Unit 2, Lesson 9
Which Variable to Solve for? (Part 2)
Algebra 1
9.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this Warm-up, students are given a simple equation in three variables and are prompted to rearrange it to solve for a particular variable. In each question, only the value of one variable is given, so students need to manipulate the equation even when some quantities are unknown. The work here prepares students to rearrange other variable equations later in the lesson.
As students work, look for those who substitute the given value before rearranging and those who first isolate the variable of interest before substituting. Invite them to share their approaches during class discussion.
Launch
NoneActivity
NoneIn an earlier lesson, you saw the equation , which relates the number of vertices, faces, and edges in a Platonic solid.
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Write an equation that makes it easier to find the number of vertices in each of the Platonic solids described:
- An octahedron (shown here), which has 8 faces.
- An icosahedron, which has 30 edges.
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A Buckminsterfullerene (also called a “Buckyball”) is a polyhedron with 60 vertices. It is not a Platonic solid, but the numbers of faces, edges, and vertices are related the same way as those in a Platonic solid.
Write an equation that makes it easier to find the number of faces that a Buckyball has if we know how many edges it has.
Student Response
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Building on Student Thinking
Some students may be unclear what it means to write an equation that "makes it easier to find the number of vertices (or faces)." Remind them of the work in an earlier activity. In the post-parade clean-up activity, for instance, they wrote the equation to quickly find the length of the road section that each volunteer would clean up, , if there were volunteers. They wrote to quickly find the number of volunteers, , if each volunteer were to clean up miles.
Activity Synthesis
Select previously identified students to share their responses and strategies. Record and display for all to see the steps they take to rearrange the equations. Emphasize how each step constitutes an acceptable move and how it keeps the equation true.
Make sure students see that we can either substitute known values into the given equation before rearranging it, or we can rearrange the equation first before substituting known values. In the examples here, it doesn't matter which way it is done. Ultimately, we were solving for in the first question and for in the second question.
Explain that there will be times when one strategy might be more helpful than the other, as students will see in subsequent activities.
9.2
Activity
Instructional Routines
Aspects of Mathematical ModelingMaterials
Activity Narrative
This activity encourages students to write an equation in two variables to represent a constraint and then to solve for each of the variables. One motivation for rearranging the equation is to find an expression that, when entered into a calculator or computer, can then be used to quickly find the value of one quantity given the value of the other.
The digital version of this activity includes instruction for using GeoGebra spreadsheets to perform calculations. As they use spreadsheet technology to create mathematical models, test them, and solve problems, students engage in aspects of mathematical modeling (MP4).
Launch
Arrange students in groups of 3–4 and provide access to devices with spreadsheet or graphing technology.
Give students a few minutes of quiet time to think about the first question and then time to discuss their responses with their group. Ask students to pause for a class discussion, and ensure that everyone is using a correct equation before proceeding.
When students have responses for the second question, invite them to share how they found the number of cars that can be shipped if the cargo already has some number of trucks. Next, ask for the expressions they wrote to find the number of cars that can fit if there are trucks. Record the expressions for all to see.
Tell students that we can test the expressions by using a calculator or a computer. Consider demonstrating how to use technology to calculate the number of cars given the number of trucks. For examples that use a table, sliders, and a spreadsheet, see the digital version of this activity. You may need to prepare alternate instructions if using different technology in the classroom.
Tell students that their job in the last question is to find the number of trucks when the number of cars is known. Encourage them to test their equations using available technology.
Action and Expression: Provide Access for Physical Action. Support effective and efficient use of tools and assistive technologies. To use spreadsheets or graphing technology, some students may benefit from a physical copy of written directions. Include step-by-step directions for how to enter equations, find the value of one variable by entering the value of another, and test equations.
Supports accessibility for: Organization, Memory, Attention
Supports accessibility for: Organization, Memory, Attention
Activity
NoneAn automobile manufacturer is preparing a shipment of cars and trucks on a cargo ship that can carry 21,600 tons.
The cars weigh 3.6 tons each, and the trucks weigh 7.5 tons each.
