check for Understanding
On the count of three, ask students to call out the
correct letter to the following:
When finding the percent of increase, the
denominator of the fraction needed is the
A) original amount (Yes)
B) new amount (No, this is the number used to
subtract)
C) amount calculated from subtracting (No, this is
the numerator)
Modeling
In 2004, the Port moved more than 5.8 million
Percentages
| 11
Percentages Lesson 2
Lesson cont’d
twenty-foot container units (TEUs). In 2005, the
Port moved more than 6.7 million TEUs. What was
the percent of increase?
1. 6.7 – 5.8 = 0.9
2. 0.9 is what % of 5.8?
0.9
5.8
4. 5.8 0.9 = 0.155
3. 5. 16%
Check for Understanding
Is the following statement true or false?
When finding the percent of increase, the first step
is to add the original amount to the new amount.
False – it is subtracted from
Guided Practice
From 2004 to 2005, the Port saw more “calls” by the
newest generation of 8,000-TEU mega-ships. In
2004, the Port container count was approximately
2.94 million TEUs. In 2005, the Port container
count was approximately 3.3 million TEUs. What
was the percent of increase in the overall inbound
containers?
Tell students to work the problem in their notes,
then write the answer on their white board.
1. 3.3 – 2.94 = 0.36
2. 0.36 is what % of 2.94?
0 .36
3.
2 .94
4. 2.94 = 0.122
5. 12%
Input
The Port of Long Beach is committed to improving
the environment. (Show picture of “Green Port.”)
Cargo-handling equipment has been part of the
12 | Percentages
Percentages Lesson 2
Lesson cont’d
emissions reduction program at the Port. The
amount of NOx (nitrogen oxide) emissions went
from approximately 36 gm/ton of cargo in 2002 to
21 gm/ton in 2005.
What is the percent of decrease?
1. Calculate the decrease. 36 – 21 = 15
2. Rewrite the problem: decrease is what % of the
original number.
15 is what % of 36?
3. Write a fraction representing the situation.
15/36
4. Divide the denominator into the numerator to
three decimal places.
≈.416
5. Write the percent.
≈ 42%
Check for Understanding
In partners, students ask each other: How is
finding the percent of decrease different from
finding the percent of increase?
How is finding the percent of decrease similar to
finding the percent of increase?
Guided Practice
The Port monitors the daily Air Quality Index
(AQI). Show chart on web site:
http://polb.airsis.com/SummaryYesterday.aspx
Information on air quality is collected every hour.
Using the graph below, find the percent of
decrease from 12:00 A.M. of 0.023 ppm to 9:00
A.M. of 0.005 ppm.
Percentages
| 13
Percentages Lesson 2
Lesson cont’d
Students should work the problem in their notes,
then write the answer on their white board.
1. 0.023 – 0.005 = 0.018
2. 0.018 is what % of 0.023?
0 .018
3.
0 .023
4. 0.023 0 .018 = 0.782
5. 78%
Closure
Have students write in their notes:
A summary of how to find the percent of increase.
A summary of how to find the percent of decrease.
Check your notes with your neighbor and adjust
yours if needed.
14 | Percentages
Distance Calculations
Distance Formula
Lesson 3
Lesson three
Terminal Objective
Students will solve real world cost problems
by using the distance formula.
Content Standard
Reference:
Geometry 15: Students use
the Pythagorean Theorem to
determine distance and find
missing lengths of sides of right
triangles.
Geometry 17: Students prove
theorems using coordinate
geometry, including the
midpoint of a line segment, the
distance formula, and various
forms of equations of lines and
circles.
Materials
Distance PowerPoint
Time Required
1 class
Distance Formula
| 15
Introduction of Lesson
Anticipatory Set:
Distance Formula
Lesson 3
The Cartesian coordinate system was originally
developed by the French philosopher and
mathematician René Descartes. You may recognize
his famous quote, “I think therefore, I am.” Legend
has it that Descartes first devised the coordinate
system while watching a bug crawl across the
ceiling. His development of the coordinate plane
helped to bring together the two disciplines of
Algebra and Geometry. The distance formula is an
application of this plane. We will use the distance
formula to calculate costs associated with shipping
cargo to the Port of Long Beach.
Student Objective:
Students will use the distance formula to calculate
the fuel costs of transporting goods to and from the
Port of Long Beach.
Purpose:
The coordinate system allows us to study the
distance between two points on a plane. We can
use this information to solve real world problems
such as calculating costs associated with shipping
cargo to and from the Port of Long Beach.
Lesson
Input
Review the basics.
The Plane
The coordinate plane has four quadrants. The
quadrant the point occupies is dependent upon the
x and y coordinates (x, y).
16 | Distance Formula
Lesson cont’d
Plotting Points
We plot points according to the ordered pair, (x, y).
Quadrant I
(+,+)
Distance Formula
Quadrant II
(- ,+)
Lesson 3
Y
X
Quadrant IV
(+,-)
Quadrant III
(-,-)
The x-coordinate tells us how far from the origin to
move, left or right. The y-coordinate tells us how
far to move from the origin, up or down.
Modeling
Plot point (5,4)
(5, 4)
x
Check for Understanding
Draw a coordinate plane. Label your axis.
Plot the following points.
A. (-3,-3) B. (4,2)
Compare your work with your neighbors.
Distance Formula
| 17
Lesson cont’d
Input
Lesson 3
The quadrant the point is in will depend upon the
signs of x and y. For example, the coordinate of
point a is (4, -3), therefore point A would be located
in quadrant IV.
Y
II
I
Distance Formula
6
4
2
2
4
6
III
X
IV
VI
Modeling
Plot point A, (4, -3), and determine which quadrant
the point falls in. The point falls in Quadrant 4.
y
x
(4, -3)
Check for Understanding
Plot the following points and identify the quadrant
that they are in.
A. (-2,3) Answer: Quadrant II
B. (-4, 0) Answer: B is on the x–axis.
C. (2, -4) Answer: Quadrant IV
18 | Distance Formula
Lesson cont’d
Guided Practice
Lesson 3
Plot the following points and identify the
quadrant they are in.
Check your answers with your neighbors.
(5,-4) (Q IV)
(0, -3) (y-axis)
(2, 1) (Q I)
(-4, -5) (Q III)
Distance Formula
1.
