Alabama: Algebra I with Probability Math Standards

EXPLAIN HOW THE MEANING OF RATIONAL EXPONENTS FOLLOWS FROM EXTENDING THE PROPERTIES OF INTEGER EXPONENTS TO THOSE VALUES, ALLOWING FOR AN ADDITIONAL NOTATION FOR RADICALS USING RATIONAL EXPONENTS.

SELECT AN APPROPRIATE METHOD TO SOLVE A SYSTEM OF TWO LINEAR EQUATIONS IN TWO VARIABLES.

• A1.10.a Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.
• A1.10.b Contrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods.

CREATE EQUATIONS AND INEQUALITIES IN ONE VARIABLE AND USE THEM TO SOLVE PROBLEMS IN CONTEXT, EITHER EXACTLY OR APPROXIMATELY. EXTEND FROM CONTEXTS ARISING FROM LINEAR FUNCTIONS TO THOSE INVOLVING QUADRATIC, EXPONENTIAL, AND ABSOLUTE VALUE FUNCTIONS.

CREATE EQUATIONS IN TWO OR MORE VARIABLES TO REPRESENT RELATIONSHIPS BETWEEN QUANTITIES IN CONTEXT; GRAPH EQUATIONS ON COORDINATE AXES WITH LABELS AND SCALES AND USE THEM TO MAKE PREDICTIONS. LIMIT TO CONTEXTS ARISING FROM LINEAR, QUADRATIC, EXPONENTIAL, ABSOLUTE VALUE, AND LINEAR PIECEWISE FUNCTIONS.

REPRESENT CONSTRAINTS BY EQUATIONS AND/OR INEQUALITIES, AND SOLVE SYSTEMS OF EQUATIONS AND/OR INEQUALITIES, INTERPRETING SOLUTIONS AS VIABLE OR NONVIABLE OPTIONS IN A MODELING CONTEXT. LIMIT TO CONTEXTS ARISING FROM LINEAR, QUADRATIC, EXPONENTIAL, ABSOLUTE VALUE, AND LINEAR PIECEWISE FUNCTIONS.

GIVEN A RELATION DEFINED BY AN EQUATION IN TWO VARIABLES, IDENTIFY THE GRAPH OF THE RELATION AS THE SET OF ALL ITS SOLUTIONS PLOTTED IN THE COORDINATE PLANE. NOTE: THE GRAPH OF A RELATION OFTEN FORMS A CURVE (WHICH COULD BE A LINE).

DEFINE A FUNCTION AS A MAPPING FROM ONE SET (CALLED THE DOMAIN) TO ANOTHER SET (CALLED THE RANGE) THAT ASSIGNS TO EACH ELEMENT OF THE DOMAIN EXACTLY ONE ELEMENT OF THE RANGE.

• A1.15.a Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Note: If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x.
• A1.15.b Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Limit to linear, quadratic, exponential, and absolute value functions.

COMPARE AND CONTRAST RELATIONS AND FUNCTIONS REPRESENTED BY EQUATIONS, GRAPHS, OR TABLES THAT SHOW RELATED VALUES; DETERMINE WHETHER A RELATION IS A FUNCTION. EXPLAIN THAT A FUNCTION F IS A SPECIAL KIND OF RELATION DEFINED BY THE EQUATION Y = F(X).

COMBINE DIFFERENT TYPES OF STANDARD FUNCTIONS TO WRITE, EVALUATE, AND INTERPRET FUNCTIONS IN CONTEXT. LIMIT TO LINEAR, QUADRATIC, EXPONENTIAL, AND ABSOLUTE VALUE FUNCTIONS.

• A1.17.a Use arithmetic operations to combine different types of standard functions to write and evaluate functions.
• A1.17.b Use function composition to combine different types of standard functions to write and evaluate functions.

SOLVE SYSTEMS CONSISTING OF LINEAR AND/OR QUADRATIC EQUATIONS IN TWO VARIABLES GRAPHICALLY, USING TECHNOLOGY WHERE APPROPRIATE.

EXPLAIN WHY THE X-COORDINATES OF THE POINTS WHERE THE GRAPHS OF THE EQUATIONS Y = F(X) AND Y = G(X) INTERSECT ARE THE SOLUTIONS OF THE EQUATION F(X) = G(X).

• A1.19.a Find the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. Note: Include cases where f(x) is a linear, quadratic, exponential, or absolute value function and g(x) is constant or linear.

REWRITE EXPRESSIONS INVOLVING RADICALS AND RATIONAL EXPONENTS USING THE PROPERTIES OF EXPONENTS.

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, USING TECHNOLOGY WHERE APPROPRIATE.

COMPARE PROPERTIES OF TWO FUNCTIONS, EACH REPRESENTED IN A DIFFERENT WAY (ALGEBRAICALLY, GRAPHICALLY, NUMERICALLY IN TABLES, OR BY VERBAL DESCRIPTIONS). EXTEND FROM LINEAR TO QUADRATIC, EXPONENTIAL, ABSOLUTE VALUE, AND GENERAL PIECEWISE.

DEFINE SEQUENCES AS FUNCTIONS, INCLUDING RECURSIVE DEFINITIONS, WHOSE DOMAIN IS A SUBSET OF THE INTEGERS.

• A1.22.a Write explicit and recursive formulas for arithmetic and geometric sequences and connect them to linear and exponential functions.

IDENTIFY THE EFFECT ON THE GRAPH OF REPLACING F(X) BY F(X) + K, K ∙ F(X), F(K ∙ X), AND F(X + K) FOR SPECIFIC VALUES OF K (BOTH POSITIVE AND NEGATIVE); FIND THE VALUE OF K GIVEN THE GRAPHS. EXPERIMENT WITH CASES AND EXPLAIN THE EFFECTS ON THE GRAPH, USING TECHNOLOGY AS APPROPRIATE. LIMIT TO LINEAR, QUADRATIC, EXPONENTIAL, ABSOLUTE VALUE, AND LINEAR PIECEWISE FUNCTIONS.

