This Warm-up prompts students to make sense of a problem before solving it by observing an image and familiarizing themselves with a context and the mathematics that might be involved. The same image will be seen in a following activity.
Launch
Arrange students in groups of 2. Display the graph for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions.
Activity
None
What do you notice? What do you wonder?
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the graph. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
6.2
Activity
Standards Alignment
Building On
Addressing
8.EE.B
Understand the connections between proportional relationships, lines, and linear equations.
This activity introduces students to the vertical intercept for a line and how to interpret it in context. They investigate the vertical intercept and slope together and observe what happens when their values are switched.
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Launch
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the diagram, without revealing the questions.
For the first read, read the problem aloud then ask, “What is this situation about?” (Lin made a graph to track the number of pages she read for a summer reading assignment, but after 4 days, her graph doesn’t match the page she is on.) Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured (Lin has already read 30 pages. She will read 40 pages each day following. She is on page 70 after day 1 and reaches page 190 after day 4).
After the third read, reveal the question “Why doesn’t Lin’s reading progress match her graph?” and ask, “What are some ways one might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read (Lin has already read 30 pages).
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of slope and intercepts. Terms may include: “vertical intercept” and “y-intercept.” Supports accessibility for: Conceptual Processing, Language
Activity
None
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 she made on her graph. After day 4, Lin reaches page 190, which does not match the point she made on her graph. Lin is not sure what went wrong since she knows she followed her reading plan. Why doesn’t Lin’s reading progress match her graph?
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to introduce the term “vertical intercept.” Begin by inviting 1–2 students to share why Lin’s reading progress does not match the graph she made.
Then define the vertical intercept as the point where a line crosses the vertical axis. Note for students that sometimes “-intercept” is used to refer to the numerical value of the -coordinate in situations where the name of the variable graphed on the vertical axis is . “Vertical intercept” can also be used to refer to this numerical value. Ask students:
“What is the vertical intercept for the graph Lin made? What does it represent in this context?” (The vertical intercept is 40. It represents that Lin had initially read 40 pages.)
“What should the vertical intercept be?” (It should be 30 because Lin had already read only 30 pages of the book.)
“What is the slope of the line in the graph Lin drew? What does it represent?” (The slope is 30. It means that Lin read 30 pages each day after the first.)
“What should the slope be?” (The slope should be 40 because Lin’s plan was to read 40 pages each day.)
If time allows, have students draw a line that matches Lin’s reading plan and progress on the same coordinate plane.
6.3
Activity
Standards Alignment
Building On
7.RP.A.2.a
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
This activity focuses on interpreting the slope of a graph and where it crosses the -axis in context. Students sort different graphs and descriptions. They match each graph with a situation it could represent, and then use the context to interpret the meaning of the slope and the vertical intercept. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2 and distribute pre-cut cards. Allow students to familiarize themselves with the representations on the cards:
Give students 1 minute to place all the cards face up and start thinking about possible ways to sort the cards into categories.
Pause the class and select 1–3 students to share the categories they identified.
Discuss as many different categories as time allows.
Attend to the language that students use to describe their categories, giving them opportunities to describe their graphs and descriptions more precisely. Highlight the use of terms like “slope” and “vertical intercept.” After a brief discussion, invite students to continue with the activity.
Activity
None
Your teacher will give you a set of cards containing descriptions of situations and graphs. Match each situation with a graph that represents it. Record your matches and be prepared to explain your reasoning.
Student Response
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Activity Synthesis
The focus of this discussion is on the interpretation of the slope and the vertical intercept. For each situation, invite students to share how they determined the matching graph.
The slopes of the 6 lines given on the situation cards are all different, so the matching part of the task can be accomplished by examining just the slopes. Then for each situation, discuss:
“What does the slope mean in this situation?”
Situation A: the cost per month of the streaming service
Situation B: an increase of 1 in side length adds 4 to the perimeter
Situation C: the amount of money Diego adds to his piggy bank each week
Situation D: the amount of money Noah adds to his piggy bank each month
Situation E: the amount of money Elena adds to her piggy bank each day
Situation F: the cost per month for internet service
“What is the vertical intercept in this situation and what does it represent?”
Situation A; 40, the cost of the tablet was \$40.
Situation B; 0, the perimeter of a square with side length 0 is 0.
Situation C; 10, Diego initially had \$10 in his piggy bank.
Situation D; 40, Noah started out with \$40 in his piggy bank.
Situation E; 9, Elena originally had \$9 in her piggy bank.
Situation F; 0, Lin’s mom paid no money before paying for internet service.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their reasoning. For example, “I noticed so I matched. . . .” or “and are the same/alike because . . . .” Encourage students to challenge each other when they disagree. Advances: Speaking, Representing
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code matching quantities in the problem stem and diagram. Supports accessibility for: Visual-Spatial Processing
Lesson Synthesis
The purpose of this discussion is to review how the slope and vertical intercept are related to the graph of a line drawn on a set of axes.
Display the graph for all to see and explain that it shows how Jada and Lin are saving some of the money they earn in the summer helping out their neighbors to use during the school year. Jada starts by putting \$20 into a savings jar and plans to save \$10 a week. Lin starts by putting \$10 into a savings jar and plans to save \$20 a week. Here are graphs of how much money each of them will save if they follow their plan:
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. Lin line, y intercept = 10. Slope = 20. Jada line, y intercept = 20, slope= 10.
Invite students to share strategies for how to determine which graph represents which person. Emphasize the following ideas:
The vertical intercept represents how much money each girl starts with and can be seen where the line meets the vertical axis.
Since Lin starts out with less money, her vertical intercept will be lower than Jada’s.
The slope represents the rate of change for each girl, or in this situation, the amount of money saved each week.
Since Lin plans on saving more money per week, her line will have a greater slope and her graph will look steeper.
When students are in agreement, label each line with the person it represents (After 1 week, Lin is the top line and Jada is the bottom line). As students share their ideas about vertical intercepts, label the vertical intercepts on the graph. If time allows and students would benefit from an additional demonstration, show how to calculate the slope for one of the lines by drawing a slope triangle connecting two points such as and on the graph of Jada’s line. The slope of 10 matches that Jada is planning to save \$10 each week.
Student Lesson Summary
Lines drawn on a coordinate plane have a slope and a vertical intercept. The vertical intercept indicates where the graph of the line meets the vertical axis. Since the vertical axis is often referred to as the -axis, the vertical intercept is often called the “-intercept.” 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.
The vertical intercept for the graph is at the point and means the initial cost for the phone was \$200.
A slope triangle connecting the two points and can be used to calculate the slope of this line. The slope of 50 means that the phone service costs \$50 per month in addition to the initial \$200 for the phone.
Standards Alignment
Building On
Addressing
Building Toward
8.EE.B
Understand the connections between proportional relationships, lines, and linear equations.
