• Middle School Mathematics Grade 6 Unit 3


    Subject: Mathematics
    Grade: 6
    Timeline: 59
    Unit 3 Title: Variables, Algebraic Equations and Ratios

    Unit Overview: 
     
    To appropriately prepare students for the Variables and Patterns CMP2 textbook investigations, the common core supplemental lessons (1 and 2) on ratios, unit rate(s), number properties and an introduction to algebraic equations will be completed first. This will allow a solid framework for their seventh grade work as well as the textbook’s investigations which is the first of the CMP2 algebra strand. These specific common core lessons are designed to offer students their first glimpse at writing, reading, and evaluating expressions in which letters stand for numbers as well as understanding what a ratio is and its functional purpose and use.  These are a concept rarely touched upon in previous units.  
     
    During the second portion of this combination unit, students will explore three ways of representing a changing situation: in the narrative, with a data table, and with a graph.  These three methods of organizing and recording data are revisited throughout the unit.  They are compared to one another to elicit the strengths of each presentation.  Students also begin to write symbolic equations as a shorter, quicker way to give a summary of the relationship between two variables.  The goal is for all students to make progress in understanding and being able to think and reason with all major useful forms of representation (Variables and Patterns pg. 3). 

    Unit Objectives:
     
    At the end of this two part unit, students should have mastered the following skills and abilities:
    • Calculating unit rates of given math situations
    • Use ratio reasoning to convert measurement units.
    • Write expressions that record operations with numbers and with letters that stand for numbers.
    • Evaluate expressions, including those arising from real-world formulas for specific values of the variables and identify when (and if) two expressions are equivalent.
    • Use the distributive property to express the sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor.
    • Apply the properties of operations to generate equivalent expressions.
    • Identify quantitative variables in situations
    • Recognize situations where changes in variables are related in useful patterns
    • Describe patterns of change shown in words, tables, graphs of data
    • Construct tables and graphs to display relations among variables, including, but not limited to, independent and dependent variables.
    • Observe relationships between two quantitative variables as shown in a table, graph or equation and describe how the relationship can be seen in each of the other forms of representation.
    • Use algebraic symbols to write rules and equations relating variables.
    • Use tables, graphs and equations to solve problems 

    Focus Standards:
     
    PA.CCSS.Math.Content.CC.2.2.6.B.1  Apply and extend previous understandings of arithmetic to algebraic expressions. (6.EE.1, 6.EE.2.a, 6.EE.2.b, 6.EE.2.c, 6.EE.3, 6.EE.4)
    PA.CCSS.Math.Content.CC.2.2.6.B.2 Understand the process of solving a one-variable equation or inequality and apply to real-world and mathematical problems. (6.EE.5, 6.EE.6, 6.EE.7, 6.EE.8)
    PA.CCSS.Math.Content.CC.2.1.6.D.1 Understand ratio concepts and use ratio reasoning to solve problems. (6.RP.1, 6.RP.2, 6.RP.3.a, 6.RP.3.b, 6.RP.3.c, 6.RP.3.d)

    Mathematical Practice Standards:    
     
    #1 Make sense of problems and persevere in solving them. 
     
    Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.
     
    #2 Reason abstractly and quantitatively. 
     
    Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.
     
    #3 Construct viable arguments and critique the reasoning of others. 
     
    Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
     
    #4 Model with mathematics. 
     
    Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
     
    #6 Attend to precision. 
     
    Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.