2.
3.
4.
Input
We will see that the distance formula is the
Pythagorean theorem.
Y
(x
(x
x 22, ,yyy22) )
Di stance
a
c 2 = a 2 + b2
b
(x
x 1 ,,yy
y1 )
(x
1 1)
x
If a2 + b2 = c2 where a and b are the lengths of
the legs and c is the length of the hypotenuse and
given p1 (x1,y1) and p2 (x2,y2):
Then a is the difference between x1 and x2 and b
is the difference between y1 and y2, and c is the
length between the two points (the distance).
(x 2 , y2 )
Distance = c
(y2 - y1)= b
a =x2 – x 1
(x1, y1)
Distance Formula
| 19
Lesson cont’d
Modeling
Lesson 3
Using the following graph, find the length of the
legs of the triangle (sides a and b), find the length
of the hypotenuse (side c).
(4, 6)
6
d
Distance Formula
4
2
(1, 2)
y2 - y 1
x2 – x1
2
4
x2 – x1
=4–1
=3
y2 - y1
=6–2
=4
d = ( x2 - x1 ) 2 + ( y2 - y1 ) 2
d=
( 3) 2 + ( 4 ) 2
d = 25
d =5
Check for Understanding
6
4
2
2
20 | Distance Formula
4
Lesson cont’d
Using the graph, calculate the distance between
the two points.
Lesson 3
p1 (1, 5) and p2 (4, 2)
d = ( 4 - 1) 2 + ( 2 - 5) 2
Distance Formula
d = ( 3) 2 + ( - 3) 2
d = 9+9
d = 18
d =3 2
= 4.2
Hands question: I got the same answer - raise your
right hand. I did not - raise your left.
Guided Practice
Calculate the distance between the two points.
1. (-4,3) and (7 , - 5)
D = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
D = (7 − (−4)) 2 + (−5 − 3) 2
D = 112 + (−8) 2
D = 121 + 64
D = 185 = 13.6
2. (3, 8) and (10, -12)
D = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
D = (10 − 3) 2 + (−12 − 8) 2
D = 7 2 + (−20) 2
D = 49 + 400
D = 449 = 21.2
Distance Formula
| 21
Lesson cont’d
Input
Lesson 3
d 2 = ( x2 - x1 ) 2 + ( y2 - y1 ) 2
d 2 = ( x2 - x1 ) 2 + ( y2 - y1 ) 2
Distance Formula
d = ( x2 - x1 ) 2 + ( y2 - y1 ) 2
The distance formula is a useful application of
the coordinate plane. It tells us the length of the
segment connecting two points.
y
Y
P 2 (x 2 ,y 2 )
dis ta nce
P 1 (x 1 ,y 1 )
x
Input
We will use the distance formula to calculate the
distance (on a two dimensional map) that the
container ships must travel from China to Long
Beach. We will then calculate the approximate fuel
cost of these voyages.
Long Beach
China
22 | Distance Formula
Lesson cont’d
Lesson 3
Since we are using a coordinate plane, these
calculations will be approximations as the
distance formula is a planar tool, and the earth
itself is round.
Modeling
Distance Formula
The Hanjin Amsterdam, a cargo ship, has just
finished unloading and loading cargo at the Port
of Shanghai (point S).
It is setting sail for the Port of Long Beach,
represented by point L. Using the data on the
figure, calculate the distance the Hanjin Amsterdam
must sail to reach the Port of Long Beach.
S
(-3414.8, 2152)
L
(3490.8, 2331.5)
D = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
D = (−3414.8 − 3490.8) 2 + (2152 − 2331.5) 2
D = (−6905.6) 2 + (−179.5) 2
D = 47,687,311.36 + 32,220.25
D = 47,719,531.61
D = 6907.9mi
Distance Formula
| 23
Lesson cont’d
Check for Understanding
Distance Formula
Lesson 3
The OOCL Los Angeles, a cargo ship, is sailing from
Zhanjiang to Honolulu and then on to Long Beach.
Using the information on the following graph,
calculate the distance the ship will traverse. Write
your answer on your white board. Z to H:
Zhanjiang
(-2465, 2234.7)
Long Beach
Honolulu
(3490.8, 2231.5)
(1423.5, 1470.7)
D = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
D = (1,423.5 − (−2,465)) 2 + (1,470.7 − 2,234.7) 2
D = 3,888.52 + (−764) 2
D = 15,120,432.25 + 583,696
D = 15,704,128.25
D = 3,962.8mi
Check for Understanding
From H to LB
D = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
D = (3490.76 − 1423.5) 2 + (2231.5 − 1470.7) 2
D = 2067.26 2 + (760.8) 2
D = 4,273,563.9 + 578,816.64
D = 4,852,380.5
D = 2202.8mi
24 | Distance Formula
Lesson cont’d
Find the total distance of the ship’s journey.
Lesson 3
Total Distance:
3962.8 + 2202.8=6165.6 mi
Total distance the ship traveled is approximately
6,165.6mi.
Distance Formula
Input
Using data from 2005 we can estimate the cost per
mile for shipping between China and Long Beach
as $87.50 per mile.
Modeling
Use this information and your answer from the
last problem to calculate the total cost of fuel for
the OOCL Los Angeles to sail from Zhanjiang to
Honolulu and then on to Long Beach.
Solution:
Total Distance: 6165.6
Approximate cost of fuel per mile: $87.50
(6165.6)(87.5)=$539,490
The approximate cost of fuel for the trip from
Zhanjiang, China to Honolulu, HI and then to
Long Beach, CA was $539,490.
Check for Understanding
Turn to your neighbor and explain to them how
you calculated the fuel cost for the OOCL Los
Angeles.
Distance Formula
| 25
Lesson cont’d
Distance Formula
Lesson 3
Guided Practice
Long Beach
(3490.76, 2231.5)
Shanton
(-4009.2,1545.5)
In 2006 fuel costs rose to approximately $171.86
per mile. Calculate the total distance the OOCL
Los Angeles will travel if it departs out of Shanton,
docks in Long Beach and then returns to Shanton.
Calculate the approximate fuel cost of the
roundtrip voyage.
D = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
D = (3490.8 − (−4009.2)) 2 + (2231.5 − 1545.5) 2
D = (7500) 2 + (686) 2
D = 56,250,000 + 470,596
D = 56,720,596
D = 7531.3mi
Round Trip: (7,531.3)(2)= 15062.6mi
Cost: 171.86(15062.6)= $2,588,658
26 | Distance Formula
Closure
Distance Formula
Lesson 3
Write a summary of today’s lesson, focusing on
the real-world application of the distance formula.
Looking at the answers for the last two problems,
how can shipping costs affect consumers and
producers in the United States?
Distance Formula
| 27
Linear Depreciation
Terminal Objective
Students will solve depreciation word
problems by writing linear equations.
Content Standard
Reference:
Algebra 5: Students solve
multi-step problems, including
word problems, involving
linear equations and linear
inequalities in one variable and
provide justification for each
step.
Algebra 8: Students
understand the concepts of
parallel lines and perpendicular
lines and how their slopes are
related. Students are able
to find the equation of a line
perpendicular to a given line
that passes through a given
point.
Materials
Linear Depreciation
Powerpoint
Time Required
1 class
Linear Depreciation
| 29
Linear Depreciation Lesson 4
Lesson four
Introduction of Lesson
Anticipatory Set:
We can use linear equations to determine an
object’s value after years of use.
Linear Depreciation Lesson 4
The Port of Long Beach invests millions of dollars
in the building of infrastructure and purchasing
of equipment. As it ages, infrastructure and
equipment lose value or depreciate.
Student Objective:
Students will use linear equations to find the
depreciation of equipment at the Port of Long
Beach.
Purpose:
Mathematics is a symbolic language that we use
to represent and study the world around us. In
this lesson you will use algebra to model simple
depreciation as used in business. Using math to
study real world problems will provide a better
understanding of the uses of math outside of the
classroom.
Lesson
Keyword
1. Slope - is the rate of change
of y with respect to x
Input
Before students can model using a linear equation,
they must remember how to write a linear
equation. Begin by reviewing slope and two ways
to use points to write a linear equation.
The slope is the rate of change of y with respect to x.
Students can visualize slope as the steepness of the
line. (Have them think of a really steep hill they
would have to walk up or down)
30 | Linear Depreciation
Lesson cont’d
Modeling
On this graph, with each single unit increase in x,
there is a single unit increase in y. Slope = rise .
run
M
S teepness
=1
1
1
Linear Depreciation Lesson 4
x
y
Input
As the absolute value of the slope becomes larger,
the line becomes steeper, moving toward vertical.
Modeling
M
=2
2
1
Input
x
y
As absolute value of the slope becomes smaller,
the line becomes flatter, moving toward
horizontal.
Modeling
m=½
1
2
x
y
Linear Depreciation
| 31
Lesson cont’d
Check for Understanding
Have students ask their neighbor:
The slope represents the __________ of a line.
Answer: steepness
Linear Depreciation Lesson 4
As the absolute value of the slope becomes larger,
___________________. Answer: the line becomes
steeper, moving toward vertical.
True or False: A line with slope of 1/8 will be
flatter, moving toward horizontal. Answer: True
Input
A line with a positive slope is drawn up and to the
right.
A line with a negative slope is drawn down and to
the right.
Modeling
+
x
-
Input
y
If points (x1, y1) and (x2,y2) are two points on a
non-vertical line, then the slope of the line is given
by the equation:
m=
32 | Linear Depreciation
y2 - y1
x2 - x1
Lesson cont’d
Modeling
The slope is the change in the y over the change in
the x.
y2 - y1
x2 - x1
y 2-y 1
x 2-x 1
x
Linear Depreciation Lesson 4
m=
y
Check for Understanding
Find the slope of the line that passes through the
points (5, -7) and (2, 4).
m=
4 - ( - 7)
2- 5
m=
11
-3
1. Th
e slope of the line that passes through the
given points is
m=
11
-3
2. Th
is line is drawn ______________. Answer:
down and to the right
3. I s this line steep –going toward vertical, or is
it flatter – going toward horizontal? Answer:
steep –going toward vertical.
Guided Practice
Find the slope of the line that passes through the
points (3,10) and (-3, 8).
Linear Depreciation
| 33
Lesson cont’d
m=
8 - 10
- 3- 3
-2
-6
1
m=
3
m=
Linear Depreciation Lesson 4
Discuss the following questions with your
neighbor:
Is this line drawn up or down to the right?
Answer: Drawn up and right
Is this line “steeper, going toward vertical” or is
it “flatter, going toward horizontal”? Answer:
flatter, going toward horizontal.
Input
The slope of a vertical line is undefined. If you
select two points on a vertical line and solve for the
slope, the result will be a zero in the denominator.
e slope of a horizontal line is zero. If you select
Th
two points on a horizontal line and solve for the
slope, the result will be a zero in the numerator.
Modeling
m=undefined
x
m=0
y
34 | Linear Depreciation
Lesson cont’d
Input
Two lines are parallel if they have the same slope,
m1= m2.
Two lines are perpendicular if the slopes of the two
lines are negative reciprocals and their product is
-1.
x
m1= m2
y
Linear Depreciation Lesson 4
Modeling
Keywords
1. Parallel - two lines with the
same slope
2. Perpendicular - slopes
of two lines are negative
reciprocals and their product
is -1
Check for Understanding
L1 passes through the points (1,-2) and (4,2),
L2 passes through the points (-1,-2) and
(3,6). Determine whether lines are parallel,
perpendicular or neither.
y2 - y1 m = y2 - y1
2
x2 - x1
x2 - x1
6 - ( - 2)
2 - ( - 2) m2 = 3 - ( - 1)
m1 =
4- 1
8
m2 =
4
4
m=
=
m
2
2
3
m
1
≠
m
2
=
The slope of L1 is 4/3 and the slope of L2 is 2. Are
these lines parallel?
m1 =
Raise your right hand if you think the answer is
yes, and your left if you think the answer is no.
Answer: No, the slopes are not the same.
