### Clarifications

*Clarification 1:*Instruction includes the use of tables, tape diagrams and number lines.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**6

**Strand:**Algebraic Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Customary units
- Metric units
- Rate
- Unit Rate

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In elementary grades, students developed understanding of fraction concepts and operations, often in the context of the relationship that relates parts to wholes. In grade 6, students solve problems involving ratio relationships. In grade 7, students will solve problems involving proportional relationships. This benchmark is the culmination of the MA.6.AR.3 standard. It combines all of the concepts developed in the previous four benchmarks and applies them to a variety of mathematical and real-world contexts, including conversions of units within a measurement system. In grade 7, students are expected to convert units across different measurement systems.- Instruction includes a variety of problem types including comparison, mixtures, lengths and conversions. Students should work within the same measurement system, for example, standard measurement to standard measurement or metric system to metric system.
- It is not the expectation of this benchmark to solve problems using proportions; however, problem types will naturally start to build the foundation of proportions and proportional relationships.
- Instruction includes the use of bar models, tables and number lines to help students see relationships and relevance to real-world application
*(MTR.2.1)*.- For example, 3 toy cars cost Mikela $5 at the local superstore. Mikela needs 18 toy cars. How much will it cost Mikela to buy the cars?
- Bar Models
- Number Lines
- Tables

- Bar Models

- For example, 3 toy cars cost Mikela $5 at the local superstore. Mikela needs 18 toy cars. How much will it cost Mikela to buy the cars?
- Discussion about why students used certain methods to solve specific problems can help students see the patterns for determining their own processes in the future
*(MTR.2.1, MTR.5.1, MTR.7.1)*. - Instruction includes many opportunities for converting between units in the same measurement system, including conversions of areas and volumes and conversions that are commonly used in the science classroom.

### Common Misconceptions or Errors

- Some students may struggle to determine a process for solving in real-world context problems. Instruction highlights strategies for students to build perseverance by modifying methods as needed and analyze the problem in a way that makes sense
*(MTR.1.1)*.

### Strategies to Support Tiered Instruction

- Teacher provides opportunities for students to comprehend the context or situation by engaging in questions (writing questions on flash cards to reuse will be useful) such as:
- What do you know from the problem?
- What is the problem asking you to find?
- Can you create a visual model to help you understand or see patterns in your problem?

- Instruction includes using colored pencils to identify the units in each corresponding portion of a ratio and identifying the units before writing the numerical values.
- Instruction includes the use of a three-read strategy. Students read the problem three different times, each with a different purpose.
- First, read the problem with the purpose of answering the question: What is the problem, context, or story about?
- Second, read the problem with the purpose of answering the question: What are we trying to find out?
- Third, read the problem with the purpose of answering the question: What information is important in the problem?

- Instruction highlights strategies for students to build perseverance by modifying methods as needed and analyzing the problem in a way that makes sense.

### Instructional Tasks

*Instructional Task 1*

**(MTR.7.1)**On the first day of camp, Sara hiked 20 miles in 5 hours.

- Part A. At this rate, how long would it take Sara to hike 12 miles? Show and explain how you determined this.
- Part B. At this same rate, how many miles could Sara hike in 2 hours? Show and explain how you determined this.

*Instructional Task 2*

**(**MTR.7.1**)**Chris and Jenny are comparing two similar punch recipes. Each recipe calls for cranberry juice and ginger ale but in different amounts. The tables below show the amounts of cranberry juice and ginger ale for four different quantities of punch.

- Part A. Is the ratio of the punch that is cranberry juice the same in each of Chris’s recipes given in his table? Explain how you determined your answer.
- Part B. Is the ratio of the punch that is cranberry juice the same in each of Jenny’s recipes given in her table? Explain how you determined your answer.
- Part C. Is the ratio of the punch that is cranberry juice the same in Chris’s recipes as it is in Jenny’s recipes? If not, whose punch has a greater concentration of cranberry juice? Explain how you determined your answer.

### Instructional Items

*Instructional Item 1*

A recent study found that parking lots for offices should have a ratio of 6 spaces for every 1000 square feet of floor space. If a new office building has 19,000 square feet of floor space, how many spaces should there be in the parking lot?

*Instructional Item 2*

Jessica made 8 out of 24 free throws. Bob made 5 out of 20 free throws. Who has the highest free throw ratio?

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

## Perspectives Video: Teaching Idea

## Problem-Solving Tasks

## Tutorials

## STEM Lessons - Model Eliciting Activity

In this Model Eliciting Activity, MEA, 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. Students will find the unit rate of the lunacrete mixes. Finally, students will write a letter to the company defending their process giving reasons and data.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

In this Model Eliciting Activity (MEA), students will learn how to use very different pieces of information and data to select the best "Bottymals" for a company that wants to manufacture them and place them on the market. The MEA includes information about animal/insect anatomy (locomotion), manufacturing materials used in robotics, and physical science of the 6th grade level. Extensive information is provided to students, thus pre-requisites are minimal.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

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.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

In this Model Eliciting Activity, MEA, 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. Experienced extras earn more money per hour than novice extras; however, novice extras need more time to shoot the commercial than experienced extras. 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.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Students will be helping Lily Rae find the most efficient delivery route by using speed and distance values to calculate the shortest time to make it to all of her customers.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

In this Model Eliciting Activity, MEA, students will analyze data to decide what blender to use, the number of times the recipes are used and the total ingredients needed.

## 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.

## Student Resources

## Perspectives Video: Professional/Enthusiast

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

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

The purpose of this task is converting square units. Use the information provided to answer the questions posed. This task asks students to critique Jada's reasoning.

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

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

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

## Tutorials

In this video, watch as we solve this word problem using what we know about equivalent ratios.

Type: Tutorial

In this video, a ratio is given and then applied to solve a problem.

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

## Parent Resources

## Perspectives Video: Professional/Enthusiast

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

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

The purpose of this task is converting square units. Use the information provided to answer the questions posed. This task asks students to critique Jada's reasoning.

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

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

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

There are two aspects to fluency with division of multi-digit numbers: knowing when it should be applied, and knowing how to compute it. While this task is very straightforward, it represents the kind of problem that sixth graders should be able to recognize and solve relatively quickly. Easily recognizing contexts that require division is a necessary conceptual prerequisite to more complex modeling problems that students will be asked to solve later in middle and high school.

This task also has a natural carryover to work with ratios and rates, so students should also be building connections between these kinds of division problems and finding unit rates.

Type: Problem-Solving Task