# Cluster 1: Analyze proportional relationships and use them to solve real-world and mathematical problems. (Major Cluster)Archived Export Print

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.7.RP.1
Title: Analyze proportional relationships and use them to solve real-world and mathematical problems. (Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Ratios & Proportional Relationships

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.7.RP.1.AP.1a
Solve one-step problems involving unit rates associated with ratios of fractions.
MAFS.7.RP.1.AP.2a
Identify the rate of change/proportional relationship of a linear equation that has been plotted as a line on a coordinate plane.
MAFS.7.RP.1.AP.2b
Identify lines plotted on a coordinate plane that represent a proportional relationship.
MAFS.7.RP.1.AP.3a
Solve word problems involving ratios.
MAFS.7.RP.1.AP.3b
Find percentages in real-world contexts.

## Related Resources

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

## Educational Games

Estimator Four:

In this activity, students play a game of connect four, but to place a piece on the board they have to correctly estimate an addition, multiplication, or percentage problem. Students can adjust the difficulty of the problems as well as how close the estimate has to be to the actual result. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers (100s). This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

Estimator Quiz:

In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, or percentages of large numbers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

## Educational Software / Tool

Free Graph Paper:

A variety of graph paper types for printing, including Cartesian, polar, engineering, isometric, logarithmic, hexagonal, probability, and Smith chart.

Type: Educational Software / Tool

## Formative Assessments

Tiffany‘s Tax:

Students are asked to calculate the amount of sales tax and total price, given prices of individual items to purchase.

Type: Formative Assessment

Unit Rate Length:

Students are asked to write ratios and unit rates from fractional values.

Type: Formative Assessment

Unit Rate Area:

Students are asked to convert a ratio of mixed numbers to a unit rate and explain its contextual meaning.

Type: Formative Assessment

Identifying Constant of Proportionality in Equations:

Students are asked to identify and explain the constant of proportionality in three different equations.

Type: Formative Assessment

Teacher to Student Ratios:

Students are asked to graph four ordered pairs given in context and decide if the variables they represent are proportionally related.

Type: Formative Assessment

Constant of Proportionality Trip:

Students are asked to identify and explain the constant of proportionality given a verbal description and a diagram representing a proportional relationship.

Type: Formative Assessment

Comparing Unit Rates:

Students are asked to compute unit rates from values that include fractions.

Type: Formative Assessment

Computing Unit Rates:

Students are asked to compute and interpret unit rates in two different ways from values that include fractions and mixed numbers.

Type: Formative Assessment

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

Type: Formative Assessment

Gasoline Prices:

Students are given gasoline prices from a year ago and today and are asked to calculate the percent change.

Type: Formative Assessment

Finding Fees:

Students are asked to complete a multi-step percent problem.

Type: Formative Assessment

Writing An Equation:

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

Type: Formative Assessment

Graphs of Proportional Relationships:

Students are asked to identify the graph of a proportional relationship.

Type: Formative Assessment

Babysitting Graph:

Students are given a graph that models the hourly earnings of a babysitter and are asked to interpret ordered pairs in context.

Type: Formative Assessment

Finding Constant of Proportionality:

Students are asked to determine the constant of proportionality using a table and a graph.

Type: Formative Assessment

Deciding If Proportional:

Students decide if two variables are proportionally related based on data given in a table.

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

Estimating: Counting Trees:

This lesson unit is intended to help you assess how well students are able to:

• Solve simple problems involving ratio and direct proportion.
• Choose an appropriate sampling method.
• Collect discrete data and record them using a frequency table.

Includes worksheets and student work examples, including specific feedback and analysis of misconceptions

Type: Formative Assessment

## Lesson Plans

Guiding Grids: Math inspired self-portraits:

In this lesson students will create a proportional self portrait from a photo using a gridded drawing method and understand how a grid system can help accurately enlarge an image in a work of art.

Students will use mathematical concepts of scale, proportion and ratio, to complete their artwork and learn about the work of artist, Chuck Close.

Type: Lesson Plan

Read about a recent uncovering of mammoths to engage students in a discussion of radioactive dating. This is the first lesson in a unit of 4 lessons that integrates science, math, and computer science standards to teach the concept of half-lives and radioactive dating.

Type: Lesson Plan

Netting 4 Bugs:

This is a STEM challenge in which students design and create a net to collect macroinvertebrates in simulated streams. Then students analyze the quality of their nets by the amount of macroinvertebrates they are able to collect. After testing, they will redesign to improve their nets. The final test will be done by evaluating a simulated stream's water quality. Students will conduct a simulated bioassessment of a stream by sampling macroinvertebrates and evaluating a stream's water quality using a pollution tolerance index. They learn about the human impact on waterways and the importance of using aquatic macroinvertebrates to monitor water quality.

Type: Lesson Plan

Real Life Tax, Tip, and Discount!:

Students calculate the tax, tip, and discount in real world situations.

Type: Lesson Plan

Wolves of Yellowstone - Ecology & Human Impact:

In this MEA, students will decide how many wolves to introduce into Yellowstone National Park's ecosystem. The number of wolves could influence many factors, from the tourism industry to local farming businesses, as well as the populations of other species in the area. Students must choose to introduce the number of wolves they feel will be most beneficial to the preservation of Yellowstone National Park as determined by the mission statement of Yellowstone and the National Park Service.

