Brianna HaroA unit lesson on constructing a bridge over the Bering Strait.

Constructing an Overseas Bridge

[Constructing an Overseas Bridge] 2

Mathematical GoalsThis lesson unit is designed to assess a student’s ability to research and apply multiple math

concepts:

-Solving for arc length-Finding volume of irregular shaped objects-Budgeting for large projects

Common Core State StandardsThis lesson relates to the following Standards for Mathematical Content in the Common

Core State Standards for Mathematics:

CCSS.MATH.CONTENT.HSG.MG.A.3

Apply geometric methods to solve design problems.

CCSS.MATH.CONTENT.HSG.GMD.A.3

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

CCSS.ELA-LITERACY.RST.9-10.7

Translate quantitative or technical information expressed in words in a text into visual

form and translate information expressed visually or mathematically into words.

CCSS.ELA-LITERACY.SL

Comprehension and Collaboration/Presentation of Knowledge and Ideas

This lesson also relates to the following Standards for Mathematical Practice in the Common

Core State Standards for Mathematics, with a particular emphasis on Practices 2, 3, and 4.

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

6. Attend to precision.

IntroductionThe unit is structured in the following way:

[Constructing an Overseas Bridge] 3

- Students will be assessed on level of understanding of various mathematical concepts.

After reviewing work, students will be grouped homogeneously based upon skill and

understandings.

- Students will work collaboratively to first choose a type of bridge to build. Students will

read informational text on the various types of bridges, weighing the pros and cons of

each. The group will choose a type of bridge to build.

- Students will choose the best path to take over the water based upon distance and

structural support. Students will read about structural support of their bridge and

approximate how many pillars, if any, they need to cross the water.

- Students will determine the size of their bridge, both the driving area, as well as an

approximate amount of material they need based upon the type of bridge and they path

they have chosen.

- Students will research the financial budget of building such a bridge, applying an

expected value to their own bridge.

- Students will research cultural and technical aspects of building their bridge, making an

argument why their bridge is important as well as stating any other important aspects of

this bridge that should be considered.

- Students will give a proposal to a board. Each person will vote upon the best bridge idea,

offering critiques and questions for the various proposals at time of presentation.

Materials Required- Textbook and notes for solving distance, arc length, and volume.

- Bridge Types document

- Financial budget for bridges document

- Maps of the Bering strait

Time NeededAssessment will be a day of reviewing recent geometry and older algebra, such as graphing

equations, graphing inequalities, and solving and creating one and two variable equations.

A day of instruction and grouping will be needed.

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Students will be given no more than three days to research types of bridges, listing their pros

and cons.

Students will spend two days looking at geography, deciding on best path, and estimating the

type of support their bridges need. This section will require looking at the elevation and the

coordinates of the location the bridge will touch down on.

No more than three days are required for determining the size of the bridge, including width,

length, height, and volume of pillars if applicable. This section should include an idea of how

much weight the bridge should hold.

A day is needed to research the financial cost of their type of bridge.

An additional 3-4 days should be used to find an approximate cost for their own bridge, as

well as a graph of the cost of various bridges per square meter. Finding a line of best fit, students

will determine the cost of their own bridge.

Students will research the various issues in building a bridge across the Bering Strait and

come up with arguments against this in a section that should not last longer than three days.

4-5 days should be given to allow students time to wrap up their information and create a

presentation with all of their information and graphs.

Presentations depend upon class size. 2-3 days should be enough for every group to give a

fair argument for why their bridge should be chosen.

The entire unit should take around 23 days.

Before the LessonAssessment Task: Two-Column Proofs

Students will complete a handout of sample problems in class the day before the unit starts.

The handout will have the following questions and should be completed individually. Calculators

are allowed, but all work must be shown.

1. Student A loves the crust on a pizza and wants the longest piece of crust. Student B

wants the pizza with the shortest crust. The pizza has a radius of 20 cm. The measure

of the angle of Pizza X is 30, while the measure of the angle of Pizza Y is 20. Which

pizza should Student A choose and how much longer is this crust than the other?

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2. A woman wants to walk a tightrope between two buildings. She lays a grid over a map

and determines that a point on top of the starting building lies on the coordinates (-2,3)

and the ending point on the other building lies on (5,-8). She also determines that each

unit is approximately 100 feet. How much feet of rope will she have to walk across?

3. a) A roommate wants to prank his friend and fill his bedroom with ping-pong balls.

The bedroom is 10ft. by 12 ft. and the ceiling is 8ft. tall. A ping-pong ball has a radius

of 1.5478 in. How many ping-pong balls will it take to fill the room? (Hint: 1 cubic in.

