8th Grade MathematicsSystems and Functions, Unit 4a, March 10th – April 4th

ORANGE PUBLIC SCHOOLS

OFFICE OF CURRICULUM AND INSTRUCTION

OFFICE OF MATHEMATICS

CURRICULUM GUIDE

8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Table of ContentsI. Unit Overview p. 2

II. Curriculum Guide p. 3

III. Content Overview p. 4

IV. Structure of the Modules p. 5

V. Common Core Standards p. 6

VI. Connections to Mathematical Practices p. 7

VII. Vocabulary p. 8

VIII. Potential Misconceptions p. 9

IX. Teaching Multiple Representations p. 10-11

X. Extensions and Sources p. 12

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8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Unit OverviewIn this unit, students will ….

Understand that the solution to a system of equations is the point of intersection when the equations are graphed and as a coordinate pair that satisfies both equations

Determine the solution to a system of equations algebraically and by graphing Understand that parallel lines do not share a solution and co-linear lines have infinitely

many solutions Identify linear and nonlinear functions in graphs, equations, and tables Apply real world situations to the appropriate graphs Compare the slopes and y-intercepts of functions Create linear functions based on real world situations

Enduring Understandings

There are several methods for solving systems of equations The number of solutions to a system of equations can vary A function can be represented in a table, graph, or equation Changes in varying quantities are often related by patterns

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8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Curriculum GuideMonday, 3/10 Tuesday, 3/11 Wednesday, 3/12 Thursday, 3/13 Friday, 3/14Lesson 1 Introductory Task

Lesson 1Guided Practice, Problem 1

Lesson 1 Guided Practice, Problem 2

Lesson 1 Collaborative Practice (The Intersection of Two Lines)

Lesson 1 Collaborative Practice, Graphing

Monday, 3/17 Tuesday, 3/18 Wednesday, 3/19 Thursday, 3/20 Friday, 3/21Lesson 1 Independent Practice

Lesson 2 Introductory Task

Lesson 2 Guided Practice

Lesson 2 Collaborative Practice

Lesson 2 Independent Practice

Monday, 3/24 Tuesday, 3/25 Wednesday, 3/26 Thursday, 3/27 Friday, 3/288.EE.8 Summative Problem

Lesson 3 Introductory Task and Guided Practice

Lesson 3 Collaborative Practice

Lesson 3 Independent Practice

Lesson 3 Summative Tasks

Monday, 3/31 Tuesday, 4/1 Wednesday, 4/2 Thursday, 4/3 Friday, 4/4Lesson 4 Introductory Task and Guided Practice

Lesson 4 Collaborative and Independent Practice

Review/Test Review/Test SGO Review Activities

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8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Content OverviewRationale

As referenced in the PARCC Model Content Frameworks, students should build on previous 7th grade work with proportions, unit rates, and graphing in preparation for deeper learning and understanding of linear relationships. Students have been working with linear equations informally since kindergarten. Building a strong foundation in earlier Ratios and Proportions and Expressions and Equations standards will aid in student success in grade 8.

Expressions and Equations questions constitute 25% of the possible points on the Grade 8 New Jersey Assessment of Skills and Knowledge. This unit plan incorporates carefully selected tasks to allow students to build concepts, practice procedures, and apply knowledge in real world and mathematical situations with systems of equations and functions.

To ensure quality materials were given to students and teachers, Common Core Modules were specially designed. These Modules were developed for use in grades 3 – 8 and Algebra I as response to the need for rich problem solving tasks that satisfy higher levels of cognitive demand. We took cues from the National Mathematics Advisory Panel’s Final Report (2008) in recognizing the mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic (i.e., quick and effortless) recall of facts.(p. xiv). The Modules contain a variety of well-selected items (offering a triad of conceptual, procedural, real world and mathematical problem solving). Students experience a steady diet of these types of problems throughout each of the Modules. In an effort to provide the District with mathematical content that is narrow, focused and deep, each Module is centered on the Critical Areas specified by the Common Core State Standards for Mathematics (CCSSM) and the Priority Areas denoted within the Partnership for Assessments of Readiness for College and Careers (PARCC) Framework.

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8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Structure of the Modules

The Modules embody 3 integrated frameworks that promote the development of conceptual and problems solving skills and computational fluency. The conceptual framework of the Modules builds from the concrete to the pictorial to the abstract (and the constant blending of each) to help students develop a deeper understanding of mathematics. The Modules also reference a multiple representations framework that encourages teachers to present content in multiple modalities to support flexible thinking. These frameworks go beyond concrete representation (i.e. manipulatives) to promote the realistic representation of concepts addressed in multiple settings. Lastly, the Modules embody a ‘gradual release’ framework that encourages teachers to progress from whole group to collaborative and finally to an independent practice format.

INTRODUCTORY TASKS The Introductory Tasks serve as the starting point for the referenced standard and are typically either diagnostic, prerequisite or anticipatory in nature.

GUIDED PRACTICE Serves for additional teacher guided instruction for students who need the additional help. The tasks can be modeled with students.

