Part 1: Come up with and describe two scenarios: one that models a direct variation situation and one that models an inverse variation situation. Do not state explicitly which scenario is which, but provide at least four data pairs for each situation. Your classmates will have to determine which of the scenarios is a direct variation and which is an inverse variation, and the value of k for each.
Part 2: View and comment on the work of at least two other students. Determine which of the scenarios is a direct variation and which is an inverse variation, and calculate the value of k for each. Be sure to explain how you knew that the situation was either a direct or inverse variation as well as any potential problems you might see or commendations that are deserved.
Part 1: Come up with and describe two scenarios: one that models a direct variation situation and one that models an inverse variation situation. Do not state explicitly which scenario is which, but provide at least four data pairs for each situation. Your classmates will have to determine which of the scenarios is a direct variation and which is an inverse variation, and the value of k for each.
Part 2: View and comment on the work of at least two other students. Determine which of the scenarios is a direct variation and which is an inverse variation, and calculate the value of k for each. Be sure to explain how you knew that the situation was either a direct or inverse variation as well as any potential problems you might see or commendations that are deserved.
Scenario 1: The speed at which a car travels on a highway compared to the amount of time it takes to travel a certain distance.
Data pairs:
- Speed: 60 mph, Time: 2 hours
- Speed: 80 mph, Time: 1.5 hours
- Speed: 50 mph, Time: 3 hours
- Speed: 70 mph, Time: 2.5 hours
Scenario 2: The number of workers needed to complete a task compared to the amount of time it takes to complete the task.
Data pairs:
- Workers: 4, Time: 8 hours
- Workers: 5, Time: 6.4 hours
- Workers: 2, Time: 16 hours
- Workers: 8, Time: 4 hours
Part 2:
Student 1:
Scenario: "The amount of water used in a household compared to the number of people living in the household"
Data pairs:
- People: 2, Water usage: 500 gallons
- People: 4, Water usage: 1000 gallons
- People: 3, Water usage: 750 gallons
- People: 5, Water usage: 1250 gallons
I can tell that this is a direct variation because as the number of people in the household increases, so does the amount of water that is used. To find k, I can divide any of the pairs of data. For example:
500/2 = 250 = k
So the equation would be: water usage = 250 * number of people
Potential problem: It's important to note that there are other factors that could affect water usage beyond the number of people in a household, such as house size or water conservation practices.
Student 2:
Scenario: "The amount of money earned by a store compared to the number of customers that visit the store".
Data pairs:
- Customers: 200, Money earned: $4000
- Customers: 300, Money earned: $3000
- Customers: 100, Money earned: $6000
- Customers: 250, Money earned: $4500
I can tell that this is an inverse variation because as the number of customers in the store increases, the amount of money earned decreases. To find k, I can multiply any of the pairs of data. For example:
200 * 4000 = 800000 = k
So the equation would be: money earned = 800000/customers
Potential problem: There could be a variety of factors that affect the amount of money earned by a store beyond the number of customers, such as the prices of goods or overhead costs.
Scenario 1: Direct Variation
In this scenario, let's consider the relationship between the number of hours worked and the amount of money earned.
Data pairs:
1. (2 hours, $20)
2. (4 hours, $40)
3. (6 hours, $60)
4. (8 hours, $80)
Explanation: As the number of hours worked increases, the amount of money earned also increases proportionally. For every additional two hours of work, the amount earned also increases by $20. The value of k, which represents the constant of proportionality, is 10.
Scenario 2: Inverse Variation
In this scenario, let's consider the relationship between the speed of a car and the time it takes to complete a certain distance.
Data pairs:
1. (60 mph, 2 hours)
2. (30 mph, 4 hours)
3. (20 mph, 6 hours)
4. (15 mph, 8 hours)
Explanation: As the speed of the car decreases, the time taken to complete the distance increases. The product of speed and time remains constant for each data pair. For example, in the first data pair, the product is 120 miles (60 mph * 2 hours). The value of k for this scenario is 120.
Part 2:
Student A's scenario is a direct variation because as the number of hours worked increases, the amount of money earned also increases proportionally. The value of k would be the additional amount earned for each additional hour worked. Student B's scenario is an inverse variation because as the speed of the car decreases, the time taken to complete the distance increases. The value of k is the constant product of speed and time.
In Student A's work, the data pairs are consistent with a direct variation. However, it would have been helpful to provide a statement specifying the relationship being represented. In Student B's work, the data pairs are consistent with an inverse variation. The calculations for the product of speed and time are accurate. The commendation goes to both students for providing clear and consistent data pairs.
One potential problem in Student A's work is the lack of a specific statement identifying the nature of the relationship. This could have made it easier for the reader to determine whether it is a direct or inverse variation. In Student B's work, it would have been helpful to include units for the speed and distance to provide better clarity.
Overall, both students did a good job identifying direct and inverse variation scenarios and providing appropriate data pairs.
Part 1:
Scenario 1:
Let's consider the number of hours a person sleeps and the quality of their sleep. The data pairs are as follows:
Number of Hours Slept (x) | Quality of Sleep (y)
-----------------------------------------------
4 | 7
6 | 9
8 | 11
10 | 13
Scenario 2:
Now, let's consider the speed of a car and the time it takes to reach a particular destination. The data pairs are as follows:
Speed of Car (x) | Time Taken (y)
--------------------------------
30 | 10
50 | 6
70 | 4
90 | 3
Note that I have not indicated which scenario represents direct variation and which represents inverse variation.
Part 2:
Student 1:
Based on the provided scenarios, it seems that Scenario 1 represents direct variation and Scenario 2 represents inverse variation.
For Scenario 1, we can see that as the number of hours slept increases, the quality of sleep also increases. This indicates a direct relationship between the two variables. To find the value of k, we can choose any pair of data points and divide the y-value by the x-value. Let's take the first pair: (4, 7).
k = y/x = 7/4 = 1.75
Therefore, the value of k for Scenario 1 is 1.75.
For Scenario 2, we can observe that as the speed of the car increases, the time taken to reach the destination decreases. This indicates an inverse relationship between the two variables. To find the value of k, we can again choose any pair of data points and multiply the x-value by the y-value. Let's take the first pair: (30, 10).
k = x*y = 30*10 = 300
Therefore, the value of k for Scenario 2 is 300.
Student 2:
I also agree with Student 1's analysis.
In Scenario 1, the quality of sleep increases directly with the number of hours slept. This indicates a direct variation. The value of k is determined by dividing the y-value by the x-value, just like Student 1 explained. So, k is indeed 1.75.
In Scenario 2, the time taken to reach the destination decreases as the speed of the car increases. This suggests an inverse variation. The value of k is found by multiplying the x-value by the y-value, as Student 1 mentioned. So, k is indeed 300.
Overall, both students correctly identified the direct and inverse variation scenarios and calculated the values of k accurately. No commendations or problems arise from their explanations.
Note: It is important to review other students' explanations to gain a better understanding of the concept and check for any potential errors.