- Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
- Solve unit rate problems including those involving unit pricing and constant speed.
*For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?* - Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
- Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
- Understand the concept of Pi as the ratio of the circumference of a circle to its diameter.

^{1}See Table 2 Common Multiplication and Division Situations)

### Clarifications

**Examples of Opportunities for In-Depth Focus**

When students work toward meeting this standard, they use a range of reasoning and representations to analyze proportional relationships.

**Subject Area:**Mathematics

**Grade:**6

**Domain-Subdomain:**Ratios & Proportional Relationships

**Cluster:**Level 2: Basic Application of Skills & Concepts

**Cluster:**Understand ratio concepts and use ratio reasoning to solve problems. (Major Cluster) -

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.

**Date Adopted or Revised:**02/14

**Date of Last Rating:**02/14

**Status:**State Board Approved

**Assessed:**Yes

**Assessment Limits :**

Rates can be expressed as fractions, with “:” or with words. Items may involve mixed units within each system (e.g. convert hours/min to seconds). Percent found as a rate per 100. Quadrant I only for MAFS.6.RP.1.3a.**Calculator :**No

**Context :**Allowable

**Test Item #:**Sample Item 1**Question:**Tom knows that in his school 10 out of every 85 students are left-handed. There are 391 students in Tom’s school.How many students in Tom’s school are left-handed?

**Difficulty:**N/A**Type:**EE: Equation Editor

**Test Item #:**Sample Item 2**Question:**The standard length of film on a film reel is 300 meters. On the first day of shooting a movie, a director uses 30% of the film on one reel.How long is the strip of film that was used?

**Difficulty:**N/A**Type:**EE: Equation Editor

**Test Item #:**Sample Item 3**Question:**In a circle, which ratio is equivalent to ?

**Difficulty:**N/A**Type:**MC: Multiple Choice

## Related Courses

## Related Access Points

## Related Resources

## Assessments

## Educational Games

## Formative Assessments

## Lesson Plans

## Original Student Tutorial

## Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

## Perspectives Video: Teaching Ideas

## Problem-Solving Tasks

## Student Center Activity

## Teaching Ideas

## Tutorials

## Video/Audio/Animations

## Virtual Manipulatives

## Worksheets

## STEM Lessons - Model Eliciting Activity

In this lesson students will utilize mathematical computation skills involving percentages and critical thinking skills to select the best tire deals advertised.

Students will create a working model that can determine the best regolith to binder solution for a settlement on Mars. The students are contacted by a company that requests their services. Students will read about, study and create their own lunarcrete (moon concrete). Students will work as a team to evaluate the provided data and determine which solution is most effective. Finally students will write a letter to the company defending their process giving reasons and data.

This MEA requires students to formulate a comparison-based solution to a problem involving choosing the BEST daycare based upon safety, playground equipment, meals, teacher to student ratio, cost, holiday availability and toilet training availability. 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. Students will receive practice on calculating a discount, finding the sum of the discounts, working with ratios and ranking day cares based on the data given.

In this MEA, the students will compare data to decide which school would be the best for a couple's son who is transferring into the county.

Students will help create a championship volleyball team by selecting 4 volleyball players to be added to open positions on the team. The students will use quantitative (ratios and decimals) and qualitative data to make their decisions.

Students will design specific nose cones for a water bottle rocket. They will test them to find out and rate which one is most effective in terms of accuracy, speed, distance, and cost effectiveness. This information will be used as criteria for a company that designs nose cones for orbitary missions.

Students will learn about how investing in education affects the economy by interpreting data and writing a persuasive letter to the Chamber of Commerce.

This Model Eliciting Activity (MEA) is written at a 6th grade level.

This MEA asks the students to decide on a lawn mower that will provide the Happy Lawns: Lawn Care Service with the best value for their money. Students are asked to rank order the lawn mowers in term of gas tank capacity, customer rating, speed, amount of time the mower takes to cut an acre of grass, shipping, and cost of the lawn mower. Students must provide a "Best Value" lawn mower to the company owner and explain how they arrived at their solution.

Students will analyze data to decide what blender to use, what size cups for adults, total ingredients needed, and create a variable that supports how many amounts and the total ounces of smoothies made.

