Unit Title: Say It With Symbols Suggested Time: 18 Days (75 – 90 minute Blocks)

Enduring understanding (Big Idea): Represent patterns and relationships in symbolic forms; Determine when different symbolic expressions are mathematically equivalent; write algebraic expressions in useful equivalent forms; combine symbolic expressions using algebraic operations; analyze expressions or equations to determine the patterns of change in tables and graphs that the equation represents; solve linear using symbolic reasoning; use algebraic reasoning to validate generalizations and conjectures

Essential Questions: What expressions or equation represents the pattern or relationship in a context? Can you write an equivalent expression for a given expression to provide new information about a relationship? What operations can transform a given equation or expression into an equivalent form that can be used to answer a question? How can symbolic reasoning help confirm a conjecture?

Unit Plans

Common Core Standards Alignment

Connection to 2003 Standards

Investigation 1
Equivalent Expressions
Problems 1.1,1.2, 1.3, 1.4
Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.7 - Solve linear equations in one variable.

8.EE.7a - Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a andb are different numbers).

8.EE.7b - Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Goal 2.01(ACE Questions); 5.04
This is also a skill to be maintained from 7th grade. (recalling of the volume formulas)

Investigation 2
Combining Expressions
Problems 2.1, 2.2
Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.7 - Solve linear equations in one variable. 8.EE.7a - Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a andb are different numbers).

8.EE.7b - Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Define, evaluate, and compare functions 8.F.2- Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and linear function represented by an algebraic expression, determine which function has the greater rate of change.

Goal 5.01a; 5.01c; 5.01d; 5.04

Investigation 3
Solving Equations
Problems 3.1, 3.2
Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations.8.EE.7 - Solve linear equations in one variable. 8.EE.7b - Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Goal 2.01 (ACE Questions); 5.04
This is also a skill to be maintained from 7th grade. (recalling of the volume formulas)

Investigation 4
Looking Back at Functions
Problems 4.1, 4.3 (modified)
Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations.8.EE.7b - Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Use functions to model relationships between quantities 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. 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 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.

Goal 5.01a; 5.01b; 5.01c; 5.01d
Goal 2.01(ACE Questions); 5.04
This is also a skill to be maintained from 7th grade. (recalling of the volume formulas)

Prior Knowledge: Using the appropriate order of operations in evaluating expressions and writing symbolic sentences; using parentheses and properties of real numbers to communicate effectively. Writing and interpreting symbolic sentences. Reasoning with equivalent expressions. Solving linear using tables, graphs, and simple symbolic rules. Modeling and solving problems.

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

Essential Terms Developed in This Unit

Useful Terms Referenced in This Unit

Terms Developed in Previous Unit

Equivalent expressions

roots

algebraic expression order of operations

commutative property parabola
of addition

commutative property patterns of change
of multiplication

Equations of circles, linear equations, and quadratic equations Say it with Symbols explores the topic that beginning algebra used to focus on almost exclusively: the use of symbols. In a “traditional” algebra curriculum, students are asked to spend most of their time learning to manipulate symbols without getting a chance to think about what the symbols actually mean. CMP emphasizes the meaning behind the symbols in order to help students build their own understanding of the basics of algebra and to give them a reason to believe in the usefulness of algebra as an aid in problem solving. For an in-depth explanation of unit goals, specific questions to ask your student and examples of core concepts from the unit, go to Say it with Symbols. Online resources for Say it with Symbols*

Goals of the Unit • Model situations with symbolic statements • Write equivalent expressions • Determine if different symbolic expressions are mathematically equivalent • Interpret the information equivalent expressions represent in a given context • Determine which equivalent expression to use to answer particular questions • Solve linear equations involving parentheses • Solve quadratic equations by factoring • Use equations to make predictions and decisions • Analyze equations to determine the patterns of change in the tables and graphs that the equation represents • Understand how and when to use symbols to display relationships, generalizations, and proofs

