Math Grade 6 Ratios and Proportional Relationships …bboextra.org/CGuides/Middle School/First Nine Weeks...Math/Grade 6 Curriculum Guide Unit 1: Ratios and Proportional Relationships

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NMSI-Laying the Foundation ⑥ BCS “Big 6” Strategies

This curriculum guide is designed to support teachers in the implementation of the Alabama Course of Study Standards. You are encouraged to use this document tosupport your planning and daily instructional practices. It is not a substitution for lesson plans.This guide was created by Sharon Muhammad, BCS Middle School Math Curriculum Writer.

This document is property of Birmingham City Schools. Do not reproduce without permission.

Math/Grade 6

Curriculum Guide

Unit 1: Ratios and Proportional Relationships

Suggested Pacing: Seven to Eight- 90 minute class periods

Unit At A Glance:

This unit will focus on student fluency in understanding ratio concepts and the use of ratioreasoning to solve problems academically and in real world situations. It will focus onreinterpretation of fractions as ratios and creating ratios and proportions from given wordproblems. The use of ratio and reasoning to solve real-world problems and mathematicalproblems will also be a focus. Fluent calculation of numbers with decimals using mathoperations within standard algorithm will be a focus.

*In order to access all available resources, click on the hyperlink by holding down the “Ctrl” and “Enter” keyssimultaneously.

ACOS/CCRS Standard:

7. Find the greatest common factor of two whole numbers less than or equal to 100 and the leastcommon multiple of two whole numbers less than or equal to 12. Use the distributive property toexpress a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of twowhole numbers with no common factors. (6-NS4)

1. Understand the concept of a ratio, and using ratio language to describe a ratio relationshipbetween two quantities. (6-RP1)

3. Use ratio and reasoning to solve real world and mathematical problems, e.g. by reasoning

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about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.(6-RP3)

a. Make tables of equivalent ratios relating quantities with whole number measurements, findmissing values in the tables, and plot the pairs of values on the coordinate plane. Use tables tocompare ratios. (6-RP3a)

b. Solve unit rate problems including those involving unit pricing, and constant speed. (6-RP3b)Example: If it took seven hours to mow 4 lawns, then at that rate, how many lawns could bemowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100; solve problems involving finding the whole,given a part and the percent. (6-RP3c)

d. Use ratio and reasoning to convert measurement units; manipulate and transform unitsappropriately when multiplying or dividing quantities. (6-RP3d)

6. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithmfor each operation. (6-NS3)

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

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7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Essential Question(s):

How can I understand the concept of a ratio, and use ratio language to describe a ratiorelationship between two quantities?

How can I use ratio concepts and ratio reasoning to solve problems mathematically and in thereal world?

How can I solve unit rate problems including those involving unit pricing, and constant speed?

How can I fluently use the operations in standard algorithm to solve problems?

How can use estimation in a strategic manner to solve problems?

Essential Question Page 3 Glencoe Math Your Common Core Edition CCSS

“How do you use equivalent rates in the real world?

Learning Objective(s):I can express ratios and rates as fractions.

I can change units of measure in the customary system.

I can use ratio tables to represent and solve problems involving equivalent ratios.

I can graph ordered pairs in ratios tables to solve problems.

I can write decimals as fractions or mixed numbers and vice versa.

I can write percents as fractions and vice versa.

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I can write decimals as percents.

I can compare and order fractions, decimals, and percents.

I can estimate and find percent of numbers.

I can solve percent problems involving finding the whole or part.

I can add and subtract decimals.

I can estimate the product of decimals and judge the reasonableness of the results.

I can estimate and find products of decimals and whole numbers.