Given a tight budget, students need to find the number of people that can be hired to film a soda commercial. Students will make the selection using a table that contains information about two types of extras. The union extra earns more money per hour than the non-union extra; however, the non-union extra needs more time to shoot the commercial than the union extra. In addition, students will select the design that would be used for the commercial taking into account the area that needs to be covered and the aesthetic factor.

This MEA requires students to formulate a comparison-based solution to a problem involving choosing the best neighborhood for Ms. Jasmine to purchase a house. 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.

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

The main problem students will need to solve is helping Lily Rae Wridenhoud find a route that will afford her the quickest time, least distance and highest customer satisfaction rating. Students will be given a map of all the streets leading around the neighborhood and customer rating (smiley faces). Students will need to use a ruler to figure out distances as well as decide elevation numbers on the topographic map. Then they will write out the route they have chosen to give Lily, and write a short explanation as to why this is the quickest and least distance traveled. Students will then be asked to look over their findings and be informed that some of the old clients have canceled the paper delivery and a few new paper clients have signed on. Does their new route still fit their findings?

The students will rank the local produce markets by using qualitative and quantitative data. The students will have to calculate unit rates and compare and order them.

Students will choose the best location for a family relocating and will consider all of the factors to make the best decision.

In this MEA, students will use their knowledge of weather and climate to select a location for a camera crew to visit in order to get high quality video footage of severe weather such as thunderstorms, blizzards, tornadoes, or hurricanes. The decision will be made using data about important weather factors such as air pressure, humidity, temperature, wind direction, and wind speed.

The purpose of this lesson is to solve real-world and mathematical problems using ratio and rate reasoning. Students will also use equivalent forms of decimals, percent applications to solve problems. They will write arguments to support claims with clear reasons and relevant evidence. Students will engage effectively in a range of collaborative discussions.

## MFAS Formative Assessments

Students are given the prices of three different quantities of cereal and are asked to determine which is the best buy.

Students are asked to solve rate problems given the time it takes each of two animals to run different distances.

Students are asked to explain the relationship between the circumference and diameter of a circle in terms of pi.

## Original Student Tutorials Mathematics - Grades 6-8

Learn how to identify explicit evidence and understand implicit meaning in a text.

You will be able to organize information in a table and write ratios equivalent to a given ratio in order to solve real-world and mathematical problems.

## Student Resources

## Original Student Tutorial

Learn how to identify explicit evidence and understand implicit meaning in a text.

You will be able to organize information in a table and write ratios equivalent to a given ratio in order to solve real-world and mathematical problems.

Type: Original Student Tutorial

## Educational Games

This fun and interactive game helps practice estimation skills, using various operations of choice, including addition, subtraction, multiplication, division, using decimals, fractions, and percents.

Various levels of difficulty make this game appropriate for multiple age and ability levels.

*Addition/**Subtraction:* The addition and subtraction of whole numbers, the addition and subtraction of decimals.

*Multiplication/Division: *The multiplication and addition of fractions and decimals.

*Percentages: *Identify the percentage of a whole number.

*Fractions: *Multiply and divde a whole number by a fraction, as well as apply properties of operations.

Type: Educational Game

This is a fun and interactive game that helps students practice ordering rational numbers, including decimals, fractions, and percents. You are planting and harvesting flowers for cash. Allow the bee to pollinate, and you can multiply your crops and cash rewards!

Type: Educational Game

## Problem-Solving Tasks

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. The total number of pennies minted either in a single year or for the last century is phenomenally large and difficult to grasp. One way to assess how large this number is would be to consider how far all of these pennies would reach if we were able to stack them one on top of another: this is another phenomenally large number but just how large may well come as a surprise.

Type: Problem-Solving Task

This problem asks the student to find a 3% sales tax on a vase valued at $450.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The purpose of this task is converting square units. Use the information provided to answer the questions posed. Since this task asks students to critique Jada's reasoning, it provides an opportunity to work on Standard for Mathematical Practice MAFS.K12.MP.3.1 - Construct Viable Arguments and Critique the Reasoning of Others.

Type: Problem-Solving Task

Students are asked to use a ratio to determine how much money Jim and Jesse had at the start of their trip.

