Unit 4, Lesson 10
On or Off the Line?
Grade 8
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This Warm-up prompts students to compare four graphs. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three graphs that go together and can explain why. Next, tell students to share their response with their group and then together to find as many sets of three as they can.
Which three go together? Why do they go together?
<p>A graph in the x y plane. Three lines that do not intersect. One line crosses the x axis to the left of the origin and the y axis below the origin. Another line crosses the y axis above the origin and the x axis to the right of the origin. The third line crosses the y axis above the origin and the x axis to the right of the origin. </p>
<p>A graph in the x y plane. Two intersecting lines. One line crosses the x axis to the left of the origin and the y axis below the origin. Another line crosses the y axis above the origin. </p>
<p>A graph in the x y plane. Three lines that intersect at a single point. One line crosses the y axis above the origin. Another line crosses the x axis to the right of the origin and the y axis below the origin. The third line crosses the x axis to the right of the origin.</p>
<p>A graph in the x y plane. Three lines. There are 3 points of intersection between two lines each. One line crosses the y axis above the origin. Another line crosses the x axis to the left of the origin and the y axis above the origin. The third line crosses the x axis to the right of the origin.</p>
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Because there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “intersection” or “parallel,” and to clarify their reasoning as needed. Consider asking:
MLR7: Compare and Connect
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In earlier lessons, students have set two expressions equal to one another to find a common value where both expressions are true (if it exists). A system of two equations asks a similar question: At what common pair of values are both equations true? In this activity, students focus on a context involving coins and use multiple representations to think about the context in different ways. The goal of this activity is not for students to write equations or learn the language “system of equations,” but rather to investigate the mathematical structure with two stated facts using familiar representations and context while reasoning about what must be true.
Monitor for students who use these representations to solve the last problem:
Before looking at the task, tell students, “I have money worth $2 in my pocket. What might be in my pocket?” Students will likely guess that you have two $1 bills, but ask what else it might be. Some answers could be 8 quarters, 200 pennies, a $2 bill, or 20 nickels, 2 quarters, and 5 dimes.
Read the problem context together. Ensure that students understand that we know that Jada has exactly $2 in her pocket, that she has only quarters and dimes, and that she has exactly 17 coins. Give 1–2 minutes for students to read and complete the first problem. Display the table for all to see, and ask students for values to fill in the table.
Give students 5–7 minutes of quiet work time to finish the remaining problems, and follow that with a whole-class discussion.
Select work from students with different strategies, such as those described in the Activity Narrative, to share later.
Jada told Noah that she has $2 worth of quarters and dimes in her pocket and 17 coins all together. She asked him to guess how many of each type of coin she has.
Here is a table that shows some combinations of quarters and dimes that are worth $2. Complete the table.
| number of quarters | number of dimes |
|---|---|
| 0 | 20 |
| 4 | |
| 0 | |
| 5 | |
Here is a graph of the relationship between the number of quarters and the number of dimes when there are a total of 17 coins.
<p>Graph of points in the x y plane, origin O, with grid. Horizontal axis, number of quarters, scale 0 to 24 by 1s. Vertical axis, number of dimes, scale 0 to 24 by 1s. Points plotted are 0 comma 17, 1 comma 16, 2 comma 15, 3 comma 14, 4 comma 13, 5 comma 12, 6 comma 11, 7 comma 10, 8 comma 9 labeled A, 9 comma 8, 10 comma 7, 11 comma 6, 12 comma 5, 13 comma 4, 14 comma 3, 15 comma 2, 16 comma 1, and 17 comma 0.</p>
The goal of this discussion is to focus students on the last problem and what must be true based on the two facts that they know: The coins total $2 and there are exactly 17 coins. Begin the discussion by asking students about some things that they know cannot be true about the coins in Jada’s pocket and how they know. Students may respond that Jada cannot have 20 dimes and 0 quarters because that is not 17 coins or other variations where either the coins do not total $2, there are not exactly 17 coins, or neither are true.
Display 2–3 approaches/representations from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different representations. Here are some questions for discussion:
Ensure that students understand that their solution is one where both facts are true.
We studied linear relationships in an earlier unit. We learned that values of
For example, let’s plan the base rocks for a terrarium. We have
Because 5 pounds of river rocks cost $4.00 and 10 pounds of unpolished rocks cost $5.00, we know that
The line shown is the graph of the equation. Notice that there are 2 points shown that are not on the line. What do they mean in the context?
<p>The graph of a line in the x y plane. The line slants downward and right, crosses the y axis at 18, and passes through the point 5 comma 10. Two additional points, 9 comma 16 and 1 comma 14, are labeled on the graph.</p>
The point
Suppose we also know that the river rocks and unpolished rocks together weigh 15 pounds. That means that
If we draw the graph of this equation on the same coordinate plane, we see it passes through 2 of the 3 labeled points:
<p>The graph of two intersecting lines in the x y plane. The first line slants downward and right, crosses the y axis at 18, and passes through the point 5 comma 10. The second line slants downward and to the right and passes through the points 1 comma 14 and 5 comma 10. An additional point, 9 comma 16, is labeled on the graph.</p>
The point
In general, if we have 2 lines in the coordinate plane and we have their corresponding equations,
Help us improve by sharing suggestions or reporting issues.
Students may wonder why the last row of the table is blank. Tell them they can enter any values that make sense in the context and that are not already in the table.
Clare and Andre are making signs for all the lockers as part of the decorations for the upcoming spirit week. Yesterday, Andre made 15 signs and Clare made 5 signs. Today, they need to make more signs. Each person's progress today is shown in the coordinate plane.
<p>Graph of two lines in the x y plane, origin 0, with grid. Horizontal axis, time in minutes, scale 0 to 110, by 5s. Vertical axis, number of completed signs, scale 0 to 70 by 5s. A line crosses the y axis at 5 and passes through the points A and B. Another line crosses the y axis at 15 and passes through the points C and A. Starting at the origin, point A is 8 unit to the right and 5 units up. Starting at the origin, point B is 15 units to the right and 8 point 5 units up. Starting at the origin, point C is 0 units to the right, and 3 units up. Starting at the origin, point D is 20 units to the right and 12 units up.</p>
| point | what it says | Clare | Andre |
|---|---|---|---|
| At 40 minutes, I have 25 signs completed. | |||
| At 75 minutes, I have 42 and a half signs completed. | |||
| At 0 minutes, I have 15 signs completed. | |||
| At 100 minutes, I have 60 signs completed. |