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

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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!

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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?

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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?

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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?

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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?

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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 ________________________

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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:

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Appendix

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76 | Appendix

Appendix

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