Unit Title: The Shapes of Algebra Suggested Time: 25 days (75 to 90 minute Blocks)

Enduring understanding (Big Idea): Write and use equations of circles; 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 conditions; Solve systems of linear equations by graphing, by substituting, and by combining equations; Graph linear inequalities; Describe the points that lie in regions determined by linear; Use systems of linear equations to solve problems; Choose strategically the most efficient solution method for a given system of linear equation.

Essential Questions:What patterns relate the coordinates of points on lines and curves? What patterns relate the points whose coordinates satisfy linear equations? Does the problem involve an equation or an inequality? Does the problem call for writing and/or solving a system of equations? If so, what method would be useful for solving the system? Are there systematic methods that can be used to solve any systems of linear equations?

Unit Plans

Common Core Standards Alignment

Connection to 2003 Standards

Investigation 1
Equations for Circles and Polygons
Problems 1.1,1.2
Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.8.b- 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.

Goal 3.02 This was a Geometry I goal

Investigation 2
Linear Equations and Inequalities
Problems 2.1, 2.2, 2.3
Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations. 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.

Investigate patterns of association in bivariate data. 8.SP.3- Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr. as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

Goal 5.02; 5.03

Investigation 3
Equations with Two or More Variables
Problems 3.1, 3.2, 3.3
Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations:

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.

Define, evaluate, and compare functions. 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.

Investigate patterns of association in bivariate data. 8.SP.3- Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr. as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

Goal 5.01a; 5.01b; 5.01c; 5.01d; 5.02; 5.03

Investigation 4
Solving Systems of Linear Equations Symbolically
Problems 4.1, 4.2, 4.3, 4.4
Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations: 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. 8.EE.8c - Solve real-world and mathematical problems leading to two linear equations in two-variables. Define, evaluate, and compare functions. 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.

Goal 5.01a; 5.01b; 5.01c; 5.01d; 5.02; 5.03

Investigation 5 (MOVED to Algebr for Common Core)
Linear Inequalities
5.1, 5.2, 5.3
Math Reflections

Prepares for A-REI-12

Goal 5.01a; 5.01b; 5.01c; 5.01d; 5.02; 5.03

Prior Knowledge: Thinking about shapes. Working with coordinates. Finding midpoints of line segments. Formulating, reading, and interpreting symbolic rules. Working with the triangle inequality. Solving problems in geometric and algebraic contexts. Solving linear equations.

Mathematical Practices Standards for Common Core 1-Make sense of problems and persevere in solving them 2-Reason abstractly and quantitatively 3-Construct viable arguments and critique the reasoning of others 4-Model with mathematics 5-Use appropriate tools strategically 6-Attend to precision 7-Look for and make use of structure 8-Look for and express regularity in repeated reasoning

Shapes of Algebra

Linear inequalities, systems of equations

The final unit of eighth grade CMP Algebra capitalizes on the strong connections between algebra and geometry to extend students’ understanding and skill in preparation for Geometry and Advanced Algebra. Students will work with equations for lines and curves. They will develop an understanding of how systems of equations and inequalities can help solve problems. Students extend their earlier work in algebra and geometry by making connections between them. For example, students connect the idea of the Pythagorean Theorem to the coordinate equation for a circle, and connect properties of polygons to slopes of lines. A student who has successfully completed eighth grade CMP has completed the same material as Algebra I and will be placed in Geometry as his or her next math course. For an in-depth explanation of unit goals, specific questions to ask your student and examples of core concepts from the unit, go to Shapes of Algebra. Online resources for Shapes of Algebra*

Unit Technology Tips Goals of the Unit • Write and use equations of circles • 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

Unit Title: The Shapes of AlgebraSuggested Time: 25 days (75 to 90 minute Blocks)Enduring understanding (Big Idea):Write and use equations of circles; 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 conditions; Solve systems of linear equations by graphing, by substituting, and by combining equations; Graph linear inequalities; Describe the points that lie in regions determined by linear; Use systems of linear equations to solve problems; Choose strategically the most efficient solution method for a given system of linear equation.Essential Questions:What patterns relate the coordinates of points on lines and curves? What patterns relate the points whose coordinates satisfy linear equations? Does the problem involve an equation or an inequality? Does the problem call for writing and/or solving a system of equations? If so, what method would be useful for solving the system? Are there systematic methods that can be used to solve any systems of linear equations?Unit PlansCommon Core Standards AlignmentConnection to 2003 StandardsInvestigation 1Equations for Circles and Polygons

