Lesson 6

More Linear Relationships

6.1: Growing (5 minutes)


This warm-up 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. 


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.

3 diagrams. diagram 1, 1 yellow, 3 blue, 1 red. diagram 2, 4 yellow, 6 blue, 2 red. diagram 3, 9 yellw, 9 blue, 3 red.
  1. 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?

  2. 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)


In the previous lesson, students analyzed the graph of a linear, non-proportional 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 whole-group discussion. 

In the whole-group 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.


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 whole-class discussion. 

Student Facing

Your teacher will give you 6 cards describing different situations and 6 cards with graphs.

  1. Match each situation to a graph.
  2. Pick one proportional relationship and one non-proportional relationship and answer the following questions about them.

    1. How can you find the slope from the graph? Explain or show your reasoning.
    2. Explain what the slope means in the situation.
    3. 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.

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: vertical intercept, y-intercept. Invite students to suggest language or diagrams to include on the display that will support their understanding of these terms.
Supports accessibility for: Memory; Language
Representing, Conversing: MLR7 Compare and Connect. As students share their matches with the class, call students’ attention to the different ways the vertical intercept is represented graphically and within the context of each situation. Take a close look at Graphs 2 and 3 to distinguish what the 40 represents in each corresponding situation. Wherever possible, amplify student words and actions that describe the correspondence between specific features of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness; Support sense-making

6.3: Summer Reading (10 minutes)


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.


Work time followed by whole-class discussion.

Representation: Internalize Comprehension. Use MLR6 Three Reads to supporting reading comprehension of the word problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (Lin’s reading assignment). Use the second read to identify quantities and relationships. Ask students what can be counted or measured without focusing on the values. Listen for and amplify the important quantities that vary in relation to each other in this situation: first 30 pages and 40 pages each day. After the third read, ask students to brainstorm possible strategies to answer the question, “What does the vertical intercept mean in this situation?”
Supports accessibility for: Language; Conceptual processing
Speaking: MLR8 Discussion Supports. Use this to amplify mathematical uses of language to communicate about vertical intercepts, slope, and constant rate. Invite students to use these words when describing their ideas. Ask students to chorally repeat phrases that include these words in context.
Design Principle(s): Support sense-making, 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.

Graph of line. Points plotted on line include 1 comma 70 and 4 comma 160.
  1. Sketch a line showing Lin's original plan on the axes.
  2. What does the vertical intercept mean in this situation? How do the vertical intercepts of the two lines compare?
  3. 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:

Graph of line in quadrant 1
  • “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: Cool-down - Savings (5 minutes)


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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:

Graph of 2 lines in quadrant 1. Horizontal axis, time in weeks, scale 0 to 12, by 1’s. Vertical axis, amount saved in dollars, scale 0 to 100, by 20’s.

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.