# Math Bridge Resources

This collection of lessons was developed to bridge the learning gap for students who have successfully completed Math CST 4 and would like to pursue the Math Science (SN) stream. These resources include interactive videos, as well as supporting documents that identify and explore topics that are not covered in the CST 4 program. *(Additional lessons are in development.)*

PLEASE NOTE: No sign-in is required to view these lessons. Simply click the arrow on the lower left-hand side of the slide to play. Use the arrows on the lower right-hand side of the slide to advance the slides.

LEARN Design and Development Team: Author/Lead Teacher: Peggy Drolet, BEd; Editors: Steve Rossy, MD + Tasha Ausman, PhD; and Instructional Designer: Kristine Thibeault, MEd.

# Factoring 1: Introduction and review of Greatest Common Factors (GCFs)

This video introduces students to the building blocks of polynomials for the Math SN4 option. Geared toward students who already have a solid grasp of polynomials and exponents from Secondary 3, this video reviews Greatest Common Factor (GCF).

**Learning Goal:**

-To be able to factor polynomials by finding the common factor

**Related Textbook Pages:**

*Visions*, *Science, Student Book Volume 1, Secondary Cycle Two*, *Year Two, *Les Editions CEC Inc, 2009, **pp 148-177**

**Additional Resources:**

**Khan Academy: Factoring Polynomials - Common Factor 1**

**Khan Academy: Factoring Polynomials - Common Factor 2**

# Factoring 2: Factoring by differences of two perfect squares

This video continues the methods of factoring necessary for the Math SN4 option: factoring the difference of two perfect squares.

**Learning Goal:**

-To be able to factor polynomials by using second degree algebraic identity: the difference of two perfect squares.

**Related Textbook Pages:**

*Visions*, *Science, Student Book Volume 1, Secondary Cycle Two*, *Year Two, *Les Editions CEC Inc, 2009, **pp 148-177**

**Additional Resources:**

**Khan Academy: Factoring Differences of Squares with Two Variables**

# Factoring 3: Factoring by grouping

This video continues the methods of factoring necessary for the Math SN4 option: factoring by grouping.

**Learning Goal:**

-To be able to factor polynomials by grouping terms

**Related Textbook Pages:**

*Visions, Science, Student Book Volume 1, Secondary Cycle Two, Year Two*, Les Editions CEC Inc, 2009, **pp 148-177**

**Additional Resources:**

**Khan Academy: Factor by Grouping**

# Factoring 4: Factoring trinomials with a value of 'a’ equal to 1 (“easy” trinomials)

In this lesson, students are introduced to “easy” trinomials. Building upon students’ knowledge of the process of FOIL from Secondary 3, students will learn how to work backwards to find two binomials leading to a simple trinomial with an a value of 1. As well, this lesson reminds students to remove any GCFs first, so that factoring might take place.

**Learning Goal:**

-To be able to factor second degree trinomials

**Related Textbook Pages:**

*Visions, Science, Student Book Volume 1, Secondary Cycle Two, Year Two*, Les Editions CEC Inc, 2009, **pp 148-177**

Additional Resources:

**Khan Academy: Factoring Quadratics Examples**

# Factoring 5: Factoring trinomials with a value of ‘a’ that is greater than 1

In this lesson, the decomposition method is introduced whereby trinomials in general form with an a value greater than 1 can be factored. The approach is to first find any GCFs and remove them, multiply the first and last terms to find a desired target value, and then determine which combination of integers results in the value of b using addition or subtraction. This video relies on prior knowledge of factoring by grouping.

**Learning Goal:**

-To be able to factor polynomials that contain decomposable second degree trinomials

# Long Division: Dividing polynomials by monomials and binomials

The purpose of this video is to learn how to employ the principles of long division, originally taught in elementary school, to polynomials. This lesson first reminds you how to divide a polynomial by a monomial, and then draws upon the steps in long division to divide a binomial into larger polynomials (second and third degree).

**Learning Goal:**

-To be able to divide a polynomial by another polynomial (with or without a remainder)

**Related Textbook Pages:**

*Visions, Science, Student Book Volume 1, Secondary Cycle Two, Year Two*, Les Editions CEC Inc, 2009, **pp 148-177**

**Additional Resources:**

**Dividing a Polynomial by a Binomial**

# Rational Expressions 1: Simplifying rational expressions

In this introductory lesson to Rational Expressions (REs), students will build upon their knowledge of factoring to, firstly, define rational expressions, and secondly, understand basic simplification. Using the concept of Greatest Common Factor (GCF), several examples are worked through, emphasizing how a common term can be removed, leaving a final answer that cannot be reduced further.

