Unit 5, Lesson 5
More Linear Relationships
Accelerated 7
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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.
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.
What do you notice? What do you wonder?
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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.
MLR6: Three Reads
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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.
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).
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
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 “
“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.
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
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
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
A slope triangle connecting the two points
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