07-Math 7a-CC3 Ch4

This chapter is designed to accomplish several objectives:

• To have students find connections between the four representations: graphs, tables, patterns, and rules.
• To challenge teams with difficult problems that require them to think deeply about the relationships between the different representations.
• To begin a focus on writing equations from word problems, which is continued in Chapter 5.
• To introduce students to solving systems of equations where both equations are in y = mx + b form.

At the beginning of Section 4.1, teams are challenged with a difficult, non-linear pattern for which they generate multiple representations (graph, table, tile pattern, and rule) and for which they predict the shape and area of the 100^thfigure. Then they focus on finding connections between different representations in the web (shown at right).

From this non-linear challenge, the chapter returns to linear patterns to help students focus on the similarities between representations of a pattern. For example, when students notice that a geometric tile pattern grows by two tiles from figure to figure, they will make a connection to how that growth is represented in the rule (the coefficient of x), in the table (each y-value is 2 more than the one before), and the graph (where points “step up” or “step-down” each time, shown with growth triangles). By the end of Section 4.1, students should be able to change any representation to any other for linear patterns.

Note that the web will help students keep track of which connections they have developed throughout the chapter. From Chapter 3, students know how to make the following connections: rule → table, graph → table, pattern → table, table → rule, and table → graph.

Connections that are explored in Section 4.1 are shown in the diagram below. The dashed lines represent connections that are explored, but are not complete.

As students move into Chapter 4, they should be starting to use some of the Mathematical Practices with more regularity. It should only take a gentle reminder from you to attend to precision in their communication with each other. They should be more comfortable constructing viable arguments and critiquing the reasoning of others as you encourage discourse during class. Students should now want to make sense of the problems that you ask them to attempt and they should be starting to show more and more perseverance in solving them.

In this chapter, you will guide students to look for and make use of the mathematical structure a linear equation as they use repeated reasoning to make connections and build understanding.

8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.