# Lesson 9

Slopes and Equations for All Kinds of Lines

## 9.1: Which One Doesn’t Belong: Pairs of Lines (5 minutes)

### Warm-up

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).

### Launch

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.

### Student Facing

Which one doesn’t belong?

### Activity Synthesis

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.

## 9.2: Toward a More General Slope Formula (15 minutes)

### Activity

The purpose of this activity is to compute the slopes of different lines to get familiar with the formula “subtract $$y$$-coordinates, subtract $$x$$-coordinates, then divide.” Students first compute slopes for some lines with positive slopes, and then special attention is drawn to the fact that a line has a negative slope. Students attend to what makes the same computations have a negative result instead of a positive result.

### Launch

Have partners figure out the slope of the line that passes through each pair of points.

1. $$(12,4)$$ and $$(7,1)$$ (answer: $$\frac35$$)

2. $$(4,\text{-}11)$$ and $$(7,\text{-}8)$$ (answer: 1)

3. $$(1,2)$$ and ($$600,3)$$ (answer: $$\frac{1}{599}$$)

4. $$(37,40)$$ and $$(30,33)$$ (answer: 1)

Ask students to share their results and how they did it. Students may say they just found the difference between the numbers; making this more precise is part of the goal of the discussion for this activity. Ask students to complete the questions in the task and share their responses with a partner before class discussion. Provide access to graph paper and rulers.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary.  Include the following terms and maintain the display for reference throughout the unit: slope. Invite students to suggest language or diagrams to include on the display that will support their understanding of this term.
Supports accessibility for: Memory; Language

### Student Facing

1. Plot the points $$(1,11)$$ and $$(8, 2)$$, and use a ruler to draw the line that passes through them.
2. Without calculating, do you expect the slope of the line through $$(1,11)$$ and $$(8, 2)$$ to be positive or negative? How can you tell?
3. Calculate the slope of this line.

### Student Facing

#### Are you ready for more?

Find the value of $$k$$ so that the line passing through each pair of points has the given slope.

1. $$(k,2)$$ and $$(11,14)$$, slope = 2

2. $$(1,k)$$ and $$(4,1)$$, slope = -2

3. $$(3,5)$$ and $$(k,9)$$, slope = $$\frac12$$

4. $$(\text-1,4)$$ and $$(\text-3,k)$$, slope = $$\frac{\text{-}1}{2}$$

5. $$(\frac {\text{-}15}{2},\frac{3}{16})$$ and $$(\frac {\text{-}13}{22},k)$$, slope = 0

### Anticipated Misconceptions

Students may struggle with operations on negative numbers and keeping everything straight if they try to take a purely algorithmic approach. Encourage them to plot both points and reason about the length of the vertical and horizontal portions of the slope triangle and the sign of the slope of the line, one step at a time. Sketching a graph of the line is also useful for verifying the sign of the line’s slope.

It is common for students to calculate a slope and leave it in a form such as $$\frac {\text{-}3}{\text{-}5}$$. Remind students that this fraction is representing a division operation, and the quotient would be a positive value.

### Activity Synthesis

When using two points to calculate the slope of a line, care needs to be taken to subtract the $$x$$ and $$y$$ values in the same order. Using the first pair as an example, the slope could be calculated either of these ways: $$\displaystyle \frac{4-1}{12-7}=\frac35$$ $$\displaystyle \frac{1-4}{7-12}=\frac {\text{-}3}{\text{-}5}=\frac35$$ But if one of the orders were reversed, this would yield a negative value for the slope when we know the slope should have a positive value.

It is worth demonstrating, or having a student demonstrate an algorithmic approach to finding the negative slope in the last part of the task. Draw attention to the fact that keeping the coordinates “in the same order” results in the numerator and denominator having opposite signs (one positive and one negative), so that their quotient must be negative. It might look like: $$\displaystyle \frac{11-2}{1-8}=\frac{9}{\text{-}7}=\text{-}\frac{9}{7}$$

Ask students how sketching a graph of the line will tell them whether the slope is positive or negative. They should recognize that when it goes up, from left to right, the slope is positive, and when it goes down, then the slope is negative. Using a sketch of the graph can also be helpful to judge whether the magnitude of the (positive or negative) slope is reasonable.

Representing, Speaking: MLR7 Compare and Connect. Use this support as students calculate the slope of the line between the points $$(1, 11)$$ and $$(8, 2)$$. Invite students to demonstrate their strategy using a visual or numerical representation. In pairs or groups, ask students to compare their strategies. Ask students to discuss how the strategies are the same and/or different, and then share with the whole class. This will help students connect how different approaches led to the same result of a negative slope by keeping coordinates “in the same order.”
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness

## 9.3: Making Designs (20 minutes)

### Activity

The goal of this activity is for students to recognize information that determines the location of a line in the coordinate plane, and to practice distinguishing between positive and negative slopes. In this activity, one partner has a design that they verbally describe to their partner, who then tries to draw it. The purpose of this activity is to provide an environment where students have to describe or interpret the slope and locations of several lines. (Students are not expected to communicate by saying the equations of the lines, though there is nothing stopping them from doing so.) Students take turns describing and interpreting by doing this two times with two different designs.

