Author:
Connecticut Department of Education
Subject:
Mathematics
Material Type:
Unit of Study
Level:
Middle School
Grade:
6
Provider:
CT State Department of Education
Provider Set:
CSDE - Public
Tags:
  • Ratios and Rates
  • Language:
    English
    Media Formats:
    Text/HTML

    Education Standards

    Unit 5 Overview: Ratios and Rates

    Unit 5 Overview: Ratios and Rates

    Overview

    Unit Overview/Summary - FOCUS: This unit focuses on the Ratios and Proportional Relationships domain.  Learning in this unit will enable students to: 

    •    Understand ratio concepts and use ratio reasoning to solve problems. 

    All of the learning in this unit is major work of the grade. 

    Relevant Standards:

    Major work of the grade is in bold. 

    The standards that are to be addressed during this unit of study include: 

    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.” 

    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.  

    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.” (Expectations for unit rates in this grade are limited to non-complex fractions.) 

    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. 

    1. 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. 
    2. 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? 
    3. Find a percent of a quantity as a rate per 100 (e.g., 30 percent of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. 
    4. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. 

     

     

    Examples and Explanations:

    6.RP.A.1   A ratio is a comparison of two quantities which can be written as a to b, \( {a\over b}\) or a:b.  

    A rate is a ratio where two measurements are related to each other. When discussing measurement of different units, the word rate is used rather than ratio. Understanding rate, however, is complicated and there is no universally accepted definition. When using the term rate, contextual understanding is critical. Students need many opportunities to use models to demonstrate the relationships between quantities before they are expected to work with rates numerically.  

    A comparison of 8 black circles to 4 white circles can be written as the ratio of 8:4 and can be regrouped into 4 black circles to 2 white circles (4:2) and 2 black circles to 1 white circle (2:1).  

                                                            

    Students should be able to identify all these ratios and describe them using “For every…., there are …” 

    6.RP.A.2  A unit rate compares a quantity in terms of one unit of another quantity. Students will often use unit rates to solve missing value problems. Cost per item or distance per time unit are common unit rates, however, students should be able to flexibly use unit rates to name the amount of either quantity in terms of the other quantity. Students will begin to notice that related unit rates are reciprocals as in the first example. It is not intended that this be taught as an algorithm or rule because at this level, students should primarily use reasoning to find these unit rates.  

    In Grade 6, students are not expected to work with unit rates expressed as complex fractions. Both the numerator and denominator of the original ratio will be whole numbers.  

    Examples:  

    On a bicycle you can travel 20 miles in 4 hours. What are the unit rates in this situation, (the distance you can travel in 1 hour and the amount of time required to travel 1 mile)?  

    Solution: You can travel 5 miles in 1 hour written as\({5 M i {} \over 1 Hr}\)and it takes \({1 {} \over 5}\)of a hour to travel each mile written as  \(\frac{\frac{1}{5}{}{}}{1 Mi}\)

    Students can represent the relationship between 20 miles and 4 hours.  

                                                  

    A simple modeling clay recipe calls for 1 cup corn starch, 2 cups salt, and 2 cups boiling water. How many cups of corn starch are needed to mix with each cup of salt? 

     

    6.RP.A.3  Examples:  

    Using the information in the table, find the number of yards in 24 feet.  

     

    There are several strategies that students could use to determine the solution to this problem.  

    Add quantities from the table to total 24 feet (9 feet and 15 feet); therefore the number of yards must be 8 yards (3 yards and 5 yards).  

    Use multiplication to find 24 feet:  

        1) 3 feet x 8 = 24 feet; therefore 1 yard x 8 = 8 yards, or  

        2) 6 feet x 4 = 24 feet; therefore 2 yards x 4 = 8 yards.  

     

    Compare the number of black to white circles. If the ratio remains the same, how many black circles will you have if you have 60 white circles?  