Type: Lesson Plan

STEM-Water Filtration:

This is a STEM-Engineering Design Challenge lesson. Students will go through the process of creating a water filtration system using their knowledge of the impact that humans have on the Earth and percent change.

Type: Lesson Plan

Irrigation Station:

This STEM lesson, complete with a design challenge, helps students design, build, and test irrigation methods. Students will incorporate and develop math skills through solving proportions as they work in teams to solve an engineering challenge.

Type: Lesson Plan

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

Bubble Burst Corporation's Chewing Gum Prototypes:

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.

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

Are Corresponding Leaf Veins Proportional to Leaf Height?:

Students will measure the length of different sized leaves and corresponding veins to determine proportionality.  Students will graph their results on a coordinate grid and write about their results.

Type: Lesson Plan

Prom Preparations:

Students will make decisions concerning features of their prom. Students will perform operations with percent and decimals to solve real-world problems involving money.

Type: Lesson Plan

Laura’s Babysitting Job:

In this 7th grade MEA Laura Banks requests a consulting firm, JJ Consulting, to help her make a decision on an employer. Students are to use the data table to calculate unit rates (nightly rate and hourly rate) and then rank her choices and write a recommendation with the procedure used to come up with the ranking.

Type: Lesson Plan

HOORAH!! Pizza For Lunch:

The principal of Central Middle School is thinking of adding pizza to the lunch menu on Mondays and Fridays but needs help deciding the costs per slice and what students think is important about the pizza. After the students' initial decision about the pizza the principal remembers that there is a delivery charge.The students must revisit their decision and do additional calculations to see if their original process still works.

Type: Lesson Plan

Students at a local middle school are interested in attending a basketball tournament in Orlando. There is an entrance fee and hotel costs to consider. Students must calculate the total cost and the cost per student to attend the tournament. Each hotel has different qualities that could influence the students' choice of which hotel is best for their team.

Type: Lesson Plan

Summer Camp Fun:

In this problem, students will work in groups to rank summer camps. They must first calculate the discounted price of each camp by applying a discount percentage. Then they must calculate the number of weeks they can attend each camp based on the discounted price and predetermined budget. There are 5 students going to the camp. Students will be given a data set to help them develop a procedure for ranking the camps. In their teams, they will write a letter giving their procedures and explanation of the strategy they used. Students will practice adding, subtracting, multiplying and dividing numbers to the thousands and will work with percentage discounts. Rubrics are included to help evaluate student work.

Type: Lesson Plan

Which van is the best buy?:

The students will have to decide which van is the "best buy" for a family. They will have to figure monthly payments and will also use critical thinking skills to decide which is the best van to purchase.

Type: Lesson Plan

How Does It Compare?:

This lesson addresses part a. of the Standard. It introduces proportions and addresses solving proportion problems with ratio tables. Students will be able to identify whether a statement shows proportionality or is simply two non-proportional ratios. The focus will be on recognizing visual proportions as being equivalent values and will lead into recognizing ratios represented as fractions as equivalent or non-equivalent.

Type: Lesson Plan

Developing a Sense of Scale:

This lesson unit is intended to help you assess whether students recognize relationships of direct proportion and how well they solve problems that involve proportional reasoning. In particular, it is intended to help you identify those students who use inappropriate additive strategies in scaling problems, which have a multiplicative structure, rely on piecemeal and inefficient strategies such as doubling, halving, and decomposition, and have not developed a single multiplier strategy for solving proportionality problems and see multiplication as making numbers bigger, and division as making numbers smaller.

Type: Lesson Plan

Distinguishing Between Proportion and Non-proportion Situations:

This lesson is from the Mathematics Assessment Resource Service (MARS) collection of the Mathematics Assessment Project's (MAP) Classroom Challenges, and involves a review task, lesson task, and assessment task along with a PowerPoint.

In this lesson, students identify when two quantities vary in direct proportion to each other, distinguish between direct proportion and other functional relationships, and solve proportionality problems using efficient methods.

Type: Lesson Plan

Increasing and Decreasing Quantities by a Percent:

This lesson unit is intended to help you assess how well students are able to interpret percent increase and decrease, and in particular, to identify and help students who have the following difficulties:

• translating between percents
• decimals and fractions
• representing percent increase and decrease as multiplication
• recognizing the relationship between increases and decreases

Type: Lesson Plan

Family Restaurant:

This MEA requires students to formulate a comparison-based solution to a problem involving finding the best choice on purchasing cooking ingredients for a family who runs a restaurant considering different aspects. Students are provided the context of the problem, a request letter from a client asking them to provide a recommendation, and data relevant to the situation. Students utilize the data to create a defensible model solution to present to the client.

Type: Lesson Plan

Estimating: Counting Trees:

This lesson unit is intended to help you assess how well students are able to solve simple problems involving ratio and direct proportion, choose an appropriate sampling method, collect discrete data, and record their data using a frequency table.

Type: Lesson Plan

Students will learn how to calculate markup, markdown, percent increase and percent decrease. Through the use of sales "ad" inserts in newspapers and store flyers, students will understand how these concepts apply to them in real-world situations.

