2: Linear Relations

2.1: Most of the expectations in the Linear Relations strand of the Grade 9 applied and academic courses are common to the two courses. Some of these expectations may require review since the concepts and skills in this strand are critical for student success in the Analytic Geometry strand of this transfer course.

Function Machines 2 (Functions, Tables, and Graphs)
Linear Functions
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A

3: Analytic Geometry

3.1: Overall Expectations

3.1.1: demonstrate an understanding of the characteristics of a linear relation;

Function Machines 2 (Functions, Tables, and Graphs)
Linear Functions
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

3.1.2: determine the relationship between the form of an equation and the shape of its graph with respect to linearity and non-linearity;

Function Machines 2 (Functions, Tables, and Graphs)
Linear Functions
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

3.1.3: determine, through investigation, the properties of the slope and y-intercept of a linear relation;

Distance-Time Graphs
Function Machines 2 (Functions, Tables, and Graphs)
Linear Functions
Point-Slope Form of a Line - Activity A
Slope - Activity B
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

3.1.4: solve problems involving linear relations.

Function Machines 2 (Functions, Tables, and Graphs)
Linear Functions
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

3.2: Understanding Characteristics of Linear Relations

3.2.1: design and carry out an investigation or experiment involving relationships between two variables, including the collection and organization of data, using appropriate methods, equipment, and/or technology (e.g., surveying; using measuring tools, scientific probes, the Internet) and techniques (e.g.,making tables, drawing graphs) (Sample problem: Design and perform an experiment to measure and record the temperature of ice water in a plastic cup and ice water in a thermal mug over a 30 min period, for the purpose of comparison. What factors might affect the outcome of this experiment? How could you design the experiment to account for them?);

Correlation

3.2.2: construct equations to represent linear relations derived from descriptions of realistic situations, and connect the equations to tables of values and graphs, using a variety of tools (e.g., graphing calculators, spreadsheets, graphing software, paper and pencil) (Sample problem: Construct a table of values, a graph, and an equation to represent a monthly cellphone plan that costs $25, plus $0.10 per minute of airtime.);

Function Machines 2 (Functions, Tables, and Graphs)
Linear Functions
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

3.2.3: determine the equation of a line of best fit for a scatter plot, using an informal process (e.g., using a movable line in dynamic statistical software; using a process of trial and error on a graphing calculator; determining the equation of the line joining two carefully chosen points on the scatter plot).

Slope-Intercept Form of a Line - Activity A

3.3: Investigating the Relationship Between the Equation of a Relation and the Shape of Its Graph

3.3.1: determine, through investigation, the characteristics that distinguish the equation of a straight line from the equations of nonlinear relations (e.g., use a graphing calculator or graphing software to graph a variety of linear and non-linear relations from their equations; classify the relations according to the shapes of their graphs; connect an equation of degree one to a linear relation);

Function Machines 2 (Functions, Tables, and Graphs)
Linear Functions
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

3.3.2: identify, through investigation, the equation of a line in any of the forms y = mx + b, Ax + By + C = 0, x = a, y = b;

Defining a Line with Two Points
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Standard Form of a Line

3.3.3: express the equation of a line in the form y = mx + b, given the form Ax + By + C = 0.

Defining a Line with Two Points
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Standard Form of a Line

3.4: Investigating the Properties of Slope

3.4.1: determine, through investigation, various formulas for the slope of a line segment or a line (e.g., m = rise/run, m = the change in y/the change in x or m = delta y/delta x, m = (y2 - y1)/(x2 - x1), and use the formulas to determine the slope of a line segment or a line;

Distance-Time Graphs
Point-Slope Form of a Line - Activity A
Slope - Activity B
Slope-Intercept Form of a Line - Activity A

3.4.2: identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b;

Defining a Line with Two Points
Function Machines 2 (Functions, Tables, and Graphs)
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

3.4.3: determine, through investigation, connections between slope and other representations of a constant rate of change of linear relation (e.g., if the cost of producing a book of photographs is $50, plus $5 per book, then the slope of the line that represents the cost versus the number of books produced has a value of 5, which is also the rate of change; the value of the slope is the value of the coefficient of the independent variable in the equation of the line,C = 50 + 5p, and the value of the first difference in a table of values);

Distance-Time Graphs
Distance-Time and Velocity-Time Graphs
Function Machines 2 (Functions, Tables, and Graphs)
Linear Functions
Point-Slope Form of a Line - Activity A
Slope - Activity B
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

3.4.4: identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism, perpendicularity), using graphing technology to facilitate investigations, where appropriate.