Key Vocabulary: Academic Describe Analyze Support

Content Specific

Ratio Rate Unit Rate Quantity Fraction Coordinate Plane Greatest Common Factor Least Common Multiple

Assessments:Formative Assessments will gauge the student’s fluency and proficiency levels during theinstruction of this unit. Use of observation allows the teacher to gauge student’s level of priorknowledge and current understanding of ratio concepts, use of ratio reasoning to solve problemsand fluently use math operations to calculate decimals. Teacher facilitated discussions willfurther evidence prior knowledge and level of current understanding of unit rate which isassociated with ratios. The teacher should utilize Formative Pencil-Paper Assessments throughinstruction of this unit to gauge current understanding. The data from this assessment will allowthe teacher to proactively monitor and adjust instruction as needed. 3,2,1 stimulates students’reflection on the days’ lesson and helps them process that learning. At the close of daily lessons,

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students will state 3 things learned, 2 things they would like to learn more about, and ask 1question related to the lesson. Think-Pair-Share should be a resource to effectively gaugelearning within group activities. Students will discuss, with a partner, what they have learnedbefore sharing with the whole class. ⑥ Quickwrite allows students to write and response to aspecific prompt for a short period of time. For example, students should be instructed to drawfour triangles, two squares and five circles. Ask students to write the ratio triangles to totalshapes.

Formative assessments can be used to adjust teaching and learning while they are happeningwithin the classroom. Formative assessments can inform both teachers and students wheninstructional adjustments should be made and helps teachers determine next steps during thelearning process.

Summative assessments can be used to gauge students’ learning relative to content standards.They are given to determine what students have mastered. Some examples of summativeassessments include, but are not limited to the following: state assessments, district benchmarks,weekly tests, end of unit tests, and end of term exams

Summative Assessment I: Pre-Test, August 25- September 5, 2014Post-Test, October 1-15, 2014

Common Core Review p. 14

Instructional Consideration:Before:The teacher should begin the unit with pre-assessment/probe resources which can be found atwww.mathisfun/ratios, or other online math resources. Assessment resources can also be foundin the text book Glencoe Math Your Common Core Edition CCSS Student Edition and Math YourCommon Core Edition CCSS, Assessment Masters. The teacher should utilize the “Are Youready?” component of this unit located on page 6 to gauge student proficiency levels of divisionwith whole number skills and simplifying fractions. These skills are essential to gainingstrategies for fluent calculation of ratio and rate problems. Instruction should transition to the“Vocabulary Start-up” activity on page 7. Students should also participate in Response Cardactivities in groups of 4 with GCF and LCM related vocabulary on them to build readingcomprehension of math content vocabulary for this unit. The Unit Project Preview titled PeopleEverywhere on page 2 of the textbook Glencoe Math Your Common Core Edition CCSS 1 should

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be completed by students to allow interaction with real-world problem solving involving ratios.Students should be allowed to work in groups and choose two populations at their school. Thestudents should then choose a characteristic, such as sports or number of siblings. A comparisonshould then be made of the two characteristics. The results should then be shown on a doublebar graph. Before instruction the teacher should engage in peer collaboration as the unit beginsand during instruction of the unit. Collaboration can occur with leveled math team members oncampus or teacher peers off campus. The teacher collaboration should include discussion of bestinstructional strategies for transitioning students through the unit. Emphasis should be placed onteaching strategies which will lead to fluent and proficient calculation of unit rates or rates byusing ratios or proportions. Students should also gain strategies which will allow them tofluently solve word problems involving: unit rate, unit price, and constant speed.During:The teacher should facilitate student discussion of ratios by defining what ratios are:

Ratios

A ratio compares values.

A ratio says how much of one thing there is compared to another thing.

There are 3 blue squares to 1 yellow square

www.mathisfun/ratios

Teacher facilitated student discussion should also include the definition and explanations ofproportions:

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Proportions

Proportion says that two ratios (or fractions) are equal.

1/3, 1:3, 1 to 3, 1 out of 3, are the same as:

Pre-AP strategies to consider (explain)Differentiation:

Example:

So 1-out-of-3 is equal to 2-out-of-6

Instruction should transition to gaining strategies to find the GCF and LCM:

Greatest Common Factor

A common factor is a number that is a factor of two or more numbers.