Monitor for students who use language of slope and vertical or horizontal intercepts to communicate the location of each line. Invite these students to share during the discussion. There are many other ways students might communicate the location of each line, but the recent emphasis on studying slope and intercepts should make these choices natural.

The two designs in the blackline masters look like this:

You will need the Info Gap: Making Designs blackline master for this activity.

Thanks to Henri Picciotto for permission to use these designs.

Teacher Notes for IM 6–8 Math Accelerated
This activity is optional. Use this activity to give students additional practice communicating precise information about the location of a line.

### Launch

Tell students they will describe some lines to a partner to try and get them to recreate a design. The protocol is described in the student task statement. Consider asking a student to serve as your partner to demonstrate the protocol to the class before distributing the designs and blank graphs.

From the blackline master that you have copied and cut up ahead of time, give one partner the design, and the other partner a blank graph. Arrange the room to ensure that the partner drawing the design cannot peek at the design anywhere in the room. Once the first design has been successfully created, provide the second design and a blank graph to the other student in each partnership.

Representation: Provide Access for Perception. Display or provide students with a physical copy of the written directions and read them aloud. Check for understanding by inviting students to rephrase directions in their own words. Keep directions visible throughout the activity.
Supports accessibility for: Language; Memory
Conversing: Use this modified version of MLR4 Information Gap to give students an opportunity to describe (orally) the graph of a line using formal or informal language. Circulate and listen for common words or phrases students use to describe the designs. Record this language and display for all to see. Encourage students to borrow words or phrases from the display as needed.
Design Principle(s): Cultivate conversation

### Student Facing

Your teacher will give you either a design or a blank graph. Do not show your card to your partner.

If your teacher gives you the design:

1. Look at the design silently and think about how you could communicate what your partner should draw. Think about ways that you can describe what a line looks like, such as its slope or points that it goes through.
2. Describe each line, one at a time, and give your partner time to draw them.
3. Once your partner thinks they have drawn all the lines you described, only then should you show them the design.

If your teacher gives you the blank graph:

1. Listen carefully as your partner describes each line, and draw each line based on their description.
3. Do not show your drawing to your partner until you have finished drawing all the lines they describe.

When finished, place the drawing next to the card with the design so that you and your partner can both see them. How is the drawing the same as the design? How is it different? Discuss any miscommunication that might have caused the drawing to look different from the design.

Pause here so your teacher can review your work. When your teacher gives you a new set of cards, switch roles for the second problem.

### Activity Synthesis

After students have completed their work, ask students to discuss the process of communicating how to draw a line. Some guiding questions:

• "What details were important to pay attention to?"
• "How did you use coordinates to help communicate where the line is?"
• "How did you use slope to communicate how to draw the line?"
• "Were there any cases where your partner did not give enough information to know where to draw the line? What more information did you need?"

Students might notice that the lines with the same slope can be described in terms of translations (for example, line $$b$$ is a vertical translation of line $$a$$ down two units). This is an appropriate use of rigid motion language which recalls work done earlier in this unit. Students might also describe $$b$$ as parallel to $$a$$ and containing the point $$(0,3)$$. Finally, some students might use equations to communicate the location of the lines. All of these methods are appropriate. Keep the discussion focused on describing each line using slope and coordinates for each individual line on its own.

## 9.4: All the Same (15 minutes)

### Activity

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 $$y = mx + b$$.

Teacher Notes for IM 6–8 Math Accelerated
The activity narrative references students’ work in “previous lessons.” Due to the sequence of IM 6–8 Math Accelerated, the referenced work is in both previous lessons and the previous activities of this lesson.

### Launch

Allow students quiet think time. Instruct them to pause their work after question 2 and discuss which equation makes sense and why. Tell students to resume working and pause again after question 4 for discussion. Discuss why the equations only contain one variable and what this means about the relationship between the quantities represented by $$x$$ and $$y$$. After students complete questions 5 and 6, ask if they can think of some real-world situations that can be represented by vertical and horizontal lines.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills. For example, present one question at a time.
Supports accessibility for: Organization; Attention

### Student Facing

1. Plot at least 10 points whose $$y$$-coordinate is -4. What do you notice about them?

2. Which equation makes the most sense to represent all of the points with $$y$$-coordinate -4? Explain how you know.

$$x=\text-4$$

$$y=\text-4x$$

$$y=\text-4$$

$$x+y=\text-4$$

3. Plot at least 10 points whose $$x$$-coordinate is 3. What do you notice about them?

4. Which equation makes the most sense to represent all of the points with $$x$$-coordinate 3? Explain how you know.

$$x=3$$

$$y=3x$$

$$y=3$$

$$x+y=3$$

5. Graph the equation $$x=\text-2$$.

6. Graph the equation $$y=5$$.

### Launch

Allow students quiet think time. Instruct them to pause their work after question 2 and discuss which equation makes sense and why. Tell students to resume working and pause again after question 4 for discussion. Discuss why the equations only contain one variable and what this means about the relationship between the quantities represented by $$x$$ and $$y$$. After students complete questions 5 and 6, ask if they can think of some real-world situations that can be represented by vertical and horizontal lines.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills. For example, present one question at a time.
Supports accessibility for: Organization; Attention

