Strand: RATIOS AND PROPORTIONAL RELATIONSHIPS (7.RP)

Analyze proportional relationships and use them to solve real-world and mathematical problems (Standards 7.RP.1-3).
• Anna in D.C.
The purpose of this task is to give students an opportunity to solve a multi-step percentage problem that can be approached in many ways.
• Art Class, Assessment Variation
This task is part of a set of three assessment tasks for 7.RP.2.
• Art Class, Variation 1
Given a table about paint mixtures students are asked to answer questions about the mixture proportions and plot points on a plane to represent each mixture.
• Art Class, Variation 2
Given a table about paint mixtures students are asked to answer questions about the mixture proportions and write an equation that relates y, the number of parts of yellow paint, and b, the number of parts of blue paint for each of the different shades of paint on the table.
This task is part of a set of three assessment tasks for 7.RP.2.
The purpose of this task is for students to find a unit rate in a context where two quantities are in a proportional relationship and to draw the graph of that proportional relationship.
• Buying Protein Bars and Magazines
The task reads "Tom wants to buy some protein bars and magazines for a trip. He has decided to buy three times as many protein bars as magazines. Each protein bar costs \$0.70 and each magazine costs \$2.50. The sales tax rate on both types of items is 6½%. How many of each item can he buy if he has \$20.00 to spend?"
• Chapter 4 - Mathematical Foundation (UMSMP)
This is Chapter 4 of the Utah Middle School Math: Grade 7 textbook. It provides a Mathematical Foundation for Proportional Relationships and Solving Problems.
• Chapter 4 - Student Workbook (UMSMP)
This is Chapter 4 of the Utah Middle School Math: Grade 7 student workbook. It focuses on these topics: Proportional Relationships and Solving Problems.
• Chess Club
This problem includes a percent increase in one part with a percent decrease in the remaining and asks students to find the overall percent change. The problem may be solved using proportions or by reasoning through the computations or writing a set of equations.
• Cider versus Juice - Variation 1
This task asks students to compute multiple unit rates, aligning with standard 7.RP.A.1. The problem also has a real-world context, which requires students to compare two rates in different units in order to reach a conclusion on buying two different products.
• Cider versus Juice - Variation 2
The goal of this task is to apply proportional reasoning to determine which of two ways of buying apple juice/cider is a better deal. This task is a second variation to 7.RP.A.1, 7.RP.A.2.b Cider Versus Juice - Variation 1. This version offers a less directed approach to one of the questions posed in that task.
• Climbing the steps of El Castillo
The purpose of this task is for students to solve a straight-forward problem involving a proportional relationship in a context. In order to solve the problem, students must assume that the steps are of uniform height, which looks reasonable given the picture.
• Comparing Years
This task asks students to compare two quantities and calculate the percent decrease between the larger and smaller value.
• Cooking with the Whole Cup
While the task as written does not explicitly use the term "unit rate," most of the work students will do amounts to finding unit rates. A recipe context works especially well since there are so many different pair-wise ratios to consider.
• Double Discounts
The goal of this problem is to calculate percent decreases in the context of several (sequential) discounts.
• Drill Rig
The purpose of this task is to provide a context for multiplying and dividing signed rational numbers, providing a means for understanding why the signs behave the way they do when finding products.
• Dueling Candidates
The goal of this task is to have students examine some properties of ratios (and fractions) in an important real world context. Students will gain practice working with ratios while investigating some of the complexities of voting theory.
• Estimating: Counting Trees
This lesson unit is intended to help educators assess how well students are able to solve simple problems involving ratio and direct proportion, choose an appropriate sampling method, and collect discrete data and record them using a frequency table.
• Finding a 10% increase
Students are asked to complete this task: "5,000 people visited a book fair in the first week. The number of visitors increased by 10% in the second week. How many people visited the book fair in the second week?"
• Framing a House - student task
This task has students recreate house plans on graph paper and then determine how many linear feet of wall plate material will be needed.
• Friends Meeting on Bikes
Students are asked to complete this task: "Taylor and Anya are friends who live 63 miles apart. Sometimes on a Saturday, they ride toward each other's houses on their bikes and meet in between. One day they left their houses at 8 am and met at 11 am. Taylor rode at 12.5 miles per hour. How fast did Anya ride?"
• Gotham City Taxis
The purpose of this task is to give students an opportunity to solve a multi-step ratio problem that can be approached in many ways.
• Grade 7 Math Module 1:Ratios and Proportional Relationship (EngageNY)
In this 30-day Grade 7 module, students build upon sixth grade reasoning of ratios and rates to formally define proportional relationships and the constant of proportionality. Students explore multiple representations of proportional relationships by looking at tables, graphs, equations, and verbal descriptions. Students extend their understanding about ratios and proportional relationships to compute unit rates for ratios and rates specified by rational numbers. The module concludes with students applying proportional reasoning to identify scale factor and create a scale drawing.