Type: Problem-Solving Task

Students are asked to determine the percent of the area of a store covered by a security camera. Then, students are asked to determine the "best" place to position the camera and support their answer.

Type: Problem-Solving Task

Use the information provided to find out the original price of Selina's shirt. There are several different ways to reason through this problem; two approaches are shown.

Type: Problem-Solving Task

This problem is the fifth in a series of seven about ratios. At first glance the problem may look to be beyond MAFS.6.RP.1.3, which limits itself to "describe a ratio relationship between two quantities." However, even though there are three quantities (the number of each candidates' votes), they are only considered two at a time.

Type: Problem-Solving Task

This is the sixth problem in a series of seven that use the context of a classroom election. While it still deals with simple ratios and easily managed numbers, the mathematics surrounding the ratios are increasingly complex. In this problem, the students are asked to determine the difference in votes received by two of the three candidates.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The purpose of this task is to have students convert multiple currencies to answer the problem. Students may find the CDN abbreviation for Canada confusing. Teachers may need to explain the fact that money in Canada is also called dollars, so to distinguish them, we call them Canadian dollars.

Type: Problem-Solving Task

Use the information provided to find out what percentage of Dana's lot won't be covered by the house.

Type: Problem-Solving Task

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. The first solution relies more on reasoning about the meaning of multiplication and division, while the second solution uses units to help keep track of the steps in the solution process.

Type: Problem-Solving Task

Students are asked to use knowledge of rates and ratios to answer a series of questions involving time, distance, and speed.

Type: Problem-Solving Task

Given a ratio, students are asked to determine how much of each ingredient is needed to make concrete.

Type: Problem-Solving Task

This problem provides an interesting geometric context to work on the notion of percent. Two different methods for analyzing the geometry are provided: the first places the two squares next to one another and then moves one so that they overlap. The second solution sets up an equation to find the overlap in terms of given information which reflects the mathematical ideas described in cluster MAFS.6.EE.2 - Reason about and solve one-variable equations and inequalities.

Type: Problem-Solving Task

Students are asked apply knowledge of ratios to answer several questions regarding speed, distance and time.

Type: Problem-Solving Task

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Tutorials

Let's solve this word problem using what we know about equivalent ratios.

Type: Tutorial

In this example, we are given a ratio and then asked to apply that ratio to solve a problem. No problem!

Type: Tutorial

You are asked to find the percent when given the part and the whole.

Type: Tutorial

This video demonstrates how to find percent of a whole number.

Type: Tutorial

You're asked to find the whole when given the part and the percent.

Type: Tutorial

It's nice to practice conversion problems, but how about applying our new knowledge of percentages to a real life problem like recycling? Hint: don't forget your long division!

Type: Tutorial

This video demonstrates solving a unit price problem using equivalent ratios.

Type: Tutorial

This video talks about what percent really means by looking at a 10 by 10 grid.

Type: Tutorial

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

Type: Tutorial

## Video/Audio/Animations

Percentages are one method of describing a fraction of a quantity. the percent is the numerator of a fraction whose denominator is understood to be one-hundred.

Type: Video/Audio/Animation

Ratio errors confuse one of the coaches as two teams face off in an epic dodgeball tournament. See how mathematical techniques such as tables, graphs, measurements and equations help to find the missing part of a proportion.

Atlantean Dodgeball addresses number and operations standards, the algebra standard, and the process standard, as established by the National Council of Teachers of Mathematics (NCTM). It guides students in:

- Understanding and using ratios and proportions to represent quantitative relationships.
- Relating and comparing different forms of representation for a relationship.
- Developing, analyzing, and explaining methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
- Representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

Type: Video/Audio/Animation

## Virtual Manipulatives

This virtual manipulative allows the student to enter any two of the three quantities involved in percentage computation: the whole, a part and the percent. This manipulative can also be used for the discussions of relations among fractions, decimals, ratios and percentages.

Type: Virtual Manipulative

Using this virtual manipulative, students apply dimensional analysis (AKA factor-label method or unit-factor method) to solve unit conversion problems. There is also the opportunity to create your own unit conversion problems.