Type: Lesson Plan

Percent of Change:

Students will investigate percent of change in real-world situations and will differentiate between an increase or a decrease. The students will use a formula to find the percent of change.

Type: Lesson Plan

Car Shopping:

This MEA requires students to formulate a comparison-based solution to a problem involving finding the best decision on purchasing official vehicles for school district considering different aspects. Students are provided the context of the problem, a request letter from a client asking them to provide a recommendation, and data relevant to the situation. Students utilize the data to create a defensible model solution to present to the client.

Type: Lesson Plan

In Whose Best Interest is Interest?:

The students will explore real world examples of interest rates. Students will explore loan rates, CD rates and compare benefits of different rates versus different terms of loans. Students will use the formula for simple interest.

Type: Lesson Plan

Back to the Past with the Geologic Time Scale:

This lesson introduces the geologic time scale and the concept of time segments being divided by major events in Earth's history. It gives students an opportunity to place various fossils into appropriate periods, observe the change in the complexity of fossils and draw conclusions regarding the change. Students complete a brace map including the eras and periods showing their understanding of parts to the whole within the geologic time scale. On day 2, students research an organism of their choice and trace it back to their most basic relative. Students then create a final product, such as a brochure, timeline or a poster, demonstrating the change of the organism over time. Students will be provided with a rubric that will guide them while they work on the final product.

Type: Lesson Plan

Pop-A-Tag: Buying Clothes at a Thrift Shop:

Students will participate in a simulation where they purchase clothing items and determine the total cost of their purchases, including tax and any discounts offered.

Offered as a cooperative group activity, they will simulate being the customer and the sales person. The activity can be modified for independent work, however, the move toward real-world problem solving and Mathematical Practices suggest that it be an activity involving discussion and collaboration between students.

Type: Lesson Plan

Pricing The Twelve Days of Christmas:

Students will discover how much the items in the classic song, "The Twelve Days of Christmas," would cost in the current year; and then they will update the list for modern times.

Type: Lesson Plan

Recognizing Proportional Relationships to Develop Sense of Scale:

This 90-minute lesson (15-minute pre-lesson, 60-minute lesson and 15-minute follow up lesson or homework) asks students to analyze proportional relationships to solve real world and mathematical problems. The examples use recipes, paint, and buildings. Students begin by working individually, then in pairs or threes, and then as a whole class. Student will need calculators, large sheets of paper to make a poster and the lesson materials.

Type: Lesson Plan

Making a Scale Drawing:

Objective: Students will create a detailed scale drawing. Context: Students have used tools to measure length, solve proportions, and interpret scale drawings. They will continue to use ratio and proportion in the study of similar figures, percent, and probability.

Type: Lesson Plan

Increasing and Decreasing Quantities by a Percent:

This lesson unit is intended to help you assess how well students are able to interpret percent increase and decrease, and in particular, to identify and help students who have the following difficulties:

• Translating between percents, decimals, and fractions.
• Representing percent increase and decrease as multiplication.
• Recognizing the relationship between increases and decreases.

Type: Lesson Plan

Savvy Shopping:

This is the second part of a lesson on this standard. You can find the first part under Markup and Make Money. In Savvy Shopping students will go shopping at a store of their peer and buy items. If it is discounted, they will have to calculate the revised price. They will then find the total price including the tax.

Type: Lesson Plan

For Students by Students:

Students are presented with the task of evaluating several types of fabric based on each of its characteristics. They need to analyze their current uniform needs and decide by choosing which type of fabric will best fit their uniform needs. Then they have to write a report explaining the procedure they used to analyze their choices, reasoning for their ranking and make the requested recommendations.

Type: Lesson Plan

Scientific calculations from a distant planet:

Students will act as mathematicians and scientists as they use models, observations and space science concepts to perform calculations and draw inferences regarding a fictional solar system with three planets in circular orbits around a sun. Among the calculations are estimates of the size of the home planet (using a method more than 2000 years old) and the relative distances of the planets from their sun.

Type: Lesson Plan

Have you ever heard the question, "Why do I have to learn this?" This lesson answers that question. This lesson requires the students to apply their knowledge in real world scenarios but does not teach a basic conceptual understanding of percents. The teacher may use the whole lesson or select particular problems.

Type: Lesson Plan

Stock Market MEA:

Students will calculate percents and ratios to determine which stock would be the best to invest into. They will be provided a selection of stocks and their cost, growth potential, risk and expert analysis. After developing a procedure for choosing the best stock, they will present their findings orally to the class, describing the analysis that they conducted during their procedure.

Type: Lesson Plan

"Are My Values in Direct Variation?":

Students will learn that a direct variation is a proportional relationship that can be represented by a table, a graph, or an equation. They will also be able to recognize if they are dealing with a direct variation by the table, graph, or equation.

Type: Lesson Plan

Installing Tile Floor:

This MEA requires students to formulate a comparison-based solution to a problem involving finding the best plan for installing tile floor considering different aspects. Students are provided the context of the problem, a request letter from a client asking them to provide a recommendation, and data relevant to the situation. Students utilize the data to create a defensible model solution to present to the client.