    Concepts - Students will know: 
    • The concept of unit rate associated with the ratio a : b with b not equaling 0.
    • Equivalent ratios and expressions (parts, how to write, solve)
      • When you may need to write an expression with more than one variable.
    • How to use ratios to convert measurement units
    • That algebraic expressions can be applied to previously taught concepts to simplify 
    • Properties of numbers and the order of operations and their ability to be equivalent in different forms.
    • The parts of expressions in mathematical terms
    • Quantitative variables in situations
    • How to represent the relationship between independent and dependent variables via graphs, tables, and equations.
    • Situations where changes in variables are related in useful patterns
    • Patterns of change shown in words, tables, graphs of data
    • Tables and graphs displaying relations among variables
    • Relationships between two quantitative variables as shown in a table, graph or equation and describe how the relationship can be seen in each of the other forms of representation.
    • Algebraic symbols to write rules and equations relating variables.
    • How to use tables, graphs and equations to solve problems
    Competencies -Students will be able to:
    • Use rate language in the context of a ratio relationship
    • Successfully solve ratio problems by:
    • Selecting the correct ratio
    • Determining which operation to perform.
    • Find missing values in equivalent ratio tables
    • Collect experimental data and place it into a table
    • Plot pairs of values on a coordinate plane.
    • Utilize the correct conversion factors and operation to convert between customary and metric units.
    • Calculate the amount of a set number in one system on measurement into another system (i.e 2 inches equals ________ centimeters)
    • Translate mathematical situations into algebraic expressions that can be simplified for given values of the variable using the correct operation and inverse operation to solve.
    • Find the perimeter of a square using an algebraic expression
    • rewrite expressions, using the properties of numbers and the order of operations, in equivalent forms.
    • Identify equivalent expressions in different forms
    • Label, using mathematical language, the parts of an expression
    • Identify which variable is independent and dependent in a real world mathematical situation.
    • Define how an independent variable affects a dependent variable.  
    • Solve one and two step variable equations.
    • Collect experimental data and organize it in a table
    • Identify patterns and relationships between variables using information in tables, graphs and equations.
    • Create a coordinate graph from data in a table and a table from a coordinate graph
    • Compare table and graph representations of the same data
    • Predict data values between and beyond plotted points
    • Make decisions using tables and graphs
    • Tell the “story” shown in a graph
    • Write a one- and two-step equations 
    • Represent relationships between       
    • Variables 

    Assessments:
    • Unit Exams:
      • Part I will cover the Common Core lessons 1 &2
      • Part II will cover Variables and Patterns Investigations 1 – 3.
    • Strategic Math Shell Response(s) sheets
    • Daily informal assessment on opener as whole class
    • Collection/Grading of homework problems to monitor ongoing progress.
    • “Do Now” mathematical review/introductory questions in the beginning of class daily
    • Optional “exit slips” at the end of class, lessons, etc.
    • Reflection questions at the end of each investigation to assess students’ comprehension of key concepts.

    Elements of Instruction:
     
    As stated on the common core state standards initiative website (http://www.corestandards.org/Math/Content/5/introduction), Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume.
     
    Therefore, fifth grade does not lead into the concepts for this unit as discussed in the above sections.  In fact, the concepts of this unit are introduced in and developed during the sixth grade year to create a solid foundation from which students will draw upon in the coming academic years when they will be expected to expand their working knowledge with these terms, procedures, etc.  As students are unfamiliar with this branch of mathematics, there are several misconceptions and struggles for students in this unit.  These could include some or all of the following:
    1. Selecting the wrong conversion number
    2. Confusing the independent and dependent variables when labeling tables and graphs
    3. Describing how variables are related to coordinate pairs
    4. Writing expressions when given a scenario that accurate reflect the intended variables
    5. Correctly expanding the distributive property to solve according to the order of operations
    6. Comparing data reported in different forms (i.e. data in a table verses data on a graph)
    7. Remembering that a coefficient next to a variable denotes the operation of multiplication
    8. Describing patterns of change.
    Knowing your students levels and thought processes throughout each lesson during discussion, cooperative learning, and independent work will allow the teacher to prevent or correct these as or when they arise. 

    Differentiation:
     
    Each lesson has differentiation options for each portion of the lesson.  Additional differentiation options are listed with directions and student masters in the Teacher’s Guide, located under the Bits and Pieces lll section:
    • Special Need Handbook (adapting instruction/lessons)
    • Investigations 1-5  Adapted Practice Questions
    • Adapted Unit Assessment
    • Student Notebooks
    • Spanish Additional Practice, Skills Worksheet and Assessment

    Interdisciplinary Connections:
    • Mathematical Reflections
    • “Did You Know?” sections
    • phschool.com and web codes
    • “Connections” homework questions
    • The real-world context embedded in lesson problems
    • Learning Target 
    • Students will have an opportunity to  Explore the lesson (independently, with a partner, small group)

    Additional Resources / Games:
     
    CMP2 and/or the Erie School District provide the following additional resources to aid students in achieving mathematical success. 
    • Additional Practice worksheets per investigation
    • Skills Review worksheets to target key components of each investigation
    • Parent letter to be sent home prior to beginning the unit to share with parents the skills, goals, and expectations of the coming unit.
    • Assessment Resources workbook with extra test items (multiple choice, essay, open ended, question bank, etc)
    • Investigation specific pre-generated notebooks that include tables, graphs, problem numbers, and all other items students may need to complete the investigation and all its parts.  Students are provided with one per unit.
    • Reflection questions at the end of each investigation to assess students’ comprehension of key concepts. 
    • phschool.com and web codes
    • Transparencies of models, graphs, etc used within lesson(s)