In this activity, a system of equations is partially given in words, but key elements are provided only in the graph. Students have worked before with lines that represent a context. Now they must work with two lines at the same time to determine whether a point lies on one line, both lines, or neither line.
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Launch
Arrange students in groups of 2. Introduce the context of making signs for spirit week. Use Co-Craft Questions to orient students to the context and to elicit possible mathematical questions.
Display only the problem stem and related image, without revealing the questions. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
Invite several partners to share one question with the class, and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as the meaning of the points and lines in context.
Reveal the question, “Based on the lines, mark the statements as true or false for each person.” Give students 1–2 minutes to compare the revealed question to their own question and those of their classmates. Invite students to identify similarities and differences by asking, “Is there a main mathematical concept that is present in both your questions and those provided? If so, describe it.”
Tell students to complete the table one row at a time, with one person responding for Clare and the other responding for Andre. Give students 2–3 minutes to finish the table, and follow that with a whole-class discussion.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Suggest that students answer all situations for Clare first and then consider Andre's situations. Check in with students to provide feedback and encouragement after each chunk. Ensure that students understand the instructions and how to fill in the table. Supports accessibility for: Attention, Social-Emotional Functioning
Activity
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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>
Which line represents Andre and which represents Clare?
Based on the lines, mark the statements as true or false for each person.
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.
Student Response
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Building on Student Thinking
Activity Synthesis
Display the graphs from the Task Statement. The goal of this discussion is for students to realize that points that lie on a line make that situation true. If a point is on more than 1 line, such as at an intersection, then it makes all of those situations true. Ask students:
“Is the point on either line? What does that mean in this situation?” (It is on Clare’s line. It means that, 20 minutes after they worked today, Clare has made a total of 20 signs.)
“Imagine a point that is not on either line. What does that information tell us about this point?” (It means that it represents a situation that is not true for either Andre or Clare.)
“If a point is on a line, what does it mean? If the point is the intersection of 2 lines, what does that mean?” (A point on a line means that it represents a situation that is true for the situation represented by the line. The intersection point represents a situation that is true for both situations represented by the lines.)
Invite groups to share their reasoning about points A–D. Conclude by pointing out to students that, in this context, there are many points true for Clare and many points true for Andre but only one point true for both of them. Future lessons will be about how to figure out that point.
13.2
Activity
Standards Alignment
Building On
Addressing
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
In this task, students find and graph a linear equation given only the graph of another equation, information about the slope, and the coordinates where the lines intersect. The purpose of this task is to check student understanding about the point of intersection in relationship to the context, while applying previously learned skills of equation writing and graphing.
Launch
Tell students to close their books or devices (or to keep them closed). Display the image for all to see.
Four blank number lines. Each number line has a ladybug facing right and an ant facing left. Number line 1 is labeled ladybug start on the left end and 0 seconds. The ladybug is to the left of the number line and the ant is to the right of the number line. Number line 2 is labeled 2 seconds. The ladybug is at the 8th tick mark from the right. The ant is at the 16th tick mark from the left. Number line 3 is labeled 4 seconds. The ladybug is at the 16th tick mark from the right. The ant is at the 32nd tick mark from the left. Number line 4 is labeled 6 seconds. The ladybug is at the 24th tick mark. The ant is not shown in the number line.
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. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the image.
Tell students to open their books or devices and display the graph from the Task Statement. Explain that the graph is about a different ant and ladybug. Give 3–4 minutes for students to complete the problems, and follow that with a whole-class discussion.
Activity
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A different ant and ladybug are a certain distance apart, and they start walking toward each other. The graph shows the ladybug’s distance from its starting point over time and the labeled point indicates when the ant and the ladybug pass each other.
<p>Graph of a line, origin O, with grid. Horizontal axis, time in seconds, scale 0 to 5, by 1’s. Vertical axis, distance in centimeters, scale 0 to 24, by 2’s. The line passes through the origin and the point 2 point 5 comma 10.</p>
The ant is walking 2 centimeters per second.
Write an equation representing the relationship between the ant’s distance from the ladybug’s starting point and the amount of time that has passed.
If you haven’t already, draw the graph of your equation on the same coordinate plane.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to ensure that all students understand both how the labeled point in the task statement relates to the context and how to write and graph an equation from the given information.
Invite 2–3 groups with different approaches to share their solutions. Here are some questions for discussion:
“How does the ant’s rate of change show up in each method?” (It is the slope, which is the coefficient of in the equation and is related to the steepness of the line in the graph.)
“How can each method be used to find the starting distance between the 2 bugs?” (In one method, I counted back 2 cm per second to find the starting position. In the other method, I counted forward 2 cm per second to find the line, then drew the line and looked at where it crosses the vertical axis.)
If students struggled to graph the ant’s path, you may wish to conclude the discussion by asking students for different ways to add the graph of the ant’s distance onto the coordinate plane. For example, some students may say to use the equation figured out in the first problem to plot points and then draw a line through them. Other students may suggest starting from the known point, , and “working backward” to figure out that 1 second earlier at 1.5 seconds, the ant would have to be 12 centimeters away because .
13.3
Activity
Standards Alignment
Building On
Addressing
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
In previous lessons, students encountered equations with a single variable that had infinitely many solutions. In this activity, students interpret a system with infinitely many solutions. A race is described using different representations (a table and a description in words). Students graph the relationships given by the descriptions and notice that the lines overlap so that both relationships are true for any pair of values along the graphed line.
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
Allow students 7–10 minutes of silent work time followed by a whole-class discussion.
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the table, without revealing the questions.
For the first read, read the problem aloud, and then ask, “What is this situation about?” (friends racing bikes). 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 (the distance and times for 2 points in Jada’s race).
After the third read, reveal the question: “Graph the relationship between distance and time for Jada’s bike race.” Ask, “What are some ways that we might get started on this?” Invite students to name some possible starting points, referring to quantities from the second read (plot the points).
Activity
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Elena and Jada are racing 100 meters on their bikes. Both racers start at the same time and ride at constant speed. Here is a table that gives information about Jada’s bike race:
time from start (seconds)
distance from start (meters)
6
36
9
54
Graph the relationship between distance and time for Jada’s bike race. Make sure to label and scale the axes appropriately.
Elena travels the entire race at a steady 6 meters per second. On the same set of axes, graph the relationship between distance and time for Elena’s bike race.
Who won the race?
Student Response
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Building on Student Thinking
Activity Synthesis
The key point for discussion is to connect what students observed about the graph that they made to the concept of “infinitely many solutions” encountered in earlier lessons. Graphically, students see that there are situations where two lines align “on top of each other.” We can interpret each point on the line as representing a solution to both Elena’s and Jada’s equations. If is the time from start and is the distance from the start, the equation for Elena is . The equation for Jada is also . Every solution to Elena’s equation is also a solution to Jada’s equation, and every solution to Jada’s equation is also a solution to Elena's equation. In this way, there are infinitely many points that are solutions to both equations at the same time.
Ensure that students clearly understand that just because there are infinitely many points that are solutions, it does not mean that any pair of values will solve both Elena’s and Jada’s equations. In this example, the pair of values must still be related by the equation . So, pairs of values like , and are all solutions, but is not.
Lesson Synthesis
Tell students to think about ordered pairs that make relationships true. Ask:
“When using graphs, where are the points whose coordinates do not make a given relationship true? (Points that are off of the line do not make the given relationship true. They can be above or below the line.)
“A line goes through the point and has a -intercept of . What is an equation for this line?” ()
“A second line also goes through the point but has a different -intercept. What does the point represent for these lines?” (The pair of values that is true in both situations.)
Student Lesson Summary
The solutions to an equation correspond to points on its graph. For example, if Car A is traveling 75 miles per hour and passes a rest area when , then the distance in miles it has traveled from the rest area after hours is
The point is on the graph of this equation because it makes the equation true (). This means that 2 hours after passing the rest area, the car has traveled 150 miles.
If you have 2 equations, you can ask whether there is an ordered pair that is a solution to both equations simultaneously. For example, if Car B is traveling toward the rest area, and its distance from the rest area is
We can ask if there is ever a time when the distance of Car A from the rest area is the same as the distance of Car B from the rest area. If the answer is yes, then the solution will correspond to a point that is on both lines.
Graph of 2 lines, origin O, with grid. Horizontal axis, time in hours, scale 0 to point 22, by point 0 2’s. Vertical axis, distance in miles, scale 0 to 14, by 2’s. One line passes through the origin and the point 0 point 1 comma 7 point 5. Another line crosses the y axis at 14 and passes through the point 0 point 1 comma 7 point 5
Looking at the coordinates of the intersection point, we see that Car A and Car B will both be 7.5 miles from the rest area after 0.1 hours (which is 6 minutes).
Now suppose another car, Car C, also passes the rest stop at time and travels in the same direction as Car A, also going 75 miles per hour. It's equation is also . Any solution to the equation for Car A is also a solution for Car C, and any solution to the equation for Car C is also a solution for Car A. The line for Car C is on top of the line for Car A. In this case, every point on the graphed line is a solution to both equations, so there are infinitely many solutions to the question, “When are Car A and Car C the same distance from the rest stop?” This means that Car A and Car C are side by side for their whole journey.
When we have two linear equations that are equivalent to each other, like and , we get 2 lines that are right on top of each other. Any solution to one equation is also a solution to the other, so these 2 lines intersect at infinitely many points.
In general, if we have 2 lines in the coordinate plane and we have their corresponding equations,
The coordinates of a point on a line make that equation true.
The coordinates of a point off of a line make that equation false.
The coordinates of a point that is the intersection of the 2 lines make both equations true.
Standards Alignment
Building On
Addressing
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