The greatest of the common factors of two or more numbers is called the Greatest Common

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Factor. (GCF)

Let's start with an Example: ...

Greatest Common Factor of 12 and 16

1. Find all the Factors of each number,2. Circle the Common factors,3. Choose the Greatest of those

So ... what is a "Factor"?

Factors are the numbers you multiply together to get another number:

A number can have many factors:

Factors of 12 are 1, 2, 3, 4, 6 and 12 ...

... because 2 × 6 = 12, or 4 × 3 = 12, or 1 × 12 = 12.

Least Common Multiple

The least number that is a multiple of two or more whole numbers is the Least Common Multiple(LCM) of the numbers.

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Let's start with an Example...

Least Common Multiple of 3 and 5:

• List the Multiples of each number,

• Find the first Common (same) value

The Least Common Multiple of 3 and 5 is 1515 is a common multiple of 3 and 5, and is the smallest (least) common multiple

So ... what is a "Multiple"?

We get a multiple of a number when we multiply it by another number (such as multiplyingby 1, 2, 3, 4, 5, etc). Just like the multiplication table.

Here are some examples:

The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,...

The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...

What is a "Common Multiple"?

Say we have listed the first few multiples of 4 and 5: the common multiples are those that arefound in both lists:

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The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,...

The teacher should facilitate discussion about decimals and the use or math operations to

calculate them:

Definition of Decimal

The word "Decimal" really means "based on 10" (From Latin decima: a tenth part).

We sometimes say "decimal" when we mean anything to do with our numbering system, but a"Decimal Number" usually means there is a Decimal Point.

And that is a Decimal Number!

We can continue with smaller and smaller values, from tenths, to hundredths, and so on, like inthis example:

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The ratios are the same, so they are in proportion.

Students will activate prior knowledge through the use of open ended questioning by the

teacher, visual cues, tactile activities, and manipulative activities such as fruit, stars and planets,

cookies, candy. These manipulatives can easily be downloaded through google image searches

and laminated with adhesive backs placed on them for use on the board. Students will engage in

frequent skill building of using strategies to solve mathematical problems involving ratios and

proportions through work sheet tasks which can be found at www.mathisfun/ratios and

www.educationworld.com. Group activity involving the strategies ⑥ 3,2,1 and ⑥ Think-Pair-

Share should be used often during instruction of this unit.

Students will perform rate to ratio conversions using tactile/ manipulative activities, watch ratioand rate calculation in real world via in-class video https://www.khanacademy.org/math/pre-algebra/rates-and-ratios/ratios_and_proportions/v/introduction-to-ratios--new-hd-version , andread literature which is ratio, rate, and quantity embedded. This literature should include utilitybills http://www.homosassawater.com/images/Blank-Customer-Water-Bill.jpg , businessmagazines, and the daily newspaper. Students will make tables of equivalent ratios relatingquantities with whole number measurements. Students will plot pairs of values on the coordinate

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plane from completed tables and find missing values where needed on the tables. Resources forskills practice of these mathematical strategies are located in the “Practice and Apply” sectionof the text book p. 11, for this unit. Students will use tables to compare ratios and will solve rateproblems involving unit, pricing, and constant speed. Assessments will gauge fluency andproficiency during progression of instruction of problem solving strategies. Introduction to real-world situations should begin with the examples shown on p. 3 of Glencoe Math Your CommonCore Edition, CCSS Teacher’s Edition, entitled Math in the Real World, which depicts aCheetah running at full speed. This mini-lesson allows the students to gain information from thetext about Cheetahs with mathematical facts embedded. The students are then prompted tocalculate the answer to the word problem embedded in the factual text. Students should beallowed to engage in Turn and Talk to solve the problem.

