# Engineering in the K-12 Classroom: Math and Science Education for the 21st-Century Workforce

## CONTENTS OF CURRICULUM UNIT 12.04.06

## The Road to Bridge Design

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## Classroom Activity 1

### Objective:

Students will build a suspension bridge using a building kit and determine the standard equation of the parabola created by the main cable. Students will also complete a laboratory report and determine a hypothesis showing higher level thinking.### Procedure:

Students will work in teams as described above to build a suspension bridge.Students will know the following:

Vertex form:

- · The main forces of the suspension bridge are tension and compression.
- · The tension is supplied by the cables and the compression is supplied by the towers. The cables hold up the span or road by being attached to the towers. This causes the weight of the traffic to be transferred to the towers by the attachment of the cables to it and then to the ground by the towers.
- · It is the main cables of a suspension bridge that form the parabola.
- · Students should be able to define: parabolic graph, focus, directrix, vertex, axis of symmetry
- · Students should know these parabolic equations:
- Standard form:
y= ax^{ 2 }+bx+c

*y = a(x - h)*where (h, k) is the vertex

^{ 2 }
Intercept form:
*
y = a(x - p)(x - q)
*

Students will locate the vertex of the suspension cable of the bridge they built using a kit by using a ruler and derive the value of
*
a
*
to write the parabolic equation in standard form.

*
Deriving the value of a
*
: It is important to note that a parabola is formed by a locus of points that is equidistant from the focus and the directrix. The vertex is found midway from the focus to the directrix. This is useful information when the students are trying to compute
*
a
*
since
*
p
*
of the ancients conic form of a parabolic equation is the distance between the vertex and the focus or from the vertex to the directrix.

Compare the modern vertex form of the parabolic equation to the older conics form of the parabolic equation from ancient times which is
*
4p(y - k)=(x - h)
^{
2
}
*
where (h, k) is the vertex.

Show how the vertex form translates to the conics form and yields the value of
*
a
*
.

*
y = a(x - h)
^{
2
}
+k
*

y - k = a(x - h)
^{
2
}

(y - k)/a = (x - h)
^{
2
}

Replace 1/
*
a
*
with 4
*
p
*
from the conics form, so
*
4p(y - k)=(x - h)
^{
2
}
*
and the vertex form becomes the conics form therefore, 1/

*a*= 4

*p*and

*a*= 1/4

*p*where p is the distance between the vertex and the focus and is also the distance between the vertex and the directrix. Note:

*2p*is the distance between the focus and the directrix.

### Sample Lesson

One way of executing this lesson using the above described teaching strategies is to first have students discuss general information about bridges during a report out with attention to covering essential topics as tension, compression, and use of cables in a suspension bridge. Have students discover that the main cable is a parabola. If basic information is not discovered by the students then it must be given.

Remind students of the three parabolic equations already studied and listed above.

Next, show students how a parabola is formed by using a locus of points from a point (focus) and a line (directrix). Inform students that the parabola is created by points equidistant from the point and the line. Note that if the line is below the point then the parabola opens upward and
*
a
*
is positive. If the line is above the point then the parabola opens downward and
*
a
*
is negative. Do not emphasize to the students that the distance

between the vertex and the focus and the vertex to the directrix are equal. This is for them to realize later when using the ancients conic form of the parabolic equation.

Provide students with more background information by introducing the ancient's conic form for the parabolic equation using the solids of their time. Hence, the conic form of the parabolic equation. The cone would be appropriate.

See if students can establish a relationship between the modern vertex form of the parabolic equation and the conic form. If not, then a hint might be appropriate. See if students will recognize the need to translate the vertex form to the conic form. If not, provide guidance or hints before having students translate the vertex form into the conic form. They should find that
*
1/a
*
and
*
4p
*
are equal and that p is the distance from the vertex of the cable of the bridge to the deck or road by association of the focus and directrix demonstration. This would serve as an application to the real world. Appropriate guidance may be needed.

Students can compute p measuring their bridge using a ruler or tape measure, find
*
a
*
, and write the equation in standard form.

### Assessment

Laboratory Report

The most important part of the laboratory report is the hypothesis or problem statement. This is where students may become creative or express a curiosity about the suspension bridge as they are working on it. For example, students may decide to look for patterns by lengthening and contracting the main cable to see how or where it affects the parabolic equation. Also, in order to write a hypothesis students show they understand the initial mathematical concept. See appendix 1 for a sample Laboratory Report outline but any will suffice.