- Write an equation that represents the weight constraint of a shipment. Let be the number of cars and be the number of trucks.
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For one shipment, trucks are loaded first, and cars are loaded afterward. (Even though trucks are bulkier than cars, a shipment can consist of all trucks as long as it is within the weight limit.)
Find the number of cars that can be shipped if the cargo already has:
- 480 trucks
- 1,500 trucks
- 2,736 trucks
- trucks
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For a different shipment, cars are loaded first, and then trucks are loaded afterward.
- Write an equation that you could enter into a calculator or a spreadsheet tool to find the number of trucks that can be shipped if the number of cars is known.
- Use your equation and a calculator or a computer to find the number of trucks that can be shipped if the cargo already has 1,000 cars. What if the cargo already has 4,250 cars?
Student Response
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Building on Student Thinking
Activity Synthesis
When finding the number of cars, , given trucks, students may arrive at the equation by generalizing the calculation that they performed when the number of trucks was a numerical value. Likewise, in the last question they may arrive at by using some numerical values for and generalizing the process.
While this strategy is expected and perfectly reasonable, make sure that students also see that we can arrive at the same expression for and for by rearranging the equation .
Highlight that we can solve for when we know the number of trucks and want to compute the number of cars, and solve for when we know the number of cars and want to find the number of trucks.
9.3
Activity
Materials
NoneActivity Narrative
This activity gives students another opportunity to write and rearrange equations in two variables and to do so in context.
Unlike in previous activities, students no longer start by computing one quantity given numerical values of the other quantity and then generalizing the process. Instead, they are prompted to write one equation, solve for each variable, and interpret the solution. Students also articulate why one model might be more helpful than the other under a certain circumstance. Along the way, students reason quantitatively and abstractly (MP2) and engage in aspects of modeling (MP4).
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
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 image, without revealing the questions.
- For the first read, read the problem aloud then ask, “What is this situation about?” (A city department has money to fix roads this year.) 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 (amount of money available, cost to resurface 1 mile of a 2-lane road, average worker’s salary, number of miles of road to resurface).
- After the third read, reveal the first question 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. (Find how much it will cost to pay the workers and how much it will cost to resurface the roads, then add them together to equal the budget.)
Representation: Access for Perception. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content.
Supports accessibility for: Language
Supports accessibility for: Language
Activity
NoneThe Department of Streets of a city has a budget of \$1,962,800 for resurfacing roads and hiring additional workers this year.
The cost of resurfacing a mile of 2-lane road is estimated at \$84,000. The average starting salary of a worker in the department is \$36,000 a year.
- Write an equation that represents the relationship between the miles of 2-lane roads the department could resurface, , and the number of new workers it could hire, , if it spends the entire budget.
- Take the equation you wrote in the first question and:
- Solve for . Explain what the solution represents in this situation.
- Solve for . Explain what the solution represents in this situation.
- The city is planning to hire 6 new workers and to use its entire budget.
- Which equation should be used to find out how many miles of 2-lane roads it could resurface? Explain your reasoning.
- Find the number of miles of 2-lane roads the city could resurface if it hires 6 new workers.
Student Response
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Building on Student Thinking
Some students may have trouble sorting out the given information because of unfamiliarity with the context or with certain terms. Ask them to explain the setup of the problem as best they understand it, and then point out any information that might be missing.
Activity Synthesis
Verify that the equations students wrote do correctly isolate each variable. If students wrote expressions in different forms for a variable (for instance, and for ), discuss how the expressions are equivalent.
If time permits, consider asking students to use their equations to answer these questions:
- "How many people could be hired if the department is resurfacing 16 miles of roads?" (17 people) "3 miles?" (47 people)
- "How many miles of roads could be resurfaced if the department is hiring 2 new workers?" (22.5 miles) "If no new workers are hired?" (About 23.37 miles)
Make sure students understand that solving for makes it possible to quickly find the number of people that the department could hire given some miles of road to be resurfaced (and while sticking to the budget). Similarly, solving for makes it possible to quickly find the miles that could be resurfaced given any number of new hires.