Linear Depreciation
| 35
Lesson cont’d
Guided Practice
L1 passes through the points (-2,5) and (4,2),
L2 passes through the points (-1,-2)and
(3,6). Determine whether lines are parallel,
perpendicular or neither.
y 2 − y1
x 2 − x1
2−5
m1 =
4 − (−2)
−3
m1 =
6
1
m1 = −
2
Linear Depreciation Lesson 4
m1 =
y 2 − y1
x 2 − x1
6 − (−2)
m2 =
3 − (−1)
8
m2 =
4
m2 = 2
m2 =
The slope of L1 is -1/2 and the slope of L2 is 2.
Are these lines parallel? Answer: No.
Why? Answer: The slopes are not the same.
Check for Understanding
The slope of L1 is 4/3 and the slope of L2 is 2.
Are these lines perpendicular?
Raise your right hand if you think the answer is
yes, and your left if you think the answer is no.
Answer: No, they are not negative reciprocals.
Lines L1 and L2 are neither parallel nor
perpendicular.
To be parallel, the slopes have to be equal.
To be perpendicular, the slopes have to be negative
reciprocals (the product of the two slopes will equal
-1).
36 | Linear Depreciation
Lesson cont’d
Explain to your neighbor how you determine
whether two lines are perpendicular or parallel.
Guided Practice
The slope of L1 is -1/2 and the slope of L2 is 2.
Are these lines perpendicular?
1
× 2 = −1
2
Linear Depreciation Lesson 4
−
Answer: Yes
Why? Answer: The product of the two slopes is -1.
The numbers are negative reciprocals.
Input
We can write the equation of a line if we know two
points on the line or a point on the line and the
slope of the line.
We can use the Point–Slope Form which is given
by the equation:
y – y1 = m(x – x1)
Modeling
Find the equation of the line that passes through
the points (4, 5) and (6, -1).
Step 1: Find the slope.
−1 − 5
6−4
−6
m=
2
m=
Step 2: Using point (4,5) and m= -3, substitute the
values into the equation.
y – y1 = m(x – x1).
Linear Depreciation
| 37
Lesson cont’d
y – 5=-3(x – 4)
y – 5=-3x +12
y =-3x + 17
Check for Understanding
Linear Depreciation Lesson 4
Find the equation of the line that passes through
the point (2,3) and (-4, -6) using the Point Slope
equation, y – y1 = m(x – x1)
- 6- 3
- 4- 2
-9
m=
-6
3
m=
2
m=
y - y1 = m( x - x1 )
y - 3 = 23 ( x - 2)
y - 3 = 23 x - 3
y = 23 x
Guided Practice
Write the equation of the line passing through the
points (5,1) and (-6, -4).
- 4- 1
- 6- 5
-5
m=
- 11
5
m=
11
m=
y - y1 = m( x - x1 )
y - 1 = 115 ( x - 5)
y - 1 = 115 x -
38 | Linear Depreciation
25
11
y = 115 x -
25
11
+1
y=
x-
25
11
+ 11
11
y = 115 x -
14
11
5
11
Lesson cont’d
Input
Sometimes we are given the slope and a point.
Modeling
Find the equation of the line that passes through
the point (6, -2) and has a slope of 2.
Linear Depreciation Lesson 4
y – (-2) = 2(x – 6)
y + 2 = 2x – 12
y = 2x - 14
Check for Understanding
Write the equation of the line with a slope of zero
and passing through the point (6,9).
y – y1 = m (x – x1)
y – 9 = 0 (x – 6)
y–9=0
y=9
Is this line vertical or horizontal? Why? Answer:
It is horizontal because the slope is zero.
Input
Another way to find the equation of a line is by
using the slope intercept form:
y =mx + b
Given two points we can find the slope of the line.
With the slope and a point we solve for b.
Once we have m and b we substitute the values in
to the equation.
Modeling
1. Find the slope of points (1,3) and (4,-6).
Linear Depreciation
| 39
Lesson cont’d
( y2 - y1 )
( x2 - x1 )
- 6- 3
m=
4- 1
-9
m=
3
m= - 3
m=
Linear Depreciation Lesson 4
2. Find b
y = mx + b
3 = - 3(1) + b
3 = - 3+b
6 =b
3. Substitute
y = mx + b
y = - 3x + 6
Check for Understanding
Using the slope intercept form of the linear
equation, find the equation of the line that passes
through the point (4,1) and (5,-3).
y 2 − y1
x 2 − x1
− 3 −1
m=
5−4
m = −4
m=
y = mx + b
1 = −4(4) + b
1 = −16 + b
b = 17
y = mx + b
y = −4 x + 17
Guided Practice
Find the equation of the line passing through the
given points. Use the specified method.
40 | Linear Depreciation
Lesson cont’d
A) Point-Slope, (4,7) and (-2, -3).
Linear Depreciation Lesson 4
y 2 − y1
x 2 − x1
−3−7
m=
−2−4
− 10
m=
−6
5
m=
3
m=
y − y1 = m( x − x1)
5
y − 7 = ( x − 4)
3
5 20
y−7 = x−
3
3
5 20
y = x− +7
3
3
5 20
3
y = x − + 7( )
3
3
3
5 20 21
y = x− +
3
3 3
5 1
y = x+
3 3
B) Slope – Intercept for (-4, -6) and (5, 8).
y 2 − y1
x 2 − x1
8 − (−6)
m=
5 − (−4)
14
m=
9
m=
y=
y = mx + b
14
− 6 = (−4) + b
9
− 56
−6 =
+b
9
126
−6+
=b
9
9 56
− 6( ) +
=b
9
9
− 54 56
+
=b
9
9
2
b=
9
Keyword
1. Depreciation - the decrease
or loss in value of capital due to
age, wear or market conditions
14
2
x+
9
9
Input
Students will be writing linear equations for the
straight line method of linear depreciation to
study the depreciation of capital investments
made by the Port of Long Beach.
Depreciation is the decrease or loss in value of
capital due to age, wear or market conditions. In
accounting it is the allowance made for a loss in
the value of capital.
Linear Depreciation
| 41
Lesson cont’d
Modeling
In 2002 the Port of Long Beach purchased a ZPMC
Crane for Pier T for the amount of $6,811,461.73.
The crane is to be depreciated over 15 years with
a scrap value of $0. Write an expression that will
calculate the value of the crane at the end of year
(t). What is the value of the crane in 2007?
Linear Depreciation Lesson 4
Two coordinates of the form (time, value).
