Unit 7, Lesson 9
Modeling with Systems of Inequalities in Two Variables
Integrated Math 1
9.1
Warm-up
10 mins
An Enjoyment Quotient
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
NoneStudent Task Statement
Complete 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
20 mins
Custom Trip
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
9.3
Activity
20 mins
Design Your Own Trip
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.