standard_description: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
problem_activity: Activity
In the previous activity, the system of equations was represented in words, a table, and a graph. In this activity, the system of equations is partially given in words, but key elements are only provided in the graph. Students have worked with lines that represent a context before. 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.
Teacher Notes for IM 6–8 Math Accelerated
This activity is optional. Use this activity if time allows to preview some of the work ahead with systems of equations or if students need additional practice making connections between points on and off graphed lines as they relate to a context. Due to the sequence of IM 6–8 Math Accelerated, this activity is in a different lesson so the reference to the “previous activity” in the narrative should be disregarded.
Launch
Arrange students in groups of 2. Give students 1 minute to read the problem and answer any questions they have about the context. 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 followed by whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students within the first 2–3 minutes of work time to ensure that they have understood the directions. If students are unsure how to begin, suggest that they consider each statement for Clare first, and then for Andre.
Supports accessibility for: Organization; Attention
Writing, Speaking: MLR5 Co-Craft Questions.
Display the graph that shows Clare and Andre’s progress of making signs, without showing the table. Ask students to write down possible mathematical questions that can be asked about the situation, and then share with a partner. As students discuss, listen for and amplify questions that notice and wonder about the points A, B, C, and D; the location of these points; and why the points were possibly highlighted and what it means in context of the situation. Finally, reveal the table from the activity or use questions generated by the students and allow time to work on them. This will help students make sense of a graph with two lines at the same time by generating questions using mathematical language.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Student Facing
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.
[IMG]
Based on the lines, mark the statements as true or false for each person.
| point |                     what it says                     | Clare | Andre |
|-------|------------------------------------------------------|-------|-------|
| \(A\) |      At 40 minutes, I have 25 signs completed.       |       |       |
| \(B\) | At 75 minutes, I have 42 and a half signs completed. |       |       |
| \(C\) |       At 0 minutes, I have 15 signs completed.       |       |       |
| \(D\) |      At 100 minutes, I have 60 signs completed.      |       |       |
Student Response
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Student Facing
Are you ready for more?
4 toothpicks make 1 square
7 toothpicks make 2 squares
10 toothpicks make 3 squares
[IMG]
Do you see a pattern? If so, how many toothpicks would you need to make 10 squares according to your pattern? Can you represent your pattern with an expression?
Student Response
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Activity Synthesis
Display the graphs from the task statement. The goal of this discussion is for students to realize that points that lie on one line can be interpreted as statements that are true for Clare, and points that lie on the other line as statements that are true for Andre. Ask students:
“What is true for Clare and Andre after 20 minutes?” (Clare has 15 signs completed and Andre has 20 signs completed.)
“What, then, do you know about the point
\((20, 15)\)
and the equation for Clare's graph?” (The point
\((20, 15)\)
is a solution to the equation for Clare's graph.)
“What do you know about the point
\((20, 20)\)
and the equation for Andre's graph?” (The point
\((20, 20)\)
is a solution to the equation for Andre's graph.)
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.


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