Image


These pre-calculus lessons are meant to bridge the gap between Math SN5 and Calculus 1. Many students who go directly from Math SN5 to Calculus 1 have difficulty with some of the algebraic procedures that are essential to Calculus but are not covered in the Quebec SN5 curriculum. These lessons provide a solid foundation that students will need in order to be successful in Calculus. (Additional lessons are in development.)

The primary objectives of these lessons are to provide:

• Review and consolidation of essential secondary high school topics covered in the Math SN4/SN5 courses.

• Additional algebra instruction that picks up where Math SN4/SN5 leave off, and covers topics essential to successfully study Calculus 1 and 2.

• Enrichment, which may include introductory Calculus topics, such as Limits & the Derivative of a Function.

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: Audrey McLaren, BSc, DipEd; Editor: Steven Rossy, MD; and Instructional Designer: Kristine Thibeault, MEd

______________________________________________________________________________________________
 

The Factoring Series gives a solid foundation for the concept and need for factoring. Each video lesson provides practice on the basics as well as more challenging combinations of several factoring types.

 

Factoring 1: Introduction to Factoring & GCF

Factoring 1 covers the definition of factoring and introduces GCF factoring, including rational exponents and grouping to form a binomial common factor.

 

Factoring 2: Trinomials

Factoring 2 reviews trinomials both with and without leading coefficient 1, as well as perfect square trinomials

 

Factoring 3: Difference of Squares

Factoring 3 covers difference of squares and also discusses common difficulties, such as recognizing when an expression is prime.
 


Additional Resources:

Geogebra Practice: Factoring Trinomials (Slide 23)

Geogebra Practice: Copy of Factoring Trinomials (Slide 37)

Geogebra Practice: Factoring a Difference of Two Squares and Perfect Squares (Slide 16)

Factoring 4: Sum and Difference of Cubes

Factoring 4 introduces factoring a sum and difference of squares, again reinforcing the difference between that and (a ∓  b)3.
 


Additional Resources:

Geogebra Practice: Factoring Sum/Difference of Cubes (Slide 27)

Factoring 5: Cubics

Factoring 5 covers factoring a cubic expression by using a combination of trial and error and long division.


Additional Resources:

Geogebra Practice: Factors, Remainders and Cubic Graphs (Slide 27)


The Rational Expressions Series reviews the basic procedures, but also provides practice that includes many factoring types covered in the factoring series.

Rational Expressions 1: Simplifying and Restrictions

Lesson 1 explains how to simplify a single rational expression, and gives the rationale for stating restrictions on a rational expression.

 


Additional Resources:

Geogebra Practice: Sorting Rational and Non-rational Expressions (Slide 22)

Geogebra Practice: Simplifying Rational Expressions and Stating Restrictions (Slide 81)

Rational Expressions 2: Adding and Subtracting

This lesson covers adding and subtracting rational expressions in many different situations. It is intended to give a preview of partial fractions.

Rational Expressions 3: Multiplying and Dividing

In addition to covering multiplication and division of rational expressions, some background is given as to why the usual division algorithm is used.

_________________________________________________________________________________________________

Slope of a Line

This lesson gives a visual definition of slope of a line, as well as a formula to calculate slope. GeoGebra is used to demonstrate the concept, to provide the student with practice, and to encourage the opportunity to explore independently.

 


Additional Resources:

GeoGebra: Demonstrating slope and its properties (as per slide 1)

Practice GeoGebras related to slope (as per slide 2)

 

Parallel and Perpendicular Lines

This is a demonstration of  the arithmetic relationships between slopes of lines that are parallel or perpendicular to each other.  GeoGebra is used to demonstrate the concept, to provide the student with practice, and to encourage the opportunity to explore interactively and independently.
 


Additional Resources:

GeoGebra: Demonstrating slopes of parallel lines (as per slide 1)

GeoGebra: Demonstrating slopes of perpendicular lines (as per slide 2)

GeoGebra: An Interactive investigation of  slopes of perpendicular lines (as per slide 3)

 

 

Finding the Rule of a Linear Function

This lesson outlines a procedure for finding the standard algebraic representation of a linear function given various types of information about the line. Included are several worked out examples which progress from simple (given slope and y intercept) to more complex (given 2 points, or the equation of perpendicular line and y intercept). GeoGebra is used to provide the student with practice and immediate feedback.

 


Addtional Resources:

GeoGebra: Unlimited practice finding equation of a line given slope and one point (as per slide 1)

GeoGebra: Unlimited practice finding equation of a line given two points (as per slide 2)

GeoGebra: Unlimited practice finding equation of a line involving parallel and perpendicular lines (as per slide 3)

 

Linear functions: Standard Form & General Form

Outlines procedures for transforming an equation to/from the standard form
y = ax + k and Ax + By + C = 0.

Graspable Math and GeoGebra are used to demonstrate the concept, and to provide the opportunity to explore independently.

 


Additional Resources:

Graspable Math: An  interactive example of how to convert from general form to slope-intercept form (as per slide 1)

Graspable Math: An interactive example of how to convert from slope-intercept form to general form (as per slide 2)

GeoGebra: Unlimited practice converting linear equation forms (as per slide 3)

 


The Exponent Rules Boot Camp Series is made up of five lessons that are meant to be viewed in sequence. Each lesson builds on the properties covered in those prior to it. Each Boot Camp ends with examples that gradually spiral through all the exponent rules, thus enabling students to simplify more and more complex expressions, both numerical and algebraic.

Exponent Boot Camp 1: Product and Quotient Properties

Exponent Boot Camp 1 offers an intuitive explanation of the two rules with which most students are already familiar:

xn . xm = xn+m and xn ÷ xm = xn-m

Several examples result in fractional, zero, and negative exponents, all of which are more deeply examined in later lessons.

