Unit Overview



|Unit Overview |

|Content Area: Math |

|Unit Title: Linear Functions and Systems of Equations Unit: 5 |

|Target Course/Grade Level: Eighth Grade Timeline: 3/4 weeks |

|Unit Summary: Students continue the study of linear functions. Students identify constant rates of change and determine proportional and |

|non-proportional relationships. They interpret the slope and x- and y-intercepts when graphing a linear equation representing a real-world |

|situation. Through the use of tables and graphs, students represent, analyze, and solve real-world problems related to linear equations and |

|systems of linear equations. |

|Primary interdisciplinary connections: Language Arts and Technology |

|9.1 21st-Centuries Life & Career Skills |

|Standard 9.1 All students will demonstrate the creative, critical thinking, collaboration, and problem-solving skills needed to function |

|successfully as both global citizens and workers in diverse ethnic and organizational cultures. |

|Strand: A. Critical Thinking and Problem Solving |

|B. Creativity and Innovation |

|C. Collaboration, Teamwork and Leadership |

|Content Statement: |

|9.1.8: A The ability to recognize a problem and apply critical thinking skills and problem |

|solving skills to solve the problem is a lifelong skill that develops over time. |

|9.1.8: B Gathering and Evaluating knowledge and information from a variety of sources, |

|including global perspective, fosters creativity and innovative thinking. |

|9.1.8: C Collaboration and team work enable individuals or groups to achieve common goals |

|with greater efficiency. |

|Leadership abilities develop over time through participation in group and or teams that |

|that are engaged in challenging or competitive activities. |

|21st Century themes and skills: Critical Thinking and Problem Solving, Collaboration, Teamwork and |

|Leadership, Creativity and Innovation |

|Mathematical Practices: |

|8.MP.1 Make sense of problems and persevere in solving them. |

|8.MP.2 Reason abstractly and quantitatively. |

|8.MP.3 Construct viable arguments and critique the reasoning of others. |

|8.MP.4 Model with mathematics. |

|8.MP.5 Use appropriate tools strategically. |

|8.MP.6 Attend to precision. |

|8.MP.7 Look for and make use of structure. |

|8.MP.8 Look for and express regularity in repeated reasoning. |

|Learning Targets |

|Domain: Expressions and Equations |

|Cluster: Understand the connections between proportional relationships, lines, and linear equations. Analyze and solve linear equations and |

|pairs of simultaneous linear equations. Use functions to model relationships between quantities. |

|Standard # | Standards |

|8.EE.5 |Understand the connections between proportional relationships, lines, and linear equations. |

|8.EE.6 |Use similar triangles to explain why slope m is the same between any two distinct points on a non-vertical line in the |

| |coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line |

| |intercepting the vertical axis at b. |

|8.EE.8 |Analyze and solve pairs of simultaneous linear equations. |

|8.EE.8a |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. |

|8.EE.8b |Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. |

| |Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot |

| |simultaneously be 5 and 6. |

|8.EE.8c |Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given |

| |coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line |

| |through the second pair. |

|8.F.3 |Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of |

| |functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side |

| |length is not linear because its graph contains the points (1,1), (2,4), and (3,9), which are not on a straight line. |

|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 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. |

|9.1.8.A.1 |Develop strategies to reinforce positive attitudes and productive behaviors that impact critical thinking and |

| |problem-solving skills. |

|9.1.8.A.2 |Implement problem-solving strategies to solve a problem in school or the community. |

|9.1.8.B.2 |Assess data gathered to solve problems for which there are varying perspective (e.g., cross cultural, gender specific, |

| |generational, etc.) and determine how the data can best be used to design the multiple solutions. |

|9.1.8.C.1 |Determine an individual’s responsibility for personal actions and contributions to group activities. |

|9.1.8.C.2 |Demonstrate the use of compromise, consensus and community building strategies for carrying out different task, |

| |assignments and projects. |

|9.1.8.C.3 |Model leadership skills during classroom and extracurricular activities. |

|Unit Essential Questions |Unit Enduring Understandings |

|How can linear equations be used to represent real-world situations? |Patterns and relationships can be represented graphically, numerically, |

|How can slope, rate of change, and direct variation be used to |symbolically, or verbally. |

|represent real-world situations? |Varieties of representations of linear systems of equations, including |

|When is it useful to represent an equation on a graph? |matrices, are used to model and solve real-world problems. |

|What types of real-world situations can be solved by finding the |The understanding of arithmetic, its generalization in algebra, and the |

|solution of a system of equations? |application of mathematics in the real world depends upon the |

| |utilization of numbers, variables and their symbolic representation and |

| |manipulation. |

| |Systems of equations are used to model situations involving interacting |

| |functions with the same variables. |

|Unit Learning Targets |

|Students will ... |

|Identify proportional and nonproportional linear relationships by finding a constant rate of change. |

|Find the slope of a line. |

|Use direct variation to solve problems. |

|Graph linear equations using slope and y-intercept. |

|Graph a function using the x- and y-intercepts. |

|Solve systems of equations by graphing |

|Solve systems of equations by substitution. |

|Evidence of Learning |

|Summative Assessment |

|Find the rate of change to distinguish proportional and non-proportional relationships. |

|Solve multi-step problems involving direct variation. |

|Find and interpret the slope and x- and y-intercepts when graphing a linear equation for a real-world problem. |

|Use tables, graphs, and models to represent, analyze, and solve real-world problems related to systems of linear equations. |

|Solve systems of equations by graphing and substitution. |

|Equipment needed: coordinate grids, colored pencils, rulers, Smart Board, white boards, calculators, Elmo |

|Teacher Instructional Resources: Textbook (TBD) |

|Study Island |

|Khan Academy Videos |

|Formative Assessments |

|Skill sheets |Homework |

|Quizzes/Tests |Math games |

|Student workbook |Study Island |

| |

|Integration of Technology: |

|Smart Board to play online games, utilize online resources, generate models with Smart Software. |

|Kahn Academy Videos |

|Elmo – for demonstration |

|Study Island |

|Technology Resources: |

| |

| |

| – Interactive 2.0 instructional and practice site. Students can view instructional videos and complete practice |

|modules for additional practice/remediation. |

| |

| - Web-based instruction, practice, assessment and reporting built from NJ standards. |

| |

| - IXL 8th grade online interactive activities for the students to complete |

| |

| - AAA math 8th grade – online interactive activities and problems for the student to complete. |

| |

| – Grade level material for practice, lessons, games, etc. |

|Opportunities for Differentiation: |

|Decelerate: Using table of values, the students will graph a variety of equations and describe what they have in common. What generalizations |

|can be made? |

|Given an equation, have the students make a table of values to help them find the x- and y-intercepts. |

|When graphing a system, have students color code their equations and graphs. |

| |

|Accelerated: Given a graph of a line, challenge the students to determine the slope-intercept form of the equation for the line. |

|Ask students to graph a system for which no solution exists. |

|Teacher Notes: Practice graphing and writing equations daily. |

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