• Middle School Mathematics Grade 8 Unit 5

    Subject: Mathematics
    Grade: 8
    Timeline: 16 days
    Unit 5 Title: Shapes of Algebra

    Unit Overview: 
    Algebraic ideas and techniques are powerful tools for reasoning about geometric shapes on a coordinate grid. Conversely, geometric images are useful aids to algebraic reasoning about linear equations and inequalities. This unit is designed to capitalize on the strong connections between algebra and geometry in order to extend students’ understanding and skill in several significant aspects of those two key strands in the middle grades curriculum. These are the key ideas developed by problems in The Shapes of Algebra unit. They extend earlier work with the Pythagorean Theorem by connecting it to the standard equation for circles; with properties of polygons by connecting parallel and perpendicular lines to slopes of lines and linear functions; and with solutions of linear equations by considering solutions of linear systems and equations in standard ax + by = c form, and solutions of linear inequalities. These topics are standard parts of traditional Algebra I syllabi and they are included in many standard algebra examinations. However, the Connected Mathematics approach to the topics exploits the rich connections between algebra and geometry to strengthen student understanding of problems and solution methods that are often taught and learned in quite formal and rote ways. Since students who proceed to mathematically oriented academic specialties will undoubtedly study these topics in greater detail in high school mathematics, it is important to develop the sort of conceptual understanding that will provide a solid base for future work, not simply to settle for short-term rote learning of procedures that will be quickly forgotten.

    Unit Objectives:
    • Determine if lines are parallel or perpendicular by looking at patterns in their graphs, coordinates, and equations  
    • Find coordinates of points that divide line segments in various ratios 
    • Find solutions to inequalities represented by graphs or equations 
    • Write inequalities that fit given situations 
    • Solve systems of linear equations by graphing, by substituting, and by combining equations
    • Choose strategically the most efficient solution method for a given system of linear equations 
    • Graph linear inequalities and systems of inequalities 
    • Describe the points that lie in regions determined by linear inequalities and systems of inequalities  
    • Use systems of linear equations and inequalities to solve problems 

    Focus Standards:
    PA.CCSS.Math.Content.CC.2.2.HS.D.10.  Represent, solve and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. (HSA-A.REI.3, HSA-A.REI.6, HSA-A.REI.12)
    PA.CCSS.Math.Content.CC.2.2.8.B.3. Analyze and solve linear equations and pairs of simultaneous linear equations. (8.EE.C.8B) 

    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.
    #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.
    #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.
    #8 Look for and express regularity in repeated reasoning. 
    Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

    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.
    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.  

    Formative Assessments: 
    • Informal assessments on learning targets
    • Check-up Quiz
    Summative Assessment:
    • Common Core Unit Assessment - Shapes of Algebra

    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)