Lesson 6
More Linear Relationships
6.1: Growing (5 minutes)
Warmup
This warmup encourages students to look for regularity in how the number of tiles in the diagram are growing. This relates naturally to the work that they are doing with understanding the linear relationships as two of the three patterns students are likely to observe are linear and, in fact, proportional.
Launch
Arrange students in groups of 2. Display the image for all to see and ask students to look for a pattern in the way the collection of red, blue, and yellow tiles are growing. Ask how many tiles of each color will be in the 4th, 5th, and 10th diagrams if the diagrams keep growing in the same way. Tell students to give a signal when they have an answer and strategy. Give students 1 minute of quiet think time, and then time to discuss their responses and reasoning with their partner.
Student Facing
Look for a growing pattern. Describe the pattern you see.

If your pattern continues growing in the same way, how many tiles of each color will be in the 4th and 5th diagram? The 10th diagram?

How many tiles of each color will be in the \(n\)th diagram? Be prepared to explain how your reasoning.
Student Response
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Activity Synthesis
Invite students to share their responses and reasoning. Record and display the different ways of thinking for all to see. If possible, record the relevant reasoning on or near the images themselves. After each explanation, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to what is happening in the images each time.
Time permitting, ask students which patterns represent a linear relationship? Which ones represent a proportional relationship? The patterns for the blue and red blocks are both proportional (hence linear) while the pattern for the yellow blocks is neither proportional nor linear.
6.2: Slopes, Vertical Intercepts, and Graphs (20 minutes)
Activity
In the previous lesson, students analyzed the graph of a linear, nonproportional relationship (number of cups in a stack versus the height of the stack). This task focuses on interpreting the slope of a graph and where it crosses the \(y\)axis in context. Students are given cards describing situations with a given rate of change and cards with graphs. Students match each graph with a situation it could represent, and then use the context to interpret the meaning of the slope. They find where the line crosses the vertical axis, i.e., the vertical intercept, and interpret its meaning in each situation. They also decide if the two quantities in each situation are in a proportional relationship.
Make sure that students draw the triangle they use to compute the slope. There are strategic choices that can be made to make the computation easier and more precise. Watch for students who use different triangles for the same slope computation, and ask them to share their reasoning during the wholegroup discussion.
In the wholegroup discussion at the end of the task, discuss and emphasize the meaning of the terms slope and vertical intercept or \(y\)intercept (in situations where the name of the variable graphed on the vertical axis is \(y\)).
You will need the Slopes and Graphs blackline master for this lesson.
Launch
Tell students that they will match a set of cards describing different relationships with a set of cards showing graphs of lines. The axes on the graphs are not labeled (since this could be used as an aid in the matching). Instruct students to add labels to the axes as they make their matches.
Arrange students in groups of 2. Distribute a set of 12 cards to each group. 10 minutes of group work and then wholeclass discussion.
Student Facing
Your teacher will give you 6 cards describing different situations and 6 cards with graphs.
 Match each situation to a graph.