Type: Lesson Plan

The Human Population Growth Rate:

Just how quickly is the world's human population growing? In the US and other developed countries, the current growth rate is slow compared to some developing countries where it is speeding up. There are factors that slowed down this growth rate and there are similar factors that actually speed it up. Discussing and explaining the factors that determine the fluctuation in growth rate.

The US population growth between 1950 - 2000 is 7.5 times slower than that of India. In 1950 the US had a population of 80 million which increased every ten years with 1 million.

Type: Lesson Plan

Makeover, Home Edition Part III:

This is the third lesson of the Unit, "Makeover, Home Edition."This lesson is designed to teach students how to put ideas into reality by creating and using scale drawings in the real world. In Part I (#48705) students determine backyard dimensions for fence installation. Part II (#48967) concentrates on inserting a pool and patio into this backyard. In Part III (#49025) students create a scale drawing of this backyard. Part IV (#49090) focuses on inserting a window and painting walls inside the house.

Type: Lesson Plan

Makeover, Home Edition Part II:

This is the second part of the Unit Lesson, "Makeover, Home Edition". This lesson will continue focusing on unit prices, but also incorporates area and volume as well. Part I (Makeover, Home Edition #48705) is based on creating backyard dimensions for fencing. Part III (Makeover, Home Edition #49025) will deal with creating a scale drawing of this backyard. Part IV (Makeover, Home Edition Final #49090) focuses on inserting a window and painting walls inside the house.

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

Importing Machine Parts:

This MEA requires students to formulate a comparison-based solution to a problem involving choosing the best shipping options for importing machine parts from India to US. Students are provided the context of the problem, a request letter from a client asking them to provide a recommendation, and data relevant to the situation. Students utilize the data to create a defensible model solution to present to the client.

Type: Lesson Plan

The Most Beneficial Bank:

In this Model Eliciting Activity, MEA, students will work in cooperative groups to discuss and come up with a procedure to rank the banks from best to worst by estimating the simple interest and total loan amount.

Type: Lesson Plan

Makeover, Home Edition Part I:

This is the first part of the Unit Lesson, "Makeover Home Edition". This lesson is designed to get the students excited about the unit. Students will have to think critically about fencing in their new "dream" backyard by calculating total fencing needed and choosing the most cost effective method of purchasing their fencing by comparing unit rates mathematically and graphically. Part II (#48967) will concentrate on inserting a pool and patio into this backyard. Part III (#49025) will deal with creating a scale drawing of this backyard. Part IV (#49090) focuses on inserting a window and painting walls inside the house.

Type: Lesson Plan

Best Day Care Center in the Neighborhood:

This MEA requires students to formulate a comparison-based solution to a problem involving choosing the best day care center in the neighborhood for the residents of Dream Living Housing Community. Students are provided the context of the problem, a request letter from a client asking them to provide a recommendation, and data relevant to the situation. Students utilize the data to create a defensible model solution to present to the client.

Type: Lesson Plan

Let's Rate it!:

The purpose of this lesson is to introduce rates of change to students, allowing them to explore how rates are formed, what rates are used for, and how rates can be used to solve real life problems.

Type: Lesson Plan

Math in Mishaps:

Students will explore how percents, 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

Shopping & Dining with Proportions:

In this lesson, students will take an imaginary trip to a retail store and enjoy dinner with their friends while using proportions to calculate sales tax, tips and discounts.

Type: Lesson Plan

Students will go on a "road trip" with a partner. Using the map scale they find out how far they traveled, how much gas they used, and how much the gas costs.

Type: Lesson Plan

Bargain Town, USA:

In this lesson, students will participate in a simulated real-world exploration of the relationship between fractions, decimals, and percents, by converting number forms and calculating discounted prices.

Type: Lesson Plan

The Price is Right:

In this lesson, students practice finding the unit price of pre-selected items from local grocery store ads to decide which store has the best prices. (from Beacon Learning Center)

Type: Lesson Plan

Let's Go Shopping: Calculating Percents:

In this lesson, students will participate in a simulated shopping experience where they choose items they would like to purchase from local sale advertisements. The students will be able to apply the percent formula and the percent of change formula to real world financial situations. Students will learn how to calculate percent discounts, their percent of savings, and tax. The students will analyze, compare, draw conclusions and explain in writing why specific types of discounts are the most advantageous given specific situations.

Type: Lesson Plan

Budget Committee:

In this MEA, students will take on the role as a member of the Sunshine County Budget Committee. Members will collaborate to determine the optimal sales tax rate, use that rate to calculate how much money can be used for special projects, then decide which special projects to include in the budget proposal. Students will use percentages to problem-solve in context while considering citizen input and constraints on spending.

Type: Lesson Plan

Disappearing Frogs: Percentage and Environment:

Students must explore and assess the implications various human and environmental factors are having on the yellow-legged frog population in California. Then, they must choose one avenue to attempt to help save these animals. Some options will work quickly, while others will take time to implement. However, the ones that take longer to implement are generally more likely to be effective for a longer period of time. Students will use knowledge of percentages to calculate population size and will complete research to explore the affects of human impact on the environment and the process of adaptation through natural and artificial selection.

