unit percents homework 1 answer key

Math  /  6th Grade  /  Unit 2: Unit Rates and Percent

Unit Rates and Percent

Students investigate rates and percentages by identifying the rates associated with a ratio, defining a percent as a rate per 100, and applying strategies to solve rate and percent problems.

Unit Summary

In Unit 2, 6th grade students continue and extend their study of ratios to investigate rates and percentages. Using their knowledge of ratios, students will learn to identify two rates associated with a ratio and use them as efficient strategies to solve rate problems. Students will draw on their prior knowledge of the two measurement systems, and see unit conversions as applications of ratio and rate problems. Lastly, students will define a percent as a rate per 100, and understand that the strategies you use to solve a rate problem can also be applied to solve a percent problem. Throughout the entire unit, students will draw on their ability to reason abstractly and quantitatively (MP.2). As they work with different unit rates and conversion factors, students will need to attend to the units at hand, what the quantities mean, and how all of the pieces fit together. Calculators are recommended for several lessons in order to provide students with the option to use the tool in their calculations (MP.5).

In 4th grade and 5th grade , students interpreted fractions as division problems and began to make the connection between fractions and decimals. They multiplied fractions by whole numbers and other fractions in context of real-world problems, and they reasoned about what happens to a quantity when you multiply it by a number greater than one or less than one. Students will draw on these prior skills and understandings as they make connections between unit rates and fractions, and between fractions, decimals, and percentages. 

Beyond this unit, students will revisit percentages in Unit 6 when they study equations as another strategy to solve percent problems. In 7th grade , students will solve even more complex ratio, rate, and percent problems involving, for example, tax and percent increase or decrease. Additionally, students will investigate and analyze proportional relationships between quantities, and use more efficient and abstract methods to solve problems. For example, in 7th grade , students will learn to set up and solve proportion equations, using their knowledge of unit rates and equivalent ratios. (Note that using proportions to solve ratio or rate problems is not an expectation in sixth grade).

Pacing: 19 instructional days (14 lessons, 4 flex/review days, 1 assessment day)

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The following assessments accompany Unit 2.

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

Pre-Unit Student Self-Assessment

Have students complete the Mid-Unit Assessment after lesson 7.

Use the resources below to assess student understanding of the unit content and action plan for future units.

Post-Unit Assessment

Post-Unit Assessment Answer Key

Post-Unit Student Self-Assessment

Use student data to drive instruction with an expanded suite of assessments. Unlock Pre-Unit and Mid-Unit Assessments, and detailed Assessment Analysis Guides to help assess foundational skills, progress with unit content, and help inform your planning.

Intellectual Prep

Suggestions for how to prepare to teach this unit

Before you teach this unit, unpack the standards, big ideas, and connections to prior and future content through our guided intellectual preparation process. Each Unit Launch includes a series of short videos, targeted readings, and opportunities for action planning to ensure you're prepared to support every student.

Internalization of Standards via the Post-Unit Assessment

• Standards that each question aligns to
• Strategies and representations used in daily lessons
• Relationship to Essential Understandings of unit 
• Lesson(s) that Assessment points to

Internalization of Trajectory of Unit

• Read and annotate the Unit Summary.
• Notice the progression of concepts through the unit using the Lesson Map.

Essential Understandings

• Connection to Post-Unit Assessment questions
• Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

Unit-Specific Intellectual Prep

• The Progressions for the Common Core State Standards in Mathematics 6-7, Ratios and Proportional Relationships  is valuable to read before the unit.
• UnboundEd also has a great 6th grade content guide on ratios, Ratios: Unbound, A Guide to Grade 6 Mathematics Standards .
• Read the following table that includes models used throughout the unit.