= 0.000578701 cubic ft.) Round down to the nearest integer.

b) The roommate has found two places where he can buy large bags of ping-pong

balls. One store sells bags of 2000 balls for $10. The other store sells boxes of 10,000

balls for $15. Write an equation to determine how many bags and boxes he should buy

to reach the goal. Draw a graph of this equation.

c) Only 50 bags are available to buy. How many boxes will need to be purchased to

meet ping-pong quota? How much will this cost?

Common Issues Questions/Prompts

Q1: Student uses the wrong formula

For example: solves for sector area instead of

arc length

What are our key words in this question?

Which parts-of-a-circle vocabulary word

describes the crust of a pizza?

Q2: Forgets about negative signs in the

formula

What happens when we multiply two negative

numbers together?

What happens when we add/subtract negative

numbers?

Q2: Student takes square root too soon

For example: √72+(−11)2=7+(−11)

What do you get as your final answer? Does

this answer make sense?

Where operation is a square root? Can we

rewrite this equation to have only exponents?

How can we use the order of operations here?

Q2: Student adds 100ft. to their final answer If each unit is worth 100 feet and we have

about 13 units, how many feet should we

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have?

Q3a: Student subtracts ping-pong volume from

room volume.

If I have a box full of marbles and sand, and I

take away the marbles, what is left?

Does this tell us how many marbles were in the

box to begin with?

When you want to find out how many times

something goes into something else, what

operation are you using?

Q3b: Student uses price instead of amount as

their coefficients

Are we trying to find how many ping-pong

balls to buy, or how much it will cost us?

Q3b: Student uses the same variable for both

types of purchases

Are boxes and bags the same thing?

What does a normal equation look like?

How many variables does a standard equation

have?

Q3c: Student inputs value into wrong variable What do each of our variables represent?

How many bags are we buying?

Lesson OutlineBridge Types

Students will research the various types of bridges. A helpful article is found here:

http://www.explainthatstuff.com/bridges.html but students are welcome to delve further online to

learn more. A lesson about internet credibility can come in handy here, but will prolong the

project. Suggested that students have already had a lesson in this before this project.

Students can divide the reading amongst themselves, or read as a group. Annotating the text will

be helpful here.

Suggested Activity: Gallery Walk

Put up large poster paper for each bridge type and divide into a Pros section and a Cons section.

As a group, students will take sticky notes writing the pros and cons of each type of bridge and

place them upon the posters. Students will view each other’s notes, then return as a group and

make a group decision on the type of bridge they wish to build.

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Geography

Students should research the geography around the Bering Strait. Google maps can help with a

general idea of the paths to take. Students will have to do some math involving spherical

coordinates. A helpful tool is http://boulter.com/gps/distance/ if students wish to follow the

curvature of the Earth, otherwise straight distance should be calculated with the distance

formula.

Students need to also determine how many pillar supports they may need. Since this type of math

might be beyond their level, remind them to be reasonable and think not only of the possibilities

of the bridge collapsing under too much weight, but also of the cost of building too many pillars

and maintaining them over the years. Have students look up pictures of the type of bridge they

have chosen to get an idea of how many pillars are required to hold it up.

Dimensions

Students will determine the size and height of their bridge, as well as volume of their pillars.

Encourage students to do any and all math they might think is relevant, for the more they show

in their proposal, the better chance that their bridge is picked by the board. Have students look up

how much weight different types of bridges hold to get an idea of how much their own should

hold. If there is extra time, have students look up experiments on building bridges and testing

weights and recreate those results.

Cost

Students will research the cost of building bridges. Students will have to research their own

bridge type and how much it cost to build bridges of the same type. The most useful information

would be cost per square meter. Students should graph the cost per square meter of other bridges

that are similar to their own. Drawing a line of best fit, students can project how much their

bridge will cost based upon its size.

Cultural/Economical Issues

Have students read through the Wikipedia page

https://en.wikipedia.org/wiki/Bering_Strait_crossing on the History, Technical Challenges, and

Economic Costs to understand some of the issues that they must face when building their bridge.

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Have students brainstorm arguments against each of these issues so that they are prepared to

speak on them during their proposal. The arguments can be a simple list or a one-two page paper

on how the group plans to approach these challenges.

Students should be reminded to add any financial costs of disaster preventive equipment to their

budget. This is a good time to have students wrap up on any part of the project they may have

not yet completed.

Presentation

All project materials and information will be put together in a presentation in either Powerpoint

or Prezi. Students will need to back up their information with pictures and graphs. Each group

will make a proposal to the classroom encouraging others to vote for their bridge to be built. At

the end of each presentation, students and teacher should ask questions about the bridge and any

issues or inconsistencies that may arise. Each student will individually critique each group and

then vote upon their favorite presentation.