Unit 1

Equations and Their Graphs

Algebra I 

 

Unit Length and Description:

 

40 days

 

By the end of eighth grade students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations. All of this work is grounded on understanding quantities and on relationships between them. (Mathematics Appendix A, p.17)

 

·         Reason quantitatively and use units to solve problems.

·         Interpret the structure of expressions.

·         Create equations that describe numbers or relationships.

·         Understand solving equations as a process of reasoning and explain the reasoning.

·         Solve equations and inequalities in one variable.

·         Solve systems of equations 

·         Represent and solve equations and inequalities graphically.

·         Perform arithmetic operations on polynomials.

 

Standards:

 

A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

a.    Interpret parts of an expression, such as terms, factors, and coefficients.

b.    Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

 

A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

 

A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

 

A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

 

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

 

A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

 

A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

 

A-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

 

A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

 

A-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

 

A-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

 

A-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

 

A-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

 

N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

 

N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

 

N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

 

Focus Standards of Mathematical Practice:

 

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

MP.2 Reason abstractly and quantitatively.

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

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

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

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

A-SSE.A.1

·         I can, for expressions that represent a contextual quantity, define and recognize parts of an expression, such as terms, factors, and coefficients

·         I can, for expressions that represent a contextual quantity, interpret parts of an expression, such as terms, factors, and coefficients in terms of the context

·         I can, for expressions that represent a contextual quantity, interpret complicated expressions, in terms of the context, by viewing one or more of their parts as a single entity

A-SSE.A.2

·         I can identify ways to rewrite expressions, such as difference of squares, factoring out a common monomial, regrouping, etc

·         I can identify ways to rewrite expressions based on the structure of the expression

·         I can use the structure of an expression to identify ways to rewrite it.

A-APR.A.1

·         I can identify that the sum, difference, or product of two polynomials will always be a polynomial, which means that polynomials are closed under the operations of addition, subtraction, and multiplication

·         I can define “closure”

·         I can apply arithmetic operations of addition, subtraction, and multiplication to polynomials

A-CED.A.1

·         I can solve linear and exponential equations in one variable

·         I can solve inequalities in one variable

·         I can describe the relationships between the quantities in the problem (for example, how the quantities are changing or growing with respect to each other); express these relationships using mathematical operations to create an appropriate equation or inequality to solve

·         I can create equations (linear and exponential) and inequalities in one variable and use them to solve problems

·         I can create equations and inequalities in one variable to model real-world situations

·         I can compare and contrast problems that can be solved by different types of equations (linear and exponential)

A-CED.A.2

·         I can identify the quantities in a mathematical problem or real-world situation that should be represented by distinct variables and describe what quantities the variables represent

·         I can create at least two equations in two or more variables to represent relationships between quantities

·         I can justify which quantities in a mathematical problem or real-world situation are dependent and independent of one another and which operations represent those relationships

·         I can determine appropriate units for the labels and scale of a graph depicting the relationship between equations created in two or more variables

·         I can graph one or more created equation on a coordinate axes with appropriate labels and scales

A-CED.A.3

·         I can recognize when a modeling context involves constraints

·         I can interpret solutions as viable or nonviable options in a modeling context

·         I can determine when a problem should be represented by equations, inequalities, systems of equations and/or inequalities

·         I can represent constraints by equations or inequalities, and by systems of equations and/or inequalities

A-CED.A.4

·         I can define a “quantity of interest” to mean any number or algebraic quantity (e.g. 2(a/b) = d, in which 2 is the quantity of interest showing that d must be even; πr2h/3 = Vcone and πr2h = Vcylinder showing that Vcylinder = 3* Vcone)

·         I can rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g. π * r2 can be re-written as (π *r) *r which makes the form of this expression resemble b*h)

A-REI.A.1

·         I can demonstrate that solving an equation means that the equation remains balanced during each step

·         I can recall the properties of equality

·         I can explain why, when solving equations, it is assumed that the original equation is equal

·         I can determine if an equation has a solution

·         I can classify expression by structure and develop strategies to assist in classification

·         I can explain the properties of the quantity represented by the quadratic expression

·         I can choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the original expression

·         I can explain the properties of the quantity represented by the expression

·         I can create equations and inequalities in one variable and use them to solve problems.

