Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 55960
Finding Constant of Proportionality
Students are asked to determine the constant of proportionality using a table and a graph.
Subject(s): Mathematics
Grade Level(s): 7
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, rate, unit rate, constant of proportionality
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_FindingConstantOfProportionality_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problems on the Finding Constant of Proportionality worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not understand the concept of a constant of proportionality.
Examples of Student Work at this Level
The student:
Only examines values of one variable and concludes that there is no constant of proportionality.
page 1 of 4 Adds or subtracts corresponding boat lengths and fees.
Chooses one value from the table or number from the scale of the graph and reports it as the constant of proportionality.
Questions Eliciting Thinking
What is a constant of proportionality? How did you determine the constant?
When computing the constant, do you need to also consider the daily fee? Why or why not?
What does proportional mean? How do you determine proportionality?
What does constant mean?
Can you determine the unit rate in the table (or graph)?
Instructional Implications
Provide explicit instruction on proportionality and the multiplicative relationship between values of variables that are proportionally related. Introduce the concept of the
constant of proportionality and its role in describing the relationship between variables that are proportionally related. Describe proportionality as a relationship between two
variables in which the value of one variable is a constant factor of the associated value of the other variable. Point out that since there is a multiplicative relationship
between values of variables that are proportionally related, division can be used to determine that constant factor. Model how to determine the constant of proportionality
from a table of equivalent ratios and from a graph. Have the student generate a table of equivalent ratios and identify the constant factor. Provide the student with
additional opportunities to determine the constant of proportionality using both tables and graphs. Consider implementing this task again but with a new table and graph.
Moving Forward
Misconception/Error
The student is unable to correctly find the constant of proportionality in both the table and the graph.
Examples of Student Work at this Level
The student demonstrates an understanding of the constant of proportionality by expressing it as the ratio or quotient of the daily fee to boat length for at least one of
the problems. However, the student is unable to correctly calculate the constant for one or both problems
Additionally, the student is unable to correctly interpret the constant of proportionality.
Questions Eliciting Thinking
How were you attempting to calculate the constant of proportionality in this problem? Could you use this same approach in the other problem?
What is the relationship between each boat length, its docking fee, and the constant of proportionality?
How can you use the graph to find an example of a boat length and its associated docking fee?
Instructional Implications
Confirm that the student’s approach to finding the constant is correct (i.e., finding the quotient of a docking fee and its boat length). Assist the student with finding and
correcting any calculation errors. Then guide the student to use the same approach in the other problem. If needed, help the student identify boat lengths and their
associated docking fees from the graph.
Model the process of determining the constant of proportionality in a variety of real-world contexts. Describe the constant of proportionality as a unit rate and assist the
student in interpreting constants in the context of problems. Help the student understand the meaning of unit rates by including the units of measure in the quotient.
Then guide the student to use the language of ratios to interpret the constant (e.g., say, “for each additional foot of boat length, the daily fee is increased by $2.25”).
Almost There
page 2 of 4 Misconception/Error
The student correctly calculates each constant of proportionality but is unable to interpret their meaning in the problem.
Examples of Student Work at this Level
The student determines the constant of proportionality to be 2.25 in Port Canaveral and 2.50 in Fort Lauderdale. However, the student cannot correctly explain the
meaning of the constants in either problem even with teacher prompting.
Questions Eliciting Thinking
What is the unit of measure for boat length? What is the unit of measure for the daily fee? Can you determine the unit of measure for the constants you calculated?
If the constant is 2.5, what does that mean in this problem?
How did you decide at which marina it is less expensive to dock a boat?
Instructional Implications
Model the process of determining the constant of proportionality in a variety of real-world contexts. Describe the constant of proportionality as a unit rate and assist the
student in interpreting constants in the context of problems. Help the student understand the meaning of unit rates by including the units of measure in the quotient.
Then guide the student to use the language of ratios to interpret the constant (e.g., say, “for each additional foot of boat length, the daily fee is increased by $2.25”).
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student determines the constant of proportionality to be $2.25/foot in Port Canaveral and $2.50 /foot in Fort Lauderdale by dividing one of the daily fees by its
associated length. The student can explain the meaning of the constants in context and can determine that it is less expensive to dock a boat at Port Canaveral.
Questions Eliciting Thinking
What would you expect to get if you calculated the constant using a different pair of associated values from the table? From the graph?
What point on the graph includes the constant as one of its coordinates?
page 3 of 4 Can you write an equation that shows the relationship between the boat length and the daily fee?
How would you determine the fee for a 20
-foot boat? How about a 35-foot boat?
How did you decide which marina offers the better deal?
Would a 50-foot boat still be cheaper to dock in Port Canaveral? Is there any length boat that would be cheaper to dock in Fort Lauderdale?
If you were to extend the line in the graph, would it connect to the origin? Why or why not?
Instructional Implications
Give a verbal description of a proportional relationship and challenge the student to represent the relationship with a table, graph, and equation. Encourage the student to
identify the constant of proportionality and to describe its meaning in context.
Challenge the student to write an equation representing the proportional relationship displayed in the table and then another equation representing the proportional
relationship displayed in the graph. Consider implementing MFAS task Writing An Equation (7.RP.1.2).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Finding Constant of Proportionality worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.7.RP.1.2:
Description
Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or
graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of
proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of
items purchased at a constant price p, the relationship between the total cost and the number of items can be
expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special
attention to the points (0, 0) and (1, r) where r is the unit rate.
Remarks/Examples:
Examples of Opportunities for In-Depth Focus
Students in grade 7 grow in their ability to recognize, represent, and analyze proportional relationships in various
ways, including by using tables, graphs, and equations.
page 4 of 4