     

     

     

    If 6 is 30% of a value, what is that value? (Solution: 20)  

     

    A credit card company charges 17% interest on any charges not paid at the end of the month. Make a ratio table to show how much the interest would be for several amounts. If your bill totals $450 for this month, how much interest would you have to pay if you let the balance carry to the next month? Show the relationship on a graph and use the graph to predict the interest charges for a $300 balance.  

     

     

     

     

     

    Transfer Goal: Aligned to district portrait or vision of the learner

    In this unit students will develop the skills of perseverance, reasoning, modeling, and precision. This will be accomplished through focus on the following Standards for Mathematical Practice: 

    • Make sense of problems and persevere in solving them. 

    • Reason abstractly and quantitatively. 

    • Model with mathematics. 

    • Attend to precision. 

    Grade level content is in bold 

    Coherence: 

    How does this unit build on and connect to prior knowledge and learning? 

    Students build on their previous understandings of: 

    • Fractions; 

    • Conversion of measurement; 

    • Analyzing patterns; 

    • Graphing on the coordinate plane; and 

    • Using the four operations to solve problems. 

    How does this unit prepare students for future learning?   

    The learning of this unit serves as a foundation for content that will be addressed in future years.  Specifically, students will utilize this learning to: 

    • Analyze proportional relationships; 

    • Solve real-world problems; and 

    • Reason quantitatively. 

    Essential Questions:

    Essential Questions can be approached in multiple ways. There should be no more than 2-3 essential questions and they should align with your topics. Questions can be repeated throughout a course or over years, with different enduring understandings. 

    • When is it useful to be able to relate one quantity to another? 

    • How are ratios and rates similar and different? 

    • What is the connection between a ratio and a fraction? 

    Enduring Understanding:

    The major ideas you want students to internalize and understand deeply. These understandings should be thematic in nature. They are not the end all, be all of the question. They are focused to align to the focus (unit overview). 

    Students understand: 

    • a rate is a set of infinitely many equivalent ratios; 

    • reasoning with ratios involves attending to and coordinating two quantities; and 

    • a proportion is a relationship of equality between two ratios that can be represented in a variety of ways. In a proportion, the ratio of two quantities remains constant as the corresponding values of the quantities change

    What Students Will Know:

    • A ratio expresses the comparison between two quantities. Special types of ratios are rates, unit rates, measurement conversions, and percent 

    • A ratio or a rate expresses the relationship between two quantities. Ratio and rate language is used to describe a relationship between two quantities 

    • A rate is a type of ratio that represents a measure, quantity, or frequency, typically one measured against a different type of measure, quantity, or frequency 

    • Ratio and rate reasoning can be applied to many different types of mathematical and real-life problems (rate and unit rate problems, scaling, unit pricing, statistical analysis, etc.) 

    What Students Will Do:

    • Use ratio language to describe a ratio relationship between two quantities 

    • Represent a ratio relationship between two quantities using manipulatives and/or pictures, symbols and real-life situations (a to b, a:b, or a/b)  

    • Represent unit rate associated with ratios using visuals, charts, symbols, real-life situations and rate language 

    • Use ratio and rate reasoning to solve real-world and mathematical problems 

    • Make and interpret tables of equivalent ratios 

    •  Plot pairs of values of the quantities being compared on the coordinate plane 

    • Use multiple representations such as tape diagrams, double number line diagrams, or equations to solve rate and ratio problems 

    • Solve unit rate problems (including unit pricing and constant speed) 

    • Solve percent problems, including finding a percent of a quantity as a rate per 100 and finding the whole, given the part and the percent 

    Unit Specific Vocabulary and Terminology:

    The purpose of vocabulary work should be to allow all students to access mathematics. Vocabulary is a way to provide opportunities for students to use mathematical language to communicate about how they solved a problem, describe their reasoning, and demonstrate understanding of mathematical content. Vocabulary is inclusive of key words and phrases.