·         I can rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

·         I can explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution.

·         I can construct a viable argument to justify a solution method.

·         I can solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

·         I can solve quadratic equations in one variable.

·         I can solve quadratic equations by taking square roots.

·         I can explain why the sum or product of two rational numbers is rational.

·         I can explain why the sum of a rational number and an irrational number is irrational.

·         I can explain that the product of a nonzero rational number and an irrational number is irrational.

·         I can use units as a way to understand problems and to guide the solution of multi-step problems.

·         I can choose and interpret units consistently in formulas.

·         I can define appropriate quantities for the purpose of descriptive modeling.

·         I can choose an appropriate method for solving the equation

·         I can justify solution(s) to equations by explaining each step in solving a simple equation using the properties of equality, beginning with the assumption that the original equation is equal

·         I can construct a mathematically viable argument justifying a given, or self-generated, solution method

A-REI.B.3

·         I can recall properties of equality

·         I can solve multi-step equations in one variable

·         I can solve multi-step inequalities in one variable

·         I can determine the effect that rational coefficients have on the inequality symbol and use this to find the solution set

·         I can solve equations and inequalities with coefficients represented by letters

A-REI.C.5

·         I can recognize and use properties of equality to maintain equivalent systems of equations

·         I can justify that replacing one equation in a two-equation system with the sum of that equation and a multiple of the other will yield the same solutions as the original system

A-REI.C.6

·         I can solve systems of linear equations by any method

·         I can justify the method used to solve systems of linear equations exactly and approximately focusing on pairs of linear equations in two variables

A-REI.D.10

·         I can recognize that the graphical representation of an equation in two variables is a curve, which may be a straight line

·         I can explain why each point on a curve is a solution to its equation

A-REI.D.12

·         I can identify characteristics of a linear inequality and system of linear inequalities, such as: boundary line (where appropriate), shading, and determining appropriate test points to perform tests to find a solutions set

·         I can explain the meaning of the intersection of the shaded regions in a system of linear inequalities

·         I can graph a line, or boundary line, and shade the appropriate region for a two variable linear inequality

·         I can graph a system of linear inequalities and shade the appropriate overlapping region for a system of linear inequalities

N-Q.A.1

·         I can calculate unit conversions

·         I can recognize units given or need to solve problems

·         I can use given units and the context of a problem as a way to determine if the solution to a multi-step problem is reasonable (e.g. length problems dictate different units than problems dealing with a measure such as slope)

·         I can choose appropriate units to represent a problem when using formulas or graphing

·         I can interpret units or scales used in formulas or represented in graphs

·         I can use units as a way to understand problems and to guide the solution of multi-step problems

N-Q.A.2

·         I can define descriptive modeling

·         I can determine appropriate quantities for the purpose of descriptive modeling

N-Q.A.3

·         I can identify appropriate units of measurement to report quantities

·         I can determine the limitations of different measurement tools

·         I can choose and justify a level of accuracy and/or precision appropriate to limitations on measurement when reporting quantities

·         I can identify important quantities in a problem or real-world context

 

 

 

Enduring Understandings:

 

·         Changing the way that a function is represented does not change the function, although different representations highlight different characteristics, and some may only show part of the function.

·         Algebraic and numeric procedures are interconnected and build on one another. Integration of various mathematical procedures builds a stronger foundation of finding solutions.

·         The different parts of expressions can represent certain values in the context of a situation.

·         Linear equations and inequalities can be modeled with technology.

·         Mathematical models can both clarify and distort the meaning of data.

·         Making an informed decision often involves comparing and contrasting linear relationships by solving systems of equations.

·         Integration of various mathematical procedures builds a stronger foundation of finding solutions.

 

Essential Questions:

 

·         How can I use patterns to establish relationships that will help make decisions in real-life situations?

·         How do parameters introduced in the context of the problem affect the symbolic, numeric and graphical representations of a quadratic function?

·         How can verbal, numerical, graphical and analytical representations be used to analyze and solve problems?

·         How can the solution of a system of equations be used to make decisions and predictions?

·         How might technology be used to model linear systems or inequalities? Which technology should I use? How do I decide?

·         How do changes in equations lead to changes in graphs?

·         What makes alternative algebraic algorithms both effective and efficient?