Point-Slope Form of a Line - Activity A
Slope - Activity B
Slope-Intercept Form of a Line - Activity A

3.5: Using the Properties of Linear Relations to Solve Problems

3.5.1: graph lines by hand, using a variety of techniques (e.g., graph y = 2/3x - 4 using the y-intercept and slope; graph 2x + 3y = 6 using the x- and y-intercepts);

Distance-Time Graphs
Distance-Time and Velocity-Time Graphs
Point-Slope Form of a Line - Activity A
Slope - Activity B
Slope-Intercept Form of a Line - Activity A

3.5.2: determine the equation of a line from information about the line (e.g., the slope and y-intercept; the slope and a point; two points) (Sample problem: Compare the equations of the lines parallel to and perpendicular to y = 2x - 4, and with the same x-intercept as 3x - 4y = 12.Verify using dynamic geometry software.);

Point-Slope Form of a Line - Activity A
Slope - Activity B
Slope-Intercept Form of a Line - Activity A

3.5.3: describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation (e.g., the cost to rent the community gym is $40 per evening, plus $2 per person for equipment rental; the y-intercept, 40, represents the $40 cost of renting the gym; the value of the slope, 2, represents the $2 cost per person);

Distance-Time Graphs
Function Machines 2 (Functions, Tables, and Graphs)
Linear Functions
Point-Slope Form of a Line - Activity A
Slope - Activity B
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

3.5.4: identify and explain any restrictions on the variables in a linear relation arising from a realistic situation (e.g., in the relation C = 50 + 25n,C is the cost of holding a party in a hall and n is the number of guests; n is restricted to whole numbers of 100 or less, because of the size of the hall, and C is consequently restricted to $50 to $2550).

Function Machines 2 (Functions, Tables, and Graphs)
Linear Functions
Point-Slope Form of a Line - Activity A
Slope-Intercept Form of a Line - Activity A
Using Tables, Rules and Graphs

4: Measurement and Geometry

4.1: Overall Expectations

4.1.1: solve problems involving the surface areas and volumes of three-dimensional figures;

Prisms and Cylinders - Activity A
Pyramids and Cones - Activity A
Surface and Lateral Area of Prisms and Cylinders
Surface and Lateral Area of Pyramids and Cones

4.1.2: verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.

Classifying Quadrilaterals - Activity B
Classifying Triangles
Parallelogram Conditions
Polygon Angle Sum - Activity A
Prisms and Cylinders - Activity A
Special Quadrilaterals
Triangle Angle Sum - Activity A

4.2: Solving Problems Involving Surface Area and Volume

4.2.1: solve problems involving the volumes of composite figures composed of prisms, pyramids, cylinders, cones, and spheres;

Prisms and Cylinders - Activity A
Pyramids and Cones - Activity A
Surface and Lateral Area of Prisms and Cylinders

4.2.2: determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a squarebased pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles);

Surface and Lateral Area of Prisms and Cylinders
Surface and Lateral Area of Pyramids and Cones

4.2.3: solve problems involving the surface areas of prisms, pyramids, cylinders, cones, and spheres, including composite figures (Sample problem: Break-bit Cereal is sold in a single-serving size, in a box in the shape of a rectangular prism of dimensions 5 cm by 4 cm by 10 cm. The manufacturer also sells the cereal in a larger size, in a box with dimensions double those of the smaller box. Compare the surface areas and the volumes of the two boxes, and explain the implications of your answers.);

Prisms and Cylinders - Activity A
Pyramids and Cones - Activity A
Special Quadrilaterals
Surface and Lateral Area of Prisms and Cylinders

4.2.4: identify, through investigation with a variety of tools (e.g. concrete materials, computer software), the effect of varying the dimensions on the surface area [or volume] of square-based prisms and cylinders, given a fixed volume [or surface area];

Prisms and Cylinders - Activity A
Surface and Lateral Area of Prisms and Cylinders

4.2.5: explain the significance of optimal surface area or volume in various applications (e.g., the minimum amount of packaging material; the relationship between surface area and heat loss);

Prisms and Cylinders - Activity A

4.2.6: pose and solve problems involving maximization and minimization of measurements of geometric figures (i.e., squarebased prisms or cylinders) (Sample problem: Determine the dimensions of a square-based, open-topped prism with a volume of 24 cm_ and with the minimum surface area.).

Prisms and Cylinders - Activity A
Surface and Lateral Area of Prisms and Cylinders

4.3: Investigating and Applying Geometric Relationships

4.3.1: determine, through investigation using a variety of tools (e.g., dynamic geometry software, paper folding), and describe some properties of polygons (e.g., the figure that results from joining the midpoints of the sides of a quadrilateral is a parallelogram; the diagonals of a rectangle bisect each other; the line segment joining the midpoints of two sides of a triangle is half the length of the third side), and apply the results in problem solving (e.g., given the width of the base of an A-frame tree house, determine the length of a horizontal support beam that is attached half way up the sloping sides);

Slope - Activity B