COLLABORATIVE PRACTICE Serve as small group, or partnered work. The work should promote student discourse, which allows students to make sense of problems and persevere in solving them (MP.1). Through teacher-facilitated, whole group discussion, students will have the opportunity to critique the reasoning of others (MP.3).

INDEPENDENT PRACTICE Can be used as additional in-class practice, Independent Practice, etc. This work should be reviewed and discussed. Procedural fluencies are reinforced within this section.

SUMMATIVE PROBLEMSThese problems are performance tasks that reflect an amalgamation of the skills addressed within the Module. They assesses the student’s ability to apply the skills learned in a new and non-routine context. More than one-step; problems usually require intermediate values before arriving at a solution (contextual applications). In the US, we see one step problems that require either recall or routine application of an algorithm.

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8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Common Core Standards Unit 4a

8.EE.8

Analyze and solve pairs of simultaneous linear equations.

Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

Solve real-world and mathematical problems leading to two linear equations in two variables.

CALCULATOR NOT

ALLOWED

8.F.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

CALCULATOR ALLOWED

8.F.3Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

CALCULATOR NOT

ALLOWED

8.SP.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

CALCULATOR ALLOWED

8.F.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change  and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

CALCULATOR ALLOWED

8.F.5

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

CALCULATORNOT

ALLOWED

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8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Connections to the Mathematical Practices

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Make sense of problems and persevere in solving them- 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- Students consider the units involved; attending to the meaning of quantities

3 Construct viable arguments and critique the reasoning of others- Students learn to determine domains to which an argument applies

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Model with mathematics- Students 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.

5Use appropriate tools strategically- Students analyze graphs of functions and solutions generated using a graphing

calculator, through tables, graphs, or equations

6Attend to precision- Students are careful about specifying units of measure, and labeling axes to clarify

the correspondence with quantities in a problem

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Look for and make use of structure- Students apply properties to generate equivalent expressions and solve equations.- Students examine patterns in tables and graphs to generate equations and describe

relationships.

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Look for and express regularity in repeated reasoning- By paying attention to the calculation of slope as they repeatedly check whether

points are on a line through a point given a slope, students might abstract the appropriate equation.

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8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Vocabulary

Term Definition

Function A rule of matching elements of two sets of numbers in which an input value from the first set has only one output value in the second set.

Slope

The “steepness” of a line. The slope of a line can be found directly when a linear equation is in a slope-intercept form (y=mx+b). In this form, the slope is the coefficient of x and is represented by the letter m. The slope of a line can also be found by determining the ratio of the “rise” to the “run” between two points on the graph.

Unit Rate A comparison of two measurements in which the second term has a value of one. Unit rates are used to compare the costs of items in a grocery store.

Initial Value The y – intercept in a real world situation

Linear A relationship or function that can be represented by a straight line

Nonlinear A relationship or function that cannot be represented by a straight line

System of Equations Two or more equations that together define a relationship between variables.

Infinite Solutions A system of equations where both equations share the same slope and y-intercept. Also called “co-linear” equations.

No Solutions A system of equations where both equations share the same slope but have different y-intercepts. These are parallel lines.

Solution The intersection point of a system of equations where a single point (x,y) satisfies both equations. In a graph, it is the intersection of the equations.

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8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Potential Student Misconceptions

- When solving systems of equations, students often lose sight of what the solution actually means. Students may get lost in the procedure or algebra and do not use strategies that are most convenient to the problem.

- Students incorrectly calculate or identify the slope because they have memorized a formula and do not contextualize or conceptualize the idea of slope

- Students incorrectly calculate or identify the y-intercept because they have not contextualized or conceptualized the idea of the y-intercept

- Students incorrectly plot a line. To help these students, encourage that a table of values be used to aid in the process.

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8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Teaching Multiple Representations – Grade 8 Major Work

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CONCRETE REPRESENTATIONS

Algebra Tiles

Counters

Grid/Graph Paper

Graphing Paper

8th Grade Unit 4a – Systems and Functions March 10th – April 4th

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PICTORIAL REPRESENTATIONS

Flow Charts

Unit Rate in Tables

Unit Rate in Graphs

ABSTRACT REPRESENTATIONS Equations, y = mx + b Application Problems

8th Grade Unit 4a – Systems and Functions March 10th – April 4th

Extensions and SourcesOnline Resources

http://www.illustrativemathematics.org/standards/k8- Performance tasks, scoring guides

https://www.khanacademy.org/ - Tracks student progress, video learning, interactive practice

http://www.doe.k12.de.us/assessment/files/Math_Grade_8.pdf- Common Core aligned assessment questions, including Next Generation Assessment

Prototypes

https://www.georgiastandards.org/Common-Core/Pages/Math-6-8.aspx- Special Needs designed tasks, assessment resources

http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsGRADE8_Nov2012V3_FINAL.pdf- PARCC Model Content Frameworks Grade 8

http://commoncoretools.files.wordpress.com/2011/04/ccss_progression_ee_2011_04_25.pdf- Progressions of Expressions and Equations from grades 6-8

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