(0, $6,811,461.73)
(15, $0)
( 0 - 6,81146173
, . )
(15, 0)
- 6,81146173
, .
m=
15
m = - 454, 097.45
m=
To find the equation, use the slope and one of the
points.
y - y1 = m(x-x1)
y-0 = -454,097.45 (x-15)
y= -454,097.45x + 6,811,461.73
To find the price in 2007, subtract 2007-2002 to find
the number of years of depreciation.
2007-2002 = 5
y= -454,097.45(5) + 6,811,461.73
y = $4,540,974.48
42 | Linear Depreciation
Lesson cont’d
Check for Understanding
Linear Depreciation Lesson 4
In 1984, a tractor/loader was purchased for use at the
Port of Long Beach for a price of $29,041.01. The tractor/
loader was depreciated using the straight-line method
over 8 years. Find the linear equation expressing the
tractor’s book value at the end of x years. What is the
rate of depreciation?
(0, 29,041.01) and (8,0)
0 − 29,041.01
8−0
m = −3630.13
m=
y − y1 = m( x − x1)
y − 0 = −3630.13( x − 8)
y = −3630.13x + 29,041.01
Check for Understanding
The linear equation expressing the crane’s value at the
end of x years is given by
y= -3,630.13x + 29,041.01
What is the rate of depreciation?
-3,630.13.
What was the value of the crane in 1987? (1987-1984=3)
y=-3,630.13x + 29,041.01
y=-3,630.13(3)+29,041.01
y=18,150.62
Guided Practice
A truck scale purchase at a cost of $151,999.75 in 1986
has a scrap value of $0 at the end of 10 years.
(0,151,999.75) and (10,0)
If the straight-line method of depreciation is used,
Linear Depreciation
| 43
Lesson cont’d
A) Find the rate of depreciation.
m= -15,199.98
B) Find the linear equation expressing the book
value of the scale at the end of x years.
y= -15,199.98x + 151,999.75
Linear Depreciation Lesson 4
C) Find the book value at the end of 7 years.
y = 45,600.03
Closure
Write a brief paragraph explaining the method for
writing simple depreciation equations. Include an
explanation of depreciation.
Share your paragraph with your neighbor.
44 | Linear Depreciation
Worksheet
Lesson four
Linear Depreciation Student Worksheet
Linear Depreciation Lesson 4
Solve
Find the slope of the line that passes through the given points
and determine if the lines are parallel, perpendicular or neither.
1. L1 (1, - 2) and (-3, 10)
2. L1 (-2, 5) and (4, 2)
L2 (1,5 and (-1, 1)
L2 (-1, -2) and (3,6)
3. Find the equation of a horizontal line that passes through the
point (-5, 2).
4. Find an equation of the vertical line that passes through the
point ( -4, 5).
5. Find an equation of the line in slope-intercept form that passes
through the point (3, -4) and has a slope of 5.
6. Find an equation of a line that passes through the points (-1, 3)
and (2, 9). Put the equation in slope-intercept form.
Linear Depreciation
| 45
7. Find an equation of a line in slope-intercept form that passes
through the points (6, -2) and (4, 7) using the Slope-Intercept
Linear Depreciation Lesson 4
equation.
8. In 1983, the Port of Long Beach built the Arco Oil Terminal for
a cost of $8,040,683.08 to be depreciated over its useful life
of 40 years. Given the scrap value is $0, write the equation
representing the linear depreciation of the terminal. What is
the rate of depreciation? What is the book value of the terminal
at the end of 2007?
9. The NH Terminal Railroad was built in 2002 for a cost of
$1,084,139.68. The railroad will be depreciated over 15 years
to a final book value of $0.
Write the linear expression that
represents the value of the railroad at time (t). What is the rate
of depreciation? What will be the book value of the railway in
2010?
10. What is depreciation? Describe how you applied algebra to
the above world problems to derive the linear depreciation
equation.
46 | Linear Depreciation
Answers
Lesson four
Linear Depreciation Student Worksheet
10 - ( - 2)
- 3- 1
12
L1 =
-4
L1 = - 3
L1 =
L2 =
L2 =
2.
1- 5
6 - ( - 2)
3 - ( - 1)
8
L2 =
4
L2 = 2
L2 =
- 1- 1
-4
-2
L2 = 2
L1 and L2 are neither parallel
nor perpendicular.
2- 5
4 - ( - 2)
-3
L1 =
6
L1 = - 21
L1 =
The lines are perpendicular as the
slopes of the lines are negative
reciprocals.
3. y = 2.
4. x = -4
5. m = 5
6.
( 3, - 4)
y - y1 = m( x - x1 )
y - ( - 4) = 5( x - 3)
y + 4 = 5x - 15
y = 5x - 19
Linear Depreciation Lesson 4
1.
9−3
2 − (−1)
6
m=
3
m=2
m=
y − y1 = m( x − x1)
y − 3 = 2( x − (−1))
y − 3 = 2x + 2
4 = 2x + 5
Linear Depreciation
| 47
7 − (−2)
4−6
9
m=
−2
7.
m=
8. Coordinates:
(0, 8,040,683.08) and (40, 0)
Linear Depreciation Lesson 4
m=
y = mx + b
9
y + 2 = − ( x − 6)
2
2 y + 4 = −9( x − 6)
2 y = −9 x + 54 − 4
2 y = −9 x + 50
9
y = − x + 25
2
9. Coordinates
(0, 1,084,139.68) and (15, 0)
0 − 1,084,139.68
15 − 0
m = −72,275.98
m=
y − y1 = m( x − x1)
y − 1,084,139.68 = −72,275.98( x − 0)
y − 1,084,139.68 = −72,275.98 x
y = −72,275.98 x + 1,084,139.68
The rate of depreciation is
$72,275.98 per year.
The book value of the railway in
2007 will be $505,931.80.
Y=-72,275.98 (8)+1,084,139.68
Y=505,931.80
0 - 8,040,68308
.
40 - 0
m = - 201,017.08
y - y1 = m( x - x1 )
y - 8,040,68308
. = - 201,017.08( x - 0)
y - 8,040,68308
. = - 201,017.08x
y = - 201,017.08x + 8,040,68308
.
The rate of depreciation is $201,017.08
per year.
