Lesson 11: Equations of All Kinds of Lines
About this lesson
In previous lessons, students have studied lines with positive and negative slope and have learned to write equations for them, usually in the form \(y=mx+b\). In this lesson, students extend their previous work 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.
Note that the equation of a vertical line cannot be written in the form \(y=mx+b\). It can, however, be written in the form \(Ax+By=C\) (with \(B\) = 0). This type of linear equation will be studied in greater detail in upcoming lessons. In this lesson, students encounter a context where this form arises naturally: if a rectangle has length \(ℓ\) and width \(w\) and its perimeter is 50, this means that \(2ℓ+2w=50\).
Lesson overview
 11.1 Warmup: Which One Doesn’t Belong: Pairs of Lines (5 minutes)

11.2 Activity: All the Same (15 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.

11.3 Activity: Same Perimeter (15 minutes)
 There is a digital applet in this activity.
 Lesson Synthesis
 11.4 Cooldown: Line Design (5 minutes)
Learning goals:
 Comprehend that for the graph of a vertical or horizontal line, one variable does not vary, while the other can take any value.
 Create multiple representations of linear relationship, including a graph, equation, and table.
 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 \(x=a\).
Learning goals (student facing):
 Let’s write equations for vertical and horizontal lines.
Learning targets (student facing):
 I can write equations of lines that have a positive or a negative slope.
 I can write equations of vertical and horizontal lines.
Required materials
 String
Required preparation:
 Take a piece of string 50 centimeters long and tie the ends together to be used as demonstration in the third activity.
Glossary:
 Access the complete Grade 8 glossary.
Standards
 This lesson builds on the standard: CCSS.7.G.AMO.7.GM.A
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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