Pick one proportional relationship and one nonproportional relationship and answer the following questions about them.
 How can you find the slope from the graph? Explain or show your reasoning.
 Explain what the slope means in the situation.
 Find the point where the line crosses the vertical axis. What does that point tell you about the situation?
Student Response
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Activity Synthesis
The slopes of the 6 lines given on the situation card are all different, so the matching part of the task can be accomplished by examining the slopes of the different lines. Invite students who have made strategic choices of slope triangles for calculating the slopes (for example, Graph 6 contains the points \((0,0)\) and \((5,20)\), which give a value of \(\frac{20}{5}\) for the slope), and ask them to share.
Next, focus the discussion on the interpretation of the point where the line crosses (or touches) the \(y\)axis. For some situations, the contextual meaning of this point is abstract. For example, in Situation B, a square with side length 0 is just a point that has no perimeter and so it “makes sense” that \((0,0)\) is on the graph. Some students may argue that a point is not a square at all but if we consider it to be a square then it definitely has 0 perimeter. In other situations, the point where the line touches the \(y\)axis has a very natural meaning. For example, in Situation A, it is the amount Lin’s dad spent on the tablet and 0 months of service: so this is the cost of the tablet.
Define the vertical intercept or \(y\)intercept as the point where a line crosses the \(y\)axis. Note that sometimes “\(y\)intercept” refers to the numerical value of the \(y\)coordinate where the line crosses the \(y\)axis. Go over each of the situations and ask students for the meaning of the vertical intercept in the situation (A: cost of the device, B: perimeter of a square with 0 side length, C: amount of money in Diego’s piggy bank before he started adding \$5 each week, D: money Noah saved helping his neighbor, E: amount of money in Elena's piggy bank before she started adding money, F: amount Lin’s mom has paid for internet service before her service begins.
Supports accessibility for: Memory; Language
Design Principle(s): Maximize metaawareness; Support sensemaking
6.3: Summer Reading (10 minutes)
Activity
Students have just learned the meaning of the \(y\)intercept for a line and have been interpreting slope in context. In this activity, they investigate the \(y\)intercept and slope together and investigate what happens when their values are switched.
In contexts like this one, the \(y\)intercept and the slope come with natural units and understanding this can help graph accurately. Specifically, the \(y\)intercept is the number of pages Lin read before she starts gathering data for the graph. The slope, on the other hand, is a rate: it’s the number of pages Lin reads per day.
Watch for students who understand the source of Lin’s error (confusing the \(y\)intercept with the slope) and invite them to share this observation during the discussion.
Launch
Work time followed by wholeclass discussion.
Supports accessibility for: Language; Conceptual processing
Design Principle(s): Support sensemaking, Optimize output (for explanation)
Student Facing
Lin has a summer reading assignment. After reading the first 30 pages of the book, she plans to read 40 pages each day until she finishes. Lin makes the graph shown here to track how many total pages she'll read over the next few days.
After day 1, Lin reaches page 70, which matches the point \((1,70)\) she made on her graph. After day 4, Lin reaches page 190, which does not match the point \((4,160)\) she made on her graph. Lin is not sure what went wrong since she knows she followed her reading plan.
 Sketch a line showing Lin's original plan on the axes.
 What does the vertical intercept mean in this situation? How do the vertical intercepts of the two lines compare?
 What does the slope mean in this situation? How do the slopes of the two lines compare?
Student Response
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Student Facing
Are you ready for more?
Jada's grandparents started a savings account for her in 2010. The table shows the amount in the account each year.
If this relationship is graphed with the year on the horizontal axis and the amount in dollars on the vertical axis, what is the vertical intercept? What does it mean in this context?
year  amount in dollars 

2010  600 
2012  750 
2014  900 
2016  1050 
Student Response
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Activity Synthesis
Ask students:
 “How did your graph compare to Lin’s?” (It is steeper but starts off at 30 instead of 40 pages.)
 “Which point does your graph have in common with Lin’s?” (The point \((1,70)\), 70 pages read after one day.)
Invite selected students to share what’s the likely source of Lin’s error (confusing the \(y\)intercept with the slope). In this context, the \(y\)intercept is the number of pages Lin read before she starts counting the days (30), and the slope is the number of pages Lin reads per day (40).
Emphasize how the \(y\)intercept and slope influence the graph of a line.
 The \(y\)intercept indicates where the line touches or crosses the \(y\)axis.
 The slope indicates how steep the line is.
Lesson Synthesis
Lesson Synthesis
Lines have a slope and vertical intercept. The vertical intercept indicates where the line meets the \(y\)axis. For example, a line represents a proportional relationship when the vertical intercept is 0.
Here is a graph of a line showing the amount of money paid for a new cell phone and monthly plan:
 “What is the vertical intercept for the graph?” (\((0,200)\))
 “What does it mean?” (There was an initial cost of \$200 for the phone.)
The slope of the line is 50 (draw a slope triangle connecting the points such as \((0,200)\) and \((2,300)\)). This means that the phone service costs \$50 per month in addition to the initial \$200 for the phone.
6.4: Cooldown  Savings (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
At the start of summer break, Jada and Lin decide to save some of the money they earn helping out their neighbors to use during the school year. Jada starts by putting \$20 into a savings jar in her room and plans to save \$10 a week. Lin starts by putting \$10 into a savings jar in her room plans to save \$20 a week. Here are graphs of how much money they will save after 10 weeks if they each follow their plans:
The value where a line intersects the vertical axis is called the vertical intercept. When the vertical axis is labeled with a variable like \(y\), this value is also often called the \(y\)intercept. Jada's graph has a vertical intercept of \$20 while Lin's graph has a vertical intercept of \$10. These values reflect the amount of money they each started with. At 1 week they will have saved the same amount, \$30. But after week 1, Lin is saving more money per week (so she has a larger rate of change), so she will end up saving more money over the summer if they each follow their plans.