After:Students will show proficiency in the calculation of unit rates and ratios, prove theirunderstanding of the concept of ratios and proportional relationships, and reasoning through unit-end project presentations and assessments. A project should include interaction with real-worldencounters with rates and ratios. Students should involve use of utility bills in their projects andpresent how calculation of rates are involved in the utility bills which are paid each month intheir homes. Students will make real-world connection to using ratio reasoning to solveproblems, calculate unit rate by using ratios or proportions, and writing ratios as fractions byutilizing site searches at www.youtube.com, www.google.com, www.discoverychannel.com, andwww.nationalgeographicmagazine.com . Students will see videos of peers interacting with ratiosand proportions in real-world context. Students should complete the end of unit project PeopleEverywhere as an independent project. Additionally, the teacher would have more time to guide

challenging activities like NMSI lesson called “Rating the Trip”. In this lesson studentsinvestigate the concept of rate. The activity could also be used for ratios and proportions.

Students will perform rate to ratio conversions using tactile/ manipulative activities, watch ratioand rate calculation in real world via in-class video https://www.khanacademy.org/math/pre-algebra/rates-and-ratios/ratios_and_proportions/v/introduction-to-ratios--new-hd-version , andread literature which is ratio, rate, and quantity embedded. This literature should include utilitybills http://www.homosassawater.com/images/Blank-Customer-Water-Bill.jpg , businessmagazines, and the daily newspaper. Students will make tables of equivalent ratios relatingquantities with whole number measurements. Students will plot pairs of values on the coordinate

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plane from completed tables and find missing values where needed on the tables. Resources forskills practice of these mathematical strategies are located in the “Practice and Apply” sectionof the text book p. 11, for this unit. Students will use tables to compare ratios and will solve rateproblems involving unit, pricing, and constant speed. Assessments will gauge fluency andproficiency during progression of instruction of problem solving strategies. Introduction to real-world situations should begin with the examples shown on p. 3 of Glencoe Math Your CommonCore Edition, CCSS Teacher’s Edition, entitled Math in the Real World, which depicts aCheetah running at full speed. This mini-lesson allows the students to gain information from thetext about Cheetahs with mathematical facts embedded. The students are then prompted tocalculate the answer to the word problem embedded in the factual text. Students should beallowed to engage in Turn and Talk to solve the problem.

After:Students will show proficiency in the calculation of unit rates and ratios, prove theirunderstanding of the concept of ratios and proportional relationships, and reasoning through unit-end project presentations and assessments. A project should include interaction with real-worldencounters with rates and ratios. Students should involve use of utility bills in their projects andpresent how calculation of rates are involved in the utility bills which are paid each month intheir homes. Students will make real-world connection to using ratio reasoning to solveproblems, calculate unit rate by using ratios or proportions, and writing ratios as fractions byutilizing site searches at www.youtube.com, www.google.com, www.discoverychannel.com, andwww.nationalgeographicmagazine.com . Students will see videos of peers interacting with ratiosand proportions in real-world context. Students should complete the end of unit project PeopleEverywhere as an independent project. Additionally, the teacher would have more time to guide

challenging activities like NMSI lesson called “Rating the Trip”. In this lesson studentsinvestigate the concept of rate. The activity could also be used for ratios and proportions.

Pre-AP strategies to consider: After completion of the “Rating the Trip” lesson. Studentswill then write a journal reflection of the day’s lesson. The students will consider what causeddifficulty in the lesson, what they found helpful.

Differentiation

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Intervention: Students will participate in small group instruction of unit areas, peer group work,and guided instruction. Students will be allowed to engage in Turn and Talk in order tocollaborate, analyze, and solve ratio and rate problems. Teacher to student instruction will beimplemented when indicated.

On Level: Students will be exposed to whole group teacher to student instruction and guidedpractice. Students will utilize ⑥3,2,1, ⑥Think, Pair, Share, and ⑥Quick Write: Ticket Out theDoor to indicate learning and questions they may have as during instruction of this unit.Students will engage in project based learning to gain relevance of real-world connections to thecontent.