• Grade 7 Math Module 4: Percent and Proportional Relationships (EngageNY)
In Module 4, students deepen their understanding of ratios and proportional relationships from Module 1 by solving a variety of percent problems. They convert between fractions, decimals, and percents to further develop a conceptual understanding of percent and use algebraic expressions and equations to solve multi-step percent problems. An initial focus on relating 100% to the whole serves as a foundation for students. Students begin the module by solving problems without using a calculator to develop an understanding of the reasoning underlying the calculations.
In order to assist educators with the implementation of the Common Core, the New York State Education Department provides curricular modules in Pre-K-Grade 12 English Language Arts and Mathematics that schools and districts can adopt or adapt for local purposes.
• Grade 7 Unit 3: Ratio and Proportional Relationships (Georgia Standards)
The units in this instructional framework emphasize key standards that assist students to develop a deeper understanding of numbers. They learn to express different representations of rational numbers (e.g., fractions, decimals, and percents), discover how to identify and explain the constant of proportionality, and represent proportional relationships and scale drawings within real-world contexts. The Big Ideas that are expressed in this unit are integrated with such routine topics as estimation, mental and basic computation. All of these concepts need to be reviewed throughout the year.
• Grid and Percent It
This lesson plans provides a 10 x 10 model so that students can understand how to solve percent problems.
• Gym Membership Plans
In this task, students are presented with two situations in a single context and asked which one represents a proportional relationship. Students are asked to understand this proportional relationship from a variety of perspectives -- a table, a graph, a verbal context, and an equation.
• Hay Bale Farmer
This lesson helps students understand volume by having them measure round and square hay bales.
• Holes
After watching a clip of the movie Holes from Walt Disney Pictures students will answer a series of questions, such as "If Stanley and X-Ray both dig one hole per day for a year, how much extra dirt will Stanley have dug than X-Ray?" and "How many times could that extra dirt fill one of the holes X-Ray digs?"
• How Fast is Usain Bolt?
This task involves a multi-step conversion between two rates, going from meters per second to miles per hour.
• Increasing and Decreasing Quantities by a Percent
This lesson unit is intended to help educators assess how well students are able to interpret percent increase and decrease.
• Inverse Proportions and Shadows in Practice
in this interactive a figure's shadow is projected on to a screen. Students then observe how the shadow changes as the figure moves farther away . Students also complete a chart that shows the distance of the figure from the light source and the height of the shadow helping them understand inverse proportions. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Inverse Proportions and Shadows in the Real World
A tour of drive-in theaters is the focus of this video. Students are asked to observe how the size of an object relates to its distance from the light source. In the classroom activity students do a hands-on experiments using a projector of other light source. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• IXL Game: Ratios and proportions
This game for seventh graders is designed to help them understand ratios and proportions, specifically by estimating population size using proportions. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
• Lincoln's Math Problem
The purpose of this task is for students to solve a multi-step problem involving simple interest. What is most interesting about this task is that it was one that Abraham Lincoln worked on in his youth (probably around the age of 17 years); it was discovered in some old papers that were authenticated as Lincoln's.
• Living Wages in CA: Ratio and Rate in the Real World
The use of infographics helps us understand the costs of basic living expenses. The classroom activity has students look at real-life examples and data to calculate whether the minimum wage in their state can be a living wage. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Manipulating Graphs
This video demonstrates how to use the slope-intercept of a line to the graph of that line. The classroom activity has them demonstrate their understanding by finding equations for a set of lines through the origin. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Measuring the area of a circle
This goal of this task is to give students familiarity using the formula for the area of a circle while also addressing measurement error and addresses both 7.G.4 and 7.RP.3.
• Mixtures
This activity will help students understand percentages and mixture problems by working with two piles of colored chips.
• Molly's Run
This task asks students to solve a problem in a context involving constant speed. This task provides a transition from working with ratios involving whole numbers to ratios involving fractions.
• Molly's Run, Assessment Variation
This task is part of a set of three assessment tasks that address various aspects of 6.RP domain and help distinguish between 6th and 7th grade expectations.
• Music Companies, Variation 1
This problem requires a comparison of rates where one is given in terms of unit rates, and the other is not. See "7.RP Music Companies, Variation 2" for a task with a very similar setup but is much more involved and so illustrates 7.RP.3.
• Music Companies, Variation 2
Given a scenario about share prices students are asked to calculate the value of individual shares, the value of groups of shares, and the difference between the two group amounts.
• One-Dimensional Scaling To Find Unknown Heights
An interactive activity helps students understand real-world application of ratios and asks them to scale a model of a T-Rex for a diorama. In the classroom activity students are asked to draw scale models. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Perfect Purple Paint II
The goal of this task is to provide a context for students to develop their ratio and proportional reasoning skills.
• Proportional Functions video
This video introduces proportional functions.
• Proportionality
The task has two main purposes. (1) Students make sense out of the definition of direct proportionality. (2) They engage in SMP 3 "Make a viable argument and critique the reasoning of others" and SMP 6 "Attend to precision".
• Ratios and Proportional Relationships (7.RP) - 7th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 7 Cluster "Analyze Proportional Relationships and Use Them to Solve Real-World and Mathematical Problems." / Standards 1, 2 and 3.
• Robot Races
Given a graph of line segments that show the distance d, in meters, that each of three robots traveled after t seconds, students are asked to answer specific questions about the graph.
