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Mathematics Grade 6

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Strand: RATIOS AND PROPORTIONAL RELATIONSHIPS (6.RP)

Understand ratio concepts and use ratio reasoning to solve problems (Standards 6.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. -
Apples to Apples

The purpose of this task is to connect students' understanding of multiplicative relationships to their understanding of equivalent ratios. -
Bag of Marbles

The purpose of this task is to help students develop fluency in their understanding of the relationship between fractions and ratios. It provides an opportunity to translate from fractions to ratios and then back again to fractions. -
Baking Bread 1

The purpose of this task is to provide students with a first informal experience with the idea of equivalent ratios. The intuitive notion that making multiple batches of the same recipe results in the same bread as a single recipe helps motivate the idea that different ratios can be "the same" in some important way. -
Baking Bread 2

The primary purpose of this task is to represent ratios of two or more quantities with parallel tape diagrams. Note that the solution to this task assumes that students have already studied equivalent ratios and understand that when you have a context with 8 units of one quantity and 2 units of another quantity, you can say the ratio is 4:1 because it is an equivalent ratio. -
Chapter 1 - Mathematical Foundations (UMSMP)

This is Chapter 1 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Ratio Relations. -
Chapter 1 - Student Workbook (UMSMP)

This is Chapter 1 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Ratio Relations. -
Chapter 2 - Mathematical Foundations (UMSMP)

This is Chapter 2 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Percent, Division with Fractions, and Measurement Conversion. -
Chapter 2 - Student Workbook (UMSMP)

This is Chapter 2 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Percent, Division with Fractions, and Measurement Conversion. -
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. -
Constant Speed

The purpose of this task is for students to learn to reason about whether or not ratios are equivalent using a diagram. -
Converting Square Units

Given the dimensions of a rectangular board, students must convert inches to feet, find the area of the board, and critique the reasoning the student in the problem uses the find the area. -
Currency Exchange

Given a scenario of a man traveling to another country and converting money students must determine the amount of the foreign currency he gets in exchange for his US dollars. -
Dana's House

In this task students are given the size of a lot on which a house is to be built. Given the square footage of the house, they must determine which percentage of the lot will be covered by the house. -
Data Transfer

This task asks the students to solve a real-world problem involving unit rates (data per unit time) using units that many teens and pre-teens have heard of but may not know the definition for. While the computations involved are not particularly complex, the units will be abstract for many students. -
Equivalent Ratios 1

The purpose of this task is for students to reason about whether two ratios are equivalent. -
Equivalent Ratios 2

The purpose of this task is to make explicit the meaning of equivalent ratios. Students create, analyze, and draw diagrams of two different sets of equivalent ratios, and then they write their own definition of "equivalent ratios" in their own words. -
Equivalent Ratios and Unit Rates

This task should come after students have done extensive work with representing equivalent ratios and understand that for any ratio a:b, the ratio sa:sb is equivalent to it for any s>0. The purpose of this task is to make explicit the fact that equivalent ratios have the same unit rate. -
Evaluating Ratio Statements

The goal of this task is to assess student understanding of ratios. The task offers five questions, some of which can be addressed using ony the given ratio, whereas others require knowledge of the total number of students. -
Exam scores

The goal of this task is to show how to apply ratio reasoning to calculate a percent. In order to do this task, students must know the meaning of percent, that is they need to know that a percent is a rate out of 100. The teacher may wish to encourage students to work with three different representations for the calculation: diagrams, ratio tables, and double number lines. -
Examining California's Prison System: Real-World Ratio

Using an infographic students look at such factors as age, gender and race to examine how the prison population in California compares to the general population. Students then apply an understanding of how they can find the value of a part by using a whole and a percent in order to look at how that can lead to recommendations for how to prevent crime. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. -
Fizzy Juice

The goal of this task is to provide an engaging context for students to work with ratios. -
Flexbook Chapter: Ratios, Proportions, and Percents

This chapter introduces students to ratios and rates, basic uses of proportions including understanding scale drawings, and percents. Also explored is the relationship among percents, decimals, and fractions. -
Fraction Conversion 2 (with percents)

When completing this lesson students will understand how to convert fractions, decimals, and percentages. -
Friends Meeting on Bicycles

Given a story about two friends who ride bikes to meet each other and the rate at which they travel, students must calculate the distance between them at specific times. -
Fruit Salad

The purpose of this task is for students to solve a contextual problem where there is a multiplicative relationship between several quantities in the context. These relationships can either be represented in a ratio table or with a linear equation. -
Games at Recess

In this task, given the scenario of students playing games at recess, students are asked to compare different aspects of the games using ratios and then writing sentences to express those ratios. -
Gianna's Job

The purpose of this task is to apply reasoning about ratios to solve a rate problem. This problem introduces a rate whose units are dollars per hour of work. Using this information, students need to make two separate calculations, one with units of dollars and the other with units of hours. -
Grade 6 Math Module 1: Ratios and Unit Rates (EngageNY)

