Answer :
To find the unit rate for a proportional relationship using a graph, it is essential to understand the precise meaning of unit rate. The unit rate in a proportional relationship is equivalent to the slope of the line representing the relationship. Here’s a step-by-step explanation to determine the valid method among the ones provided:
### Method 1
"Look at the graph of the relationship. Find the [tex]$y$[/tex]-value of the point that corresponds to [tex]$x=1$[/tex]. That value is the unit rate."
- Explanation: If the point at [tex]$x = 1$[/tex] is [tex]$(1, y)$[/tex] on the graph, then [tex]$y$[/tex] gives the unit rate. This approach works because if we have a proportional relationship [tex]$y = kx$[/tex], and when [tex]$x = 1$[/tex], then [tex]$y = k$[/tex]. Thus, the [tex]$y$[/tex]-value at [tex]$x = 1$[/tex] directly reveals the constant of proportionality (unit rate).
### Method 2
"Look at the graph of the relationship. Count the number of units up and the number of units to the right one must move to arrive at the next point on the graph. Write these two numbers as a fraction."
- Explanation: This method involves counting the rise ([tex]\(\Delta y\)[/tex]) and run ([tex]\(\Delta x\)[/tex]) between two points on the graph. The slope of the line (which represents the unit rate in a proportional relationship) is [tex]\(\frac{\Delta y}{\Delta x}\)[/tex]. In other words, the unit rate is the ratio of the vertical change to the horizontal change between any two points on the line.
### Method 3
"Look at the graph of the relationship. Find the [tex]$x$[/tex]-value of the point that corresponds to [tex]$y=2$[/tex]. That value is the unit rate."
- Explanation: This method is incorrect for determining the unit rate. Finding the [tex]$x$[/tex]-value that corresponds to [tex]$y = 2$[/tex] doesn’t necessarily give useful information about the unit rate unless we assume [tex]$x$[/tex] is proportional to [tex]$y$[/tex] with a specific constant, which is not typically the case.
### Method 4
"Look at the graph of the relationship. Find two points which have [tex]$y$[/tex]-values that are one unit apart. The unit rate is the difference in the corresponding [tex]$x$[/tex]-values."
- Explanation: This method is also incorrect. Finding the difference in [tex]$x$[/tex]-values corresponding to a unit change in [tex]$y$[/tex] does not reflect the unit rate of the proportional relationship. Instead, it should be the other way around: finding how much [tex]$y$[/tex] changes for a unit increase in [tex]$x$[/tex].
### Valid Method
Between all the methods described:
- Method 1 and Method 2 are valid methods.
- Method 1 finds the [tex]$y$[/tex]-value at [tex]$x = 1$[/tex], directly giving the unit rate.
- Method 2 determines the unit rate by calculating the slope between two points on the graph.
Thus, students can employ either Method 1 or Method 2 to correctly determine the unit rate for a proportional relationship.
### Method 1
"Look at the graph of the relationship. Find the [tex]$y$[/tex]-value of the point that corresponds to [tex]$x=1$[/tex]. That value is the unit rate."
- Explanation: If the point at [tex]$x = 1$[/tex] is [tex]$(1, y)$[/tex] on the graph, then [tex]$y$[/tex] gives the unit rate. This approach works because if we have a proportional relationship [tex]$y = kx$[/tex], and when [tex]$x = 1$[/tex], then [tex]$y = k$[/tex]. Thus, the [tex]$y$[/tex]-value at [tex]$x = 1$[/tex] directly reveals the constant of proportionality (unit rate).
### Method 2
"Look at the graph of the relationship. Count the number of units up and the number of units to the right one must move to arrive at the next point on the graph. Write these two numbers as a fraction."
- Explanation: This method involves counting the rise ([tex]\(\Delta y\)[/tex]) and run ([tex]\(\Delta x\)[/tex]) between two points on the graph. The slope of the line (which represents the unit rate in a proportional relationship) is [tex]\(\frac{\Delta y}{\Delta x}\)[/tex]. In other words, the unit rate is the ratio of the vertical change to the horizontal change between any two points on the line.
### Method 3
"Look at the graph of the relationship. Find the [tex]$x$[/tex]-value of the point that corresponds to [tex]$y=2$[/tex]. That value is the unit rate."
- Explanation: This method is incorrect for determining the unit rate. Finding the [tex]$x$[/tex]-value that corresponds to [tex]$y = 2$[/tex] doesn’t necessarily give useful information about the unit rate unless we assume [tex]$x$[/tex] is proportional to [tex]$y$[/tex] with a specific constant, which is not typically the case.
### Method 4
"Look at the graph of the relationship. Find two points which have [tex]$y$[/tex]-values that are one unit apart. The unit rate is the difference in the corresponding [tex]$x$[/tex]-values."
- Explanation: This method is also incorrect. Finding the difference in [tex]$x$[/tex]-values corresponding to a unit change in [tex]$y$[/tex] does not reflect the unit rate of the proportional relationship. Instead, it should be the other way around: finding how much [tex]$y$[/tex] changes for a unit increase in [tex]$x$[/tex].
### Valid Method
Between all the methods described:
- Method 1 and Method 2 are valid methods.
- Method 1 finds the [tex]$y$[/tex]-value at [tex]$x = 1$[/tex], directly giving the unit rate.
- Method 2 determines the unit rate by calculating the slope between two points on the graph.
Thus, students can employ either Method 1 or Method 2 to correctly determine the unit rate for a proportional relationship.