The book value of the terminal in 2007
is y = -201,017.08(24)+8,040,683.08
y=$3,216,273.16
10. W
hen a company depreciates
an asset, it is accounting for
the loss of value of the item for
business purposes.
Algebra was applied in the
calculation of the linear
equation by first determining
the coordinates to use for the
calculation of the slope. Once
we had calculated the slope
of the line, we then used the
point-slope formula to derive
48 | Linear Depreciation
the equation.
Clearance
Lesson five
data, students will be able to calculate the
possibility of a ship’s passage underneath
the Gerald Desmond bridge in the Port of
Long Beach with 80% accuracy.
25.0 Students use properties
of the number system to judge
the validity of results, to justify
each step of a procedure,
and to prove or disprove
statements.
25.1 Students use properties of
numbers to construct simple,
valid arguments (direct and
indirect) for, or formulate
counterexamples to, claimed
assertions.
25.2 Students judge the validity
of an argument according to
whether the properties of the
real number system and the
order of operations have been
applied correctly at each step.
Materials
Clearance Powerpoint
Time Required
1 class
Clearance
| 49
Lesson 5
Given a handout with real situational
Algebra I
1.0 Students identify and use
the arithmetic properties
of subsets of integers and
rational, irrational, and real
numbers.
Clearance
Terminal Objective
Content Standard
Reference:
Introduction of Lesson
Anticipatory Set:
The primary channel leading to the majority of
offloading docks in the interior of the Port of
Long Beach is spanned by the Gerald Desmond
Bridge. The bridge was originally built in 1968 with
plenty of clearance for shipping vessels to pass
underneath. Today ships are being built larger
and larger, and the ability for some ships to pass
underneath is becoming impossible. What sorts of
calculations need to happen before a ship is cleared
to pass?
Student Objective:
Lesson 5
Given real situational data, the student will be
able to calculate the possibility of a ship’s passage
underneath the Gerald Desmond bridge in the Port
of Long Beach.
Purpose:
Clearance
• To raise interest in the Port of Long Beach
• To apply basic algebra skills and logical reasoning
to a real situation with real data
• To raise interest in the importance of math in
everyday life
Lesson
Input
Provide students with background on the Gerald
Desmond Bridge.
Structural Type: Arch bridge / suspended deck
Function/usage: Road bridge connects the Ports of
Long Beach and Los Angeles with the I-710.
50 | Clearance
Lesson cont’d
Span: 5,134 ft.
Built: 1968 (replacing the previous
pontoon bridge)
Length: 1,053 feet long
Highest point: 250 feet
157 feet of clearance above the water
Lesson 5
The primary channel leading to the majority of
offloading docks in the interior of the Port of Long
Beach is spanned by the Gerald Desmond Bridge.
The bridge was originally built in 1968 with
plenty of clearance for shipping vessels to pass
underneath. Today, ships are being built larger
and larger, and the ability for some ships to pass
underneath the bridge is becoming impossible.
Clearance
Students will calculate whether a ship can pass
under the Gerald Desmond bridge given the ship’s
dimensions.
A typical ship’s dimensions:
Length – bow to stern
Width – widest part of the ship
Depth – height of the hull taken from the lowest
part under the water
TEUs – number of 20-foot containers the ship holds
Modeling
Can the Hanjin Amsterdam cargo ship pass under
the Gerald Desmond Bridge?
Ship: Length: Width: Depth:
TEUs:
Hanjin Amsterdam
279 meters 40 meters
14 meters
5,618 - stacked 5 high
Clearance
| 51
Lesson cont’d
What information given is needed to calculate total
height?
Answer: Depth and the number of stacked TEUs
What is the height of a TEU?
8.6 feet
20 feet
8 f eet
Lesson 5
What is the total height of the stacked TEUs?
5 TEUs
1
8.6 feet = 43 feet
1 TEU
Clearance
We now have feet and meters. We cannot combine
unlike terms, so…
Convert depth from meters to feet:
14 meters
1
3.28 feet = 45.9 ft.
1 meter
Depth of
the hull
+
45.9 feet
Height of the
stacked cargo = 88.9 feet
43 feet
Will it clear the Gerald Desmond Bridge which has
157 feet of clearance?
YES!
Modeling
Can the Neptune Amber cargo ship pass under the
Gerald Desmond Bridge?
52 | Clearance
Lesson cont’d
Ship:
Length: Width: Depth: TEUs:
Neptune Amber
231 meters
32.2 meters
12.5 meters
2,314 – stacked 6 high
What is the total height of the stacked TEUs?
6 TEUs
1
8.6 feet = 51.6 feet
1 TEU
We now have feet and meters. We cannot combine
unlike terms, so…
3.28 feet = 41 feet
1 meter
Depth of
the hull
+
41 feet
Height of the
stacked cargo = 92.6 feet
51.6 feet
Clearance
12.5 meters
1
Lesson 5
Convert depth from meters to feet:
Will it clear the Gerald Desmond Bridge which has
157 feet of clearance?
YES!
Check for Understanding
1) T
o determine whether a ship will pass under
a bridge, the length of the ship is needed.
Answer: False
2) When calculating the total height of the TEUs,
the number of TEUs stacked above the ship’s
rail must be counted. Answer: True
Clearance
| 53
Lesson cont’d
Modeling
Can the Zim Atlantic cargo ship pass under the
Gerald Desmond Bridge?
Ship:
Length:
Width:
Depth:
TEUs:
Zim Atlantic
253.7 m
32.2 m
19.2 m
3,429 – stacked 8 high
Depth of
the hull
+
63 feet
Height of the
stacked cargo = 131.8 feet
68.8 feet
Lesson 5
Will it clear the Gerald Desmond Bridge which has
157 feet of clearance?
YES!
Clearance
Modeling
Can the MSC Texas Majuro cargo ship pass under
the Gerald Desmond Bridge?
Ship:
Length:
Width:
Depth:
TEUs:
MSC Texas Majuro
321 m
42.8 m
20.2 m
8,238 – stacked 8 high
Depth of
the hull
+
66.3 feet
Height of the
stacked cargo = 135.1 feet
68.8 feet
Will it clear the Gerald Desmond Bridge which has
157 feet of clearance?
YES, but it is getting pretty tight…
54 | Clearance
Lesson cont’d
Modeling
This is one of the largest vessels currently in
existence.
