# Grades 9/10/11 Linear Relations and Analytic Geometry

Linear and non-linear relations. Solving linear equations and linear systems. Analytic geometry and statistics.

This is one of seven strands of the CEMC Grade 9/10/11 courseware. The other strands and more information about this courseware is available on the Grade 9/10/11 homepage.

The alternative formats for some lessons are currently under construction. These will be completed in the coming months.

### Units

## Unit 1: Linear Equations

This lesson will focus on solving one- and two-step equations involving rational coefficients.

This lesson will focus on solving multi-step linear equations involving rational coefficients. It will also solve linear equations using cross-multiplication.

This lesson will look at solving linear equations in application based problems. Some examples will have equations given and in others the equation will need to be developed before solving.

This lesson will solve problems algebraically involving rate, ratio, proportion, and percent using cross-multiplication.

This lesson will look at rearranging equations and solving in terms of one variable. It will also rearrange formulas to solve in terms of one variable.

In this lesson, we will learn to solve linear inequalities. We will solve application problems with and without the linear inequality given.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 2: Characteristics of Linear Relations

In this lesson, we will explore characteristics of a linear relationship between two variables using patterns, tables of values, and graphs. A variable is identified as being independent or dependent and comparisons are made between linear and non-linear relations.

In this lesson, we will continue to identify properties of a linear relation. We will define rate of change and initial value. We will then determine the rate of change and initial value from tables and graphs and use these values to develop a corresponding linear equation. In addition, we will calculate first differences in a table of values and use this to identify a relation as linear or non-linear.

In this lesson, we will define the terms direct variation and partial variation. We will use these definitions to identify which of the two variations a linear relationship represents.

In this lesson, we will introduce the terms slope and \(y\)-intercept. We will identify the slope and \(y\)-intercept in various representations of a linear relation. In addition, the idea of linear families will be explored.

In this lesson, we will look at various ways to graph linear relations by hand.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 3: Connecting Various Representations of Linear Relations

In this lesson, we will use the properties of a linear relation to help us solve for unknown values in various representations of the relation.

In this lesson, we will look at connecting various representations of a linear relation.

In this lesson, we will look at changing the conditions of a linear situation represented by a graph and explore how the changes affect the graph and equation of the line.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 4: Properties of Slope

In this lesson, we will use the idea of rise and run to develop and work with the slope formula for a linear relation.

In this lesson, we will algebraically determine the equation of a line in the form \(y=mx+b\) given a \(y\)-intercept and a point, given a slope and a point, and given two points.

In this lesson, we will learn about the properties of slope for both parallel and perpendicular lines.

In this lesson, we will investigate horizontal and vertical lines.

## Unit 5: Equations of Linear Relations and Problem Solving

In this lesson, we will look at various forms of a linear equation. We will rearrange the equations from one form to another and solve problems involving the different forms of a linear equation.

In this lesson, we will compare the properties of linear relations to those of non-linear relations. We will use these properties to classify a relation as linear or non-linear.

In this lesson, we will consider application problems that encompass ideas and concepts presented throughout the linear relations units. These ideas will be extended to include identifying and interpreting a point of intersection graphically and setting restrictions on variables within a given context.

In this lesson, we will create graphs given a scenario and describe events given a graph. This lesson will include slope calculations that relate to average speed.

## Unit 6: Solving Linear Systems of Equations

In this lesson, we will begin looking at solving linear systems of two equations by graphing.

In this lesson, we will look at solving a system of equations algebraically. We will use the methods of substitution and elimination.

In this lesson, we will model various descriptions of linear systems. We will use the developed model to solve problems, answer questions, and interpret the meaning of the solution within the given context.

## Unit 7: Properties of Line Segments and Using Analytic Geometry to Verify Geometric Properties

In this lesson, we will develop and use the formulas to calculate the midpoint and length of a line segment with defined endpoints on a Cartesian plane.

In this lesson, we will problem solve using the concepts of slope, length, and midpoint of a line segment. We will consider problems involving lines and then problems involving triangles.

In this lesson, we will investigate the properties and characteristics of various quadrilaterals. We will verify some of these properties using analytic geometry using a multi-step strategy where necessary.

In this lesson, we will develop the equation of a circle. We will use the equation to sketch a circle and solve application problems.

## Unit 8: Data Management and Statistics

In this lesson, we will interpret the meaning of points on scatter plots, construct scatter plots, and draw curves or lines of best fit. In addition, we will determine the equation of a line of best fit.

In this lesson, we will learn how to pose problems, identify variables associated with a problem and formulate a hypothesis. We will learn how to carry out an investigation to determine if a relationship exists between two variables.

In this lesson, we will learn about different types of data and about bias in data collection. We will also examine different sampling techniques.

In this lesson, we will examine various ways to organize and display data. We will interpret graphs and identify potential problems with the display of data related to bias.

In this lesson, we will look at where probability is used in society.