Type: Virtual Manipulative

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

## Parent Resources

## Problem-Solving Tasks

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. The total number of pennies minted either in a single year or for the last century is phenomenally large and difficult to grasp. One way to assess how large this number is would be to consider how far all of these pennies would reach if we were able to stack them one on top of another: this is another phenomenally large number but just how large may well come as a surprise.

Type: Problem-Solving Task

This problem asks the student to find a 3% sales tax on a vase valued at $450.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The purpose of this task is converting square units. Use the information provided to answer the questions posed. Since this task asks students to critique Jada's reasoning, it provides an opportunity to work on Standard for Mathematical Practice MAFS.K12.MP.3.1 - Construct Viable Arguments and Critique the Reasoning of Others.

Type: Problem-Solving Task

Students are asked to use a ratio to determine how much money Jim and Jesse had at the start of their trip.

Type: Problem-Solving Task

Students are asked to determine the percent of the area of a store covered by a security camera. Then, students are asked to determine the "best" place to position the camera and support their answer.

Type: Problem-Solving Task

Use the information provided to find out the original price of Selina's shirt. There are several different ways to reason through this problem; two approaches are shown.

Type: Problem-Solving Task

This problem is the fifth in a series of seven about ratios. At first glance the problem may look to be beyond MAFS.6.RP.1.3, which limits itself to "describe a ratio relationship between two quantities." However, even though there are three quantities (the number of each candidates' votes), they are only considered two at a time.

Type: Problem-Solving Task

This is the sixth problem in a series of seven that use the context of a classroom election. While it still deals with simple ratios and easily managed numbers, the mathematics surrounding the ratios are increasingly complex. In this problem, the students are asked to determine the difference in votes received by two of the three candidates.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The purpose of this task is to have students convert multiple currencies to answer the problem. Students may find the CDN abbreviation for Canada confusing. Teachers may need to explain the fact that money in Canada is also called dollars, so to distinguish them, we call them Canadian dollars.

Type: Problem-Solving Task

Use the information provided to find out what percentage of Dana's lot won't be covered by the house.

Type: Problem-Solving Task

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. The first solution relies more on reasoning about the meaning of multiplication and division, while the second solution uses units to help keep track of the steps in the solution process.

Type: Problem-Solving Task

Students are asked to use knowledge of rates and ratios to answer a series of questions involving time, distance, and speed.

Type: Problem-Solving Task

Given a ratio, students are asked to determine how much of each ingredient is needed to make concrete.

Type: Problem-Solving Task

This problem provides an interesting geometric context to work on the notion of percent. Two different methods for analyzing the geometry are provided: the first places the two squares next to one another and then moves one so that they overlap. The second solution sets up an equation to find the overlap in terms of given information which reflects the mathematical ideas described in cluster MAFS.6.EE.2 - Reason about and solve one-variable equations and inequalities.

Type: Problem-Solving Task

Students are asked apply knowledge of ratios to answer several questions regarding speed, distance and time.

Type: Problem-Solving Task

In this activity students calculate the ratio of chocolate to cereal when making a cake. Students then use that ratio to calculate to amount of chocolate and cereal necessary to make 21 cakes.

Type: Problem-Solving Task

## Video/Audio/Animation

Ratio errors confuse one of the coaches as two teams face off in an epic dodgeball tournament. See how mathematical techniques such as tables, graphs, measurements and equations help to find the missing part of a proportion.

Atlantean Dodgeball addresses number and operations standards, the algebra standard, and the process standard, as established by the National Council of Teachers of Mathematics (NCTM). It guides students in:

- Understanding and using ratios and proportions to represent quantitative relationships.
- Relating and comparing different forms of representation for a relationship.
- Developing, analyzing, and explaining methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
- Representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

Type: Video/Audio/Animation

## Virtual Manipulatives

This virtual manipulative allows the student to enter any two of the three quantities involved in percentage computation: the whole, a part and the percent. This manipulative can also be used for the discussions of relations among fractions, decimals, ratios and percentages.

Type: Virtual Manipulative

Using this virtual manipulative, students apply dimensional analysis (AKA factor-label method or unit-factor method) to solve unit conversion problems. There is also the opportunity to create your own unit conversion problems.

Type: Virtual Manipulative