Exponent Boot Camp 2: Negative and Zero Exponents

Exponent Boot Camp 2 re-examines the quotient property, this time in rational form, in order to illuminate the meaning of the zero exponent and the negative exponent.  Several examples result in rational exponents, which are more fully developed in a later lesson.

 

Exponent Boot Camp 3: Power of a Power

Exponent Boot Camp 3 is an intuitive explanation of the power of a power property (xn)m = xnm .

Examples involve applying this along with several of the properties covered so far, as well as performing a change of base in order to enable the power of a power property.
 

Exponent Boot Camp 4: Power of a Product

The property (ab)n = anbn is presented and incorporated into examples involving both like and non-like bases.
 


Additional Resources:

GeoGebra: Infinite Practice on Simplifying Expressions with Exponents

GeoGebra: Infinite Practice on Simplifying Rational Expressions with Exponents

Exponent Boot Camp 5: Rational Exponents

This lesson uses the power of a power property to develop the meaning of a1/n, then expands that to the general case of  am/n.
 


Additonal Resouces:

GeoGebra: Infinite Practice on Rational Exponents


The Trigonometric Identity Series is comprised of nine lessons that cover what the main trigonometric identities are and what they are used for. It also gives plenty of algebraic practice within the context of proving more complex identities. Most of these topics are covered in Quebec’s grade 11 curriculum, but topics 7, 8, and 9 are not.

Trig Identities 1: Introduction, Quotient and Reciprocal Identities

This lesson gives a context for the meaning and use of trigonometric identities. The quotient and reciprocal identities are introduced, and used to algebraically simplify trigonometric expressions.
 

Trig Identities 2: Pythagorean Identities

The unit circle is used to develop the basic Pythagorean identity, which is then built upon to develop and simplify other trigonometric expressions. Examples include combinations of Pythagorean, quotient, and reciprocal identities.

 

Trig Identities 3: Using Identities to Find Trigonometric Ratios

This lesson steps out of the pure algebra of identities to do some numerical calculations. Exact values for the trigonometric ratios of various angles in the unit circle are found using the Pythagorean, quotient, and reciprocal identities.

 

Trig Identities 4: Proving Trigonometric Identities

In this lesson, the traditional form and procedure of a proof of a trigonometric identity is presented. The examples are fairly simple in this lesson, but subsequent lessons will delve into more complex ones.

 

Trig Identities 5: The Algebra of Identities

Proving trigonometric identities typically involves fairly complex algebraic procedures, which are addressed in this lesson and the next one. In this lesson, examples include combining fractions and factoring.


 

Trig Identities 6: More Algebra of Identities

This second lesson on the algebra of identities covers multiplying numerator and denominator by a conjugate.


 

Trig Identities 7: The Sum and Difference Identities

This lesson presents the identities for the sum and difference of angles, and offers a geometric proof of the sum identities.
 


Additional Resources: 

GeoGebra:  Proof of Sum Identities

Trig Identities 8: The Double Angle and Half Angle Identities

The sum identities are used to develop the double angle identities for sin and cos, which are then used to develop the half angle identities.

 

Trig Identities 9: Related Angle Identities

This lesson demonstrates visually, as opposed to algebraically, the trigonometric relationships between angles such as θ and θ + π.

 


Additional Resources:

GeoGebra: Related Angles


The Radicals Series: Radicals are not typically covered in Quebec high schools, but are important for the study of limits in Calculus. These lessons begin with numerical examples and expand to algebraic ones in order to develop the algebra necessary to study Calculus.

Radicals 1: Simplifying Radical Expressions

This lesson covers radical simplification by connecting square roots to exponents, and using exponent properties to simplify.
 


Additional Resources:

GeoGebra: Infinite Practice Simplifying Radical Numbers

Radicals 2: Adding/Subtracting Radical Expressions

In this lesson, the complete steps to radical addition and subtraction are explained, both with and without simplification, and involving monomial and binomial radicals.
 

Radicals 3: Multiplying/Dividing Radical Expressions

This lesson includes multiplication of monomial and binomial radicals, as well as special cases of squaring binomial radicals and the product of conjugates.


Additional Resources:

GeoGebra: Infinite Practice - Multiplying/Dividing Radicals (Slide 24)

Graspable Math Examples - Multiplying Radical Binomials (Slide 26)

GeoGebra: Infinite Practice - Squaring a Monomial Radical (Slide 39)

GeoGebra: Infinite Practice - Multiplication of Conjugates (Slide 56)

Radicals 4: Rationalization

In this lesson, rationalization of both denominators and numerators is outlined, and some groundwork is laid for its use in finding limits. 


Additional Resources:

GeoGebra: Puzzle with Extra Practice


The Limits Series introduces students to an important concept for Calculus - Limits. All of this content is covered in any typical Calculus 1 course. It is included here because of the connections to other lessons in our Pre-Calculus course, such as factoring and rationalization. 

Limits 1: Introduction

This VoiceThread lesson introduces the concept and definition of a limit, as well as how to use a table, graph, or substitution to evaluate a limit.

 

Limits 2: Properties of Limits

In this lesson the properties of limits (additive, multiplicative, etc.) are outlined and explained.

 

Limits 3: Limits and Factoring

In this lesson, factoring is added to the list of possible methods for finding limits.

Limits 4: Limits and Rationalization

In this video lesson, rationalization of numerators or denominators is reviewed and demonstrated as another tool to evaluate limits.
 

_________________________________________________________________________________________________

Partial Fractions 1: Distinct Linear Factors

This lesson covers the decomposition of a rational expression into expressions with distinct linear factors.


Additional Resources:

Geogebra: Unlimited Practice on Partial Fractions