In this unit students will:
• graph proportional relationships;
• interpret unit rate as the slope;
• compare two different proportional relationships represented in different ways;
• use similar triangles to explain why the slope is the same between any two points on a non-vertical line;
• derive the equation y = mx for a line through the origin;
• derive the equation y = mx + b for a line intercepting the vertical axis at b; and
• interpret equations in y = mx + b form as linear functions.
In this unit, distance time problems serve the purpose of illustrating how the rates of two objects can be represented, analyzed, and described in different ways: graphically and algebraically. Students create representative graphs and the meaning of various points. They then compare the same information when represented in an equation.
By using coordinate grids and various sets of three similar triangles, students prove that the slopes of the corresponding sides are equal, thus making the unit rate of change equal. After proving with multiple sets of triangles, students generalize the slope to y = mx for a line through the origin and y = mx + b for a line through the vertical axis at b.
In Grade 8, the focus is on linear functions, and students begin to recognize a linear function from its form y = mx + b. Students also need experiences with nonlinear functions, including functions given by graphs, tables, or verbal descriptions but for which there is no formula for the rule, such as a girl’s height as a function of her age. Students learn that proportional relationships are part of a broader group of linear functions, and they are able to identify whether a relationship is linear. Nonlinear functions are included for comparison.