Type: Lesson Plan

How does scale factor affect the areas and perimeters of similar figures?:

In this lesson plan, students will observe and record the linear dimensions of similar figures, and then discover how the values of area and perimeter are related to the ratio of the linear dimensions of the figures.

Type: Lesson Plan

Markup and Make Money:

In this lesson students will create their own store with at least 15 items to sell. They will begin with a discussion and then learn about markup. They will use their knowledge to calculate prices and create a display for their store. This is the first of 2 lessons on this standard with the following lesson, Savvy Shopping, Resource ID 48879, allowing students to shop in their peer's store to calculate discount and tax.

Type: Lesson Plan

What happened to my money? Part 1:

In this lesson, students will extend their understanding of percentages to problem solve with taxes, in context, while learning about some of the different types of taxes.

Type: Lesson Plan

What happened to my money? Part 2:

In this lesson, students will extend their understanding of percentages to problem solve with taxes, in context, while exploring how taxes impact local communities.

Type: Lesson Plan

## Original Student Tutorials

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

Estimating Tax and Tip:

Follow Hailey and Kenna as they estimate tips and sales tax at the mall, restaurants, and the hair salon in this interactive tutorial.

Type: Original Student Tutorial

Math at the Mall: Markups and Markdowns:

Let's calculate markups and markdowns at the mall and follow Paige and Miriam working in this interactive tutorial.

Type: Original Student Tutorial

Simple Interest:

Calculate simple interest and estimate monthly payments alongside a loan officer named Jordan in this interactive tutorial.

Type: Original Student Tutorial

Taxes, Fees, and Commission:

Explore sales tax, fees, and commission by following a customer service representative named Julian in this interactive tutorial.

Type: Original Student Tutorial

The Percent Times: Percent Increase and Decrease:

Learn to solve percent change problems involving percent increases and decreases in in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

Statistical Sampling Results in setting Legal Catch Rate :

Fish Ecologist, Dean Grubbs, discusses how using statistical sampling can help determine legal catch rates for fish that may be endangered.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Unit Rate: Spring Water Bottling:

Nestle Waters discusses the importance of unit rate in the manufacturing process of bottling spring water.

Type: Perspectives Video: Professional/Enthusiast

Unit Rate and Florida Cave Formation:

How long does it take to form speleothems in the caves at Florida Caverns State Parks?

Type: Perspectives Video: Professional/Enthusiast

The Science and Math Behind Sour Fizzy Candy:

Master candymaker Wes Raley describes the process and science behind making sour fizzy candy.

Type: Perspectives Video: Professional/Enthusiast

Pizza Pi: Area, Circumference & Unit Rate:

How many times larger is the area of a large pizza compared to a small pizza?  Which pizza is the better deal? Michael McKinnon of Gaines Street Pies talks about how the area, circumference and price per square inch is different depending on the size of the pizza.

Type: Perspectives Video: Professional/Enthusiast

Amping Up Violin Tuning with Math:

Kyle Dunn, a Tallahassee-based luthier and owner of Stringfest, discusses how math is related to music.

Type: Perspectives Video: Professional/Enthusiast

Have a need for speed? Get out your spreadsheet! Race car drivers use algebraic formulas and spreadsheets to optimize car performance.

Type: Perspectives Video: Professional/Enthusiast

Building Scale Models to Solve an Archaeological Mystery:

An archaeologist describes how mathematics can help prove a theory about mysterious prehistoric structures called shell rings.

Type: 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

Coffee Mathematics: Ratios and Total Dissolvable Solids:

Math - the secret ingredient for an excellent cup of coffee!

Type: Perspectives Video: Professional/Enthusiast

A Golden Crown?:

This is a challenge problem that includes an assessment rubric. It involves working with volume, mass, and density. The setting is historical, modeling the Archimedes Golden Crown problem, when Archimedes proved that the king's crown was not pure gold.

Anna in D.C.:

The purpose of this task is to give students an opportunity to solve a challenging multistep percentage problem that can be approached in several different ways. Students are asked to find the cost of a meal before tax and tip when given the total cost of the meal. The task can illustrate multiple standards depending on the prior knowledge of the students and the approach used to solve the problem.

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.

Sharing Prize Money:

Students are asked to determine how to distribute prize money among three classes based on the contribution of each class.

Sand Under the Swing Set:

The 7th graders at Sunview Middle School were helping to renovate a playground for the kindergartners at a nearby elementary school. City regulations require that the sand underneath the swings be at least 15 inches deep. The sand under both swing sets was only 12 inches deep when they started. The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?

Art Class, Assessment Variation:

Art Class, Variation 1:

Students are asked to use ratios and proportional reasoning to compare paint mixtures numerically and graphically.

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.

Use the information provided to answer the questions regarding Carlos and his bananas

This is a task where it would be appropriate for students to use technology such as a graphing calculator or GeoGebra, making it a good candidate for students to engage in Standard for Mathematical Practice 5 Use appropriate tools strategically. A variant of this problem is appropriate for 8th grade; see Coffee by the Pound.

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?

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.

Comparing Years:

Students are asked to make comparisons among the Egyptian, Gregorian, and Julian methods of measuring a year.

Cooking with the Whole Cup:

Students are asked to use proportional reasoning to answer a series of questions in the context of a recipe.

