### Examples

Gordy is taking a trip from Tallahassee, FL to Portland, Maine which is about 1,407 miles. On average his SUV gets 23.1 miles per gallon on the highway and his gas tanks holds 17.5 gallons. If Gordy starts with a full tank of gas, how many times will he be required to fill the gas tank?**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**7

**Strand:**Algebraic Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Proportional Relationships
- Rate
- Unit Rates

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students solved mathematical and real-world problems involving ratios, rates and unit rates, including comparisons, mixtures, ratios of lengths and conversions within the same measurement system. In grade 7, students solve real-world problems involving proportional relationships. In grade 8, students will solve real-world problems involving linear relationships.- This benchmark is a culmination of the work students have been doing throughout MA.7.AR.4.
- Instruction for this benchmark includes opportunities to compare two different proportional relationships to each other.
- Allow various methods for solving, encouraging discussion and analysis of efficient and effective solutions
*(MTR.4.1).*

### Common Misconceptions or Errors

- Students may confuse the dependent and independent variables when graphing. To address this conception, instruction includes the understanding that the independent variable depends on the given context. Additionally, independent variables are not always on the $x$-axis and the dependent variables are not always on the $y$-axis.
- For example, if one has a proportional relationship between feet and meters, they can graph feet either on the $x$-axis or the $y$-axis. Which one that is dependent depends on the context. For instance, if one is given feet and converting to meters, then feet would be independent and meters would be dependent.

### Strategies to Support Tiered Instruction

- Teacher provides opportunities for students to comprehend the context or situation by engaging in questions.
- 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?
- Which quantity do you want to consider as the independent variable?
- Which quantity do you want to consider as the dependent variable?

- Instruction includes the use of a three-read strategy. Students read the problem three different times, each with a different purpose.
- 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?

- Instruction includes the understanding that the independent variable depends on the given context. Additionally, independent variables are not always the $x$-axis and the dependent variable are not always the $y$-axis.
- For example, if one has a proportional relationship between feet and meters, they can graph feet either on the $x$-axis or the $y$-axis. Which one that is dependent depends on the context. For instance, if one is given feet and converting to meters, then feet would be independent and meters would be dependent.

### Instructional Tasks

*Instructional Task 1*

**(MTR.4.1)**Patsy is making shortbread cookies using the ingredients below.

- Part A. This recipe makes 16 cookies, but Patsy needs 5 dozen. How much of each ingredient will she need to make the 5 dozen cookies she needs?
- Part B. Once Harrison tasted Patsy’s shortbread cookies, he ordered 7 dozen for a birthday party. If Patsy originally started with 4 cups of flour, 2 cups of powdered sugar and 16 tablespoons of butter, how much more (if any) will she need of each ingredient to complete Harrison’s order?
- Part C. After the party, Jeb shared his recipe which calls for 2 cups of flour and 1$\frac{\text{3}}{\text{4}}$ cup of powdered sugar. Since adding powdered sugar to cookies should make them sweeter, Jeb claims his larger ratio of powdered sugar to flour will produce sweeter cookies. Is this statement correct?

### Instructional Items

*Instructional Item 1*

A couple is taking a horse and carriage ride through Central Park in New York City. After 8 minutes, they had traveled $\frac{\text{1}}{\text{2}}$ mile.

- Part A. Create a graph to represent the proportional relationship between miles traveled and the number of minutes they are on the carriage.
- Part B. Use this graph to determine how long will it take to complete the 2.5 mile ride around the park.

