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Study GuideLinear equations

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Worksheet/Answer keyLinear equations

Worksheet/Answer keyLinear equations

MD.MA.8.EE. Expressions and Equations (EE)

Understand the connections between proportional relationships, lines, and linear equations.

8.EE.5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

8.EE.5.1. Ability to relate and compare graphic, symbolic, numerical representations of proportional relationships.

8.EE.5.2. Ability to calculate constant rate of change/slope of a line graphically.

8.EE.6. Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical 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.6.1. Ability to understand that similar right triangles (provide diagram of graphical notation) can be used to establish that slope is constant for a non-vertical line (see 8.G.1).

MD.MA.8.F. Functions (F)

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 a linear function represented by an algebraic expression, determine which function has the greater rate of change.

8.F.2.2. Ability to calculate slope/rate of change of a line graphically from a table or verbal description.

Use functions to model relationships between quantities.

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.4.1. Ability to calculate and interpret constant rate of change/slope from a scenario, table, graph, or two points.

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.

8.F.5.1. Ability to distinguish rate of change within an interval of a function.

8.F.5.2. Ability to interpret directionality and steepness of the graph of a function.

8.F.5.3. Ability to sketch a graph given algebraic context or a scenario (slope and initial value).

MD.MA.AI. Algebra I

Unit 1: Relationships between Quantities and Reasoning with Equations

HSA-SSE.A. Interpret the structure of expressions.

A.SSE.1a. Interpret expressions that represent a quantity in terms of its context – Interpret parts of an expression, such as terms, factors, and coefficients.

A.SSE.1a.1. Ability to make connections between symbolic representations and proper mathematics vocabulary.

A.SSE.1b. Interpret expressions that represent a quantity in terms of its context – 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.1b.1. Ability to interpret and apply rules for order of operations.

Unit 2: Linear and Exponential Relationships

HSF-IF.B. Interpret functions that arise in applications in terms of a context.

F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F.IF.6.1. Knowledge that the rate of change of a function can be positive, negative or zero.

F.IF.6.3. Ability to identify the rate of change from multiple representations.

HSF-IF.C. Analyze functions using different representations.

F.IF.7a. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases – Graph linear and quadratic functions and show intercepts, maxima, and minima.

F.IF.7a.1. See the skills and knowledge that are stated in the Standard.

HSF-LE.A. Construct and compare linear, quadratic, and exponential models and solve problems.

F.LE.1a. Distinguish between situations that can be modeled with linear functions and with exponential functions – Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals.

F.LE.1a.1. See the skills and knowledge that are stated in the Standard.

HSF-LE.B. Interpret expressions for functions in terms of the situation they model.

F.LE.5. Interpret the parameters in a linear or exponential function in terms of a context.

F.LE.5.1. Ability to interpret the slope and y-intercept of a linear model in terms of context.

Unit 3: Descriptive Statistics

HSS-ID.C. Interpret linear models.

S.ID.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S.ID.7.1. See the skills and knowledge that are stated in the Standard.

Unit 4: Expressions and Equations

HSA-SSE.B. Write expressions in equivalent forms to solve problems.

A.SSE.3a. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression – Factor a quadratic expression to reveal the zeros of the function it defines.

A.SSE.3a.1. Ability to connect the factors, zeros and x-intercepts of a graph.

A.SSE.3a.2. Ability to connect the factors, zeros and x-intercepts of a graph.

Unit 5: Quadratic Functions and Modeling

HSF-IF.B. Interpret functions that arise in applications in terms of a context.

F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F.IF.6.2. Knowledge that the rate of change of a function can be positive, negative, zero or can have no change.

F.IF.6.3. Ability to identify the rate of change from multiple representations.

MD.MA.AII. Algebra II

Unit 1: Polynomial, Rational, and Radical Relationships

HSF-IF.C. Analyze functions using different representations.

F.IF.7c. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases – Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

F.IF.7c.2. Ability to identify key features of a function: max, min, intercepts, zeros, and end behaviors.

Unit 3: Modeling with Functions

HSF-IF.B. Interpret functions that arise in applications in terms of a context.

F.IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.4.4. Knowledge of the key features of linear, exponential, polynomial, root, absolute value, piece-wise, simple rational, logarithmic and trigonometric functions.

F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F.IF.6.4. Ability to apply this skill to linear, quadratic, polynomial, root and simple rational functions.

HSF-LE.B. Interpret expressions for functions in terms of the situation they model.

F.LE.5. Interpret the parameters in a linear or exponential function in terms of a context.

F.LE.5.1. Ability to interpret the slope and y-intercept of a linear model in terms of context.

MD.MA.G. Geometry

Unit 4: Connecting Algebra and Geometry Through Coordinates

HSG-GPE.B. Use coordinates to prove simple geometric theorems algebraically.

G.GPE.4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle.

G.GPE.4.1. Ability to use distance, slope and midpoint formulas…

G.GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

G.GPE.6.1. Ability to use the slope formula.