Maryland CCRG Algebra Task Project
Building a Suspension Bridge
Common Core Standard
F.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k)
for specific values of k (both positive and negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects on the graph using technology.
Include recognizing even and odd functions from their graphs and algebraic expressions for
them. Use transformations of functions to find models as students consider increasingly more
complex situations.
For F.BF.B.3, note the effect of multiple transformations on a single graph and the common
effect of each transformation across function types.
MP1:
MP2:
MP3:
MP7:
MP8:
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Algebra II, Unit 4
The Task
The Maryland Transportation Authority is in discussion on building a new suspension bridge to
connect Maryland’s west coast to the eastern shore. This bridge
will alleviate traffic on the Chesapeake Bay Bridge as well as
allow commuters to get across the bay from another destination.
The state has chosen three locations for the proposed bridge,
each with a different distance across the water. The three
distances are 2.2 miles, 2.55 miles, and 2.8 miles respectively.
This bridge design is similar to that of the Golden Gate Bridge in
San Francisco. The bridge will have two towers close to each
shore and one suspension in between them. The suspension
touches the bridge midway between the towers. Due to ships
passing below the bridge, the minimum height of the bridge’s
midpoint will be 150 feet.
One architect who is designing the bridge wants to use the
equation
y = 0.00001432(x – 6600)2 + 200 to model the suspension where y is the height in feet and x is
the horizontal distance in feet of the bridge.
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Maryland CCRG Algebra Task Project
1. According to the equation given, which locations above will be feasible for this bridge
design?
2. Does this bridge meet height specifications? If so, what is the bridge’s height? How are you
able to calculate this value?
3. Another location has been submitted for the new bridge. The distance across the water at this
location is 1.5 miles. Write a possible equation to model the suspension of the bridge.
Explain the different values of your equation and why it works for the distance given.
4. A co-worker of the architect agrees with the first bridge design. The state transportation
authority was confused when he submitted his blueprint. His equation to model the
suspension was y = 0.00001432(x – 6600)2, but his design plan looked the same. How could
he have come up with a different equation than the original architect and still have the same
plans?
Facilitator Notes
1. Have students work in groups to investigate the problem. Have students develop a
strategy for finding a solution. Once each group has a solution, have groups compare
solutions and strategies. (Look for evidence of MP1, MP2, and MP3.)
2. Once each group has a solution, have groups compare solutions and strategies. Have
students discuss the benefits and drawbacks of the strategies they used. Make sure to
connect strategies to key mathematical terminology. (Look for evidence of MP3, MP7,
and MP8.)
Solutions
While there are many approaches students may use to solve this task, here are a few
sample answers:
Students will have to utilize their knowledge of quadratic and its transformations in order to
answer the follow up questions to this problem. I will use the general equation for a quadratic
y = a (bx – h)2 + k to demonstrate possible student solution paths.
1. According to the equation given, which locations above will be feasible for this bridge
design?
Students will have to calculate the distance across the bridge to answer this first question.
Students will need to recognize that the distances given in the in the original problem are in
miles and the function to represent the suspension of the bridge are in feet. Below are possible
methods students could use to answer this question.
Method 1: By using the function y = 0.00001432(x – 6600)2 + 200, students should recognize
that the vertex of this quadratic will be (6600, 200) based on the definition of a quadratic
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Maryland CCRG Algebra Task Project
equation in vertex form. Therefore this will be the coordinates for part of the suspension that is
touching the bridge. Since the midpoint of the bridge is 6600 feet, the bridge will be 13,200 feet
or 2.5 miles long between its towers. Since the introduction said the towers are close to the
shore, we can assume this bridge is a little longer than 2.5 miles. So the locations that are 2.55
miles long and 2.8 miles long will be feasible for this bridge design.
Method 2: Students can use technology to find the vertex of this quadratic and then follow the
same steps as above. Using technology allows students to see the transformation of this
quadratic with the values given in the function.
Using technology students can also find the y-intercept of this function by substituting in the
value of 0 in the equation. This will yield an answer of 823.78 feet. Students can use other
methods to see where this parabola has this y-value again and that will also give the distance
between the towers.
2. Does this bridge meet height specifications? If so, what is the bridge’s height? How are
you able to calculate this value?
Students who understand the values of vertex form could answer this question right away.
Knowing that the “k” value of a quadratic equation vertically shifts the graph, the vertex will be
200 feet. Since the suspension touches the bridge at its midpoint, the bridge will be 200 feet tall
therefore meeting height specifications.
Technology could also be used as in the previous question to see how tall the bridge is at its
midpoint.
3. Another location has been submitted for the new bridge. The distance across the water
at this location is 1.5 miles. Write a possible equation to model the suspension of the
bridge. Explain the different values of your equation and why it works for the distance
given.
This question is very open-ended and allows students to explore imputing different values in a
quadratic function to make this work. Answers should vary between students and will promote
good discussion on how solutions were found. Some strategies students could use is making the
“h” value less in order to shorten the bridge. Other groups may use their knowledge of
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Maryland CCRG Algebra Task Project
stretching and shrinking parabolas and increase the “a” and “b” values accordingly to fit this
location.
Teachers can put students in groups and assign them different ways to do this or have students
explore options on their own.
4. A co-worker of the architect agrees with the first bridge design. The state
transportation authority was confused when he submitted his blueprint. His equation to
model the suspension was y = .00001432(x – 6600)2, but his design plan looked the same.
How could he have come up with a different equation than the original architect and
still have the same plans?
Students will recognize that the “k” value is missing in this equation. This would eliminate the
vertical shift therefore having the bridge in the water and not meeting height specifications.
Answers will vary between students. Some may find that this architect used the bridge as his xaxis where the other architect used the water level. Students can graph this equation and visually
see that his graph represent just the bridge and not the space below. This can lead to class
discussion on what the y-axis represents and other piece of the graph.
The co-workers design where the
bridge is used as the x-axis, leaving
no space below the suspension.
This question will reinforce vertical
shifts in functions.
First architects design using the
water for his x-axis. You can see
the space below the suspension
which would represent the
bridge, air and water below.
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information on this license, refer to: http://creativecommons.org/licenses/by-nc/3.0/.