Unit 4, Lesson 9
Modeling with Systems of Inequalities in Two Variables
Algebra 1
9.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
This Warm-up gives students a chance to get an idea of how to add subjective values to a modeling situation. In later activities for this lesson, students model planning a trip using different methods of transit. One consideration for the methods of transit is which method is generally preferred. Students should keep their values in the table for an activity later in the lesson.
Launch
Tell students that they are going to explore different methods of transit, such as riding the bus, renting an electric bike, or walking.
Ask students whether they would prefer to take a bus or walk. Would a trip in which they take the bus halfway and walk the rest be as good as one in which they ride the bus 80% of the way and walk the rest?
Display the table, and give students 1–2 minutes to notice and wonder about what they see. Then, invite students to share their observations and questions.
Tell students, “Jada has assigned values to her level of enjoyment with different modes of transit. Walking is set at a baseline level of 1, and the other methods are given values relative to the baseline. For example, Jada enjoys riding a bike twice as much as walking and being in a bus half as much as walking.”
Ensure that students understand that greater numbers mean that those methods are preferred over methods with lower numbers.
Activity
NoneComplete the table with your own relative values, giving walk the same baseline value of 1.
| Jada's enjoyment per mi | your enjoyment per mi | |
|---|---|---|
| bus | 0.5 | |
| train | 1.5 | |
| bike rental | 2 | |
| scooter rental | 0.8 | |
| walk | 1 | 1 |
| car | 1.3 |
Student Response
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Building on Student Thinking
Activity Synthesis
Display a table with additional columns, and invite 2–4 students to share their responses. Clarify their understanding by asking questions such as:
- “Why do you think this student gave this mode of transit a value greater than 1?” (They like using that mode of transit more than walking.)
- “Did anyone use a value less than 1? What does that mean?” (Yes. It means that they like walking more than that mode of transit.)
- “Could you rank Jada’s favorite methods of transit from least favorite to most?” (Yes. She likes riding the bus least, then riding a scooter a little more, then walking, then taking a car, then the train, and likes riding a bike the most.)
- “What considerations did you use to rank your enjoyment riding a bus?” (On the one hand, it is better for the environment to use public transit, and it is easy to ride without having to pay attention. On the other hand, it doesn’t go directly where I want, and I’d prefer not to be around so many people.)
9.2
Activity
Materials
NoneActivity Narrative
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 prompts them to reason quantitatively and abstractly (MP2).
This activity uses the Three Reads math language routine to support reading comprehension and sense making.
Launch
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the diagram, without revealing the questions.
- For the first read, read the problem aloud, and then ask, “What is this situation about?” (transport to a destination). 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 (different transportation methods, cost, time, emissions, enjoyment).
- After the third read, reveal the question and Tyler’s inequalities, and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referring to quantities from the second read (compare the values listed in the inequalities to values given in the table).
To help students interpret the variables in the given inequalities as representing the number of miles traveled on each mode of transit, ask them to use the table to write an expression to represent the total emissions if they went miles on the train and miles on a scooter. Students should see that the expression is .
Ask for an expression representing the total enjoyment score Tyler would have for the same distances on those modes of transit ().
Arrange students in groups of 2. Ask them to analyze and answer the questions about one student’s trip (either Tyler’s or Jada’s). If time permits, the groups could analyze the other trip.
Give students a few minutes of quiet work time and time to share their thinking with their partner. Follow with a whole-class discussion.
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, highlight values from the table, equations, and graphs in corresponding colors as they discover the connections. Encourage students to label regions of the graph and label variables in addition to color coding to reinforce connections.
Supports accessibility for: Visual-Spatial Processing
Supports accessibility for: Visual-Spatial Processing
9.3
Activity
Materials
Activity Narrative
This activity is designed to give students opportunities to use their understandings from this unit to perform mathematical modeling.
The travel 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 because the 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).
Lesson Synthesis
Allow enough time for students to present their trips. Consider a gallery walk as a way for students to share their display and to ask and answer questions.