Students begin their sixth grade year investigating the concepts of ratio and rate. They use multiple forms of ratio language and ratio notation, and formalize understanding of equivalent ratios. Students apply reasoning when solving collections of ratio problems in real world contexts using various tools (e.g., tape diagrams, double number line diagrams, tables, equations and graphs). Students bridge their understanding of ratios to the value of a ratio, and then to rate and unit rate, discovering that a percent of a quantity is a rate per 100. The 35 day module concludes with students expressing a fraction as a percent and finding a percent of a quantity in real world concepts, supporting their reasoning with familiar representations they used previously in the module. -
Grade 6 Mathematics

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 6 Unit 2: Rate, Ratio, and Proportional Reasoning Using Equivalent Fractions (Georgia Standard

In this unit, students will gain a deeper understanding of proportional reasoning through instruction and practice, develop and use multiplicative thinking, develop a sense of proportional reasoning, develop the understanding that ratio is a comparison of two numbers or quantities, find percents using the same processes for solving rates and proportions and solve real-life problems involving measurement units that need to be converted. -
Grade 6 Unit 4: One Step Equations and Inequalities (Georgia Standards)

In this unit students will: Determine if an equation or inequality is appropriate for a given situation. Solve mathematical and real-world problems with equations. Represent real-world situations as inequalities. Interpret the solutions to equations and inequalities. Represent the solutions to inequalities on a number line. Analyze the relationship between dependent and independent variables through the use of tables, equations and graphs. -
Gross Domestic Product: Unit Rates in the Real World

Using interactive maps from KQED students will examine the economic divide in European countries by looking at GDP. In the activity data from the maps is used to compare GDP among other countries. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. -
Hippos Love Pumpkins

The purpose of this task is for students to find unit rates in different situations involving unusual units. Most students are familiar with miles per hour, but students are unlikely to have encountered the idea of pumpkins per hippo or goats per pizza. By working with unusual (even silly) units, students must reason abstractly and quantitatively in order to answer the questions because they can't rely on their experience with the situation to guide them through it. -
Hunger Games versus Divergent

This is an engaging introductory lesson for a unit on ratio and proportional relationships. -
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. -
IXL Game: Ratios, proportions, and percents

This game helps sixth graders understand ratios, proportions, and percents, specifically percents of numbers and money amounts. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use. -
Jim and Jesse's Money

This task reads "Jim and Jesse each had the same amount of money. Jim spent $58 to fill the car up with gas for a road-trip. Jesse spent $37 buying snacks for the trip. Afterward, the ratio of Jims money to Jesse's money is 1:4. How much money did each have at first?" -
Kendall's Vase - Tax

For this task students are given this problem: "Kendall bought a vase that was priced at $450. In addition, she had to pay 3% sales tax. How much did she pay for the vase?" -
Life Expectancy: Finding Ratio Relationships

This lesson using infographics examines how health factors such as obesity and hypertension have changed in the U.S. over the past decades. The classroom activity has students work with the mathematical concept of statistical analysis as they compare the life expectancy of men and women. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. -
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. -
Mangos for Sale

The purpose of this task is to generate a classroom discussion about ratios and unit rates in context. -
Many Ways to Say It

The purpose of this task is to help students understand and use ratio language. -
Mixing Concrete

Given that the ratio of sand and cement of 5 : 3 is needed to make concrete, students must determine how many cubic feet of each are needed to make 160 cubic feet of concrete mix? -
Mixtures

This activity will help students understand percentages and mixture problems by working with two piles of colored chips. -
Overlapping Squares

In this task students are given a drawing showing two overlapping congruent squares. They must determine the area of the overlap. -
Painting a Barn

Given the dimensions of a barn, the square footage covered by a gallon of paint, and the price of the paint, students must find the cost of painting the barn and explain their work. -
Party Planning

The goal of this task is to provide a ratio problem which can be solved efficiently with a wide variety of techniques. While it could be used at many points in a ratio unit (with or without additional instructions on which technique to apply) one possible use of the task is as a summative assessment. -
Pennies to Heaven

The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task. -
Perfect Purple Paint I

The goal of this task is to provide a good context for engaging students in reasoning about ratios.The teacher may wish to use this task to demonstrate or introduce some of the different representations of ratios (ratio table, double number line, graphing points in the coordinate plane). The numbers are small so that the focus can be on the methods and not performing arithmetic. -
Price per pound and pounds per dollar

This task could be used by teachers to help students develop the concept of unit rates. Its purpose is to help students see that when you have a context that can be modeled with a ratio and associated unit rate, there is almost always another ratio with its associated unit rate (the only exception is when one of the quantities is zero), and to encourage students to flexibly choose either unit rate depending on the question at hand. -
Ratio of Boys to Girls

This is the task for this activity: "The ratio of the number of boys to the number of girls at school is 4:5. What fraction of the students are boys? If there are 120 boys, how many students are there altogether?" -
Ratios and Proportional Relationships (6.RP) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - Ratios and Proportional Relationships -
Real World Ratio and Rate Problem: Bianca's Fifty Percent Solution

