• Middle School Mathematics Grade 8 Unit 2

    Subject: Mathematics
    Grade: 8
    Timeline: 27 days
    Unit 2 Title: Thinking With Mathematical Models

    Unit Overview: 
    Thinking With Mathematical Models is the introduction to functions and modeling; finding the equation of a line; inverse functions and inequalities.

    Unit Objectives:
    At the end of this unit, all students must:
    • Recognize linear and non-linear patterns from verbal descriptions, tables and graphs and describe those patterns using words and equations.
    • Write equations to express linear patterns appearing in tables, graphs, and verbal contexts.
    • Write linear equations when given specific information, such as two points or a point and the slope of a line. 
    • Approximate linear data patterns with graph and equation models.
    • Solve linear equations
    • Develop an informal understanding of inequalities.
    • Write equations describing inverse variation.
    • Use linear and inverse variation equations to solve problems and to make predictions and decisions.
    • Find the slope of a right triangle.

    Focus Standards:
    PA.CCSS.Math.Content.CC.2.2.7.B.3. Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. (7.EE.B.4b)
    PA.CCSS.Math.Content.CC.2.1.8.C.1. Define, evaluate, and compare functions. (8.F.A.2, 8.F.A.3
    PA.CCSS.Math.Content.CC.2.1.8.C.2. Use concepts of functions to model relationships between quantities. (8.F.B.4, 8.F.B.5)
    PA.CCSS.Math.Content.CC.2.2.8.B.2. Understand the connections between proportional relationships, lines, and linear equations. (8.EE.B.5)
    PA.CCSS.Math.Content.CC.2.2.8.B.3. Analyze and solve linear equations and pairs of simultaneous linear equations. (8.EE.7)

    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:
    • How tables and graphs represent data.
    • Relationships between variables.
    • How to use data patterns to make predictions.
    • Similarities and differences between linear and nonlinear relationships.
    • The line of best fit is used to show a linear trend.
    • How to use mathematical models to answer questions about linear relationships.
    • Strategies for writing linear equations from verbal, numerical, or graphical information.
    • How to solve linear equations with approximation and exact reasoning methods.
    • How to write inequalities to represent “at most” situations. 
    • Situations that can be modeled using inverse variation.
    • Similarities and differences between inverse variation and linear relationships. 
    • Slope m is the same between any two distinct points on a non-vertical line in a coordinate plane using similar triangles. 
    Competencies -Students will be able to:
    • Make tables and graphs to represent data.
    • Describe relationships between variables.
    • Use data patterns to make predictions.
    • Compare and contrast linear and nonlinear relationships.
    • Fit a line to data that shows a linear trend.
    • Use mathematical models to answer questions about linear relationships.
    • Practice effective strategies for writing linear equations from verbal, numerical, or graphical information.
    • Develop skill in solving linear equations with approximation and exact reasoning methods. 
    • Write inequalities to represent “at most” situations. 
    • Explore situations that can be modeled by inverse variation.
    • Compare inverse variation with linear relationships.
    • Find the slope of a right triangle.   

    Formative Assessments: 
    • Informal assessments on learning targets
    • Check-up Quiz 1
    • Partner Quiz
    Summative Assessment:
    • Common Core Unit Assessment- Thinking With Mathematical Models

    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.

    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)