Modeling with Systems of Inequalities in Two Variables
26.1: A Solution to Which Inequalities? (5 minutes)
This warm-up gives students a quick exposure to the inequalities \(x>0\), \(x\geq 0\), \(y>0\), and \(y\ge0\), so that they are prepared to deal with them later in this lesson. It also reinforces the idea of thinking carefully about whether the points on the boundary lines of a solution region are included in the solution set.
Is the ordered pair \((5.43,0)\) a solution to all, some, or none of these inequalities? Be prepared to explain your reasoning.
Invite students to share their responses. Then, display a blank four-quadrant coordinate plane for all to see.
- "If we were to graph the solutions to \(x>0\), what would the region look like?" (We would shade the right side of \(y\)-axis.) "Is the \(y\)-axis included in the solution region?" (No)
- "What about the graph of the solutions to \(y>0\)?" (We would shade the upper side of the \(x\)-axis). "Is the \(x\)-axis included in the solution region?" (No)
- "What about the graph of the solutions to the system \(x>0\) and \(y>0\)?" (The solution region would be the upper-right section of the graph, where the other two regions overlap.)
Remind students that this upper-right region of the coordinate plane is called the first quadrant.
26.2: Custom Trail Mix (20 minutes)
In this activity, students use their insights from the unit to analyze and interpret a set of mathematical models and a set of data in context. Each situation involves more than two constraints, and can therefore be represented with a system with more than two inequalities.
Interpreting and connecting the inequalities, the graphs, and the data set (which involves decimals) prompts students to make sense of problems and persevere in solving them (MP1), and to reason quantitatively and abstractly (MP2).
Give students a moment to skim through the task statement and familiarize themselves with the given information. Ask them to be prepared to share one thing they notice and one thing they wonder. Invite students to share their observations and questions.
Then, to help students interpret the variables in the given inequalities as representing the number of grams of each ingredient, ask them to use the table to write an expression to represent the total amount of fiber if they had \(a\) grams of almonds and \(b\) grams of raisins. Students should see that the expression is \(0.07a + 0.05b\).
Next, ask for an expression representing the total amount of sugar for the same amounts of almonds and raisins (\(0.21a+0.60b\)).
Arrange students in groups of 2. Ask them to analyze and answer the questions about one student's trail mix (either Tyler's or Jada's). If time permits, the groups could analyze the other trail mix.
Give students a few minutes of quiet work time and time to share their thinking with their partner. Follow with a whole-class discussion.
Supports accessibility for: Visual-spatial processing
Here is the nutrition information for some trail mix ingredients:
|calories per gram (kcal)||protein per gram (g)||sugar per gram (g)||fat per gram (g)||fiber per gram (g)|
Tyler and Jada each designed their own custom trail mix using two of these ingredients. They wrote inequalities and created graphs to represent their constraints.
- \(x+y > 50\)
- \(4.76x + 6.67y \le 400\)
- \(0.67x + 0.07y < 30\)
- \(0.14w + 0.03z > 4\)
- \(0.61w + 0z \le 15\)
Use the inequalities and graphs to answer these questions about each student's trail mix. Be prepared to explain your reasoning.
- Which two ingredients did they choose?
- What do their variables represent?
- What does each constraint mean?
- Which graph represents which constraint?
- Name one possible combination of ingredients for their trail mix.
Focus the discussion on the connections between the graphs and the inequalities, and on the inequalities \(x>0\) and \(y>0\). Ask questions such as:
- “How did you know which ingredients each person used?” (By matching the coefficients in two of the inequalities to the nutritional values in the table.)
- “The table shows the same values for some nutrients. How can you tell which one Tyler or Jada chose?” (The coefficients of \(x\) and \(y\) in one inequality and those in the other inequality must be for the same two ingredients.)
- “Why do you think Jada and Tyler both included the inequalities \(x>0\) and \(y>0\)?” (There cannot be only one ingredient, so both \(x\) and \(y\) must be greater than 0.)
- “How do those inequalities affect the graph of the solution region?” (They limit the solution region to the first quadrant.)
- “Jada and Tyler each wrote five inequalities. Could all five form a single system?” (Yes) What does it mean to have a system with five inequalities?" (There are five constraints that must be met. The solutions to the system satisfy all five constraints simultaneously.)
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
26.3: Design Your Own Trail Mix (20 minutes)
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.
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.)
Supports accessibility for: Attention; Social-emotional skills
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.
- 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.
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?
Design Principle(s): Optimize output; Cultivate conversation
Allow enough time for students to present their trail mix recipes. Consider a gallery walk as a way for students to share their display and to ask and answer questions.