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. This can be done by making a table, which helps illustrate the pattern of taxi rates for different distances traveled and with a little persistence leads to a solution which uses arithmetic. It is also possible to calculate a unit rate (dollars per mile) and use this to find the distance directly without making a table.

Finding a 10% Increase:

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?

Friends Meeting on Bikes:

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

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. This problem can be thought of in several ways; in particular, this problem also provides an opportunity for students to work with the "How many in one group?'' interpretation of division.

Molly's Run, Assessment Variation:

Use the information provided to find out how long it will take Molly to run one mile.

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 "Music Companies, Variation 2" for a task with a very similar setup but is much more involved and so illustrates .

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.

Robot Races:

Students should use information provided to answer the questions regarding robot races.

Robot Races, Assessment Variation:

Students must use interpret a graph to answer questions regarding speeds of racing robots.

Sale!:

Students are asked to determine which sale option results in the largest percent decrease in cost.

Selling Computers:

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. How many computers must the sales team sell to receive the bonus? Explain your reasoning.

Sore Throats, Variation 1:

Students are asked to decide if two given ratios are equivalent.

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.

Tax and Tip:

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. How much should you leave for the waiter? How much will the total bill be, including tax and tip?

The purpose of this task is for students to calculate the percent increase and relative cost in a real-world context. Inflation, one of the big ideas in economics, is the rise in price of goods and services over time. This is considered in relation to the amount of money you have.

Track Practice:

This activity asks the student to use unit rate and proportional reasoning to determine which of two runners is the fastest.

Two-School Dance:

The purpose of this task is to see how well students students understand and reason with ratios.

Lifting a Lion:

"Students will work in groups to solve a real-world problem presented by the book: How Do You Lift A Lion? Using a toy lion and a lever, students will discover how much work is needed to raise the toy lion. They will use proportions to determine the force needed to lift a real lion" from TI World Math.

SeaWorld Snack Shop - SeaWorld Classroom Activity:

In this problem-solving activity, challenges students to take on the role of a Food Services Manager placing orders for a snack shop at Sea World. To solve the problem they will use data and proportional reasoning to make predictions and communicate findings.

## Teaching Ideas

A Penny Saved is a Penny at 4.7% Earned:

There are lots of ways to receive income, and lots of ways to spend it. In this EconomicsMinute teaching idea, students will develop two budgets, or plans, to help them decide how to allocate their income.

Type: Teaching Idea

Statistics and Shopping:

This lesson is designed to develop students' understanding of taking percentages related to multiple markdowns and sale prices when shopping. This lesson provides links to discussions and activities related to the topic as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

Type: Teaching Idea

Scaling the Pyramids:

This web page features activities that compare the Great Pyramid to such modern structures as the Statue of Liberty and the Eiffel Tower. In the first activity, students use a template to construct a scale model of the Great Pyramid. They must find the scale heights for the tallest building in their neighborhood or for their height. In the remaining activity, students are given the dimensions for two other pyramids and challenged to create models.

Type: Teaching Idea

Top Speed At Sea-SeaWorld Classroom Activity:

In this activity, the students will calculate the top speeds of two dolphin species (killer whales and striped dolphin) and compare them to several marine animals' speeds.

Type: Teaching Idea

## Tutorials

Proportion Word Problem:

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

Type: Tutorial

Interpreting Graphs of Proportional Relationships:

This video shows how to recognize and understand graphs of proportional relationships to find the constant of proportionality.

Type: Tutorial

Solving a Proportion with an Unknown Variable :

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

Setting up Proportions to Solve Word Problems:

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

Determining Rates with Fractions:

This video demonstrates finding a unit rate from a rate containing fractions.

Type: Tutorial

Rate Problem With Fractions:

Watch as we solve a rate problem finding speed in meters per second using distance (in meters) and time (in seconds).

Type: Tutorial

Percent Word Problem:

Learn how to find the full price when you know the discount price in this percent word problem.

Type: Tutorial

Direct and Inverse Variation:

This video provides assistance with understanding direct and inverse variation.

Type: Tutorial

Using the Proportion Method to Solve Percent Problems:

This site explicitly outlines the steps for using the proportion method to solve three different kinds of percent problems. It also includes sample problems for practice determining the part, the whole or the percent.

Type: Tutorial

Converting Speed Units:

In this lesson, students will be viewing a Khan Academy video that will show how to convert ratios using speed units.

Type: Tutorial

## Unit/Lesson Sequences

Drawing to Scale: Designing a Garden:

In this lesson (or series of lessons), students interpret and use scale drawings to plan a garden layout. Students start by producing their own layout and then work together to refine their garden design. The activity requires that students use short rules (rulers), meter rules (meter sticks), string, protractors, scissors, glue, card, plain paper, graph paper, and colored pencils. Students work individually for 20 minutes, engage in a 100-minute lesson (or two 50-minute lessons), and complete a 10-minute follow up lesson or homework.

Type: Unit/Lesson Sequence

Direct and Inverse Variation:

"Lesson 1 of two lessons teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). Lesson 2 teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases." from Insights into Algebra 1 - Annenberg Foundation.

