## Slope

Materials: graph paper for students, large graph paper or transparent grids for overhead projector, marker or grease pencil

Preparation: Create and display large charts or an overhead transparency showing these graphs. Keep the equations covered for now.

Slope        Two-step Linear Equations
with Rational Numbers.

Prerequisite Skills and Concepts: Students must know how to plot points on a coordinate plane and how to generate points from a linear equation . Students must also have a clear idea of the definitions of vertical and horizontal.

• Say: Describe the best terrain for riding a bicycle fast. Compare this to the best terrain for riding for hours at a time. Compare this to the best terrain for riding a bike for a heavy workout.
This should start a discussion about relative steepness. As students make their points, introduce the term slope.

• Say: In mathematics, slope describes both the steepness and the direction of that steepness. It is the ratio that describes the relationship between the vertical change and the horizontal change in a line.

• Show students graph A (with the equations covered).

• Ask: How are these graphs alike? How are they different?
Students should notice that:
• the lines all tend upward from left to right.
• the lines appear to be parallel. (You can prove that they are parallel by noticing that they all form diagonals of the squares they pass through. This means they all cut a transversal at the same angle, which makes them parallel.)
• they can be described as having the same steepness.

• Uncover the equations for graph A.

• Say: These graphs all have the same slope. How are their equations alike? How are they different?
Help students to notice that:
• the equations are all in the same form: y = an expression involving x.
• in none of the equations is x multiplied by a number.
• the equations differ in what's added to or subtracted from x.

• Show students graph B (with the equations covered).

• Ask: How are these graphs alike? How are they different?
Students should notice that:
• the lines all tend upward from left to right.
• the lines are clearly not parallel.
• the lines all pass through the point (0, 0)
• they can not be described as having the same steepness.

• Uncover the equations for graph B.

• Say: These graphs all have different slopes, but they all pass through the same point. How are their equations alike? How are they different?
Help students to notice that:
• the equations are all in the same form: y = an expression involving x.
• in two of the equations, x is multiplied by a positive number, a different one in each case.
• in none of the equations is anything added to or subtracted from x.

• Show students graph C (with the equations covered).

• Ask: How are these graphs alike? How are they different?
Students should notice that:
• the lines all tend downward from left to right.
• the lines appear to be parallel (and can be proven to be).
• they can be described as having the same steepness.

• Uncover the equations for graph C.

• Say: These graphs all have the same slope. How are their equations alike? How are they different?
Help students to notice that:
• the equations are all in the same form: y = an expression involving x.
• in all of the equations, x is multiplied by the same negative number
• the equations differ in what's added to or subtracted from x.

• Show students graph D (with the equations covered).

• Ask: How are these graphs alike? How are they different?
Students should notice that:
• the lines all tend downward from left to right.
• the lines are clearly not parallel.
• the lines all pass through the point (0, 0)
• they can not be described as having the same steepness.

• Uncover the equations for graph D.

• Say: These graphs all have different slopes, but they all pass through the same point. How are their equations alike? How are they different?
Help students to notice that:
• the equations are all in the same form: y = an expression involving x.
• in all of the equations, x is multiplied by a negative number, a different one in each case.
• in none of the equations is anything added to or subtracted from x.

• Say: An equation that looks like y = mx + b is in slope-intercept form. Why do you think that is?
Discuss the similarities and differences among the 12 equations and graphs students have just analyzed. Students should speculate that it's the number added to or subtracted from the x term that tells where the line will cross the y axis. They should also notice that the coefficient of x (the number that multiplies it) determines the slope of the line.