Module 4

Can you spot the parabolic features in the images shown? The Gateway Arch in St. Louis, the Eiffel Tower in Paris, and the Golden Gate Bridge in San Francisco are all monuments of human achievement in architecture.

• The Gateway Arch was built to celebrate the westward expansion of the United States and to celebrate the pioneer spirit of its citizens.
  
  
• The Eiffel Tower, the universal symbol for romance, was constructed to commemorate the 100th anniversary of the French Revolution and served as the stunning centerpiece of the 1889 World’s Fair in Paris.
  
  
• The Golden Gate Bridge, an equally well-known structure and declared as one of the modern wonders of the world, is a suspension bridge. Its main span is the second longest suspension-bridge span in the United States. The bridge’s cables resemble parabolas.

In the last module you studied quadratic functions and their properties. In this module you will explore quadratic equations and inequalities. You will learn multiple strategies for solving quadratic equations, systems of quadratic equations, and quadratic inequalities. You will see how parabolas are used in architecture and how the mathematics of quadratic functions can be used to model a variety of problems in both building design and construction.

 

In this module you will investigate the following questions:

• How are quadratic equations and inequalities used to solve problems in design and architecture?
  
  
• Why is it important to be able to solve a problem using a variety of different strategies?

To investigate these questions, you will focus on the lessons and questions in the table.

 

Lesson	Topic	Lesson Questions
1	Solving Quadratic Equations by Graphing	How are the roots of quadratic equations related to the graphs of their corresponding quadratic functions?
		In what ways can you solve a quadratic equation by graphing?
2	Solving Quadratic Equations by Factoring	How can previous patterns of factoring be extended to polynomial expressions?
		How does the zero-product property relate to solving quadratic equations by factoring?
3	Solving Quadratic Equations by Completing the Square	In what instances would completing the square be favourable over other strategies for solving quadratic equations?
		What is the goal of completing the square, and how does this help determine the roots of a quadratic equation?
4	The Quadratic Formula	How is the quadratic formula related to the strategy of completing the square?
		How does a mathematical formula reveal the nature of the solutions to a problem?
5	Solving Quadratic Systems Graphically	Why are the points of intersection of a system of equations related to its solution?
		How does a graphical representation facilitate mathematical understanding?
6	Solving Systems of Equations Algebraically	Why do systems of equations yield the correct solutions even after they have been manipulated algebraically?
		Why is it important to be able to solve systems of equations algebraically as well as graphically?
7	Graphing Linear and Quadratic Inequalities	How are the properties of linear and quadratic inequalities in two variables related to their graphs?
		How are linear and quadratic inequalities in two variables applied in design?
8	Solving Quadratic Inequalities in One Variable	What principle is common to all strategies for solving quadratic inequalities?
		How can you tell when a problem can be modelled by a quadratic inequality in one variable?

 

Have you ever gone to a theme park or family entertainment centre? Perhaps you were brave enough to ride the loop-the-loop rollercoaster or take a chance on a scary waterslide. What does it take to design an amusement-park ride or even the amusement park itself? The Module 4 project will give you the opportunity to find out.

 

In Module 4 Project: Imagineering, you will design your own entertainment complex and solve problems related to the park’s design, construction, and maintenance. You will be the architect of your very own fun-filled adventure centre!

 

As you work through the lessons, pay attention to the contexts of the word problems you encounter. These contexts will help you understand how math relates to your project and, even more importantly, to your daily reality.