Specific Outcome: It is expected that students will:
[PS, R, V]
While students have sketched the graphs of linear relations in previous grades, this outcome is their introduction to the concept of slope.
In Achievement Indicators 3.2 and 3.3, students are expected to draw conclusions, through investigation, about lines with a:
In Achievement Indicator 3.8, students are expected to develop, through investigation, the rules for determining whether two lines are parallel or perpendicular.
Slope can be referred to as a rate of change, specifically the rate of change of the variable y with respect to the change in the variable x. If slope is discussed in context, the units for each variable should be included with the slope; e.g., if the context provides a relationship between distance in meters and time in seconds, then the rate of change of distance with respect to time should be expressed in m/s and related to the slope of the graph.
The focus of this outcome is to develop a visual understanding of slope and then, when solving problems, to connect slope to the concept of rate of change.
Achievement Indicators |
Acceptable Standard |
Standard of Excellence |
| 3.3.1 Determine the slope a line segment by measuring or calculating the rise and run. | ** | |
| 3.3.2 Classify lines in a given set as having positive or negative slopes. | ** | |
| 3.3.3 Explain the meaning of the slope of a horizontal or vertical line. | ** | |
| 3.3.4 Explain why the slope of a line can be determined by using any two points on that line. | ** | |
| 3.3.5 Explain, using examples, slope as a rate of change. | ** | |
| 3.3.6 draw a line, given its slope on the line. | ** | |
| 3.3.7 Determine another point on a line, given the slope and a point on the line. | ** | |
| 3.3.8 Generalize and apply a rule for determining whether two lines are parallel or perpendicular. | Develop a rule with an explanation for parallel lines; apply a rule for perpendicular lines. | Develop a rule with an explanation for perpendicular lines. |
| 3.3.9 Solve a contextual problem involving slope. | Solve simple contextual problems. | Solve complex contextual problems. |
Specific Outcome: It is expected that students will:
to their graphs.
[CN, R, T, V] [ICT: C6–4.3]
Students may have some prior exposure to equations having more than one variable. In this outcome, students make the connection between linear equations in two variables and the graph of a line.
When writing a linear equation in general form, e.g., Ax+ By+ C= 0, the coefficients A, B and C must be integers and, by convention, A> 0. In the case where A= 0, the equation is written as By+ C= 0, with B, C ∈ I and B > 0.
Technology [T] has been identified as one of the mathematical processes to be emphasized in completing this outcome. It is expected, however, that students will be able to graph linear relations with and without the use of technology.
In Achievement Indicator 6.3, students are expected to discover, through investigation, and generalize the strategies for graphing linear relations.
The equations and graphs of vertical, e.g., (B= 0), and horizontal , e.g., (A= 0), lines are part of this outcome.
On completion of this outcome, students are expected to relate the equation of a linear relation to a graph by comparing the characteristics as determined from the equation to the characteristics of the graph and vice versa.
Achievement Indicators |
Acceptable Standard |
Standard of Excellence |
| 3.6.1 Express a linear relation in different forms, and compare the graphs. | ** | |
| 3.6.2 Rewrite a linear relation in either slope-intercept or general form. | ** | |
| 3.6.3 Generalize and explain strategies for graphing a linear relation in slope-intercept, general or slope-point form. | Generalize and explain strategies for graphing a linear relation in slope-intercept or slope-point form. | Generalize and explain strategies for graphing a linear relation in general form. |
| 3.6.4 Graph, with and without technology, a linear relation given in slope-intercept, general or slope-point form, and explain the strategy used to create the graph. | Graph and explain the strategy used. | Graph and explain more than one strategy that could be used. |
| 3.6.5 Identify equivalent linear relations from a set of linear relations. | ** | |
| 3.6.6 Match a set of linear relations to their graphs. | ** |
Specific Outcome: It is expected that students will:
[CN, PS, R, V]
Students are expected to determine the equation of a linear relation algebraically.
Technology [T] has not been identified as one of the mathematical processes to be emphasized in completing this outcome. There may be some opportunities for students to use technology to investigate the equations of linear relations, given two or more points on the graph, but students are expected to meet this outcome without the use of technology.
Linear regression is not part of this outcome.
In Achievement Indicators 7.2, 7.3 and 7.4, students are expected to apply and explain multiple strategies to write the equation of a linear relation in each situation; e.g., given the slope and a point, students could substitute the coordinates of the point and the slope:
In Achievement Indicators 7.2, 7.3 and 7.4, students are expected to explain the reasoning used in determining their equations.
In Achievement Indicators 7.5 and 7.6, students are expected to identify any restrictions on the domain and range of the linear relation.
Achievement Indicators |
Acceptable Standard |
Standard of Excellence |
| 3.7.1 Determine the slope and y-intercept of a given linear relation from its graph, and write the equation in the form y=mx + b. | ** | |
| 3.7.2 Write the equation of a linear relation, given its slope and the coordinates of a point on the line, and explain the reasoning. | Write an equation with a partial explanation. | Write an equation with a complete explanation |
| 3.7.3 Write the equations of a linear relation, given the coordinates of two points on the line, and explain the reasoning. | Write an equation with a partial explanation. | Write an equation with a complete explanation |
| 3.7.4 Write the equation of a linear relation, given the coordinates of a point on line of the equation of a parallel or perpendicular line, and explain the reasoning. | Write an equation with a partial explanation. | Write an equation with a complete explanation |
| 3.7.5 Graph linear data generated from a context, and write the equation of the resulting line. | Give a complete solution. | Give a complete solution, including restriction on the domain and range. |
| 3.7.6 Solve a problem, using the equation of a linear relation. | Give a complete solution. | Give a complete solution, including restriction on the domain and range. |
the y-coordinate of the point where a line or curve crosses the y-axis
the value of y when x=0
the equation of a line in the form y = mx + b, where m is the slope of the line and b is the y-intercept
The equation of a line in the form Ax + By + C = 0, where A, B, and C are real numbers and by convention, A is a whole number. This means that A will always be positive.
The general form of a linear equation is Ax + By + C = 0, where A, B and C are real numbers, and A and B are not both zero. By convention A is a whole number.
To graph an equation in general form, determine the intercepts, then draw a line joining the intercepts; or convert to slope-intercept form.
Lines in the same plane that do not intersect
Lines that have the same slope but different intercepts
Two lines that intersect at right angles (90o)
Lines that have slopes that are negative reciprocals of each other
For example : Slopes a/b and -b/a are negative reciprocals