Length:
Width:
Depth:
TEUs:
320 m
41 m
29 m
7 high above rail
Depth of
the hull
+
95.1 feet
Height of the
stacked cargo = 155.3 feet
60.2 feet
Lesson 5
Will it clear the Gerald Desmond Bridge which has
157 feet of clearance?
Eeeeeking by! Is it worth the risk?
Clearance
Check for Understanding
Think, pair, share:
1) Think: What is one thing that was different
about this example (vessel #5)?
2) Pair: Take turns telling your partner what was
different.
3) Share: One person will be called on to share
with the group.
Modeling
This is another very large vessel:
Length: Width:
Depth:
TEUs:
327 m
41.8 m
32.2 m
6 high above rail
Clearance
| 55
Lesson cont’d
Depth of
the hull
+
105.6 feet
Height of the
stacked cargo = 157.2 feet
51.6 feet
Will it clear the Gerald Desmond Bridge at 157 feet
of clearance?
NO!
Modeling
Let’s try an even bigger ship.
Lesson 5
Ship:
Length:
Width:
Depth:
TEUs:
OOCL Long Beach
SX Class Vessel
323.2 m
43 m
38.8 m
8,000 + - stacked 7 high above rail
Clearance
Depth of
the hull
+
127.3 feet
Height of the
stacked cargo = 188.1 feet
60.2 feet
Will it clear the Gerald Desmond Bridge at 157 feet
of clearance?
NO!
Closure
• In your notes, list the steps used today to
determine whether or not a ship will be able to
pass under a bridge.
• Check your list with your partner.
• Add any steps to your list that you feel are
missing.
56 | Clearance
Extension Activities
How do tide changes affect the ability of ships to
pass under the Gerald Desmond Bridge?
See “Tide Table Handout” or go to the following
website for real time information: http://www.
saltwatertides.com/cgi-local/california.cgi
Extreme low tide: -1.1 feet
Extreme high tide: 7.2 feet
How would high tide affect the passage of ships
under the Gerald Desmond Bridge?
Lesson 5
Answer: High tide would shorten the clearance by
7.2 feet to 149.8 ft.
Vessel #
Height
in feet
Vessel #1
92.6
Vessel #3
131.8
Vessel #2
Vessel #4
Vessel #5
Vessel #6
Vessel #7
114.7
135.1
155.3
157.2
188.1
Passage
under
bridge
(157 feet)
Height
added to
tide in
feet
Effect of
passage
yes
121.9
same
yes
yes
yes
yes
no
no
99.8
same
139
same
142.3
162.5
164.4
195.3
Clearance
Which vessels would be affected? Make a table to
support your claim.
same
will not
pass
same
same
Vessel 5 would be affected where it was not before.
Write a statement in your notes describing the
effect of high tide on ships passing under a bridge.
How would low tide affect the passage of vessels
under the Gerald Desmond Bridge?
Clearance
| 57
Extension Activities cont’d
Which vessels do you imagine would be most
affected? Why?
Clearance
Lesson 5
Answer: Low tide would change the depth of the
water. The heaviest vessels would be most affected
because they sit further below the water line than a
lighter ship.
58 | Clearance
Handout
Lesson five
Container Ship Clearance Handout
Solve
Directions:
Calculate the following vessel clearance heights and determine
Lesson 5
whether or not they would be able to pass under the Gerald
Desmond Bridge.
1 meter = 3.28 feet
TEU height = 8.6 feet
Gerald Desmond Bridge clearance = 157 feet
1. Hull depth: 26 meters
1. Clearance height_______________
TEUs above deck: 7
Will it clear the bridge?________
If low tide is -1 foot? ___________
2. Hull depth: 26 meters
If high tide is 7 feet? ___________ 2. Clearance height ______________
TEUs above deck: 8
3. Hull depth: 29.5 meters
Clearance
Will it clear the bridge?________ If high tide is 7 feet? ___________
If low tide is -1 foot? ___________
3. Clearance height ______________
TEUs above deck: 7
Will it clear the bridge?________
Clearance
| 59
4. Hull depth: 90 feet
Lesson 5
Clearance
If high tide is 7 feet? ___________
5.Clearance height ______________
60 | Clearance
Will it clear the bridge?________
If low tide is -1 foot? ___________
TEUs above deck: 5
If low tide is -1 foot? ___________
4.Clearance height ______________
TEUs above deck: 9
5. Hull depth: 30 meter
If high tide is 7 feet? ___________
Will it clear the bridge?________
If high tide is 7 feet? ___________
If low tide is -1 foot? ___________
ANSWERS
Lesson five
Container Ship Clearance Answers
2. Hull depth: 26 meters
TEUs above deck: 8
3. Hull depth: 29.5 meters
TEUs above deck: 7
4. Hull depth: 90 feet
TEUs above deck: 9
5. Hull depth: 30 meters
TEUs above deck: 5
Will it clear the bridge? yes
If high tide is 7 feet? yes
If low tide is -1 foot? yes
2. Clearance height 154.1
Lesson 5
TEUs above deck: 7
1. Clearance height 145.5
Will it clear the bridge? yes
If high tide is 7 feet? no
If low tide is -1 foot? yes
Clearance
1. Hull depth: 26 meters
3. Clearance height 157
Will it clear the bridge? no
If high tide is 7 feet? no
If low tide is -1 foot? yes
4. Clearance height 167.4
Will it clear the bridge? no
If high tide is 7 feet? no
If low tide is -1 foot? no
5. Clearance height 141.4
Will it clear the bridge? yes
If high tide is 7 feet? yes
If low tide is -1 foot? yes
Clearance
| 61
Building a Bridge
Lesson six
Given a handout and the current bridge
dimensions, the student will be able to
estimate a plan with 75% accuracy for
building a higher bridge to accommodate
future container ships.
Algebra I
5.0 Students solve multistep
problems, including word
problems, involving linear
equations and linear
inequalities in one variable and
provide justification for each
step.
8.0 Students understand
the concepts of parallel lines
and perpendicular lines and
how those slopes are related.
Students are able to find the
equation of a line perpendicular
to a given line that passes
through a given point.
