This document provides information and tasks related to investigating parabolic curves through the design of suspension bridges and industrial packaging. It outlines three approaches - arithmetic, procedural, and conceptual - for examining parabolas in real-life applications. Students are asked to draw bridge plans showing supports, cables, and hangers on a coordinate plane and calculate related values. For packaging, students define the three investigation approaches and derive generalized equations to calculate box dimensions and capacity based on a given variable.

1. Suspension Bridges
and the Parabolic
Curve
COMPILED BY MR SOO

2. Overarching Inquiry Statement
Modelling using a logical process helps
us to understand the world around us.

3. Learning Intent:
 To examine the phenomenon of suspension bridges and see how the
parabolic curve strengthens the construction. The student will then
diagram their own bridge given a scenario and find key points using a
quadratic equation.

4. Success Criteria
 The student will be able to:
 investigate and analyze quadratic functions both algebraically and
graphically
 make connections between and among multiple representations of
functions including concrete, verbal, numeric, graphic, and algebraic.

5. Mathematical Objective(s)
 The student will be able to:
 • investigate and analyze quadratic functions both algebraically and
graphically
 • make connections between and among multiple representations of
functions including concrete, verbal, numeric, graphic, and algebraic

6. Algebraic Strands
 use symbolic algebra to represent and explain mathematical relationships
 • build new mathematical knowledge through problem solving
 • recognize and apply mathematics in contexts outside of mathematics

7. Prior Knowledge
 Students should have basic knowledge of quadratic equations and the
nature of a parabola to include the vertex form of a quadratic equation. 𝑦
= 𝑎(𝑥 − ℎ)2 + 𝑘,
 𝑤ℎ𝑒r𝑒 (ℎ, 𝑘) 𝑖s the vertex of the parabola

8. Curiosities
 You will investigate the wonder of suspension bridges and the science
behind their construction.
 You will learn about tension and compression forces and how a parabolic
curve between supports on a bridge helps achieve maximum strength.
 You will also design your own bridge and plot the points on a coordinate
plane paying special attention to the coordinates of the intersection of the
cable and hangers.

9. Questions:
 Factual
 Conceptual
 Debatable

10. Concept Map

11. Personal Inquiry
 How can we craft a personal inquiry that answers our curiosities and
connect to the overarching inquiry?

12. Worldviews
 Social Constructivist?
 Pragmatism?

13. Deductive vs inductive approaches
 What is your approach here?

14. Consequential vs Deontological
approach
 What is your orientation here?

22. Summary
 Supports: The towers of the bridge.
 Span: Distance between two towers on suspension bridge. Approaches:
The roads leading up to the bridge.
 Cable: Runs from tower to tower and connects to hangers.
 Hanger: Connects the cable to the roadway of the bridge.

23. Scenario and investigation
 A bridge has failed just as in the video. As an up and coming engineer, you
have been asked to help with the design of a new bridge. In the
preliminary stages, you are mostly concerned with the area between the
two pillars of the bridge as that is where the problem with the last bridge
seems to have originated. The bridge must span a 2000 meter section of a
bay.
 For the purposes of your early design you are to assume the following: •
 The end of the approach from the west side of the bay will represent point
(0, 40) on your coordinate plane.
 There is a 40 m drop from the end of the approaches to the body of water
below

24. Task
 Draw a plan for a bridge on the coordinate plane that includes two square
supports, at least one cable from support to support on each side of the
road, and as many hangers as the student feels necessary to attach the
road to the cables.

25. Questions
 Answer the following questions:
 1. How tall and wide are your supports?
 2. What are the coordinates of the top your supports?
 3. What is the length of your shortest hanger?
 4. What are the coordinates of the vertex of the cable that extends from support to
support?
 5. What is the equation of your parabolic cable?
 6. How many additional hangers are you using on your bridge between the two
supports?
 7. What are the intersections of your other hangers with the parabolic cable?
 8. How much steel cable will you need for all your hangers on both sides of the road?
 9. Try to estimate how much cable you will need for the parabolic cable that stretches
between the two supports.

26. Part 2 – Industrial Packaging

27. Industrial Designing
Task - Packaging
Efficiency
Parabolic Investigation Task
YEAR 9 EVEREST
WRITTEN BY MR SOO

29. Personalized Inquiry
 To evaluate and connect concepts of parabolas in real-life situations
 3 approaches to investigate: Arithmetic, Procedural and Conceptual .

38. Format: Industrial
Packaging Design
Task
BY JOHN MONASH
EXTENSION

39. Background and Abstract
In this investigation, I am to use non-linear algebra in estimating packaging dimensions
that are cost effective and sustainable.
I will use 3 methods in this parabolic investigation. They are:
1) Arithmetic – Define Arithmetic approach in algebra
2) Procedural – Define Procedural approach in algebra
3) Conceptual – Define
Each of these three methods have its merits and is dependent on the availability of
resources. However, efficiency is maximized when we employ the conceptual method as it
minimizes wastage and keeps the process sustainable. XXXXXXXXXXXX* Provide more
insights on the three methods..

41. Arithmetic Approach (Concrete
Models)
If x = 6,
length of box = 2
Width of box = 2
If x = 10,
length of box = 6
Width of box =
If x = 6,
length of box = 2
Width of box = 2
If x = 6,
length of box = 2
Width of box = 2

42. Model A (x=6)
2CM
2CM
2CM

43. What do I notice about the rectangles?
 XXXXXXX
 Are the dimensions different?
 How have they changed?
 Is there a pattern to this change?
 What is the pattern?
 Can this pattern be recorded using a set of algebraic
equations/expressions? How?

44. Capacity investigation
 Capacity of box 1 =
 Capacity of box 2 =
 Graph of capacity (y) against x: Insert graph here
 How do we determine the maximum capacity?

45. Procedural Algebraic Approach
 INSERT TABLE HERE
 Highlight your class intervals
 X values that are impossible are as follows:

46. Procedural approach Graph
 Insert graph here

47. Steps in calculating the Apex
 Note the difference between the 2 graphs
 Why is each graph important? And under what conditions do we use each
of these graphs?

48. Conceptual Algebraic Approach
 Translate the dimensions of the box into a series of mathematical
equations:
 Length?
 Width?
 Diagonal?

49. Generalization of patterns
 Translate the perimeter of the box into a generalized equation:
 XXXXXXXXXX

50. Generalization of patterns
 Translate the base area of the box into a generalized equation:
 XXXXXXXXXX

51. Generalization of patterns
 Translate the capacity of the box into a generalized equation:
 XXXXXXXXXX

52. Combine the equations to give the
capacity of the box in terms of only x

53. Graph of this relationship

57. Lesh’s Translation Model (Reflection)

58. Extension
 Parabolic investigation
- Real life application of the parabola
 Cubic investigation
 Binomial expansion investigation

59. Bibliography (links) or References
(APA7)