Linear Functions and Equations
Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate of change, so if the input, or x-coordinate, changes by a specific amount, a, the output, or y-coordinate, changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations of functions (recognizing that tabular and graphical representations are usually only partial representations), and they describe how such aspects of a function as slope and y-intercept appear in different representations. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems (NCTM, 2006, p. 20).
These resources offer a variety of ways to learn the material targeted in this Focal Point: tutorials, games, carefully crafted lessons, and online simulations. Your middle school students will also find plenty of opportunity for practice in real-world as well as imaginative scenarios.
Lines and Slope
At this site, students learn to draw a line and find its slope. Joan, a cartoon chameleon, is used throughout the tutorial to demonstrate the idea of slope visually. Background information on solving equations and graphing points is laid out clearly, followed by a step-by-step explanation of how to calculate slope using the formula. Finally, the slope-intercept form (y = mx + b) is carefully set out.
Walk the Plank
Students place one end of a wooden board on a bathroom scale and the other end on a textbook, then "walk the plank." They record the weight measurement as their distance from the scale changes and encounter unexpected results: a linear relationship between the weight and distance. Possibly most important, the investigation leads to a real-world example of negative slope. An activity sheet, discussion questions, and extensions of the lesson are included.
Writing Equations of Lines
This lesson uses interactive graphs to help students deepen their understanding of slope and extend the definition of slope to writing the equation of lines. Online worksheets with immediate feedback are provided to help students learn to read, graph, and write equations using the slope intercept formula.
Linear Function Machine
The functions produced by this machine are special because they all graph as straight lines and can be expressed in the form y = mx + b. In this activity, students input numbers into the machine and try to determine the slope and y-intercept of the line.
Algebra: Linear Relationships
Seven activities focus on generalizing from patterns to linear functions. Designed for use by mentors in after-school programs or other informal settings, these instructional materials have students work with number patterns, the function machine, graphs, and variables in realistic situations. Excellent handouts included.
The following three resources come from this subscription site; a free 30-day trial is available. Experiment with the online simulations, particularly selected for their use in teaching equations of a line. Subscriptions include inquiry-based lessons, assessment, and reporting tools.
Examine the graph of two points in the plane. Find the slope of the line that passes through the two points. Drag the points and investigate the changes to the slope and to the coordinates of the points.
Point-slope form of a line
Compare the point-slope form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response.
Compare the slope-intercept form of a linear equation to its graph. Find the slope of the line using a right triangle on the graph. Vary the coefficients and explore how the graph changes in response.
What difference does it make to the graph of a function if you change the slope or the y-intercept? Students can see the changes in the equation itself and in its graph as they vary both slope and y-intercept. Excellent visual! The activity could be used for class or small group work, depending on computer access.
Grapher: algebra (grades 6-8)
Using this online manipulative, students can graph one to three functions on the same window, trace the function paths to see coordinates, and zoom in on a region of the graph. Function parameters can be varied as can the domain and range of the display. Tabs allow the student to incorporate fractions, powers, and roots into their functions.
In this interactive game, students find the coordinates of four planets shown on the grid or locate the planets when given the coordinates. Finally, they must find the slope and y-intercept of the line connecting the planets in order to write its equation. Players select one of three levels of difficulty. Tips for students are available as well as a full explanation of the key instructional ideas underlying the game.
This complete lesson plan requires students to measure the length and width of a rectangle using both standard and nonstandard units of measure, such as pennies and beads. As students graph the ordered pairs, they discover that the ratio of length to width of a rectangle is constant, in spite of the units. This hands-on experience leads to the definition of a linear function and to the rule that relates the dimensions of this rectangle.
Looking for a real-world example of a linear function? In this lesson, students model a bungee jump using a doll and rubber bands. They measure the distance the doll falls and find that it is directly proportional to the number of rubber bands. Since the mathematical scenario describes a direct proportion, it can be used to examine linear functions.
Exploring linear data
This lesson connects statistics and linear functions. Students construct scatterplots, examine trends, and consider a line of best fit as they graph real-world data. They also investigate the concept of slope as they model linear data in a variety of settings that range from car repair costs to sports to medicine. Handouts for four activities, spread out over three class periods, are provided.
Supply and Demand
Your company wants to sell a cartoon-character doll. At what price should you sell the doll in order to satisfy demand and maintain your supply? The lesson builds from graphing data to writing linear equations to creating and solving a system of equations in a real-world setting. Discussion points, handouts, and solutions are given.
Presented with the pricing schedules from three printing companies, students must determine the least expensive way to have their algebra books printed. They compile data in tables, spreadsheets, and a graph showing three equations. Throughout the lesson, students explore the relationships among lines, slopes, and y-intercepts in a real-world setting.
Purplemath - Your Algebra Resource
Algebra modules provide free tutorials in every topic of algebra, from beginning to advanced. Lessons concentrate on "practicalities rather than the technicalities" and include worked examples as well as explanations. Of particular interest are the modules on Systems of Linear Equations and Systems-of-Equations Word Problems. A site worth visiting!