Understanding Proportional Relationships
This lesson is the first of four where students work with proportional relationships from a grade 8 perspective. Embedded alongside their work with proportional relationships, students learn about graphing from a blank set of axes. Attending to precision in labeling axes, choosing an appropriate scale, and drawing lines are skills students work with in this lesson and refine over the course of this unit and in units that follow (MP6).
The purpose of this lesson is to get students thinking about what makes a “good” graph by first considering what are the components of a graph (e.g., labels, scale) and then adding scale to graphs of the pace of two bugs. Students also graph a line based on a verbal description of a relationship and compare the newly graphed line to already graphed proportional relationships.
This lesson includes graphs with elapsed time in seconds on the vertical axis and distance traveled in centimeters on the horizontal axis. It is common for people to believe that time is always the independent variable, and should therefore always be on the horizontal axis. This is a really powerful heuristic. The problem is, it isn’t true.
In general, a context that involves a relationship between two quantities does not dictate which quantity is the independent variable and which is the dependent variable: that is a choice made by the modeler. Consider this situation: A runner is traveling one mile every 10 minutes. There is more than one way to represent this situation.
- We can say the number of miles traveled, \(d\), depends on the number of minutes that have passed, \(t\), and write \(d = 0.1t\). This way of expressing the relationship might be more useful for questions like, "How far does the runner travel in 35 minutes?"
- We can also say that the number of minutes that have passed, \(t\), depends on the number of miles traveled, \(d\), and write \(t = 10d\). This way of expressing the relationship might be more useful for questions like, "How long does it take the runner to travel 2 miles?"
These are both linear relationships. The rate of change in the first corresponds to speed (0.1 miles per minute), and the rate of change in the second corresponds to pace (10 minutes per mile). Both have meaning, and both could be of interest. It is up to the modeler to decide what kinds of questions she wants to answer about the context and which way of expressing the relationship will be most useful in answering those questions.
- Comprehend that for the equation of a proportional relationship given by $y=kx$, $k$ represents the constant of proportionality.
- Create graphs and equations of proportional relationships in context, including an appropriate scale.
- Interpret diagrams or graphs of proportional relationships in context.
Let’s study some graphs.
- I can graph a proportional relationship from a story.
- I can use the constant of proportionality to compare the pace of different animals.
constant of proportionality
In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.
In this example, the constant of proportionality is 3, because \(2 \boldcdot 3 = 6\), \(3 \boldcdot 3 = 9\), and \(5 \boldcdot 3 = 15\). This means that there are 3 apples for every 1 orange in the fruit salad.
number of oranges number of apples 2 6 3 9 5 15
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