Type: Unit/Lesson Sequence

Percents: What's the Use?:

This activity focuses on the use of percents in situations involving discounts and taxes. The students are assigned an interview to discover the use of percents in various careers. Working in pairs and using shopping catalogues, they will further their knowledge of percents by calculating discounts and taxes. To access their knowledge of percents, there is a writing activity and an assignment to create a menu with questions and an answer key.

Type: Unit/Lesson Sequence

## Virtual Manipulatives

Graphing Lines:

This manipulative will help you to explore the world of lines. You can investigate the relationships between linear equations, slope, and graphs of lines.

Type: Virtual Manipulative

Mixtures:

In this online activity, students apply their understanding of proportional relationships by adding circles, either colored or not, to two different piles then combine the piles to produce a required percentage of colored circles. Students can play in four modes: exploration, unknown part, unknown whole, or unknown percent. This activity also includes supplemental materials in tabs above the applet, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

Curve Fitting:

With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic. Best fit or adjustable fit can be displayed.

Type: Virtual Manipulative

Planet Size Comparison: Ratio:

Images of two planets selected on two drop-down menus with a display of their respective diameters and the applicable ratio.

Type: Virtual Manipulative

## Student Resources

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

## Original Student Tutorials

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

Estimating Tax and Tip:

Follow Hailey and Kenna as they estimate tips and sales tax at the mall, restaurants, and the hair salon in this interactive tutorial.

Type: Original Student Tutorial

Math at the Mall: Markups and Markdowns:

Let's calculate markups and markdowns at the mall and follow Paige and Miriam working in this interactive tutorial.

Type: Original Student Tutorial

Simple Interest:

Calculate simple interest and estimate monthly payments alongside a loan officer named Jordan in this interactive tutorial.

Type: Original Student Tutorial

Taxes, Fees, and Commission:

Explore sales tax, fees, and commission by following a customer service representative named Julian in this interactive tutorial.

Type: Original Student Tutorial

The Percent Times: Percent Increase and Decrease:

Learn to solve percent change problems involving percent increases and decreases in in this interactive tutorial.

Type: Original Student Tutorial

## Educational Games

Estimator Four:

In this activity, students play a game of connect four, but to place a piece on the board they have to correctly estimate an addition, multiplication, or percentage problem. Students can adjust the difficulty of the problems as well as how close the estimate has to be to the actual result. This activity allows students to practice estimating addition, multiplication, and percentages of large numbers (100s). This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

Estimator Quiz:

In this activity, students are quizzed on their ability to estimate sums, products, and percentages. The student can adjust the difficulty of the problems and how close they have to be to the actual answer. This activity allows students to practice estimating addition, multiplication, or percentages of large numbers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

## Perspectives Video: Professional/Enthusiasts

Building Scale Models to Solve an Archaeological Mystery:

An archaeologist describes how mathematics can help prove a theory about mysterious prehistoric structures called shell rings.

Type: 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

Anna in D.C.:

The purpose of this task is to give students an opportunity to solve a challenging multistep percentage problem that can be approached in several different ways. Students are asked to find the cost of a meal before tax and tip when given the total cost of the meal. The task can illustrate multiple standards depending on the prior knowledge of the students and the approach used to solve the problem.

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.

Sharing Prize Money:

Students are asked to determine how to distribute prize money among three classes based on the contribution of each class.

Sand Under the Swing Set:

The 7th graders at Sunview Middle School were helping to renovate a playground for the kindergartners at a nearby elementary school. City regulations require that the sand underneath the swings be at least 15 inches deep. The sand under both swing sets was only 12 inches deep when they started. The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?

Art Class, Variation 1:

Students are asked to use ratios and proportional reasoning to compare paint mixtures numerically and graphically.

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.

Comparing Years:

Students are asked to make comparisons among the Egyptian, Gregorian, and Julian methods of measuring a year.

Cooking with the Whole Cup:

Students are asked to use proportional reasoning to answer a series of questions in the context of a recipe.

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. This can be done by making a table, which helps illustrate the pattern of taxi rates for different distances traveled and with a little persistence leads to a solution which uses arithmetic. It is also possible to calculate a unit rate (dollars per mile) and use this to find the distance directly without making a table.

Finding a 10% Increase:

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?

Friends Meeting on Bikes:

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

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. This problem can be thought of in several ways; in particular, this problem also provides an opportunity for students to work with the "How many in one group?'' interpretation of division.

Molly's Run, Assessment Variation:

Use the information provided to find out how long it will take Molly to run one mile.

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 "Music Companies, Variation 2" for a task with a very similar setup but is much more involved and so illustrates .

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.

Robot Races:

Students should use information provided to answer the questions regarding robot races.

Sale!:

Students are asked to determine which sale option results in the largest percent decrease in cost.

Selling Computers:

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. How many computers must the sales team sell to receive the bonus? Explain your reasoning.

Sore Throats, Variation 1:

Students are asked to decide if two given ratios are equivalent.

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.

Tax and Tip:

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. How much should you leave for the waiter? How much will the total bill be, including tax and tip?

The purpose of this task is for students to calculate the percent increase and relative cost in a real-world context. Inflation, one of the big ideas in economics, is the rise in price of goods and services over time. This is considered in relation to the amount of money you have.

Track Practice:

This activity asks the student to use unit rate and proportional reasoning to determine which of two runners is the fastest.

