# MA.7.AR.4.4

Given any representation of a proportional relationship, translate the representation to a written description, table or equation.

### Examples

Example: The written description, there are 60 minutes in 1 hour, can be represented as the equation m=60h.

Example: Gina works as a babysitter and earns \$9 per hour. She would like to earn \$100 to buy a new tennis racket. Gina wants to know how many hours she needs to work. She can use the equation , where e is the amount of money earned, h is the number of hours worked and is the constant of proportionality.

### Clarifications

Clarification 1: Given representations are limited to a written description, graph, table or equation.

Clarification 2: Instruction includes equations of proportional relationships in the form of y=px, where p is the constant of proportionality.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Constant of Proportionality
• Proportional Relationships
• Rate
• Unit Rates

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students translated written descriptions into algebraic expressions and translate algebraic expressions into written descriptions. In grade 7, students translate any representation of a proportional relationship to a written description, table or equation. In grade 8, students will extend this work to include linear relationships.
• Instruction includes different ways of representing proportional relationships, such as tables, equations 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. The equations generated with the ratios will be unique in that they will follow the form of $y$ = $p$$x$.
• Tables

• Equations
$y$ = $\frac{\text{1}}{\text{5}}$$x$
• Graphs

• When providing a graph, be sure there are easily identifiable points for students to use in calculating the constant of proportionality.
• As students are building meaning, instruction makes connections between the different representations.
• Even though proportional relationships exist in Quadrant I, instruction includes opportunities for students to realize that the line does continue into Quadrant III but are not appropriate for the real-world situation.
• Students should be able to explain examples from the points on a graph or the numbers within the table by putting it back into the real-world context when appropriate.
• Instruction includes flexibility in understanding of the dependent and independent variables. Students can represent situations in terms of $x$ or in terms of $y$.
• For instance, within example 1 students can represent the situation as $m$ = 60$h$ or $h$ = $\frac{\text{1}}{\text{60}}$$m$.
• Students should construct verbal descriptions.
• For example, a student might describe the situation as “the number of packs of gum times the cost for each pack is the total cost in dollars.” They can use the verbal model to construct the equation.
• Students can check the equation by substituting values and comparing their results to the table. The checking process helps students revise and recheck their model as necessary (MTR.6.1).
• Provide tables of values for various proportional relationships. Ask students to look at the tables and generalize how they can find the $y$-value in the tables given any $x$-value (MTR.1.1). Have students look for patterns and assist with developing the equation $y$ = $p$$x$ where $p$ is the constant of proportionality (MTR.5.1).
• Ensure the formal development of the equation $y$ = $p$$x$ where $p$ is the constant of proportionality. Instruction supports flexibility in the variable used for the constant of proportionality. Provide practice for students to develop this equation using different variables based on given scenarios, as in Example 2.

### Common Misconceptions or Errors

• Students may neglect the scales on the axes when calculating and interpreting the constant of proportionality.
• Students may not be able to approximate the constant of proportionality from the graphs. To address this misconception, begin with graphs having easily identifiable points before moving toward problems that need approximations.
• Students may not see the connection between the constant of proportionality and the steepness of the graph. To address this misconception, provide a variety of graphs with various steepness and ask students to organize them based on increasing order of the constants of proportionality.

### Strategies to Support Tiered Instruction

• Instruction includes utilizing the $x$-and $y$-axis when determining the constant of proportionality. Teacher provides instruction on locating the values for the variables $y$ and $x$ from the axis labels, rather than counting the minor gridlines to the chosen point on the graph.
• Instruction includes utilizing graphs containing easily identifiable points on minor gridlines before moving toward graphs containing points that lie between gridlines which requires estimation to determine an appropriate constant of proportionality.
• Instruction includes the co-creation of a graphic organizer containing examples of proportional relationships with increasing levels of steepness. For each example, include a real-world scenario, a table, a graph and the constant of proportionality.
• Instruction includes using letters for variables that relate to the given scenario, such as $w$ for water.
• For students that are not able to approximate the constant of proportionality from the graphs, begin with graphs having easily identifiable points before moving toward problems that need approximations. For students that cannot see the connection between the constant of proportionality and the steepness of the graph, provide a variety of graphs with various steepness and ask students to organize them based on increasing order of the constants of proportionality.

Instructional Task 1 (MTR.5.1)
Kell works at an after-school program at an elementary school. The table below shows how much money he earned every day last week.

Mariko has a job mowing lawns that pays \$7 per hour.
• Part A. Who would make more money for working 10 hours? Explain or show work.
• Part B. Draw a graph that represents $y$, the amount of money Kell would make for working $x$ hours, assuming he made the same hourly rate he was making last week.
• Part C. Using the same coordinate axes, draw a graph that represents $y$, the amount of money Mariko would make for working $x$ hours.
• Part D. How can you see who makes more per hour just by looking at the graphs? Explain.
• Part E. Write one equation to represent the how much money Kell earns in $x$ hours and one equation to represent how much money Mariko earns in $h$ hours.

### Instructional Items

Instructional Item 1
Kelsi works as a lifeguard at the local pool. After an 8 hour day at work, she earns \$100.
• Part A. Write an equation that describes the relationship between the number of hours worked and the amount of money that she earns.
• Part B. Kelsi would like to earn \$450 to buy a new gaming system. Use your equation to determine how many hours she needs to work to buy a new gaming system.

*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.4: Given a table representation of a proportional relationship, translate the relationship into an equation or a graph.

## Related Resources

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

## Formative Assessments

Writing An Equation:

Students are asked to write an equation to represent a proportional relationship depicted in a graph.

Type: Formative Assessment

Serving Size:

Students are given the number of calories in a serving of oatmeal and are asked to write an equation that models the relationship between the size of the serving and the number of calories.

Type: Formative Assessment

## Lesson Plans

How Fast Can One Travel on a Bicycle?:

Students investigate how the pedal and rear wheel gears affect the speed of a bicycle. A GeoGebra sketch is included that allows a simulation of the turning of the pedal and the rear wheel. A key goal is to provide an experience for the students to apply and integrate the key concepts in seventh-grade mathematics in a familiar context.

Type: Lesson Plan

Cricket Songs:

Using a guided-inquiry model, students in a math or science class will use an experiment testing the effect of temperature on cricket chirping frequency to teach the concepts of representative vs random sampling, identifying directly proportional relationships, and highlight the differences between scientific theory and scientific law.

Type: Lesson Plan

The Speeding Ticket: Part 2 - Graphing Linear Functions:

This lesson allows the student to learn about dependent and independent variables and how to make the connection between the linear equation, a linear function, and its graph. The student will learn graphing relationships and how to identify linear functions.

Type: Lesson Plan

The Speeding Ticket (Part 1: Solving Linear Equations with One Variable):

"The Speeding Ticket" lesson uses real world application to create and solve linear equations and tables with one variable numerically, verbally, and algebraically. The student will also learn the relationship between the independent and dependent variables.

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

Art Class, Variation 2:

Giving the amount of paint in "parts" instead of a specific standardized unit like cups might be confusing to students who do not understand what this means. Because this is standard language in ratio problems, students need to be exposed to it, but teachers might need to explain the meaning if their students are encountering it for the first time.

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.

## MFAS Formative Assessments

Serving Size:

Students are given the number of calories in a serving of oatmeal and are asked to write an equation that models the relationship between the size of the serving and the number of calories.

Writing An Equation:

Students are asked to write an equation to represent a proportional relationship depicted in a graph.

## 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.

## Parent Resources

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