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

## Related Courses

## Related Access Points

## Related Resources

## Lesson Plans

## Perspectives Video: Experts

## Perspectives Video: Professional/Enthusiasts

## Perspectives Video: Teaching Idea

## Problem-Solving Tasks

## Tutorials

## STEM Lessons - Model Eliciting Activity

In this Model Eliciting Activity, MEA, students will help analyze, compare, and select fencing options for a college’s pool and recreation area. Students will use unit conversions, calculate total costs, and justify their recommendations to develop problem solving and critical thinking skills within a purchasing context.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

In this Model Eliciting Activity, MEA, students are asked to rank fabrics designated for a new women's volleyball team. Students will use proportional reasoning, percentages, and conversions to analyze and compare fabrics to support their rankings.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

In this Model Eliciting Activity, MEA, students will use unit rates and scoring systems to analyze and interpret data to recommend the best store from which a family restaurant should purchase its weekly non-frozen food items.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx

In this Model Eliciting Activity, MEA, students will interpret and analyze a scale drawing to provide possible routes from a teacher’s home to the school. Students will consider factors including traffic patterns, construction zones, and wait times to recommend the best route including the total distance, in miles, and estimated delay times.

In this Model Eliciting Activity (MEA), students will analyze data related to tiling rooms in a house. Students will calculate the square footage of various rooms, convert measurements to determine the amount of tile needed, and compute both the cost of the tiles and the cost of installation. They will evaluate and compare different flooring options based on cost, quality, and installation factors, and develop a procedure to recommend the best choices.

The owner of newly opened Smith Valley Farms is looking to breed the next generation of top race horses. In this MEA, students will study race horse pedigrees as well as horse racing data to determine which is the best stallion to breed with a filly. Students will have to read a horse pedigree, calculate percentages based on a data table, and complete Punnett squares to determine genetic probability.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

This activity engages the students into time scheduling, budgeting, and decision making to maximize time efficiency.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx.

This Model Eliciting Activity (MEA) presents students with the real-world problem of contaminated drinking water. Students are asked to provide recommendations for a non-profit organization working to help a small Romanian village acquire clean drinking water. They will work to develop the best temporary strategies for water treatment, including engineering the best filtering solution using local materials. Students will utilize measures of center and variation to compare data, assess proportional relationships to make decisions, and perform unit conversions across different measurement systems.

In this Model Eliciting Activity, MEA, students will act as travel agents to plan a vacation package for a family of 5. Students will apply proportional reasoning and multi-step problem-solving skills to design vacation packages that meet specific criteria and stay within a given budget.

In this Model Eliciting Activity, MEA, students will analyze and compare data for various products sold in a joke shop to make recommendations on the best, and worst, products. Students will apply weighted averages, ratios, percentages, and proportions to perform calculations that support their recommendations as well as create graphical representations to help make sense of and compare the data.

## Student Resources

## Perspectives Video: Expert

<p>It's impossible to count every animal in a park, but with statistics and some engineering, biologists can come up with a good estimate.</p>

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiast

<p>Sometimes scientists conduct a census, too! Learn how population sampling can help monitor the progress of an ecological restoration project.</p>

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

In this task students use different representations to analyze the relationship between two quantities and to solve a real world problem. The situation presented provides a good opportunity to make connections between the information provided by tables, graphs and equations. In the later part of the problem, the numbers are big enough so that using the formula is the most efficient way to solve the problem; however, creative use of the table or graph will also work.

Type: Problem-Solving Task

## Tutorials

This introductory video demonstrates the basic skill of how to write and solve a basic equation for a proportional relationship.

Type: Tutorial

Here's an introductory video explaining the basic reasoning behind solving proportions and shows three different methods for solving proportions which you will use later on to solve more difficult problems.

Type: Tutorial

This introductory video shows some basic examples of writing two ratios and setting them equal to each other. This is just step 1 when solving word problems with proportions.

Type: Tutorial

## Parent Resources

## Perspectives Video: Expert

<p>It's impossible to count every animal in a park, but with statistics and some engineering, biologists can come up with a good estimate.</p>

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiast

<p>Sometimes scientists conduct a census, too! Learn how population sampling can help monitor the progress of an ecological restoration project.</p>

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

In this task students use different representations to analyze the relationship between two quantities and to solve a real world problem. The situation presented provides a good opportunity to make connections between the information provided by tables, graphs and equations. In the later part of the problem, the numbers are big enough so that using the formula is the most efficient way to solve the problem; however, creative use of the table or graph will also work.

Type: Problem-Solving Task