Constant of Proportionality Trip

Getting Started

The student is unable to correctly identify one or both constants of proportionality.

The student:

• Understands that the relationship is proportional but does not understand the concept of a constant of proportionality.
  
  
  
   
• Can find the constant of proportionality from a double number line diagram but does not understand how to find it from a verbal description of the two variables.
  
  
  
   
• Can find the constant of proportionality from a verbal description of the two variables but does not understand how to find it from a double number line diagram.
  
  
  
   
• Attempts an incorrect calculation or chooses an incorrect value from the diagram.
  
  
  
   

What does it mean for two variables to be proportionally related?

What is a constant of proportionality? How is it related to variables that are proportional?

Could you find the unit rate? How are the constant of proportionality and the unit rate related?

Can you explain what the double number line diagram is showing?

Review what it means for two variables to be proportionally related. Introduce the concept of the constant of proportionality and its role in describing the relationship between variables that are proportionally related. Help the student recognize that if there is a proportional relationship between the variables, then there is a constant factor that relates the pairs of associated values. In other words, the value of one variable can always be found by multiplying the value of the other variable by the constant of proportionality. Demonstrate this by creating a table of values for one of the proportional relationships in this task. Then write an equation that models the relationship between the two variables. Show the student that proportional relationships can be modeled by equations of the form y = cx where c is the constant of proportionality. Provide additional opportunities for the student to find and interpret the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

Relate the constant of proportionality to the concept of a unit rate. Explain that the number of miles driven after 1 hour is the same as the constant of proportionality. Guide the student to calculate or identify the unit rate in each scenario and use his or her understanding of unit rate to interpret the meaning of the constant of proportionality.

If needed, explain the double number line diagram and how it is used to represent proportional relationships. Assist the student in determining the constant of proportionality first by identifying the number that values on the “Hours” axis can be multiplied by to get the corresponding values on the “Miles” axis. Then guide the student to identify the constant by identifying the number of miles associated with 1 hour.

Making Progress

The student is able to identify the constants of proportionality but cannot explain their meaning in context.

The student correctly identifies the constant of proportionality in each problem but:

• Does not attempt an explanation. 
  
  
  
   
• Is only able to provide an explanation of one of the constants.
  
  
  
   
• Provides an incorrect explanation. 
  
  
  
   

You identified the constant of proportionality correctly. How did you determine your answer?

Is the constant of proportionality the same as the unit rate? What is a unit rate?

What does the constant of proportionality tell you about the relationship between the variables?

Assist the student in using the context of the proportional relationship to explain the relationship between the variables in terms of the constant of proportionality. Model an explanation of the constant of proportionality for the first problem and ask the student to develop an explanation for the second problem. Relate the constant of proportionality to a unit rate and guide the student to explain its meaning in terms of the variables and their units of measure (e.g., 340 miles/5 hours = 68 miles per hour which means that the number of hours that Geoffrey traveled can be multiplied by 68 to find the distance he has travelled). Provide additional opportunities for the student to find and interpret the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

Got It

The student provides complete and correct responses to all components of the task.

The student correctly identifies each constant of proportionality, 68 in the first problem and 62 in the second problem. The student explains that in each problem, it is:

• The factor by which you multiply the number of hours travelled to get the distance travelled or
• The distance that can be traveled in one hour on average or when traveling at a constant rate.
  
  
  

How is the constant of proportionality related to a unit rate? Can you write a unit rate for each of these proportional relationships?

Where can you find the constant of proportionality on the double number line diagram?

Ask the student to write equations for each proportional relationship and graph the equations on the same set of axes. Challenge the student to compare the equations and their graphs by identifying both similarities and differences.