**Learning Goal:**

-To be able to manipulate rational expressions

**Related Textbook Pages:**

*Visions, Science, Student Book Volume 1, Secondary Cycle Two, Year Two*, Les Editions CEC Inc, 2009, **pp 148-177**

**Additonal Resources:**

**Khan Academy: Intro to Rational Expression Simplification**

**Khan Academy: Simplifying Rational Expressions: Common Binomial Factors**

# Rational Expressions 2: Multiplying and dividing rational expressions

Beginning with an introduction on multiplying fractions, this lesson first focuses on factoring out terms in the numerators and denominators of polynomial fractions. Once again, reminding ourselves that common factors can be cancelled top-to-bottom, these are removed, leaving us with a final, non-reducible solution. In division, the same method is employed, but with a poignant reminder that when dividing fractions, including polynomial ones, one must reciprocate the second term.

**Learning Goal:**

-To be able to manipulate rational expressions

**Related Textbook Pages:**

*Visions, Science, Student Book Volume 1, Secondary Cycle Two, Year Two*, Les Editions CEC Inc, 2009, **pp 148-177**

**Additional Resources:**

**Khan Academy: Multiplying and Dividing Rational Expressions Part 1**

**Khan Academy: Multiplying and Dividing Rational Expressions Part 2**

**Khan Academy: Multiplying and Dividing Rational Expressions Part 3**

**Khan Academy: Dividing Rational Expressions - Unknown Expression**

# Rational Expressions 3: Adding and subtracting rational expressions

In this lesson, we learn about adding and subtracting rational expressions (polynomials in fraction form). Students are reminded about the basics of arithmetic, whereby a common denominator must be found before any two fractions can be added or subtracted. Special emphasis is placed upon distributing the negative (subtraction sign) in the numerator when subtracting rational expressions.

**Learning Goal:**

-To be able to manipulate rational expressions

**Related Textbook Pages:**

*Visions, Science, Student Book Volume 1, Secondary Cycle Two, Year Two*, Les Editions CEC Inc, 2009, **pp 148-177**

**Additional Resources:**

**Khan Academy: Adding & Subtracting Rational Expressions: Like Denominators**

# Rational Expressions 4: Adding and subtracting rational expressions

In this lesson, the steps for adding and subtracting rational expressions are reviewed and emphasized, and several difficult examples are worked through. Try these examples alongside the step-by-step instructions, and check your work as you proceed. If you can do these questions, you have an excellent grasp of adding and subtracting rational expressions.

**Learning Goal:**

-To be able to manipulate rational expressions

**Related Textbook Pages:**

*Visions, Science, Student Book Volume 1, Secondary Cycle Two, Year Two*, Les Editions CEC Inc, 2009, **pp 148-177**

**Addtional Resources:**

**Khan Academy: Adding Rational Expressions: Unlike Denominators**

# The Greatest Integer Function (Step Function): An Introduction

This lesson introduces the concept of a new mathematical operation that involves [ ]. Students will have the opportunity to explore a situation that requires this new operation. As a result, students will learn about the Basic Greatest Integer Function by building a table of values, constructing the graph and analyzing this Basic Greatest Integer Function

# The Role of Parameters a and b in the Greatest Integer Function

# The Role of Parameters h and k in the Greatest Integer Function

In this lesson, students will review the basic Greatest Integer Function. Then, students will explore the effects of changing parameters h and k. As a result, they will learn how to build a table of values and learn how to construct a graph of a Greatest Integer Function when parameters h and k are introduced in the basic rule.

# Solving Quadratic Equations 1: The Factoring Method

In this lesson, students get a mini-review on solving a linear equation. Then, there is an emphasis on how to recognize a quadratic equation. The students will be exposed to three techniques on how to solve a quadratic equation and how to recognize what technique to use. The focus of this lesson is solving a quadratic equation using the factoring method.