Two-School Dance:

The purpose of this task is to see how well students students understand and reason with ratios.

## Tutorials

Proportion Word Problem:

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

Type: Tutorial

Interpreting Graphs of Proportional Relationships:

This video shows how to recognize and understand graphs of proportional relationships to find the constant of proportionality.

Type: Tutorial

Solving a Proportion with an Unknown Variable :

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

Setting up Proportions to Solve Word Problems:

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

Determining Rates with Fractions:

This video demonstrates finding a unit rate from a rate containing fractions.

Type: Tutorial

Rate Problem With Fractions:

Watch as we solve a rate problem finding speed in meters per second using distance (in meters) and time (in seconds).

Type: Tutorial

Percent Word Problem:

Learn how to find the full price when you know the discount price in this percent word problem.

Type: Tutorial

Direct and Inverse Variation:

This video provides assistance with understanding direct and inverse variation.

Type: Tutorial

Using the Proportion Method to Solve Percent Problems:

This site explicitly outlines the steps for using the proportion method to solve three different kinds of percent problems. It also includes sample problems for practice determining the part, the whole or the percent.

Type: Tutorial

Converting Speed Units:

In this lesson, students will be viewing a Khan Academy video that will show how to convert ratios using speed units.

Type: Tutorial

## Virtual Manipulatives

Mixtures:

In this online activity, students apply their understanding of proportional relationships by adding circles, either colored or not, to two different piles then combine the piles to produce a required percentage of colored circles. Students can play in four modes: exploration, unknown part, unknown whole, or unknown percent. This activity also includes supplemental materials in tabs above the applet, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

Curve Fitting:

With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic. Best fit or adjustable fit can be displayed.

Type: Virtual Manipulative

## Parent Resources

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

## Perspectives Video: Professional/Enthusiasts

Building Scale Models to Solve an Archaeological Mystery:

An archaeologist describes how mathematics can help prove a theory about mysterious prehistoric structures called shell rings.

Type: 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

Anna in D.C.:

The purpose of this task is to give students an opportunity to solve a challenging multistep percentage problem that can be approached in several different ways. Students are asked to find the cost of a meal before tax and tip when given the total cost of the meal. The task can illustrate multiple standards depending on the prior knowledge of the students and the approach used to solve the problem.

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.

Sharing Prize Money:

Students are asked to determine how to distribute prize money among three classes based on the contribution of each class.

Sand Under the Swing Set:

The 7th graders at Sunview Middle School were helping to renovate a playground for the kindergartners at a nearby elementary school. City regulations require that the sand underneath the swings be at least 15 inches deep. The sand under both swing sets was only 12 inches deep when they started. The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?

Art Class, Assessment Variation:

Art Class, Variation 1:

Students are asked to use ratios and proportional reasoning to compare paint mixtures numerically and graphically.

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.

Use the information provided to answer the questions regarding Carlos and his bananas

This is a task where it would be appropriate for students to use technology such as a graphing calculator or GeoGebra, making it a good candidate for students to engage in Standard for Mathematical Practice 5 Use appropriate tools strategically. A variant of this problem is appropriate for 8th grade; see Coffee by the Pound.

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?

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.

Comparing Years:

Students are asked to make comparisons among the Egyptian, Gregorian, and Julian methods of measuring a year.

Cooking with the Whole Cup:

Students are asked to use proportional reasoning to answer a series of questions in the context of a recipe.

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. This can be done by making a table, which helps illustrate the pattern of taxi rates for different distances traveled and with a little persistence leads to a solution which uses arithmetic. It is also possible to calculate a unit rate (dollars per mile) and use this to find the distance directly without making a table.

Finding a 10% Increase:

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?

Friends Meeting on Bikes:

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

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. This problem can be thought of in several ways; in particular, this problem also provides an opportunity for students to work with the "How many in one group?'' interpretation of division.

Molly's Run, Assessment Variation:

Use the information provided to find out how long it will take Molly to run one mile.

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 "Music Companies, Variation 2" for a task with a very similar setup but is much more involved and so illustrates .

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.

Robot Races:

Students should use information provided to answer the questions regarding robot races.

Sale!:

Students are asked to determine which sale option results in the largest percent decrease in cost.

Selling Computers:

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. How many computers must the sales team sell to receive the bonus? Explain your reasoning.

Sore Throats, Variation 1:

Students are asked to decide if two given ratios are equivalent.

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.

Tax and Tip:

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. How much should you leave for the waiter? How much will the total bill be, including tax and tip?

The purpose of this task is for students to calculate the percent increase and relative cost in a real-world context. Inflation, one of the big ideas in economics, is the rise in price of goods and services over time. This is considered in relation to the amount of money you have.

Track Practice:

This activity asks the student to use unit rate and proportional reasoning to determine which of two runners is the fastest.

Two-School Dance:

The purpose of this task is to see how well students students understand and reason with ratios.

## Teaching Idea

A Penny Saved is a Penny at 4.7% Earned:

There are lots of ways to receive income, and lots of ways to spend it. In this EconomicsMinute teaching idea, students will develop two budgets, or plans, to help them decide how to allocate their income.

Type: Teaching Idea

## Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative