GRADE 8 ALGEBRA CURRICULUM

REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE LITERACY

Number Systems and Number Sense

Students Will:

Recognize the different properties that hold in different number systems

Explain multiplicative and additive inverse

Identify associativity, commutativity, and distributivity, as well as identity and inverse elements, are used in arithmetic and algebraic calculations.

Describe the reasons for the different effects of multiplication by, or exponentiation of, a positive

Justify numerical relationships

Representations and Relationships

Students Will:

Interpret representations that reflect absolute value relationships

Organize and summarize a data set in a table, plot, chart, or spreadsheet; find patterns in a display of data; understand and critique data displays in the media.

CALCULATION, ALGORITHMS, AND ESTIMATION

Calculation Using Real and Complex Numbers

Students Will:

Explain the meaning and uses of weighted averages.

Calculate fluently with numerical expressions involving exponents.

Explain the exponential relationship between a number and its base 10 logarithm and use it to relate rules of logarithms.

Know that the complex number i is one of two solutions to x 2 = -1.

Add, subtract, and multiply complex numbers. Use conjugates to simplify quotients of complex numbers.

Recognize when exact answers are not always possible or practical. Use appropriate algorithms to approximate solutions to equations (e.g., to approximate square roots).

MEASUREMENT AND PRECISION

Measurement Units, Calculations, and Scales

Students Will:

Describe and interpret logarithmic relationships in such contexts as the Richter scale, the pH scale, or decibel measurements (e.g., explain why a small change in the scale can represent a large change in intensity).

Solve applied problems.

EXPRESSIONS, EQUATIONS, AND INEQUALITIES

Construction, Interpretation, and Manipulation of Expressions (linear, quadratic, polynomial, rational, power, exponential, and logarithmic)

Students Will:

Give a verbal description of an expression that is presented in symbolic form, write an algebraic expression from a verbal description, and evaluate expressions given values of the variables.

Know the definitions and properties of exponents and roots and apply them in algebraic expressions.

Factor algebraic expressions using, for example, greatest common factor, grouping, and the special product identities (e.g., differences of squares and cubes).

Use the properties of exponents and logarithms, including the inverse relationship between exponents and logarithms, to transform exponential and logarithmic expressions into equivalent forms.

Solutions of Equations and Inequalities (linear, exponential, logarithmic, quadratic, power, polynomial, and rational)

Students Will:

Write and solve equations and inequalities

Associate a given equation with a function whose zeros are the solutions of the equation.

Solve linear and quadratic equations and inequalities. Justify steps in the solutions, and apply the quadratic formula appropriately.

Solve absolute value equations and inequalities and justify.

Solve power equations (e.g., ( x + 1) 3 = 8) and equations including radical expressions (e.g., 3 x - 7 = 7), justify steps in the solution, and explain how extraneous solutions may arise.

Solve an equation involving several variables (with numerical or letter coefficients) for a designated variable. Justify steps in the solution.

FUNCTIONS

Definitions, Representations, and Attributes of Functions

Students Will:

Recognize whether a relationship is a function and identify its domain and range.

Read, interpret, and use function notation and evaluate a function at a value in its domain.

Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations.

Recognize that functions may be defined by different expressions over different intervals of their domains.

Recognize that functions may be defined recursively. Compute values of and graph simple recursively defined functions (e.g., f (0) = 5, and f ( n ) = f ( n -1) + 2).

Identify the zeros of a function and the intervals where the values of a function are positive or negative. Describe the behavior of a function as x approaches positive or negative infinity, given the symbolic and graphical representations.

Identify and interpret the key features of a function from its graph or its formula(e), (e.g., slope, intercept(s), asymptote(s), maximum and minimum value(s), symmetry, and average rate of change over an interval).

Operations and Transformations

Students Will:

Combine functions by addition, subtraction, multiplication, and division.

Apply given transformations (e.g., vertical or horizontal shifts, stretching or shrinking, or reflections about the x - and y -axes) to basic functions and represent symbolically.

Recognize Determine whether a function (given in tabular or graphical form) has an inverse and recognize simple inverse pairs.

Families of Functions (linear, quadratic, polynomial, power, exponential, and logarithmic)

Students Will:

Identify a function as a member of a family of functions based on its symbolic or graphical representation. Recognize that different families of functions have different asymptotic behavior at infinity and describe these behaviors.

Describe the tabular pattern associated with functions having constant rate of change (linear) or variable rates of change.

Lines and Linear Functions

Students Will:

Write the symbolic forms of linear functions (standard [i.e., Ax + By = C, where B ≠ 0 ], point-slope, and slope-intercept) given appropriate information and convert between forms.

Graph lines (including those of the form x = h and y = k ) given appropriate information.

Relate the coefficients in a linear function to the slope and x - and y -intercepts of its graph.

Find an equation of the line parallel or perpendicular to given line through a given point. Understand and use the facts that nonvertical parallel lines have equal slopes and that nonvertical perpendicular lines have slopes that multiply to give -1.

Exponential and Logarithmic Functions

Students Will :

Write the symbolic form and sketch the graph of an exponential function given appropriate information (e.g., given an initial value of 4 and a rate of growth of 1.5, write f ( x ) = 4 (1.5) x ).

Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and how base affects the rate of growth or decay.

Relate exponential and logarithmic functions to real phenomena, including half-life and doubling time.

Quadratic Functions

Students Will :

Write the symbolic form and sketch the graph of a quadratic function given appropriate information (e.g., vertex, intercepts, etc.).

Identify the elements of a parabola (vertex, axis of symmetry, and direction of opening) given its symbolic form or its graph and relate these elements to the coefficient(s) of the symbolic form of the function.

Convert quadratic functions from standard to vertex form by completing the square.

Relate the number of real solutions of a quadratic equation to the graph of the associated quadratic function.

Express quadratic functions in vertex form to identify their maxima or minima and in factored form to identify their zeros.

Power Functions (including roots, cubics, quartics, etc.)

Students Will :

Write the symbolic form and sketch the graph of power functions.

Express direct and inverse relationships as functions and recognize their characteristics results in multiplying y by a factor of 8).

Analyze the graphs of power functions, noting reflectional or rotational symmetry.

Polynomial Functions

Students Will :

Write the symbolic form and sketch the graph of simple polynomial functions.

Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2.

Determine the maximum possible number of zeroes of a polynomial function and understand the relationship between the x -intercepts of the graph and the factored form of the function.

MATHEMATICAL MODELING

Models of Real-world Situations Using Families of Functions (linear, quadratic, exponential and power)

Students Will:

Example: An initial population of 300 people grows at 2% per year. What will the population be in 10 years?

Identify the family of function best suited for modeling a given real-world situation [e.g., quadratic functions for motion of an object under the force of gravity or exponential functions for compound interest. In the example above, recognize that the appropriate general function is exponential (P = P 0 a t )].

Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers. In the example above, substitute the given values P 0 = 300 and a = 1.02 to obtain P = 300(1.02) t .

Using the adapted general symbolic form, draw reasonable conclusions about the situation being modeled. In the example above, the exact solution is 365.698, but for this problem, an appropriate approximation is 365.

RECOMMENDED: Use methods of linear programming to represent and solve simple real-life problems.

BIVARIATE DATA-EXAMINING RELATIONSHIPS

Scatter plots and Correlation

Students Will:

Construct a scatter plot for a bivariate data set with appropriate labels and scales.

Given a scatter plot, identify patterns, clusters, and outliers. Recognize no correlation, weak correlation, and strong correlation.

Estimate and interpret Pearson’s correlation coefficient for a scatter plot of a bivariate data set. Recognize that correlation measures the strength of linear association.

Differentiate between correlation and causation. Know that a strong correlation does not imply a cause-and-effect relationship. Recognize the role of lurking variables in correlation.

Linear Regression

Students Will:

Interpret the slope of the equation for a regression line. For bivariate data that appear to form a linear pattern, find the least squares regression line by estimating visually and by calculating the equation of the regression line.

Use the equation of the least squares regression line to make appropriate predictions.