    Often multilingual learners/English learners (MLs/ELs) are perceived as lacking academic language and needing remediation. Research shows that MLs/ELs bring standards-aligned background knowledge and experiences to the task of learning, and they need opportunities to extend their language for academic purposes. When considering the language demand of a lesson (at the word level), you can check for cognates and polysemous words. Pointing out these words to students can help them activate and build background knowledge assumed in lessons. TESOL professionals can assist with the identification of cognates and polysemous words, and they can provide guidance about the background knowledge MLs/ELs bring or may need.

    The words with an * are words that appear in the Smarter Balanced Construct Relevant Vocabulary for Mathematics intended to ensure that teachers remember to embed these terms in their instruction.

    Academic Vocabulary 

    • Equivalent 

    • Percent 

    • Rate 

    • Unit rate

    • Quantity 

    Content Vocabulary 

    • Conversion factor  

    • Proportion 

    • Ratio

    • Rational Number 

    • Tape diagram

    Vocabulary resources: 

    English Math Vocabulary Cards

    Spanish Math Vocabulary Cards

    Chinese Math Vocabulary Cards

    French Math Vocabulary Cards

    Aligned Unit Materials, Resources and Technology:

    High-quality instructional resources are critical for improving student outcomes. The alignment guidance is intended to clarify content and support understanding for clear implementation and coherence. 

    Materials selection is a local control decision and these documents have been provided from participating publishers to assist districts in implementation. Use of the materials from these publishers is not required. These aligned core programs meet expectations as reported by edReports. Strong alignment of curricula and instructional materials have the potential to support student engagement and teacher growth. 

    Opportunities for Interdisciplinary Connections:

    Science 

    • MS-PS2-4 Construct and present arguments using evidence to support the claim that gravitational interactions are attractive and depend on the masses of interacting objects 

    • MS-PS3-1 Construct and interpret graphical displays of data to describe the relationships of kinetic energy to the mass of an object and to the speed of an object 

    • MS-PS3-5 Construct, use, and present arguments to support the claim that when the kinetic energy of an object changes, energy is transferred to or from the object 

    • MS-PS4-1 Use mathematical representations to describe a simple model for waves, which includes how the amplitude of a wave is related to the energy in a wave 

    • MS-LS1-8 Gather and synthesize information that sensory receptors respond to stimuli by sending messages to the brain for immediate behavior or storage as memories 

    • MS-LS2-3 Develop a model to describe the cycling of matter and flow of energy among living and nonliving parts of an ecosystem 

    • MS-LS2-4 Construct an argument supported by empirical evidence that changes to physical or biological components of an ecosystem affect populations 

    • MS-LS2-5 Evaluate competing design solutions for maintaining biodiversity and ecosystem services 

    • MS-LS4-4 Construct an explanation based on evidence that describes how genetic variations of traits in a population affects individuals’ probability of surviving and reproducing in a specific environment. 

    • MS-LS4-6 Use mathematical representations to support explanations of how natural selection may lead to increases and decreases of specific traits in populations over time 

    • MS-ESS1-1 Develop and use a model of the Earth-sun-moon system to describe the cyclic patterns of lunar phases, eclipses of the sun and moon, and seasons 

    • MS-ESS1-2 Develop and use a model to describe the role of gravity in the motions within galaxies and the solar system 

    • MS-ESS1-3 Analyze and interpret data to determine scale properties of objects in the solar system 

    • MS-ESS3-1 Construct a scientific explanation based on evidence for how the uneven distributions of Earth’s mineral, energy, and groundwater resources are the result of past and current geoscience processes 

    • MS-ESS3-3 Apply scientific principles to design a method for monitoring, evaluating, and managing a human impact on the environment 

    • MS-ESS3-4 Construct an argument supported by evidence for how changes in human population and per-capita consumption of natural resources impact Earth’s systems 

    ISTE 

    • 1c Empowered Learner 

    Music 

    • MU:Cn11.0.6 Demonstrate understanding of relationships between music and the other arts, other disciplines, varied contexts, and daily life 

    Opportunities for Application of Learning:

    Defined Learning provides an open access online library of standards-aligned project-based lessons to help students meet the expectations of the Standards. Each project is based on a situation in a relevant career to help students connect classroom content to career pathways. This supplemental resource is available at no cost to teachers and districts. Create an account and log in to access this free resource to support your curriculum.

