In this activity, students create equations that represent horizontal and vertical lines, and then graph them. Horizontal lines can be thought of as being described by equations of the form where . Vertical lines, on the other hand, cannot be described by an equation of the form , and students must explain their reasoning when identifying an equation that can describe a vertical line (MP3).

The goal of this discussion is for students to articulate their reasoning about the equations that represent vertical and horizontal lines.

Direct students’ attention to the reference created using Collect and Display. Ask students to share their equation and reasoning for the horizontal line () and then the vertical line (). Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. Discuss: 

• "Why does the equation for the points with -coordinate -4 not contain the variable ?" ( can take any value while is always -4. The only constraint is on and there is no dependence of   on .)

• "Why does the equation for the points with -coordinate 3 not contain the variable ?" ( can take any value while is always 3. The only constraint is on and there is no dependence of on .)

• "What does this say about the relationship between the quantities represented by and in these situations?" (Changes in one do not affect the other. One is not dependent on the other.)

• "What would be some real-world examples of situations that could be represented by these types of equations?" (Everyone pays the same fee regardless of age. Bus tickets cost the same no matter how far the trip. A student remains the same distance from home as the hours pass during the school day.)

In this activity, students analyze a line and an equation defining the line in a geometric context. By finding and graphing pairs of numbers for the width and length of rectangles that all have a perimeter of 50, students observe that this set of numbers lie on a line. 

Writing an equation for this relationship introduces students to equations of the form . This form of equation will be used in later lessons and it is not necessary for students to know this general form or to manipulate this form of equation at this time. The goal is for students to see multiple ways to describe a line and how each part of each equation relates to the situation (MP2).

Monitor for students who come up with different equations for the line. An equation of the form , where is the length of the rectangle, and is its width, makes sense in this context of the perimeter of a rectangle being 50 units. An equation of the form also makes sense because the -intercept of the graph is 25 and its slope is -1.

The goal of this discussion is for students to observe how different forms of equations can reveal different information about a situation. Invite students who wrote (or equivalent) for an equation of the line to share and explain their reasoning. If no student wrote this equation, suggest it now. Invite students who wrote (or equivalent) to share and explain their reasoning. If not brought up in students’ explanations, discuss:

•  “How does the equation describe this situation?” (A rectangle has 2 sides of length and 2 sides of length and their sum must equal 50 units.)

• “How does the equation describe this line?” (The vertical intercept is 25 and the slope is -1.)

• “What does a vertical intercept of 25 mean in this situation?” (The rectangle has to have a length that is less than 25 since the width has to be positive.)

• “What are some advantages for using each form of equation?” (The first equation makes it easier to see the relationship between the length and width while the second equation makes it easier to see what the line looks like.)