### Examples

*Example:*Scott is mowing lawns to earn money to buy a new gaming system and knows he needs to mow 35 lawns to earn enough money. If he can mow 4 lawns in 3 hours and 45 minutes, how long will it take him to mow 35 lawns? Assume that he can mow each lawn in the same amount of time.

*Example:* Ashley normally runs 10-kilometer races which is about 6.2 miles. She wants to start training for a half-marathon which is 13.1 miles. How many kilometers will she run in the half-marathon? How does that compare to her normal 10K race distance?

**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

- Constant of Proportionality
- Proportional Relationships

### 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 apply that ratio reasoning to solve real-world problems involving proportions. In grade 8, students will determine if a linear relationship is also a proportional relationship and will solve problems involving proportional relationships between similar triangles.- Instruction includes making connections to comparing ratios from grade 6 as a comparison using the equal sign.
- For example, if a student can complete 7 math problems in 30 minutes and one wants to determine how many math problems they can complete in 90 minutes, they can compare the two ratios $\frac{\text{7}}{\text{30}}$ and $\frac{\text{p}}{\text{90}}$ as the equation $\frac{\text{7}}{\text{30}}$ = $\frac{\text{p}}{\text{90}}$ to determine the number of math problems.

- Instruction does not emphasize rules, like cross multiplying, when solving proportions.
- Instruction allows time for students to analyze real-world situations. Ratio and rate reasoning can be applied to many types of real-life problems, including rate and unit rate, scaling, unit pricing, and statistical analysis
*(MTR.7.1).*

### Common Misconceptions or Errors

- Students may not understand the difference between an additive relationship and a multiplicative relationship. To help address this misconception, instruction includes the understanding that proportions are multiplicative relationships.
- Students may incorrectly set up proportions with one of the ratios having incorrect numbers in the numerator and denominator.
- Students may not recognize simplified forms of ratios in order to find equivalent ratios.

### Strategies to Support Tiered Instruction

- Teacher provides instruction focused on the understanding of multiplicative relationships between two quantities in a proportional relationship.
- Teacher provides instruction on color-coding and labeling the different units when setting up a proportional relationship to ensure corresponding units are placed in the corresponding positions within the proportion.
- For example, a student can complete 7 math problems in 30 minutes. How many math problems can they complete in 90 minutes?
- Teacher co-constructs visual models with students to visualize the multiplicative relationship between quantities.
- For example, to solve the proportion, the corresponding numbers are tripled to find a missing value of 21.

- For example, to solve the proportion, the corresponding numbers are tripled to find a missing value of 21.
- Instruction includes the understanding that proportions are multiplicative relationships.

### Instructional Tasks

*Instructional Task 1*

**(MTR.3.1)**A recipe that makes 16 cookies calls for $\frac{\text{1}}{\text{4}}$ cup of sugar and $\frac{\text{2}}{\text{3}}$ cup of flour. Janelle wants to proportionally increase these amounts to get a new recipe using one cup of sugar.

Part B. How many cookies can she make with the new recipe? Explain or show your work.

*Instructional Task 2*

**(MTR.6.1,***MTR.7.1*)In buying ground beef for hamburgers, there are several packages from which to choose, as shown in the table below.

### Instructional Items

*Instructional Item 1*

Anthony is writing the place cards for his best friend’s wedding reception. If he can write 12 place cards in 5 minutes, how long will it take him to complete the entire group of 180 place cards?

**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

## Formative Assessments

## Lesson Plans

## Original Student Tutorial

## Perspectives Video: Professional/Enthusiasts

## Perspectives Video: Teaching Idea

## Problem-Solving Tasks

## STEM Lessons - Model Eliciting Activity

In this Model Eliciting Activity, MEA, students will evaluate and compare daycare centers near a neighborhood. They will develop a scoring system to assess various characteristics of each center and justify their ranking using averages, ratios, and other mathematical calculations.

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 Activities, MEA, students will calculate unit rate & circumference, compare & order decimals, convert metric units, and round decimals. Bubble Burst Corporation has developed some chewing gum prototypes and has requested the students to assist in the selection of which gum prototypes will be mass produced by using both quantitative and qualitative data to rank the prototypes for Bubble Burst Corporation.

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 analyze and interpret data to recommend the best vehicle purchases for a school district. Students will work collaboratively to perform calculations that can be used to make comparisons and create composite scores for each vehicle.

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.

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.

In this MEA, the students will be able to convert measurements within systems and between systems. They will be able to use problem solving skills to create a process for ranking orange juices for a Bed and Breakfast.

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.

In this Model Eliciting Activity, MEA, students will apply their knowledge of percentages and ratios to evaluate various stocks for investment. Students will develop a systematic procedure to recommend the best stock for investment.

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.

## MFAS Formative Assessments

Students must find proportionally equivalent values given a set of rational number quantities.

Students are asked to solve a multi-step problem involving rational numbers.

## Original Student Tutorials Mathematics - Grades 6-8

Roll up your sleeves and learn how proportions can be used in everyday life in this interactive tutorial.

## Student Resources

## Original Student Tutorial

Roll up your sleeves and learn how proportions can be used in everyday life in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiast

<p>Ceramic glaze recipes are fluid and not set in stone, but can only be formulated consistently with a good understanding of math!</p>

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

Type: Problem-Solving Task

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Type: Problem-Solving Task

Using the information provided find out how fast Anya rode her bike.

Type: Problem-Solving Task

This problem has multiple steps. In order to solve the problem it is necessary to compute: the value of the TunesTown shares; the total value of the BeatStreet offer of 20 million shares at $25 per share; the difference between these two amounts; and the cost per share of each of the extra 2 million shares MusicMind offers to equal to the difference.

Type: Problem-Solving Task

Students are asked to solve a problem using proportional reasoning in a real world context to determine the number of shares needed to complete a stock purchase.

Type: Problem-Solving Task

Students are asked to solve a multistep ratio problem in a real-world context.

Type: Problem-Solving Task

## Parent Resources

## Perspectives Video: Professional/Enthusiast

<p>Ceramic glaze recipes are fluid and not set in stone, but can only be formulated consistently with a good understanding of math!</p>

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

Type: Problem-Solving Task

In this task, students are presented with a real-world problem involving the price of an item on sale. To answer the question, students must represent the problem by defining a variable and related quantities, and then write and solve an equation.

Type: Problem-Solving Task

Using the information provided find out how fast Anya rode her bike.

Type: Problem-Solving Task

This problem has multiple steps. In order to solve the problem it is necessary to compute: the value of the TunesTown shares; the total value of the BeatStreet offer of 20 million shares at $25 per share; the difference between these two amounts; and the cost per share of each of the extra 2 million shares MusicMind offers to equal to the difference.

Type: Problem-Solving Task

Students are asked to solve a problem using proportional reasoning in a real world context to determine the number of shares needed to complete a stock purchase.

Type: Problem-Solving Task

Students are asked to solve a multistep ratio problem in a real-world context.

Type: Problem-Solving Task