Unit 5: Linear Functions

Ga DOE Unit 5 Student Frameworks

Lesson Plans

ENDURING UNDERSTANDINGS
• Patterns and relationships can be represented graphically, numerically, and symbolically.
• Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportion problems.

ESSENTIAL QUESTIONS

• How can patterns, relations, and functions be used as tools to best describe and help explain real-life relationships?
• How can the same mathematical idea be represented in a different way? Why would that be useful?
• What is the significance of the patterns that exist between the triangles created on the graph of a linear function?
• When two functions share the same rate of change, what might be different about their tables, graphs and equations? What might be the same?
• What does the slope of the function line tell me about the unit rate?
• What does the unit rate tell me about the slope of the function line?

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.

Vocabulary:

Intersecting Lines: Two lines that cross each other. Lines intersect at one point unless the lines fall directly on top of each other (in which case they are essentially the same line and are sometimes called coincidental).
Origin: The point of intersection of the vertical and horizontal axes of a Cartesian plane. The coordinates of the origin are (0, 0).
Proportional Relationships: A relationship between two equal ratios.
Slope: The "steepness" of a line. The slope of a line can be found directly when a linear equation is in slope-intercept form (y = mx + b). In this form, the slope is the coefficient of x and is represented by the letter m. The slope of a line can also be found by determining the ratio of the "rise" to the "run" between two points on the graph. In other words, slope measures how much the line rises vertically given a particular run or horizontal distance.
Unit Rate: A comparison of two measurements in which the second term has a value of 1. Unit rates are used to compare the costs of items in a grocery store.