• Middle School Mathematics Grade 8 Unit 4

    Subject: Mathematics
    Grade: 8
    Timeline: 22 days
    Unit 4 Title: Say It With Symbols

    Unit Overview: 
    Say it With Symbols Unit is an eighth grade unit that emphasizes using the properties of numbers to look at equivalent expressions and the information each expression represents in a given context, and to interpret underlying patterns that a symbolic equation or statement represents. Students look critically at each part of an expression and how each part relates to the original expression.  They examine the graph and table of an expression as well as the context the expression models.  The properties of equality and numbers are used extensively in this unit as students write and interpret equivalent expressions, combine expressions to form new expressions, predict patterns of change represented by an equation or expression, and solve equations.  This unit also helps to develop “symbol sense.”   

    Unit Objectives:
    At the end of this unit, students should have mastered the following skills and abilities:     
    • Model situations with symbolic statements
    • Write equivalent expressions
    • Determine if different symbolic expressions are mathematically equivalent
    • Interpret the information equivalent expressions represent in a given context
    • Determine which equivalent expression to use to answer particular questions
    • Solve linear equations involving parentheses
    • Solve quadratic equations by factoring
    • Use equations to make predictions and decisions
    • Use formulas to find the volume of cones, cylinders, and spheres.

    Focus Standards:
    PA.CCSS.Math.Content.CC.2.2.8.B.1. Apply concepts of radicals and integer exponents to generate equivalent expressions. (8.EE.A.1, 8.EE.A.2)
    PA.CCSS.Math.Content.CC.2.2.8.B.3. Analyze and solve linear equations and pairs of simultaneous linear equations. (8.EE.C.7b)
    PA.CCSS.Math.Content.CC.2.3.8.A.3. Apply the concepts of volume of cylinders, cones, and spheres to solve real-world and mathematical problems. (8.G.9). 

    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.
    #5 Use appropriate tools strategically. 
    Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, or a calculator. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
    #7 Look for and make use of structure. 
    Mathematically proficient students look closely to discern a pattern or structure. Mid-level students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. 

    Concepts - Students will know:
    • The equivalence of 2 or more symbolic expressions for the same situation.
    • The Distributive Property independent of a specific context.
    • The properties of real numbers as it relates to combining equivalent expressions.
    • Linear and quadratic equations and solutions to these equations.
    • Underlying patterns of change represented by symbolic statements.
    • The concept of volume in a sphere, cylinder, and cone.
    Competencies -Students will be able to:
    • Determine whether expressions are equivalent.
    • Distribute values using the Distributive Property.
    • Combine expressions to write new expressions using the properties of real numbers.
    • Solve linear equations with parentheses and solve quadratic equations by factoring.
    • Write symbolic equations to represent specific patterns of change. 
    • Use a formula to find the volume of a sphere, cylinder, and cone.  

    Formative Assessments: 
    • Informal assessments on learning targets and additional practice pages
    • Check-up Quiz 1
    • Check-up Quiz 2
    Summative Assessment:
    • Common Core Unit Assessment - Say It With Symbols

    Elements of Instruction:
    In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
    Representing and reasoning about patterns of change has been the main focus up to this point in the development of algebra.  Students have used tables, graphs, and symbols to represent relationships and to solve equations to find information or make predictions.  The properties of real numbers such as the Commutative and Distributive properties were first introduced in Accentuate the Negative and then used again in Moving Straight Ahead and Growing, Growing, Growing.  The Distributive Property was expanded in Frogs, Fleas, and Painted Cubes to include multiplication of two binomials.
    Major misconceptions by and struggles for students in this unit include:
    1. Students are deliberately presented with situations in which contextual clues can be interpreted in several ways to produce different but equivalent equations.
    2. Forgetting to distribute all figures through the expression
    3. Combining expressions and mixing up the variables (Ex: 2X + 2Y simplified as 4XY instead of staying in its most simplified form)
    4. Solving equations (not making both sides of the equation equal)
    5. Predicting linear patterns of change 

    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.
    • Special Needs Handbook
    • Unit Projects
    • Spanish Additional Practice and Skills guide
    • Strategies for English Language Learners Guide
    • “Extension” homework questions
    • District-created notebooks

    Interdisciplinary Connections:
    • Mathematical Reflections
    • “Did You Know?” sections
    • phschool.com and web codes
    • “Connections” homework questions
    • The real-world context embedded in lesson problems

    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)