Unit Title: Say It With SymbolsSuggested Time: 18 Days (75 – 90 minute Blocks)Enduring understanding (Big Idea):Represent patterns and relationships in symbolic forms; Determine when different symbolic expressions are mathematically equivalent; write algebraic expressions in useful equivalent forms; combine symbolic expressions using algebraic operations; analyze expressions or equations to determine the patterns of change in tables and graphs that the equation represents; solve linear using symbolic reasoning; use algebraic reasoning to validate generalizations and conjecturesEssential Questions:What expressions or equation represents the pattern or relationship in a context? Can you write an equivalent expression for a given expression to provide new information about a relationship? What operations can transform a given equation or expression into an equivalent form that can be used to answer a question? How can symbolic reasoning help confirm a conjecture?Unit PlansCommon Core Standards AlignmentConnection to 2003 StandardsInvestigation 1Equivalent Expressions

Problems 1.1,1.2, 1.3, 1.4

Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations.8.EE.7- Solve linear equations in one variable.8.EE.7a -Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the formx=a,a=a, ora=bresults (whereaandbare different numbers).8.EE.7b- Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.This is also a skill to be maintained from 7th grade. (recalling of the volume formulas)

Investigation 2Combining Expressions

Problems 2.1, 2.2

Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations.8.EE.7- Solve linear equations in one variable.8.EE.7a -Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the formx=a,a=a, ora=bresults (whereaandbare different numbers).8.EE.7b- Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Define, evaluate, and compare functions 8.F.2-Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and linear function represented by an algebraic expression, determine which function has the greater rate of change.Investigation 3Solving Equations

Problems 3.1, 3.2

Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations.8.EE.7- Solve linear equations in one variable.8.EE.7b- Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.This is also a skill to be maintained from 7th grade. (recalling of the volume formulas)

Investigation 4Looking Back at Functions

Problems 4.1, 4.3 (modified)

Math Reflections

Analyze and solve linear equations and pairs of simultaneous linear equations.8.EE.7b- Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Use functions to model relationships between quantities 8.F.3-Interpret the equationy = mx + bas defining a linear function, whose graph is a straight line; give examples of functions that are not linear.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 values8.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.Goal 2.01(ACE Questions); 5.04

This is also a skill to be maintained from 7th grade. (recalling of the volume formulas)

Prior Knowledge:Using the appropriate order of operations in evaluating expressions and writing symbolic sentences; using parentheses and properties of real numbers to communicate effectively. Writing and interpreting symbolic sentences. Reasoning with equivalent expressions. Solving linear using tables, graphs, and simple symbolic rules. Modeling and solving problems.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 reasoningEssential Terms Developed in This UnitUseful Terms Referenced in This UnitTerms Developed in Previous Unitcommutative property parabola

of addition

commutative property patterns of change

of multiplication

distributive property surface area

solutions to equationssolving equations

expanded form term

factored form x-intercept

functiony-intercept

linear relationship

Say it With Symbols

Equations of circles, linear equations, and quadratic equationsSay it with Symbolsexplores the topic that beginning algebra used to focus on almost exclusively: the use of symbols. In a “traditional” algebra curriculum, students are asked to spend most of their time learning to manipulate symbols without getting a chance to think about what the symbols actually mean. CMP emphasizes the meaning behind the symbols in order to help students build their own understanding of the basics of algebra and to give them a reason to believe in the usefulness of algebra as an aid in problem solving.For an in-depth explanation of unit goals, specific questions to ask your student and examples of core concepts from the unit, go to Say it with Symbols.

Online resources for*Say it with SymbolsOther online resourcesResourcesLab-Sheet

Additional Practice/Skills Worksheets

CMP2 Website –online & technology resources

Formal Assessment

Assessment Options

Parent Guide-Unit Letters

Spanish Assessment Resources

PHSchool.com

TeacherExpress CD-ROM

LessonLab Online Courses

Unit Technology Tips

Goals of the Unit• Model situations with symbolic statements• Write equivalent expressions• Determine if different symbolic expressions aremathematically equivalent• Interpret the information equivalentexpressions represent in a given context• Determine which equivalent expression to useto answer particular questions• Solve linear equations involving parentheses• Solve quadratic equations by factoring• Use equations to make predictions and decisions• Analyze equations to determine the patterns ofchange in the tables and graphs that theequation represents• Understand how and when to use symbols todisplay relationships, generalizations, and proofs## Say It With Symbols

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