Lesson 11

This warm-up prompts students to compare four pairs of lines. It invites students to explain their reasoning and hold mathematical conversations, and allows you to hear how they use terminology and talk about lines. To allow all students to access the activity, each figure has one obvious reason it does not belong. Encourage students to find reasons based on geometric properties (e.g., only one set of lines are not parallel, only one set of lines have negative slope). 

Arrange students in groups of 2–4. Display the image of the four graphs for all to see. Ask students to indicate when they have noticed one graph that does not belong and can explain why. Give students 2 minutes of quiet think time and then time to share their thinking with their group. After everyone has conferred in groups, ask the group to offer at least one reason each graph doesn’t belong.

After students have conferred in groups, invite each group to share one reason why a particular pair of lines might not belong. Record and display the responses for all to see. After each response, ask the rest of the class if they agree or disagree. Since there is no single correct answer to the question asking which shape does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, prompt students to explain the meaning of any terminology they use, such as parallel, intersect, origin, coordinate, ordered pair, quadrant or slope. Also, press students on claims of lines being parallel to one another. Ask students how they know they are parallel and highlight ideas about slope.

In previous lessons, students have studied lines with positive slope, negative slope, and 0 slope and have written equations for lines with positive and negative slope. In this activity, they write equations for horizontal lines (lines of slope 0) and vertical lines and they graph horizontal and vertical lines from equations. Students explain their reasoning (MP3).

Horizontal lines can be thought of as being described by equations of the form \(y = mx + b\) where \(m = 0\). In other words, a horizontal line can be thought of as a line with slope 0. Vertical lines, on the other hand, cannot be described by an equation of the form \(y = mx + b\).

In order to highlight the structure of equations of vertical and horizontal lines, ask students:

• "Why does the equation for the points with \(y\)-coordinate -4 not contain the variable \(x\)?" (\(x\) can take any value while \(y\) is always -4. The only constraint is on \(y\) and there is no dependence of \(x\) on \(y\).)
• "Why does the equation for the points with \(x\)-coordinate 3 not contain the variable \(y\)?" (\(y\) can take any value while \(x\) is always 3. The only constraint is on \(x\) and there is no dependence of \(y\) on \(x\).)
• "What does this say about the relationship between the quantities represented by \(x\) and \(y\) in these situations?" (Changes in one do not affect the other. One is not dependent on the other. They don’t change together according to a formula.)
• "What would be some real-world examples of situations that could be represented by these types of equations?" (Examples: You pay the same fee regardless of your age; bus tickets cost the same no matter how far you travel; you remain 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. Students find pairs of numbers for the width and length of rectangles that all have the same perimeter. Next, they draw some of the rectangles with a vertex at the origin in the coordinate plane, and discover that the set of opposite vertices lie on a line. Finally, they write an equation for the line and consider how the slope of the line relates to the changing lengths and widths of the rectangles.

Identify students who come up with different equations for the line. An equation of the form \(2\ell + 2w = 50\), where \(\ell\) is the length of the rectangle and \(w\) is its width, is natural if they are thinking about the context, namely, that the perimeter of the rectangle is 50 units. In fact, students have likely seen this equation in this context in grade 6. On the other hand, students who follow the steps in the task and draw the line connecting the rectangle vertices are likely to write an equation in the form \(w = 25 - \ell\) (or perhaps \(2w = 50 - 2\ell\)) because the \(w\)-intercept of the graph is 25 and its slope is -1. Invite students who wrote these different forms for an equation to present during the discussion.

Invite students who write \(2\ell + 2w = 50\) for an equation of the line to share their response (if no one has written this equation for the third question, ask students if it is correct). This equation has the advantage of directly modeling the context: the perimeter of a rectangle is \(2\ell + 2w\), so if the perimeter is 50, then \(2\ell + 2w = 50\). An additional advantage to this form of the equation is that every line, including horizontal and vertical lines, can be written in this form: \(\ell = 5\) is an equation for a horizontal line while \(w = 3\) is an equation for a vertical line.

Invite students who write \(\ell = 25- w\) (or some variant) to share. This equation is not apparent from the scenario, but it reveals a few interesting aspects of the problem:

• The rectangle has to have length less than 25 (since the width has to be positive)
• For each unit the length increases, the width decreases by one unit (in order to balance out when the sides of the rectangle are added to get the perimeter)

Ask students if the lengths and widths need to be whole numbers. In the next two lessons, students will encounter equations where the contexts determine what values the variables can take on. If students agree that the lengths and widths can take on any measurable value, ask how many different rectangles can be drawn. Practically speaking, the number is limited by what we can measure and draw with reasonable precision, while in theory, there are an infinite number of rectangles with perimeter 50.