The central mathematical concepts that students will come to understand in this unit

• A rate, associated with a ratio a:b, is a/b or b/a units of one quantity per 1 unit of the other quantity. For example, if a person walks 6 miles in 2 hours, the person is traveling at a rate of 3 miles per hour, or equivalently a rate of 1/3 hour per mile. 
• A percent is a rate per 100; percent problems can be solved using the same strategies that are used for other rate problems.
• There are many applications to rate, including unit price, constant speed, and measurement conversions, that can be solved using several different strategies, such as using double number lines, tables, tape diagrams, and the unit rate. 

Terms and notation that students learn or use in the unit

conversion rate

customary system

equivalent ratio

metric system

percent/ percentage

To see all the vocabulary for Unit 2, view our 6th Grade Vocabulary Glossary .

The materials, representations, and tools teachers and students will need for this unit

• Calculators (1 per student) — It is recommended that students use calculators in this lesson since some of the problems involve decimals. Students have experience with calculating with decimals from fifth grade, but they will study them more in depth and with fluency in Unit 3. Calculators will ensure the focus of the lesson is on the concepts and reasoning, rather than calculations (MP.5).
• Massachusetts Comprehensive Assessment System Grade 6 Mathematics Reference Sheet (1 per student) — We recommend the MCAS reference sheet, however, other reference sheets may suffice as well.

To see all the materials needed for this course, view our 6th Grade Course Material Overview .

Topic A: Defining Rate & Solving Rate Problems

Investigate and use rate in real-world situations.

6.RP.A.3 6.RP.A.3.B

Define rate and unit rate, and find rates from situations involving ratios. 

6.RP.A.2 6.RP.A.3.B

Find unit rates and use them to solve problems.

Compare situations using unit rates, including speed, price, and work problems.

Solve challenging problems involving unit rate.

6.RP.A.2 6.RP.A.3 6.RP.A.3.B

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Topic B: Measurement Unit Conversions

Reason with measurement units and solve unit conversion problems using different strategies.

Solve measurement unit conversion problems.

Topic C: Percent

Define percent as a rate per 100. Find percentages of 100 and 1.

Solve percent problems using benchmark fractions and percentages, with and without written work.

Convert between fractions, percent, and decimals.

Find a percent of a quantity. 

Find the whole quantity, given a part and percent.

Solve percent problems, including finding percent when given a part and a whole. 

Solve multi-step problems involving rate and percent.

6.RP.A.2 6.RP.A.3 6.RP.A.3.C 6.RP.A.3.D

Common Core Standards

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Ratios and Proportional Relationships

6.RP.A.2 — Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Expectations for unit rates in this grade are limited to non-complex fractions. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."

6.RP.A.3 — Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

6.RP.A.3.B — 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?

6.RP.A.3.C — 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.

6.RP.A.3.D — Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

Measurement and Data

4.MD.A.1 — Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …

4.MD.A.2 — Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

5.MD.A.1 — Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Number and Operations in Base Ten

5.NBT.B.6 — Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Number and Operations—Fractions

4.NF.B.4.B — Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

4.NF.B.4.C — Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

4.NF.C.6 — Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

5.NF.B.3 — Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

5.NF.B.4.A — Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

5.NF.B.5 — Interpret multiplication as scaling (resizing), by:

5.NF.B.5.A — Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

5.NF.B.5.B — Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

5.NF.B.6 — Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

6.RP.A.1 — Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."

Future Standards

Standards in future grades or units that connect to the content in this unit

Expressions and Equations

6.EE.B.7 — Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

6.EE.C.9 — Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

7.RP.A.1 — Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction ^1/2/_1/4 miles per hour, equivalently 2 miles per hour.

7.RP.A.2 — Recognize and represent proportional relationships between quantities.

7.RP.A.3 — Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

CCSS.MATH.PRACTICE.MP6 — Attend to precision.

CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Understanding and Representing Ratios

Multi-Digit and Fraction Computation

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Course: 6th grade   >   Unit 3

• Finding a percent

Finding percents

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Percents Unit | Ratio and Percent Proportions Notes for 7th Grade Math

• Google Apps™

What educators are saying

Description.

