# MA.7.AR.3.2

Apply previous understanding of ratios to solve real-world problems involving proportions.

### 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?

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
• 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.

• Instruction includes the understanding that proportions are multiplicative relationships.

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 A. Using the new recipe, how much flour will she need? Explain or show your work.
Part B. How many cookies can she make with the new recipe? Explain or show your work.

In buying ground beef for hamburgers, there are several packages from which to choose, as shown in the table below.

If Johannes needs 5 pounds of beef for his barbeque, what will he pay?

### 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

This benchmark is part of these courses.
1205020: M/J Accelerated Mathematics Grade 6 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205040: M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 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.3.AP.2: Solve simple ratio problems in real-world contexts.

## Related Resources

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

## Formative Assessments

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

Type: Formative Assessment

Reeling in Expressions:

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

Type: Formative Assessment

## Lesson Plans

Who's in the House? Part 2:

Use data from U.S. Census Bureau that shows Apportionment Population, Resident Population, and Overseas Population for 2020 & 2010 Census to create and compare ratios in this integrated lesson plan.

Type: Lesson Plan

WHO’S IN THE HOUSE? PART 1:

Students will use ratios to discuss and examine the relationship between a Census, state populations, and apportionment of representatives in the House of Representatives, in this integrated lesson plan.

Type: Lesson Plan

Students will work collaboratively to rank civic duties and responsibilities needed to keep a constitutional republic. They will utilize mathematical strategies to convert measurements of time as they calculate costs using the four operations with decimals and create an effective schedule for the ads within a budget in this model eliciting activity.

Type: Lesson Plan

Comparing Amendments:

Students will read brief summaries about different amendments ratified throughout history intended to expand civic participation, analyze voter turnout and voting age population data for presidential elections before and after the ratification of each amendment, and use percentages and ratios to rank the amendments in order of most to least effective in expanding civic participation, in this model eliciting activity.

Type: Lesson Plan

A Rocky Debate: How do coastal structures reduce rates of coastal erosion?:

Students will be tasked with analyzing various methods of protecting coasts from erosion. Students will review a dataset with logistics about each type of coastal structure. Students will rank which structures they feel should be utilized to best protect a local beachfront town. The students will write a letter to the local government to persuade them on which structure should be used. Students will be challenged to think critically, analyze information, and work collaboratively in this model eliciting activity.

Type: Lesson Plan

Comparing Amendments:

In this lesson plan, students will analyze voter turnout and voting age population data for past presidential elections to explore how various amendments broadened the opportunity for civic participation in the political process.

Type: Lesson Plan

Guiding Grids: Math inspired self-portraits:

Students will create a proportional self portrait from a photo using a gridded drawing method and learn how a grid system can help accurately enlarge an image in a work of art. Students will use the mathematical concepts of scale, proportion and ratio, to complete their artwork.

Type: Lesson Plan

Partition Point For The Queen:

Students will locate a point that partitions a line segment into a given ratio. Students will use a variety of methods; the activities range from informal student definitions and sketches to tasks using number lines and the coordinate plane.

Type: Lesson Plan

Geometree Thievery:

This geometry lesson focuses on partitioning a segment on a coordinate grid in a non-traditional and interesting format. Students will complete a series of problems to determine which farmers are telling the truth about their harvested "Geometrees."

Type: Lesson Plan

Partitioning a Segment:

In this lesson, students find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Type: Lesson Plan

Dilation Transformation:

Students identify dilations, verify that polygons are similar, and use the dilation rule to map dilations. Task cards are provided for independent practice. The PowerPoint also includes detailed illustrations for constructing a dilation using a compass and a straight edge.

Type: Lesson Plan

Partition Me:

Students will learn how to partition a segment. Turn your class into a partitioning party; just BYOGP (Bring your own graph paper).

Type: Lesson Plan

Water Troubles:

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.

Type: Lesson Plan

Math in Mishaps:

Students will explore how percentages, proportions, and solving for unknowns are used in important jobs. This interactive activity will open their minds and address the question, "When is this ever used in real life?"

Type: Lesson Plan

## Original Student Tutorial

Working With Proportions:

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/Enthusiasts

Ratios and Proportions in Mixing Ceramic Glazes:

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

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Energy and Nutrition:

Calorie-dense foods can power the human body across the ocean? Feel the burn.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

## Perspectives Video: Teaching Idea

This teacher explains how a 3D-printed quadrat can be used with an M&M sampling lesson to engage students when they explore how to use data from a random sample to draw inferences about a population.

Type: Perspectives Video: Teaching Idea

Archimedes and the King's Crown:

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.

Coupon Versus Discount:

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.

Friends Meeting on Bikes:

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

Music Companies, Variation 2:

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.

Stock Swaps, Variation 2:

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.

Stock Swaps, Variation 3:

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

## STEM Lessons - Model Eliciting Activity

Water Troubles:

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.

## MFAS Formative Assessments

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

Reeling in Expressions:

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

## Original Student Tutorials Mathematics - Grades 6-8

Working With Proportions:

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

## Student Resources

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

## Original Student Tutorial

Working With Proportions:

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

Ratios and Proportions in Mixing Ceramic Glazes:

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

Type: Perspectives Video: Professional/Enthusiast

Archimedes and the King's Crown:

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.

Coupon Versus Discount:

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.

Friends Meeting on Bikes:

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

Music Companies, Variation 2:

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.

Stock Swaps, Variation 2:

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.

Stock Swaps, Variation 3:

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

## Parent Resources

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

## Perspectives Video: Professional/Enthusiast

Ratios and Proportions in Mixing Ceramic Glazes:

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

Type: Perspectives Video: Professional/Enthusiast

Archimedes and the King's Crown:

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.

Coupon Versus Discount:

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.

Friends Meeting on Bikes:

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

Music Companies, Variation 2:

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.