Lesson 9

Slopes and Equations for All Kinds of Lines

Lesson Narrative

The purpose of this lesson is for students to extend their work with slope triangles to develop a method for finding the slope of any line given the coordinates of two points on the line. They practice finding slopes this way and use a graph in order to check their answer (especially the sign).

Next, students build on their previous experiences writing equations of lines, usually in the form \(y=mx+b\), and extend this thinking to include equations for horizontal and vertical lines. Horizontal lines can still be written in the form \(y=mx+b\), but because \(m=0\) in this case, the equation simplifies to \(y=b\). Students interpret this to mean that, for a horizontal line, the \(y\) value does not change, but \(x\) can take any value. This structure is identical for vertical lines except that now the equation has the form \(x=a\) and it is \(x\) that is determined while \(y\) can take any value. 

This lesson includes an optional activity where students consider what information is sufficient to define (and accurately communicate) the position of a line in the coordinate plane. Lines with positive and negative slope are examined as students move flexibly between coordinates of points on a line, the slope of the line, and the graph showing the “uphill” or “downhill” orientation of the line. Many methods for describing the location of the lines are available, but students need to calculate carefully and use the coordinate grid in order to communicate the positions of the line clearly (MP6).

Learning Goals

Teacher Facing

  • Comprehend that for the graph of a vertical or horizontal line, one variable does not vary, while the other can take any value.
  • Create a graph of a line using a verbal description of its features.
  • Generalize (in writing) that a set of points of the form $(x,b)$ satisfy the equation $y=b$ and that a set of points of the form $(a,y)$ satisfy the equation $x=a$.
  • Generate a method to find slope values given two points on the line.

Student Facing

Let’s figure out the slope and equations for all kinds of lines.

Required Preparation

If using the optional activity Making Designs, print and cut up slips from the blackline master for this activity.

Learning Targets

Student Facing

  • I can calculate positive and negative slopes given two points on the line.
  • I can write equations of vertical and horizontal lines.

CCSS Standards

Building On


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