**Learning Goal:**

• To be able to solve an equation of the second degree in one variable using the factoring method

**Textbook Pages:**

*Visions*, *Science, Student Book Volume 1, Secondary Cycle Two*, *Year Two, *Les Editions CEC inc, 2009, pp 148-177

**Additional Resources:**

Memory Aid Tips: Solving a Quadratic Equation

Khan Academy: Solving Quadratics by Factoring

# Solving Quadratic Equations 2: The Perfect Square Method

In this lesson, solving quadratic equations is explained using the perfect square method. Emphasis is placed on remembering that there are both positive and negative roots of any perfect square. As well, the binomial that is squared must be isolated in order to take the root of both sides.

**Learning Goal:**

** • **To be able to solve an equation of the second degree in one variable using the perfect square method

**Related Textbook Pages:**

*Visions*, *Science, Student Book Volume 1, Secondary Cycle Two*, *Year Two, *Les Editions CEC inc, 2009, pp 148-177

**Additional Resource:**

# Solving Quadratic Equations 3: The Quadratic Formula

In this voicethread, students are introduced to a new rule: the quadratic formula. This easy to apply rule for equations given in the general form can be implemented to find the zeroes of a quadratic function, particularly in cases where easy factoring is not possible or when the factors are not integers.

**Learning Goal: **

• To be able to solve an equation of the second degree in one variable using the quadratic formula

**Related Textbook Pages:**

*Visions*, *Science, Student Book Volume 1, Secondary Cycle Two*, *Year Two, *Les Editions CEC inc, 2009, pp 148-177

**Additonal Resources:**

Memory Aid Tips: Solving a Quadratic Equation

# Problem Solving Using Quadratic Equations

This lesson builds on your understanding of quadratic equations to begin solving some word problems involving geometric shapes such as the area of a triangle or equivalent areas. Students are taught how to solve for numerical answers rather than algebraic expressions for side lengths and areas.

**Learning Goal:**

• To be able to analyze situations using an equation of the second degree in one variable

**Related Textbook Pages:**

*Visions*, *Science, Student Book Volume 1, Secondary Cycle Two*, *Year Two, *Les Editions CEC inc, 2009, pp 148-177

**Additional Resources:**

Memory Aid Tips: Useful Formulas

Memory Aid Tips: Story Problems

Khan Academy: Quadratic Equations Word Problem

# Quadratic Inequalities: One Variable

This lesson introduces the concept of quadratic inequalities. Using a method that involves swapping out the inequality symbol for an equal sign, students are encouraged to choose the T-method, Quadratic Formula, or Perfect Square method to find the zeroes (solutions) of the inequality. Students are encouraged to graph the quadratic inequality as an equation first, and then determine the answer to the original inequation and write the solution set as a domain.

**Learning Goal:**

• To be able to find the solution set of a quadratic inequality in one variable

# The Law of Cosines

This lesson features a second formula that helps to solve triangles. Sine Law, which is part of the CST 10 curriculum, offers us a way to solve triangles when sides and their corresponding angles (opposite angle) are given. However, what happens when there are two sides in a triangle and the enclosed angle between them? What about when you are given three sides and no angles? How do you solve these kinds of triangles? In this lesson, students are introduced to cosine law, which enables you to solve these special cases.

**Learning Goal:**

** • **To find unknown measures in any triangle using the cosine law** **

**Related Textbook Pages:**

*Visions*, *Science, Student Book Volume 2, Secondary Cycle Two*, *Year Two, *Les Editions CEC inc, 2009, pp 180-193

**Additional Resources:**

Extra Practice Questions

Extra Practice Answers

# Problem Solving Algebra

This video focuses on exam-style questions involving rational expressions and factoring. Using function notation often seen on the ministry exam, the teacher goes through two main styles of questions in detail: purely mathematical calculations and word problems. Two geometric word problems are an excellent return to grade nine algebra, where area and volume are expressed as algebraic expressions; however, new methods such as polynomial long division and rational expressions are integrated. All answers are fully explicated for students along the way.

**Learning Goal:**

• To be able to manipulate algebraic expressions in order to analyze situations

**Related Textbook Pages:**

*Visions*, *Science, Student Book Volume 1, Secondary Cycle Two*, *Year Two, *Les Editions CEC inc, 2009, pp 148-177

**Additional Resources:**

Memory Aid Tips: Manipulates Algebraic Expressions to Analyze Situations