Problems 1.1,1.2

Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.8.b- 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.Goal 3.02This was a Geometry I goalInvestigation 2Linear Equations and Inequalities

Problems 2.1, 2.2, 2.3

Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.8 -Analyze and solve pairs of simultaneous linear equations8.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 simultaneously8.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.Investigate patterns of association in bivariate data.8.SP.3-Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr. as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.Goal 5.02; 5.03Investigation 3Equations with Two or More Variables

Problems 3.1, 3.2, 3.3

Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations:8.EE.8- Analyze and solve pairs of simultaneous linear equations8.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.Define, evaluate, and compare functions.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.Investigate patterns of association in bivariate data.8.SP.3-Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr. as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.Goal 5.01a; 5.01b; 5.01c; 5.01d; 5.02; 5.03Investigation 4Solving Systems of Linear Equations Symbolically

Problems 4.1, 4.2, 4.3, 4.4

Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations:8.EE.8-Analyze and solve pairs of simultaneous linear equations8.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 simultaneously8.EE.8b -Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.8.EE.8c -Solve real-world and mathematical problems leading to two linear equations in two-variables.Define, evaluate, and compare functions.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.Goal 5.01a; 5.01b; 5.01c; 5.01d; 5.02; 5.03Investigation 5(MOVED to Algebr for Common Core)Linear Inequalities

5.1, 5.2, 5.3

Math Reflections

Prepares for A-REI-12Goal 5.01a; 5.01b; 5.01c; 5.01d; 5.02; 5.03Prior Knowledge:Thinking about shapes. Working with coordinates. Finding midpoints of line segments. Formulating, reading, and interpreting symbolic rules. Working with the triangle inequality. Solving problems in geometric and algebraic contexts. Solving linear equations.Mathematical Practices Standards for Common Core1-Make sense of problems and persevere in solving them 2-Reason abstractly and quantitatively 3-Construct viable arguments and critique the reasoning of others 4-Model with mathematics 5-Use appropriate tools strategically 6-Attend to precision 7-Look for and make use of structure 8-Look for and express regularity in repeated reasoningShapes of Algebra

The final unit of eighth grade CMP Algebra capitalizes on the strong connections between algebra and geometry to extend students’ understanding and skill in preparation for Geometry and Advanced Algebra. Students will work with equations for lines and curves. They will develop an understanding of how systems of equations and inequalities can help solve problems.Linear inequalities, systems of equationsStudents extend their earlier work in algebra and geometry by making connections between them. For example, students connect the idea of the Pythagorean Theorem to the coordinate equation for a circle, and connect properties of polygons to slopes of lines.

A student who has successfully completed eighth grade CMP has completed the same material as Algebra I and will be placed in Geometry as his or her next math course.

For an in-depth explanation of unit goals, specific questions to ask your student and examples of core concepts from the unit, go to Shapes of Algebra.

Online resources for*Shapes of AlgebraOther online resourcesResourcesLab-Sheet

Additional Practice/Skills Worksheets

CMP2 Website –online & technology resources

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Parent Guide-Unit LettersSpanish Assessment Resources

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TeacherExpress CD-ROM

LessonLab Online Courses

Unit Technology Tips

Goals of the Unit• Write and use equations of circles• Determine if lines are parallel or perpendicularby looking at patterns in their graphs,coordinates, and equations• Find coordinates of points that divide linesegments in various ratios• Find solutions to inequalities represented bygraphs 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 solutionmethod for a given system of linear equations• Graph linear inequalities and systems ofinequalities• Describe the points that lie in regionsdetermined by linear inequalities and systems ofinequalities• Use systems of linear equations and inequalitiesto solve problems