Viewers follow Bianca as she's drawn into a store by the discounts advertised in this video from Cyberchase. While shopping she understands that while discounts are nice they still can add up when shopping. The classroom activity asks students to calculate the savings on an item when various discounts are applied. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. -
Real-World Proportional Relationships: Gender Wage Gap

Students use an infographic to understand how wages of today compare with those of 50 years ago in this lesson plan. The classroom activity helps students understand and calculate the wage gap using media salaries for men and women. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. -
Representing a Context with a Ratio

The purpose of this task is to introduce students to ratios and ratio language. -
Riding at a Constant Speed, Assessment Variation

Riding at a Constant Speed addresses aspects of 6.RP.2 "Understand the concept of a unit rate a/b associated with a ratio a:b" and 6.RP.3 "Use ratio and rate reasoning to solve real-world and mathematical problems." The numbers are chosen so that it would be easy to implement this task as a fill-in-the-blank item. -
Running at a Constant Speed

The purpose of this task is to give students experience in reasoning with equivalent ratios and unit rates from both sides of the ratio when given information about a runner and their pace. -
Same and Different

The purpose of this task is to analyze some very common contexts that can be represented by ratios and to motivate the idea of equivalent ratios for different kinds of contexts. It can also be used to introduce students to double number line diagrams. -
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. -
Security Camera

Students are given the scenario of a shop owner wants to prevent shoplifting. They are shown the shop floor plan and the rotation ability of the camera. They then must answer questions about which parts and percentages of the shop are now seen by the camera. -
Shirt Sale

In this task students are given the scenario of a student who buys a shirt at a percentage of the original price. They must calculate the original price and explain and show their work. -
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. -
Simple Unit Conversion Using Ratio Reasoning

The purpose of this instructional task is for students to use rate and ratio reasoning to solve unit conversion problems. In grade 6, unit conversion should be approached as a case of ratio reasoning, rather than a separate procedure to learn, and this task is an example of what that might look like. This task should come after students have spent time building up their understanding of equivalent ratios and are comfortable with some different representations of equivalent ratios. -
Speed Conversions

The goal of this task is to perform a unit conversion in the context of speed while also focusing on the precision of the conversion factor. Because the conversion rate is a decimal, this task should be used after students have gained some familiarity with ratio and rate reasoning. -
Sweet Tea

he purpose of this task is to help students understand why equivalent ratios of food ingredients result in mixtures that taste the same even if the individual quantities are different, and to use this understanding to determine whether two different mixtures will taste the same or not. The key move in reasoning is to compare ratios where one quantity is the same to see whether the other quantity is the same or different. -
The Escalator, Assessment Variation

This task presents a scenario about someone riding an escalator. Students are then given a series of statements such as "He traveled 2 meters every 5 seconds" and then asked to determine which of the statements are true. -
Ticket Booth

The goal of this task is to compare unit rates in a real world context. In addition to solving the problem by finding unit rates, students could also make a ratio table. -
Unit Conversions

The goal of this task is to study conversion between some volume and weight units. The focus of this task is understanding the relationship between multiplication, linear measurements, area, and volume. -
Voting for Three, Variation 1

In this first problem of three, students define the simple ratios that exist among three candidates in an election. It opens an opportunity to introduce unit rates. -
Voting for Three, Variation 2

In this problem, the total number of votes in the election and the number of votes for individual candidates is not provided. It provides the ratio of John's votes to Will's votes and enough information to compute the number of votes for Marie. -
Voting for Three, Variation 3

This is the last problem of seven in a series about ratios set in the context of a classroom election. Since the number of voters is not known, the problem is quite abstract and requires a deep understanding of ratios and their relationship to fractions. -
Voting for Two, Variation 1

This task can be used to solidify studentsâ€™ understanding of ratio tables or can be used to highlight how one can use unit rates to reason in a ratio context. -
Voting for Two, Variation 2

This task presents the problem: "John and Will ran for 6th grade class president. There were 36 students voting. John got two votes for every vote Will got. How many more votes did John get than Will?" -
Voting for Two, Variation 3

This problem, set in the context of a class election, is more than just a problem of computing the number of votes each person receives. This would be a good task type to help students see the connection between ratio tables and more abstract approaches to solving the problem. -
Voting for Two, Variation 4

This is the fourth in a series of tasks about ratios set in the context of a classroom election. Here is the first time that students will have to go beyond a ratio table to solve it. -
Walk-a-thon 1

In this task, students are given information about a context where there is a proportional relationship between two quantities in a table that has missing values. Students need to fill in the missing values, plot the corresponding points in the coordinate plane, and find the two unit rates that are associated with this proportional relationship. -
Which detergent is a better buy?

This purpose of this task is to provide a context for comparing ratios by using the example of laundry detergents, their costs, and how many loads they can do.

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.