options: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

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
This activity is designed to give students opportunities to use their understandings from this unit to perform mathematical modeling.
The trail mix context is familiar from the previous activity, but students are challenged to choose quantities, determine how to represent them, interpret and reason about them, and use the model they create to make choices. It also enables students to reflect on their model and revise it as needed (MP4).
Students are likely to want to use graphing technology, as the nutritional information involves decimals and the inequalities written would be inconvenient to graph by hand. This is an opportunity for students to choose tools strategically (MP5).
No time estimate is given here because the time would depend on decisions about the research students do and on the expectations for collaboration and presentation.
Launch
Arrange students in groups of 2–4. Provide access to Desmos or other graphing technology.
Explain the expectations for researching nutritional values, for collaboration with group members, and for presentation of student work. (If each group is presenting one response, provide each group with tools for creating a visual display. If each student is presenting a response, give each student tools for creating a visual display.)
Engagement: Develop Effort and Persistence.
Provide prompts, reminders, guides, rubrics, or checklists that focus on increasing the length of on-task orientation in the face of distractions. For each question, allot a specific amount of work time and display expectations. Display a countdown timer, along with a bulleted list of what teams or individuals are to produce to complete a given step. For instance, during question 1, display a 5 minute countdown timer and the list: “You’re finished if you. . . (1) Have your two ingredients (2) Have the nutrition information ready.”
Supports accessibility for: Attention; Social-emotional skills
Student Facing
It's time to design your own trail mix!
Choose two ingredients that you like to eat. (You can choose from the ingredients in the previous activity, or you can look up nutrition information for other ingredients.)
Think about the constraints for your trail mix. What do you want to be true about its calories, protein, sugar, fat, or fiber?
Write inequalities to represent your constraints. Then, graph the inequalities.
[IMG]
Is it possible to make trail mix that meets all your constraints using your ingredients? If not, make changes to your constraints or your ingredients and record them here.
Write a possible combination of ingredients for your trail mix.
Pause here so your teacher can review your work and give further instructions for displaying your work.
Student Response
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Activity Synthesis
Select groups to share their visual displays. Encourage students to ask questions about the mathematical thinking or design approach that went into creating the display. Here are questions for discussion, if not already mentioned by students:
What constraints did every group use?
How do the graphs of the various mixes compare?
Did anyone have to revise or change their model in order to come up with a solution they could use?
How did you use the graph to choose a recipe for your mix?
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite groups to create a visual display of their work. Students should consider what types of details (annotations, notes, diagrams, arrows, etc.) to include on their displays that will help communicate their reasoning. Begin the whole-class discussion by selecting and arranging 2–4 displays for all to see. Give students 2–3 minutes of quiet think time to interpret the displays before inviting the authors to present their work.
Design Principle(s): Optimize output; Cultivate conversation


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