This collection of resources is designed for students who are preparing to write the final CST Sec 4 C2 (mathematical reasoning) exam. These resources include videos demonstrating strategies to solve questions that typically appear on the provincial CST exam, as well as additional practice questions and accompanying solutions.

At the bottom of this page, you will also find stand-alone lessons, for general review of specific Math CST 4 topics. (Additional lessons are in development.)

LEARN Design and Development Team: Authors/Lead Teachers: Tasha Ausman, PhD + Audrey McLaren, BSc, DipEd; Editors: Audrey McLaren, BSc, DipEd + Steve Rossy, MD; and Instructional Designer: Kristine Thibeault, MEd.

#### Applications of linear functions and analytic geometry

There are many ways that linear functions are addressed on the CST exam, from working with intercepts, solving systems of equations, and finding the rule given points on a graph. The videos and questions in this section will help you solve a variety of different types of questions.

# Working with X and Y Intercepts

Sometimes questions about x and y intercepts pop up on the multiple choice section of the exam. What properties of x and y intercepts help us solve these kinds of questions? Here, a simple multiple choice question reminds about the important properties of intercepts.

# Solving Multiple Choice and Short Answer Problems Involving Parallel and Perpendicular Lines

Most often, we first see questions about parallel and perpendicular lines as a multiple choice question on the CST exam. What is the difference between a parallel and perpendicular line? If you are not sure, click on the videos below for two examples.

# Solving Long Answer Systems of Equations Questions Using the Elimination Method

In this video, we tackle the most common type of long answer problem seen on the CST exam.  In these questions, worth ten marks, a scenario is given with two cases and a third case is unknown.  Each of the first two can be written as linear equations, enabling you to find values of x and y to substitute back into the third (unknown) equation.  The method used here is called Elimination Method.

# Working with Analytic Geometry Principles in a Single Question

Here we take a look at the three main principles of analytic geometry, consolidated into a single question. This video features the use of point of division, midpoint, and distance formulas to solve for the length of a missing line segment. It is a great way to practice each of these concepts together.

# Consolidating Properties of Linear Functions and Analytic Geometry in Long Answer Questions

In this video, we put together all of the information we know about the properties of linear functions and the applications of analytic geometry to solve a long answer question in the style of the CST exam. You need to apply all of your knowledge in order to solve a ten mark question for full points!

#### Applications of Non-Linear Functions

This section highlights typical exam questions that require analysis of  non-linear functions, namely the exponential, second degree, periodic, and piecewise functions.

# Solving Long Answer Problems Involving Exponential Functions

Exponential functions show up on nearly every CST exam in the form of a long answer question. Most often taking the form of an increase in value, these functions are excellent for exploring investments and population growth. In this problem, similar to several that have appeared on previous exams, you will compare two investments using the exponential function rule.

# Solving Long Answer Problems Involving Second Degree Functions

Second degree functions are often seen as daunting for students. However, once you realize the form of the rule, they become quite manageable and easy to work with. In this long answer problem, two second degree functions are used to determine costs in three scenarios.

# Reading and Interpreting Periodic Functions

In this video, you will learn how to draw two cycles of a periodic function and to solve for values of y that correspond to large values of x often unreadable from a graph given in the question.  Similar to multiple choice questions on the CST exam, solving for values of y corresponding to large values of x involves subtracting the length of the period until you can read from the graph.

# Long Answer Problems Involving Piecewise and Step Functions

Long answer questions involving piecewise and step functions are often married together, where you have to use a “back and forth” method using each function to solve multiple elements of the question.  In this example, we practice such a method so that students are used to inserting values into piecewise functions and reading step functions to understand what they are depicting.

#### Applications of Triangle and Trigonometry Theorems

The CST course covers the properties of many different types of triangles, as well as relations that can exist between triangles. This section demonstrates typical exam questions on metric relations, trigonometric formulas, and applications of the congruency/similarity theorems.

# Solving Metric Relations Problems

One of the most common short answer questions on the CST exam, metric relations questions involve finding unknown parameters in a right angled triangle where an altitude has been drawn from the hypotenuse to the right angle. Derived from similar triangles, various unknown segments of the right angled triangle can be found.

# Solving Sine Law Problems for a Missing Side or Missing Angle, Including Obtuse Angles

One of the most common questions to recurrently appear on the CST exam is the sine law multiple choice question asking for the value of an obtuse angle. We go through this special case here, alongside a review of how to find a missing side and a missing angle in any triangle using the sine law. This is a great review video that will help you conquer long answer problems as well!