Acceleration: The unit will be extended to include additional access to real-world ratio and unitrate problem solving through project based learning, technology interaction, and site searchessuch as www.discoverychannel.com. Students will be instructed to find real world examples ofuse of ratios and proportions and performQuickwrite: Ticket s Out the Door utilizing quickresponse as they exit class. Those students who grasp the concepts quickly should be paced pastthe current lessons in small increments and monitored closely.

Accommodations: ELL/SPED

ELLA resource for ELL can be found here:WWW.wida.us/standards/CAN_DOs/<http://www.wida.us/standards/CAN_DOs/>The descriptors provide a starting point for working with ELLs. It serves as a tool for planning.The descriptors provide a continuum of English language development. As teachers becomeaware of their students’ English levels of proficiency, the descriptors provide sensory, graphicand interactive support needed to facilitate ELLs' access to content.

SPED

Accommodations should include: Proximity seating, tactile/manipulative resources, visual

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aids/flash cards, auditory aids/tape recorder, peer buddy/tutoring, highlighting, chunking, cueing,frequent comprehension checks, repeated verbal directions to check for understanding, verbalprompts, books on tape, study guides, and recorded books with appropriate pacing.

Additional accommodations may be specific to the student’s specially designed instructionwithin the IEP.

Technology: Hardware Promethean Board Promethean Clickers Document Camera LCD Projector Laptops Computers

Software

McGraw-Hill eTeacher EditionCDRom Course 1

Materials

Glencoe Math Your CommonCore Edition CCSS TeacherEdition Course 1

Glencoe Math Your CommonCore Edition CCSS StudentEdition Course 1 Volume 1

Glencoe Math Your CommonCore Edition CCSSAssessment Masters Course 1

Supplemental Resources

Tactile/ManipulativeResources

Online Resources

*The information contained in thesewebsites is for educational purposes only.We make no representations or warranties ofany kind, express or implied, about thecompleteness accuracy, reliability, suitabilityor availability with respect to the sites.

www.alex.alsde.eduwww.mathisfun/ratioswww.mathisfun/proportionswww.google.comwww.youtube.comwww.connected.mcgraw-hill.com

www.nms.org Layingthe Foundation (mini

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lesson) provideMathematics Teacherswith sample lessons andresources for Pre-APclasses.

Teacher Notes:

Students may have difficulty grasping the concept of prime factorization and may need to reviewwhat Prime Numbers are before interacting with the strategies of prime factorization. The QuickReview component on page 9 is a resource which can be used to enrich instruction of primefactorization. Similarly, students may struggle initially with finding the GCF and LCM. TheFind the Greatest Common Factor and Find the Least Common Multiple components of thetext book on pages 8 and 9 are helpful resources and can be used repeatedly until the skills aremastered.Teachers can add additional resources to adapt to their lesson plans. Teachers can also includefield trips to allow concrete knowledge development of the concepts. Field trip locations couldinclude The Space and Flight Center in Huntsville, Alabama, The Southern Museum ofFlight in Birmingham, Alabama, and The McWane Center in Birmingham, Alabama. Theselocations are excellent venues to showcase unit rate concepts in the real world which can beapplied across the curriculums of Math and Science. Teachers can learn from students. Listen tostudents as movement through instruction occurs. Use student real life interaction with theconcepts of ratio and proportional relationships to create teachable moments to enrich studentengagement.

Looking Ahead:Students will extend and apply previous knowledge to multiplication and division of fractions.They will interpret and compute quotients of fractions solve word problems involving division offractions through using visual fraction models and equations to represent the problems.

Documents

Math Grade 6 Ratios and Proportional Relationships …bboextra.org/CGuides/Middle School/First Nine Weeks...Math/Grade 6 Curriculum Guide Unit 1: Ratios and Proportional Relationships