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:
Language:
English
Media Formats:
Text/HTML

Education Standards

Unit 6 Overview: Algebraic Reasoning

Unit 6 Overview: Algebraic Reasoning

Overview

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

• Reason about and solve one-variable equations and inequalities. 

• Represent and analyze quantitative relationships between dependent and independent variables. 

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.EE.B.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 

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. 

Examples and Explanations:

6.EE.B.6 Connecting writing expressions with story problems and/or drawing pictures will give students a context for this work. It is important for students to read algebraic expressions in a manner that reinforces that the variable represents a number.  

Examples:  

  • Maria has three more than twice as many crayons as Elizabeth. Write an algebraic expression to represent the number of crayons that Maria has. (Solution: 2c + 3 where c represents the number of crayons that Elizabeth has.)  

  • An amusement park charges $28 to enter and $0.35 per ticket. Write an algebraic expression to represent the total amount spent. (Solution: 28 + 0.35t where t represents the number of tickets purchased)  

  • Andrew has a summer job doing yard work. He is paid $15 per hour and a $20 bonus when he completes the yard. He was paid $85 for completing one yard. Write an equation to represent the amount of money he earned. (Solution: 15h + 20 = 85 where h is the number of hours worked)  

  • Describe a problem situation that can be solved using the equation 2c + 3 = 15; where c represents the cost of an item.  

  • Bill earned $5.00 mowing the lawn on Saturday. He earned more money on Sunday. Write an expression that shows the amount of money Bill has earned. (Solution: $5.00 + n)  

  • The commutative property can be represented by a + b = b + a where a and b can be any rational number. 

 

6.EE.B.7 Students create and solve equations that are based on real world situations. It may be beneficial for students to draw pictures that illustrate the equation in problem situations. Solving equations using reasoning and prior knowledge should be required of students to allow them to develop effective strategies.  

Example:  

  • Meagan spent $56.58 on three pairs of jeans. If each pair of jeans costs the same amount, write an algebraic equation that represents this situation and solve to determine how much one pair of jeans cost.  

                     

  

Sample Solution: Students might say: “I created the bar model to show the cost of the three pairs of jeans. Each bar labeled J is the same size because each pair of jeans costs the same amount of money. The bar model represents the equation 3J = $56.58. To solve the problem, I need to divide the total cost of 56.58 between the three pairs of jeans. I know that it will be more than $10 each because 10 x 3 is only 30 but less than $20 each because 20 x 3 is 60. If I start with $15 each, I am up to $45. I have $11.58 left. I then give each pair of jeans $3. That’s $9 more dollars. I only have $2.58 left. I continue until all the money is divided. I ended up giving each pair of jeans another $0.86. Each pair of jeans costs $18.86 (15+3+0.86). I double check that the jeans cost $18.86 each because $18.86 x 3 is $56.58.” 

 

  • Julio gets paid $20 for babysitting. He spends $1.99 on a package of trading cards and $6.50 on lunch. Write and solve an equation to show how much money Julio has left. (Solution: 20 = 1.99 + 6.50 + x, x = $11.51) 

                                    

 

6.EE.C.9 Students can use many forms to represent relationships between quantities. Multiple representations include describing the relationship using language, a table, an equation, or a graph. Translating between multiple representations helps students understand that each form represents the same relationship and provides a different perspective on the function.  

Examples:  

  • What is the relationship between the two variables? Write an expression that illustrates the relationship.  

                                     

  

  • Use the graph below to describe the change in y as x increases by 1.  

                                    

  

  • Susan started with $1 in her savings. She plans to add $4 per week to her savings. Use an equation, table and graph to demonstrate the relationship between the number of weeks that pass and the amount in her savings account.  

  • Language: Susan has $1 in her savings account. She is going to save $4 each week.  

  • Equation: y = 4x + 1  

  • Table:                                                                   

                                                        

  • Graph:  

                                                   

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. 

  • Construct viable arguments and critique the reasoning of others 

  • Model with mathematics. 

  • Look for and make use of structure

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: 

  • Using equivalent fractions to add and subtract; 

  • Multiplication and division including multiplication and division of fractions; 

  • Analyzing patterns and relationships; 

  • Arithmetic to apply to algebraic expressions; and 

  • Multiplication and division to divide fractions by fractions. 

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: 

  • Solve problems using numerical and algebraic expressions and equations; and 

  • Understand solving equations as a process of reasoning and explain the reasoning

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.  

  • What is equivalence? 

  • How can properties of operations be used to prove equivalence? 

  • How are variables defined and used? 

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: 

  • properties of operations are used to determine if expressions are equivalent; 

  • there is a designated sequence to perform operations (Order of Operations); 

  • variables can be used as unique unknown values or as quantities that vary; and 

  • algebraic expressions may be used to represent and generate mathematical problems and real life situations.

What Students Will Know:

  • Quantities can be represented in real world problems by variables that change in relationship to each other 

  • Variables can be used to represent, write, and solve equations and inequalities for real world problems 

What Students Will Do:

  • Use variables to represent numbers and write expressions to solve real-world problems  

  • Solve real-world problems using equations with variables where variables are nonnegative rational numbers 

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 

  • Solution 

  • Substitution 

  • Variable/unknown 

Content Vocabulary 

  • Dependent Variable

  • Graph 

  • Equation 

  • Independent Variable* 

  • Inverse Operation 

  • Solution 

  • Solution Set

  • Table of Values 

  • X-axis 

  • Y-axis 

  • Relation*

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-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-ESS2-6 Develop and use a model to describe how unequal heating and rotation of the Earth cause patterns of atmospheric and oceanic circulation that determine regional climates 

  • 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-LS1-1 Conduct an investigation to provide evidence that living things are made of cells; either one cell or many different numbers and types of cells 

  • MS-LS1-2 Develop and use models to describe the parts, functions, and basic processes of cells 

  • MS-LS1-3 Use argument supported by evidence for how the body is a system of interacting subsystems composed of groups of cells 

  • MS-LS1-6 Construct a scientific explanation based on evidence for the role of photosynthesis in the cycling of matter and flow of energy into and out of organisms 

  • MS-LS1-7 Develop a model to describe how food molecules (sugar) are rearranged through chemical  and/or release energy as this matter moves through an organism

Computer Science 

  • 2-AP-11 Create clearly named variables that represent different data types and perform operations on their values 

ISTE 

  • 1c Empowered Learner 

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 

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