• Robot Races, Assessment Variation
This task is part of a set of three assessment tasks for 7.RP.2. This task asks students to "explain what a point (x,y) on the graph of a proportional relationship means in terms of the situation" and to "compute unit rates associated with ratios of fractions." Students also need to compare the speeds of the robots.
• Sale!
The purpose of this task is to engage students in Standard for Mathematical Practice 4, "Model with mathematics." The teacher might use this task after formally teaching 7.RP.1-3. Students could be given the task and asked to collaborate in small groups to solve the questions posed using all the formal instruction on ratio and proportional reasoning.
• Sand Under the Swing Set
The purpose of this task is for students to solve a contextual problem where there are multiple entry points to this geometry based concept. The student can choose to solve the problem using a scale factor or a unit rate, but must first must analyze the context of the problem to understand the situation and choose their approach. This task provides opportunities for students to reason about their computations to see if they make sense. This task could be used as an assessment question or for guided instruction on scale factoring and/or unit rate.
• Scale Models and Three-Dimensional Scaling in Practice
Students can use this interactive to explore how an object changes when enlarged by a factor of 10. They put this understanding to use in the activity when they compare two cubes of different sizes by volume and surface area. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Scale Models in the Real World
This video show scale models of railroads, dollhouses, and architecture to explain what the ratios represent. The hands-on classroom activity then has students create their own models. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Scaling
An interactive from Annenberg asks students to scale a picture by using the math strategies of multiplicative and additive relationships. Students then use those strategies to compare photocopies and rectangles in different scales. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Scaling angles and polygons
The goal of this task is to gather together knowledge and skills from the seventh grade in a context which prepares students for the important eighth grade notion of similarity.
• Scaling Up Rectangles in the Real World
By looking at murals in this video students see how artists use proportion and measurement to create them. The class then does an activity where they enlarge a drawing by using scale and then create a classroom mural. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Scaling Up Rectangles Using Simulations
This interactive allows students to create murals in different sizes by understanding two-dimensional scaling. The activity for the classroom has students use graph paper to draw and compare squares with different proportional dimensions and record the data as they change. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Selling Computers
Given this scenario: "The sales team at an electronics store sold 48 computers last month. The manager at the store wants to encourage the sales team to sell more computers and is going to give all the sales team members a bonus if the number of computers sold increases by 30% in the next month," students must determine how many computers the sales team needs to sell to get the bonus.
• Similar Figures and Unknown Heights in Practice
Students will learn about proportional reasoning in this interactive and how it can help find an unknown height as well as exploring proportional relationships among similar triangles. The classroom activity uses the interactive as a catalyst into a discussion of these math concepts. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Similar Figures in the Real World
A visit to the Louisville Slugger Museum and Factory shows students how measurement and engineering are involved in the creation of new baseball bats. The math skills of proportional reasoning and equivalent ratios are used in the classroom activity involving an oversized bat and a player. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Sore Throats, Variation 1
Given the scenario of mixing salt and water students must identify which of a set of equations best relates that information.
• Stock Swaps, Variation 2
Given the price of two stocks to be swapped, students must determine how many shares of stock they need to offer to get an even swap.
• Stock Swaps, Variation 3
Given the price of two stocks to be swapped, students must determine how many shares of stock they need to offer to get an even swap.
In this task, students are asked to solve this query: In a sale, the store reduces all prices by 25% each week. Does this mean that, after 4 weeks, everything in the store will cost \$0? If not, why not?
• Student Task: A Golden Crown?
Archimedes famously solved a problem for a King who thought his crown might be a fake. In this task, students must work out whether the crown is pure gold.
In this task, students use math to plan how to sell ice cream at a school sports event.
In this task, students use mathematics to decide which special offers give the biggest and smallest price reductions.
A store sells T-shirts at various prices and offers "Any 3 T-shirts for \$14.50". The student's task is to work out how much people have saved.
A set of short tasks for grade 7 dealing with ratios and proportion.
• Tax and Tip
Given this scenario: "After eating at your favorite restaurant, you know that the bill before tax is \$52.60 and that the sales tax rate is 8%. You decide to leave a 20% tip for the waiter based on the pre-tax amount," students must calculate the tip amount and the total bill including it.
• Temperature Change
The goal of this task is to provide a context for interpreting the expressions that match the last part of the standard 7.NS.2.b, ''Interpret quotients of rational numbers by describing real-world contexts,'' though in this case the numerator and denominator are integers. Because of the context, students will also gain experience working with rates.
The purpose of this task is for students to calculate the percent increase and relative cost in a real-world context.
• Thunder and Lightning
The purpose of this task is to work on performing unit conversions in a real world context about the speed of sound.
• Track Practice
This task asks students to find the unit rates that one can compute in a context.
• Two-School Dance
The purpose of this task is to see how well students students understand and reason with ratios.
• Walk-a-thon 2
The purpose of this task is for students to translate information about a context involving constant speed into information presented in a table and to find the time it takes to travel a unit distance as well as the distance traveled per unit time. Students then have to translate the information to equations and graphs and then use these mathematical tools to make predictions about the future.

http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.