# Solving Long Answer Problems Involving Multiple Triangles

The CST exam always features a question where triangles are seemingly “stuck” to other triangles, and an unknown value (angle or side) is stuck in the middle of a multi-triangle figure.  How do you solve one of these questions fully and completely? Sine law, metric relations, Pythagorean theorem, and SOH CAH TOA (if your teacher uses it) can all be helpful or necessary.

# Interpreting Congruency and Similarity in Triangles

Here, we go through two questions that are similar to those found on the multiple choice section of the CST exam.  A sample triangle is given and the question is asked which of the multiple choice options represents a triangle necessarily similar or congruent to the sample one. We employ the minimum conditions for congruence and similarity to find out.

# Three Ways to Find the Area of a Triangle

In this topic, we cover three ways to find area depending on what lengths or angles are given for a formula.  The most important of these three formulas is Hero’s formula, used when three sides but not angles are given and you are asked to solve for the area of a triangle.  A question involving Hero’s formula most often shows up as a short answer question on the final exam.

#### Statistics

The topics covered in the Statistics section are percentile, linear correlation, and mean deviation.

For the past several years, one of the mainstays of the CST final exam has been a short answer question about percentile.  Often a question is given with too much information, so ensure that you don’t get caught up in using information you don’t need. As well, rounding is key to getting this question correct, as percentile is never given as a decimal value.

# Linear Correlation Questions with Graphs, Values and Frequency Tables

Using a scatterplot to determine the value of a linear correlation coefficient has been a commonplace question on the short answer portion of the CST exam.  As well, learning to read frequency tables, or a table with a list of correlation coefficients show up each year on the multiple choice. This video explains how to tackle all three types of questions which are sure to appear on the final.

# Calculating Mean Deviation

A simple mean deviation question nearly always appears on the final examination short answer or multiple choice section.  In this video, we go through a technique to ensure you can calculate mean deviation, and pay particular attention to the absolute value of the differences from the mean.

# Slope of a Line

The concept of slope is introduced both visually and arithmetically. This lesson provides plenty of practice at calculating slope using the rise/run formula as well as using the coordinates of two points.

# Equation of a Line: Function Form

This lesson is about finding the rule of a linear function in the form y = mx + b. The additional resources provide unlimited practice given 3 types of info: Slope + one point (link in slide 1), two points (link in slide 2), one point + equation of parallel or perpendicular line (link in slide 3).

# Slopes of Parallel and Perpendicular Lines

This lesson examines the relationships between the slopes of parallel and perpendicular lines, and provides interactive GeoGebras for student experimentation.

# Equation of a Line: Standard and General Form

This lesson compares the two forms for the linear function: y = mx + b, and Ax + By - C = 0, and demonstrates changing from one form to another. Opportunities for interaction and practice are provided using Graspable Math & GeoGebra.

# Distance Between 2 Points on the Cartesian Plane

In this lesson tthe formula for finding the distance between two points is developed as an application of Pythagoras.

# Midpoint of a Line Segment

This lesson develops the formula for finding the coordinates of the midpoint of a line segment given the coordinates of the segment’s endpoints.

# Points of Division of a Line Segment

This lesson expands on the midpoint lesson and develops a formula for finding other points on AB, for example, a point that is ⅓ of the way or ¾ of the way from A to B.

# Special Linear Systems

Systems involving parallel lines and coincident lines are explored visually and algebraically in this lesson.

# Introduction to Trigonometry

This lesson covers naming conventions for opposite and adjacent sides in a right angled triangle, as well as the definitions for the sine, cosine, and tangent ratios.

# Using Sine to Find a Missing Side

This lesson focuses on using proportions to find missing sides, with examples that only  use the sine ratio.

# Using Cosine and Tangent to Find Missing Sides

This lesson covers using cosine and tangent ratios, as well as how to determine which of the 3 ratios to use in a given problem.

# Problem Solving With Trigonometry

In this lesson, examples involving angle of elevation and depression are solved, as well as situations in which there is no obvious right triangle.

# Finding Missing Angles Using Trigonometry

This lesson demonstrates how to use sohcahtoa and the inverse trig feature on a calculator to find missing angles given only side lengths on a right angled triangle

# The Law of Sines

Here the sine law is fully developed from the formula for the area of any triangle. Numeric examples on various types of triangles, right angled or not, are worked out.

# Hero's Formula

This lesson demonstrates how Hero’s formula is used to find the area of any triangle whose 3 side lengths are known.