A 10 day CCSS-Aligned Percents Unit including: solving real-life proportions, solving percent problems, percent of change, percent error, and simple interest. Students will apply both percent proportions and percent equations to real-world situations.

Students will practice with both skill-based problems, real-world application questions, and error analysis to support higher level thinking skills. You can reach your students and teach the standards without all of the prep and stress of creating materials!

Standards: 7.RP.2, 7.RP.3; Texas Teacher? Grab the TEKS-Aligned Proportionality Unit. Please don’t purchase both as there is overlapping content.

Learning Focus:

• use proportional relationships to solve multi-step ratio and percent problems
• solve problems involving simple interest
• apply percents to real-world situations, including percent of change and percent error

More details on what is included:

1. Unit Overviews

• Streamline planning with unit overviews that include essential questions, big ideas, vertical alignment, vocabulary, and common misconceptions.
• A pacing guide and tips for teaching each topic are included to help you be more efficient in your planning.

2. Student Handouts

• Student-friendly guided notes are scaffolded to support student learning. 
• Available as a PDF and the student handouts/homework/study guides have been converted to Google Slides™ for your convenience.

3. Independent Practice

• Daily homework is aligned directly to the student handouts and is versatile for both in class or at home practice. 

4. Assessments

• 1-2 quizzes, a unit study guide, and a unit test allow you to easily assess and meet the needs of your students.
• The Unit Test is available as an editable PPT, so that you can modify and adjust questions as needed.

5. Answer Keys

• All answer keys are included.

***Please download a preview to see sample pages and more information.***

How to use this resource:

• Use as a whole group, guided notes setting
• Use in a small group, math workshop setting
• Chunk each student handout to incorporate whole group instruction, small group practice, and independent practice.
• Incorporate our Percents Activity Bundle for hands-on activities as additional and engaging practice opportunities.

Time to Complete:

• Each student handout is designed for a single class period. However, feel free to review the problems and select specific ones to meet your student needs. There are multiple problems to practice the same concepts, so you can adjust as needed.

Is this resource editable?

• The unit test is editable with Microsoft PPT. The remainder of the file is a PDF and not editable.

Looking for hands-on classroom activities?

• Check out the corresponding Percents Activity Bundle , which includes activities like scavenger hunts, find it and fix its, mazes and more to allow students to engage and practice the concepts. Win-win!

More 7th Grade Units:

Unit 1: Rational Number Operations

Unit 2: Expressions and Equations

Unit 3: Inequalities

Unit 4: Proportional Relationships

Unit 5: Percents

Unit 6: Angles and Triangles

Unit 7: Plane Geometry and Similarity

Unit 8: Surface Area

Unit 9: Volume

Unit 10: Data and Statistics

Unit 11: Probability

More 7th Grade Activity Bundles:

Looking for more helpful teaching tips, ideas, and support? Check out Maneuveringthemiddle.com and join our online FB community MTM VIPS! 

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Licensing: 

This file is a license for ONE teacher and their students. Please purchase the appropriate number of licenses if you plan to use this resource with your team. Thank you!

Customer Service:

If you have any questions, please feel free to reach out for assistance .  We aim to provide quality resources to help teachers and students alike, so contact me before leaving feedback if you have a need. 

Maneuvering the Middle ® Terms of Use

Products by Maneuvering the Middle®, LLC may be used by the purchaser for their classroom use only. This is a single classroom license only. All rights reserved. Resources may only be posted online in an LMS such as Google Classroom, Canvas, or Schoology. Students should be the only ones able to access the resources.  It is a copyright violation to upload the files to school/district servers or shared Google Drives. See more information on our terms of use here . 

If you are interested in a personalized quote for campus and district licenses, please click here . 

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Student-Centered Math Lessons

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Percents Unit 7th Grade CCSS

A 10 day CCSS-Aligned Percents Unit including: solving real-life proportions, solving percent problems, percent of change, percent error, and simple interest. Students will apply both percent proportions and percent equations to real-world situations.

Description

Additional information.