    The tasks below provide additional opportunities to apply the content of this unit.

    Critical Consciousness for Diversity and Equity:

    Culturally relevant mathematics engages and empowers students. Opportunities for teachers to orchestrate discussions where students share not only connections to prior mathematics learned but also to their lived experiences must be provided. It is important to dig deep to find ways to link students’ home cultures and the mathematics classroom. Build authentic relationships with families through two-way, reciprocal conversations that acknowledge families’ cultures as assets for teaching and learning. As you plan to implement this unit, focus on designing experiences that have students at the center. In addition to keeping students engaged, ensure the learning experiences have a context that reflects lived experiences (mirror) or provide opportunities to view and learn about the broader world (window). 

    One crucial link to students’ home cultures is through their language. Students’ language repertoires –all the languages and language varieties they use everyday– are a valuable resource to be engaged in the mathematics classroom. This approach is referred to as a translanguaging stance. It is based on a dynamic view of bilingualism that understands individuals as having one linguistic repertoire composed of various named languages (such as English and Spanish) and/or language varieties on which they draw to make meaning.

    The following questions are intended to assist in promoting diverse voices and perspectives while avoiding bias and stereotyping:

    • How will students share their experiences with others while attending to the mathematics in the unit?
    • What opportunities are there for students to make connections from their life to the mathematics?
    • What do I know or need to learn about my students to create lessons free from bias and stereotypes?
    • In what ways can the mathematical thinking already taking place in the classroom and community be honored?
    • How is relevant background knowledge developed so that all students can access the mathematics of the unit?
    • What opportunities are there for students to use their full language repertoires during mathematical discussions and practice? Where can I create these opportunities?
    • What do I know or need to learn about students’ languages and how they use them? How can I learn this?

    * Resources to support diversity and equity in the classroom please visit the DEI Collection

    Supporting Multilingual Learners/English Learners (ML/EL):

    Mathematical symbols, expressions, and methods are not universal; ways of doing math differ across cultures. When working with diverse students, especially with those from different countries, it is important to be aware that these differences exist.

    It is also important to remember that communicating about mathematical content and practices requires complex language. Since conceptual learning and language learning are interconnected and acquired through participation in meaningful activities, the research-based strategies listed below focus on making content comprehensible (accessible) and creating opportunities for student voice, both verbal and written.

    Additional resources for ML/EL

    CELP Standards--Linguistic Supports

    ML/EL Support Collection for Math  

    This unit presents opportunity to address the following CELP Standards: 

    CELP Standard 1: Construct meaning from oral presentations and literary and informational text through grade-appropriate listening, reading, and viewing 

    CELP Standard 2: Participate in grade-appropriate oral and written exchanges of information, ideas, and analyses, responding to peer, audience, or reader comments and questions 

    CELP Standard 3: Speak and write about grade-appropriate complex literary and informational texts and topics 

    CELP Standard 4: Construct grade-appropriate oral and written claims and support them with 

    reasoning and evidence 

    CELP Standard 5: Conduct research and evaluate and communicate findings to answer questions or solve problems 

    CELP Standard 6: Analyze and critique the arguments of others orally and in writing 

    CELP Standard 7: Adapt language choices to purpose, task, and audience when speaking and writing 

    CELP Standard 8: Determine the meaning of words and phrases in oral presentations and literary and informational text 

    CELP Standard 9: Create clear and coherent grade-appropriate speech and text 

    CELP Standard 10: Make accurate use of standard English to communicate in grade appropriate speech and writing 

    The document below provides guidance on these standards that is grade appropriate and broken down by language level descriptors which will assist the teacher in making the content of the unit accessible to all students.

    Grade 6 Language Level Descriptors