Materials
Building a Bridge Powerpoint
Time Required
1 class
Building a Bridge
| 63
Building a Bridge Lesson 6
Terminal Objective
Content Standard
Reference:
Introduction of Lesson
Anticipatory Set:
Currently new vessels, or “Mega Ships,” are
being produced but they are too large to clear the
Gerald Desmond Bridge. As ships get larger, the
infrastructure of the port needs to keep up; namely
the bridge height, width/depth of waterways and
docks/wharfs.
How can we plan a new bridge that will
accommodate shipping and traffic needs?
Student Objective:
Given the current bridge dimensions, students will
be able to estimate a plan for a higher bridge to
accommodate future container ships.
Purpose:
Building a Bridge Lesson 6
• To raise interest in the Port of Long Beach
• To show a real life application of math and its
critical importance
• To apply basic algebra skills to solve a current real
problem
• To further appreciate the complexity of finding a
feasible solution
Lesson
Input
Provide students with background on the Gerald
Desmond Bridge:
Structural Type: Arch bridge / suspended deck
Function/usage: Road bridge connects the Ports
of Long Beach and Los Angeles
64 | Building a Bridge
Lesson cont’d
with the I-710.
Span: 5,134 Ft.
Built: 1968 (replacing the previous
pontoon bridge)
Length: 1,053 feet long
Highest point: 250 feet
157 feet of clearance above the water
250 fee t a t h igh es t
po int
157 fee t a t ro a d
level
G round Leve l
G round Leve l
5134
fee t
Building a Bridge Lesson 6
Currently new vessels, or “Mega Ships,” are being
produced, but they are too large to clear the
Gerald Desmond Bridge. As ships get larger, the
infrastructure of the port - namely the bridge
height, width/depth of waterways and docks/
wharves - needs to keep up.
How can we plan a new bridge that will
accommodate shipping and traffic needs?
Input
A grade (or gradient) is the pitch of a slope, and
is often expressed as “rise over run.” It is used to
express the steepness of slope of a hill, stream,
roof, railroad, or road.
This is especially important in trucking because
fully loaded “big rigs” can’t make it up a grade that
is too steep!
Building a Bridge
| 65
Lesson cont’d
157 fee t at road lev el
256 7 fee t
-2567 fee t
G round Le vel
256 7 runG round Le vel
Because the bridge is symmetrical, let’s put it in a
coordinate plane.
Then we can look at the average slope of one side of
the bridge.
Because we need to calculate slope, we need to
know the length of the base of the triangle.
157 fee t at road leve l
256 7 fee t
-2567 fee t
Building a Bridge Lesson 6
157 ris e
256 7 run
S lop e =
R ise
Run
157 rise
= 0 .061 16
256 7 r un
= 6.116% grade
For our purposes, we will use the decimal form
(0.06116) because it is equal to the fraction
and we are writing an equation.
Modeling
Considering very heavy trucks use this bridge,
we will need to build a new bridge with the same
grade (or slope).
If we raise the height of the bridge to 250 feet but
keep the same slope, what will happen to its span?
66 | Building a Bridge
Lesson cont’d
Check for Understanding
250 fee t at road leve l
Sa me sl op e
250 ris e
Sa me sl op e
x run
0.06116 =
250 rise
x r un
Guided Practice
=
250
x
Cr os s m ul tip ly
0.06 116 x = 250
0.06 116
0.06 116
So lve for “ x ”
Building a Bridge Lesson 6
0.06 116
1
x = 4087 .6 feet
x = 40 87.6 feet
s pan = 2 (40 87.6 feet)
s pan = 8175 .2 feet
250 fee t at road leve l
408 7.6 fee t
-4087.6 fee t
x
x
817 5.2 fee t
Modeling
What if we raise the height of the bridge to 275
feet?
Building a Bridge
| 67
Lesson cont’d
Check for Understanding
275 fee t at road leve l
Sa me sl op e
275 ris e
Sa me sl op e
x run
0.06116 =
275 rise
x r un
Guided Practice
0.06 116
1
=
275
x
Building a Bridge Lesson 6
0.06 116 x = 27 5
0.06 116
0.06 116
Cr
os smultiply
m ul tip ly
Cross
So lve for “ x ”
Solve for "x"
x = 4496 .4 feet
x = 4496 .4 feet
span
spa
n = 2 (4496.4 f ee t)
span
spa
n = 8992 .8 fee t
275 fee t at road leve l
449 6.4 fee t
-4496.4 fee t
x
x
899 2.8 fee t
Modeling
What if we raise the height of the bridge to 300
feet?
68 | Building a Bridge
Lesson cont’d
Guided Practice
300 fee t at road leve l
490 5.2 fee t
-4905.2 fee t
x
x
981 0.4 fee t
Closure
Building a Bridge Lesson 6
Examine the aerial photo of the bridge and its
surrounding infrastructure with the new spans.
Discuss the new problems these longer spans may
create.
Written reflection:
• What limits the height of a future bridge?
• What solutions can you imagine for that
problem?
• Are the ideas you have realistic?
Building a Bridge
| 69
Handout
Lesson six
Rebuilding the Gerald Desmond Bridge
Solve
G round Le vel
Building a Bridge Lesson 6
256 7 runG round Le vel
Given a slope of 0.06116, what would the half span and total span
be for the following bridge heights:
1. For 200 feet
Half Span _ ______________________
2. For 225 feet
Half Span _ ______________________
3. For 325 feet
Half Span _ ______________________
4. For 207 feet Half Span _ ______________________
Full Span ________________________
Full Span ________________________
Full Span ________________________
Full Span ________________________
Building a Bridge
| 71
72 | Appendix
Handout
Lesson six
Rebuilding the Gerald Desmond Bridge
Solutions
What would the half span and total span be for the following
1. For 200 feet
3270.1 ft
Half Span _ ______________________
6540.2 ft
Full Span ________________________
2. For 225 feet
3678.9 ft
Half Span _ ______________________
7357.8 ft
Full Span ________________________
3. For 325 feet
5313.9 ft
Half Span _ ______________________
10627.9 ft
Full Span ________________________
4. For 207 feet 3384.6 ft
Half Span _ ______________________
6769.1 ft
Full Span ________________________
Building a Bridge
Building a Bridge Lesson 6
bridge heights:
| 73
Appendix
| 75
76 | Appendix
Appendix
| 77