# MA.7.AR.4.2

Determine the constant of proportionality within a mathematical or real-world context given a table, graph or written description of a proportional relationship.

### Examples

Example: A graph has a line that goes through the origin and the point (5,2). This represents a proportional relationship and the constant of proportionality is .

Example: Gina works as a babysitter and earns \$9 per hour. She can only work 6 hours this week. Gina wants to know how much money she will make. Gina can use the equation e=9h, where e is the amount of money earned, h is the number of hours worked and 9 is the constant of proportionality.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Constant of Proportionality
• Origin

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students determined rates and unit rates in ratios. In grade 7, students take a broader view of a rate or unit rate as they understand it to be the constant of proportionality in a proportional relationship. In grade 8, students will expand their understanding of the constant of proportionality in proportional relationships to slope in linear relationships.
• Instruction includes different ways of representing proportional relationships, such as tables and graphs. Multiplying or dividing one quantity in a ratio by a particular factor requires doing the same with the other quantity in the ratio to maintain the proportional relationship. Graphing equivalent ratios create a straight line passing through the origin.
• Tables

• Graphs

• Starting instruction with real-world context rather than mathematical procedure allows students to reason through the meaning of the constant of proportionality (MTR.7.1). Connect prior knowledge of unit rates when developing the constant of proportionality.
• Instruction includes a connection to pi (π) as the constant of proportionality in the circumference formula within MA.7.GR.1.3.
• Problem types include positive and negative constants of proportionality.

### Common Misconceptions or Errors

• Some students reverse the order of the ratio between the two quantities in a proportional relationship.
• Students may neglect the scale(s) of the axes on a graph. To address this misconception, have students interpret the constant of proportionality in context and evaluate the reasonableness of the answer. This may prompt students to revisit the graphical representation for better details.
• Students may incorrectly believe any line represents a proportional relationship. To address this misconception, revisit the development of the equation $y$ = $p$$x$ and be sure students see all proportional relationships have the origin as a common point.

### Strategies to Support Tiered Instruction

• Instruction focuses on students’ comprehension of the context or situation by engaging in questions (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
• What do you know from the problem?
• What is the problem asking you to find?
• What are the two quantities in this problem?
• How are the quantities related to each other?
• When determining the constant of proportionality in a graph, the teacher can instruct students to interpret the coordinate point as it relates to the titles of each axis. Teacher and students can use this information to co-construct a table to clear up the misconception of misinterpreting the context of the constant of proportionality.
• Instruction includes the use of a three-read strategy. Students read the problem three different times, each with a different purpose (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
• First, read the problem with the purpose of answering the question: What is the problem, context, or story about?
• Second, read the problem with the purpose of answering the question: What are we trying to find out?
• Third, read the problem with the purpose of answering the question: What information is important in the problem?
• Teacher has students interpret the constant of proportionality in context and evaluate the reasonableness of the answer. This may prompt students to revisit the graphical representation for better details.
• Teacher revisits the development of the equation $y$ = $p$$x$ and be sure students see all proportional relationships have the origin as a common point.

Part A. The daily fee for docking a boat at a marina in Port Canaveral is proportional to the length of the boat. The table displays the fee for four different boat lengths. Find the constant of proportionality and explain what it means in the context of this problem.
Part B. The daily fee for docking a boat at a marina in Fort Lauderdale is also proportional to the length of the boat. The graph displays the relationship between the fee and the boat length. Find the constant of proportionality and explain what it means in the context of this problem.
Part C. At which marina is it less expensive to dock a boat? Explain how you determined your answer.

### Instructional Items

Instructional Item 1
After a workout at the gym, three friends made protein shakes to help in their recovery. Each protein shake contains 2 scoops of protein powder and 12 ounces of water. What is the constant of proportionality for this relationship?

Instructional Item 2
Determine the constant of proportionality for the following proportional relationships.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1205040: M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812020: Access M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.7.AR.4.AP.2: Identify the constant of proportionality when given a table or graph of a proportional relationship.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Identifying Constant of Proportionality in Equations:

Students are asked to identify and explain the constant of proportionality in three different equations.

Type: Formative Assessment

Constant of Proportionality Trip:

Students are asked to identify and explain the constant of proportionality given a verbal description and a diagram representing a proportional relationship.

Type: Formative Assessment

Babysitting Graph:

Students are given a graph that models the hourly earnings of a babysitter and are asked to interpret ordered pairs in context.

Type: Formative Assessment

Finding Constant of Proportionality:

Students are asked to determine the constant of proportionality using a table and a graph.

Type: Formative Assessment

## Lesson Plans

Beginning Linear Functions:

This lesson is designed to introduce students to the concept of slope. Students will be able to:

• determine positive, negative, zero, and undefined slopes by looking at graphed functions.
• determine x- and y-intercepts by substitution, or by examining graphs.
• write equations in slope-intercept form and make graphs based on slope/y-intercept of linear functions.

Type: Lesson Plan

Running and Rising:

In this lesson students will graph and compare two proportional relationships from different representations in contextual problems and be introduced to the constant of proportionality as the unit rate.

Type: Lesson Plan

Sir Cumference introduces Radius and Diameter:

This lesson is designed to be a fun and creative way to introduce math vocabulary (radius, diameter, and circumference) related to circles. Students will create a story board (comic strip) to retell or create a story using targeted vocabulary, and then demonstrate understanding of the relationships among radii, diameter, and circumference by completing the worksheet.

Type: Lesson Plan

Are My Values Proportional?:

Students will learn that a proportional relationship can be represented by a table, a graph, or an equation. They will also be able to determine the constant of proportionality from a table, graph, or equation.

Type: Lesson Plan

What's the Going Rate?:

Students discover that the unit rate and the slope of a line are the same, and these can be used to compare two different proportional relationships. Students compare proportional relationships presented in table and graph form.

Type: Lesson Plan

Who goes faster, earns more, drives farthest?:

Given a proportional relationship, students will determine the constant of proportionality, write an equation, graph the relationship, and interpret in context.

Type: Lesson Plan

Coffee by the Pound:

Students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of constant of proportionality/slope in the given context.

## Tutorial

Interpreting Graphs of Proportional Relationships:

This video shows how to recognize and understand graphs of proportional relationships to find the constant of proportionality.

Type: Tutorial

## MFAS Formative Assessments

Babysitting Graph:

Students are given a graph that models the hourly earnings of a babysitter and are asked to interpret ordered pairs in context.

Constant of Proportionality Trip:

Students are asked to identify and explain the constant of proportionality given a verbal description and a diagram representing a proportional relationship.

Finding Constant of Proportionality:

Students are asked to determine the constant of proportionality using a table and a graph.

Identifying Constant of Proportionality in Equations:

Students are asked to identify and explain the constant of proportionality in three different equations.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Coffee by the Pound:

Students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of constant of proportionality/slope in the given context.

## Tutorial

Interpreting Graphs of Proportional Relationships:

This video shows how to recognize and understand graphs of proportional relationships to find the constant of proportionality.

Type: Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Coffee by the Pound:

Students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of constant of proportionality/slope in the given context.