• Reviews (0)

Students will practice with both skill-based problems, real-world application questions, and error analysis to support higher level thinking skills.  You can reach your students and teach the standards without all of the prep and stress of creating materials!

Standards:   7.RP.2, 7.RP.3;  Texas Teacher?  Grab the TEKS-Aligned Proportionality Unit.  Please don’t purchase both as there is overlapping content.

Learning Focus:

• use proportional relationships to solve multi-step ratio and percent problems
• solve problems involving simple interest
• apply percents to real-world situations, including percent of change and percent error

What is included in the 7th grade ccss Percents Unit?

1. Unit Overviews

• Streamline planning with unit overviews that include essential questions, big ideas, vertical alignment, vocabulary, and common misconceptions.
• A pacing guide and tips for teaching each topic are included to help you be more efficient in your planning.

2. Student Handouts

• Student-friendly guided notes are scaffolded to support student learning.
• Available as a PDF and the student handouts/homework/study guides have been converted to Google Slides™ for your convenience.

3. Independent Practice

• Daily homework is aligned directly to the student handouts and is versatile for both in class or at home practice.

4. Assessments

• 1-2 quizzes, a unit study guide, and a unit test allow you to easily assess and meet the needs of your students.
• The Unit Test is available as an editable PPT, so that you can modify and adjust questions as needed.

5. Answer Keys

• All answer keys are included.

***Please download a preview to see sample pages and more information.***

How to use this resource:

• Use as a whole group, guided notes setting
• Use in a small group, math workshop setting
• Chunk each student handout to incorporate whole group instruction, small group practice, and independent practice.
• Incorporate our  Percents Activity Bundle  for hands-on activities as additional and engaging practice opportunities.

Time to Complete:

• Each student handout is designed for a single class period. However, feel free to review the problems and select specific ones to meet your student needs. There are multiple problems to practice the same concepts, so you can adjust as needed.

Is this resource editable?

• The unit test is editable with Microsoft PPT. The remainder of the file is a PDF and not editable.

Looking for more 7 th Grade Math Material? Join our All Access Membership Community! You can reach your students without the “I still have to prep for tomorrow” stress, the constant overwhelm of teaching multiple preps, and the hamster wheel demands of creating your own teaching materials.

• Grade Level Curriculum
• Supplemental Digital Components
• Complete and Comprehensive Student Video Library 

Click here to learn more about All Access by Maneuvering the Middle®!

Licensing: This file is a license for ONE teacher and their students. Please purchase the appropriate number of licenses if you plan to use this resource with your team. Thank you!

Customer Service: If you have any questions, please feel free to reach out for assistance .  We aim to provide quality resources to help teachers and students alike, so please reach out if you have any questions or concerns. 

Maneuvering the Middle ® Terms of Use: Products by Maneuvering the Middle®, LLC may be used by the purchaser for their classroom use only. This is a single classroom license only. All rights reserved. Resources may only be posted online in an LMS such as Google Classroom, Canvas, or Schoology. Students should be the only ones able to access the resources.  It is a copyright violation to upload the files to school/district servers or shared Google Drives. See more information on our terms of use here . 

If you are interested in a personalized quote for campus and district licenses, please click here . 

©Maneuvering the Middle® LLC, 2012-present

This file is a license for one teacher and their students. Please purchase the appropriate number of licenses if you plan to use this resource with your team. Thank you!

Customer Service

We strive to provide quality products to help teachers and students alike, so contact us with any questions.

Maneuvering the Middle® Terms of Use

Products by Maneuvering the Middle, LLC may be used by the purchaser for their classroom use only. This is a single classroom license only. All rights reserved. Resources may only be posted online if they are behind a password-protected site.

Campus and district licensing is available please contact us for pricing.

©Maneuvering the Middle LLC, 2012-present

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This resource is often paired with:

Proportionality Unit 7th Grade TEKS

Percents Activity Bundle 7th Grade

Digital Math Activity Bundle 7th Grade