### Student Facing

1. Plot at least 10 points whose $$y$$-coordinate is -4. What do you notice about them?

2. Which equation makes the most sense to represent all of the points with $$y$$-coordinate -4? Explain how you know.

$$x=\text-4$$

$$y=\text-4x$$

$$y=\text-4$$

$$x+y=\text-4$$

3. Plot at least 10 points whose $$x$$-coordinate is 3. What do you notice about them?

4. Which equation makes the most sense to represent all of the points with $$x$$-coordinate 3? Explain how you know.

$$x=3$$

$$y=3x$$

$$y=3$$

$$x+y=3$$

5. Graph the equation $$x=\text-2$$.

6. Graph the equation $$y=5$$.

### Student Facing

#### Are you ready for more?

1. Draw the rectangle with vertices $$(2,1)$$, $$(5,1)$$, $$(5,3)$$, $$(2,3)$$.
2. For each of the four sides of the rectangle, write an equation for a line containing the side.
3. A rectangle has sides on the graphs of $$x = \text-1$$, $$x = 3$$, $$y = \text-1$$, $$y = 1$$. Find the coordinates of each vertex.

### Activity Synthesis

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.)
Representing, Speaking, Listening: MLR2 Collect and Display. As students discuss which equation makes sense and why with a partner, create a 2-column table with the headings “horizontal lines” and “vertical lines”. Circulate through the groups and record student language in the appropriate column. Look for phrases such as “$$x$$ (or $$y$$) is always the same,” “$$x$$ (or $$y$$) is always changing,” and “the slope is 0.” This will help students make sense of the structure of equations for horizontal and vertical lines.
Design Principle(s): Support sense-making; Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

The purpose of this discussion is for students to consider what they actually need to know to be able to determine information about a line. Students should recognize that everything they could want to know about the pairs of numbers that make up linear relationships can be determined from relatively little starting information. Here are some possible questions for discussion:

• “What information do you need to know exactly where a line is?” (the coordinates of two points on the line or the coordinates of one point and the line’s slope)
• “Why is a single point or just the value of the slope not enough to determine where a line is?” (With just a point, the line goes through it, but could have any slope. With just a slope, we could draw a line, but it could be located anywhere. Note: it can be helpful to use a yardstick to represent “the line” in this situation as you move it around a coordinate plane on the board.)
• “How can you tell from a real-world situation that the graph of the equation that represents it will be a horizontal line? Be a vertical line? Have a negative slope?” (For horizontal and vertical lines, the key feature is that one of the two variables does not vary while the other one can take any value. On the coordinate plane, when the variable $$y$$ can take any value, it is a vertical line, and when the variable $$x$$ can take any value, it is a horizontal line.)

## Student Lesson Summary

### Student Facing

We learned earlier that one way to find the slope of a line is by drawing a slope triangle. For example, using the slope triangle shown here, the slope of the line is $$\text{-}\frac24$$, or $$\text{-}\frac12$$ (we know the slope is negative because the line is decreasing from left to right).

But slope triangles are only one way to calculate the slope of a line. Let’s compute the slope of this line a different way using just the points $$A=(1,5)$$ and $$B=(5,3)$$. Since we know the slope is the vertical change divided by the horizontal change, we can calculate the change in the $$y$$-values and then the change in the $$x$$-values. Between points $$A$$ and $$B$$, the $$y$$-value change is $$3-5=\text{-}2$$ and the $$x$$-value change is $$5-1=4$$. This means the slope is $$\text{-}\frac24$$, or $$\text{-}\frac12$$, which is the same as what we found using the slope triangle.
Notice that in each of the calculations, We subtracted the value from point $$A$$ from the value from point $$B$$. If we had done it the other way around, then the $$y$$-value change would have been $$5-3=2$$ and the $$x$$-value change would have been $$1-5=\text-4$$, which still gives us a slope of $$\text- \frac12$$. But what if we were to mix up the orders? If that had happened, we would think the slope of the line is positive $$\frac12$$ since we would either have calculated $$\frac{\text-2}{\text-4}$$ or $$\frac{2}{4}$$. Since we already have a graph of the line and can see it has a negative slope, this is clearly incorrect. It we don’t have a graph to check our calculation, we could think about how the point on the left, $$(1,5)$$, is higher than the point on the right, $$(5,3)$$, meaning the slope of the line must be negative.
Horizontal lines in the coordinate plane represent situations where the $$y$$ value doesn’t change at all while the $$x$$ value changes, meaning they have a slope of 0. For example, the horizontal line that goes through the point $$(0,13)$$ can be described in words as “for all points on the line, the $$y$$ value is always 13.” An equation that says the same thing is $$y=13$$.
Vertical lines represent situations where the $$x$$ value doesn’t change at all while the $$y$$ value changes. The equation $$x=\text-4$$ describes a vertical line through the point $$(\text-4,0)$$.