standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: Activity
This activity provides the first example in this unit of a relationship that is not proportional. The second question focuses students’ attention on the unit rates. If the relationship were proportional then regardless of the number of people in a vehicle, the cost per person would be the same. The question about the bus is to show students that they can’t just scale up from 10. Students who write an equation also see that it is not of the form
\(y = kx\)
. In a later lesson students will learn that only equations of this form represent proportional relationships.
Monitor for students who approached this problem using different representations.
Launch
Keep students in the same groups of 2. Give 5 minutes of quiet work time, followed by 5 minutes of students discussing responses with a partner, followed by a whole-class discussion.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students have accounted for the cost of the vehicle in their calculations of the total entrance cost for 4 people and 10 people.
Supports accessibility for: Memory; Organization
Student Facing
Entrance to a state park costs
$
6 per vehicle, plus
$
2 per person in the vehicle.
How much would it cost for a car with 2 people to enter the park? 4 people? 10 people? Record your answers in the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of", "people in vehicle"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["total entrance cost", "in dollars"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "2"}, {}]}, {"td": [{"_value": "4"}, {}]}, {"td": [{"_value": "10"}, {}]}]}]}]}
For each row in the table, if each person in the vehicle splits the entrance cost equally, how much will each person pay?
How might you determine the entrance cost for a bus with 50 people?
Is the relationship between the number of people and the total entrance cost a proportional relationship? Explain how you know.
Student Response
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Student Facing
Are you ready for more?
What equation could you use to find the total entrance cost for a vehicle with any number of people?
Student Response
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Anticipated Misconceptions
Some students may not account for the cost of the vehicle. They will get the following table with incorrect values and will need to be prompted to include the cost of the vehicle.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of", "people in vehicle"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["total entrance cost", "in dollars"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "2"}, {"_value": "\\$10"}]}, {"td": [{"_value": "4"}, {"_value": "\\$20"}]}, {"td": [{"_value": "10"}, {"_value": "\\$50"}]}]}]}]}
Teachers will want to circulate around the room keeping an eye out for this mistake and address it as soon as possible so that students spend most of their work time analyzing the non-proportional relationship. These diagrams may be helpful in illustrating to them that their resulting prices are including more than one vehicle. This gives them an opportunity to make sense of problems and persevere in solving them (MP1).
[IMG]
Activity Synthesis
Select students to explain why they think the relationship is or is not proportional. Reasons they may give:
The cost per person is different for different number of people in a vehicle, i.e. the quotients of the entries in each row are not equal for all rows of the table.
The ratio of people in the vehicle to total entrance cost are not equivalent ratios. You can't just multiply the entries in one row by the same constant to get the entries in another row.
Each number of people and corresponding total entrance cost is not characterized by the same unit rate. You can't multiply the entries in the first column by the same number (constant of proportionality) to get the numbers in the second column.
Students who found an equation will also note that the equation is not of the same form as other equations, but they can’t use this as a criterion until the class has established that only equations of this form represent proportional relationships. (This part of the discussion should come at the end of the next lesson, after students have analyzed lots of different equations.)


aligns: no