DISTINGUISH BETWEEN SITUATIONS THAT CAN BE MODELED WITH LINEAR FUNCTIONS AND THOSE THAT CAN BE MODELED WITH EXPONENTIAL FUNCTIONS.

• A1.24.a Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.
• A1.24.b Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.
• A1.24.c Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

CONSTRUCT LINEAR AND EXPONENTIAL FUNCTIONS, INCLUDING ARITHMETIC AND GEOMETRIC SEQUENCES, GIVEN A GRAPH, A DESCRIPTION OF A RELATIONSHIP, OR TWO INPUT-OUTPUT PAIRS (INCLUDE READING THESE FROM A TABLE).

USE GRAPHS AND TABLES TO SHOW THAT A QUANTITY INCREASING EXPONENTIALLY EVENTUALLY EXCEEDS A QUANTITY INCREASING LINEARLY OR QUADRATICALLY.

INTERPRET THE PARAMETERS OF FUNCTIONS IN TERMS OF A CONTEXT. EXTEND FROM LINEAR FUNCTIONS, WRITTEN IN THE FORM MX + B, TO EXPONENTIAL FUNCTIONS, WRITTEN IN THE FORM AB^X.

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. NOTE: KEY FEATURES INCLUDE: INTERCEPTS; INTERVALS WHERE THE FUNCTION IS INCREASING, DECREASING, POSITIVE, OR NEGATIVE; MAXIMUMS AND MINIMUMS; SYMMETRIES; AND END BEHAVIOR. EXTEND FROM RELATIONSHIPS THAT CAN BE REPRESENTED BY LINEAR FUNCTIONS TO QUADRATIC, EXPONENTIAL, ABSOLUTE VALUE, AND LINEAR PIECEWISE FUNCTIONS.

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. LIMIT TO LINEAR, QUADRATIC, EXPONENTIAL, AND ABSOLUTE VALUE FUNCTIONS.

DEFINE THE IMAGINARY NUMBER I SUCH THAT I^2 = -1.

GRAPH FUNCTIONS EXPRESSED SYMBOLICALLY AND SHOW KEY FEATURES OF THE GRAPH, BY HAND IN SIMPLE CASES AND USING TECHNOLOGY FOR MORE COMPLICATED CASES.

• A1.30.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
• A1.30.b Graph piecewise-defined functions, including step functions and absolute value functions.
• A1.30.c Graph exponential functions, showing intercepts and end behavior.

USE THE MATHEMATICAL MODELING CYCLE TO SOLVE REAL-WORLD PROBLEMS INVOLVING LINEAR, QUADRATIC, EXPONENTIAL, ABSOLUTE VALUE, AND LINEAR PIECEWISE FUNCTIONS.

USE MATHEMATICAL AND STATISTICAL REASONING WITH BIVARIATE CATEGORICAL DATA IN ORDER TO DRAW CONCLUSIONS AND ASSESS RISK.

DESIGN AND CARRY OUT AN INVESTIGATION TO DETERMINE WHETHER THERE APPEARS TO BE AN ASSOCIATION BETWEEN TWO CATEGORICAL VARIABLES, AND WRITE A PERSUASIVE ARGUMENT BASED ON THE RESULTS OF THE INVESTIGATION.

DISTINGUISH BETWEEN QUANTITATIVE AND CATEGORICAL DATA AND BETWEEN THE TECHNIQUES THAT MAY BE USED FOR ANALYZING DATA OF THESE TWO TYPES.

ANALYZE THE POSSIBLE ASSOCIATION BETWEEN TWO CATEGORICAL VARIABLES.

• A1.35.a Summarize categorical data for two categories in two-way frequency tables and represent using segmented bar graphs.
• A1.35.b Interpret relative frequencies in the context of categorical data (including joint, marginal, and conditional relative frequencies).
• A1.35.c Identify possible associations and trends in categorical data.

GENERATE A TWO-WAY CATEGORICAL TABLE IN ORDER TO FIND AND EVALUATE SOLUTIONS TO REAL-WORLD PROBLEMS.

• A1.36.a Aggregate data from several groups to find an overall association between two categorical variables.
• A1.36.b Recognize and explore situations where the association between two categorical variables is reversed when a third variable is considered (Simpson's Paradox).

DESCRIBE EVENTS AS SUBSETS OF A SAMPLE SPACE (THE SET OF OUTCOMES) USING CHARACTERISTICS (OR CATEGORIES) OF THE OUTCOMES, OR AS UNIONS, INTERSECTIONS, OR COMPLEMENTS OF OTHER EVENTS ("OR," "AND," "NOT").

EXPLAIN WHETHER TWO EVENTS, A AND B, ARE INDEPENDENT, USING TWO-WAY TABLES OR TREE DIAGRAMS.

COMPUTE THE CONDITIONAL PROBABILITY OF EVENT A GIVEN EVENT B, USING TWO-WAY TABLES OR TREE DIAGRAMS.

INTERPRET LINEAR, QUADRATIC, AND EXPONENTIAL EXPRESSIONS IN TERMS OF A CONTEXT BY VIEWING ONE OR MORE OF THEIR PARTS AS A SINGLE ENTITY.

RECOGNIZE AND DESCRIBE THE CONCEPTS OF CONDITIONAL PROBABILITY AND INDEPENDENCE IN EVERYDAY SITUATIONS AND EXPLAIN THEM USING EVERYDAY LANGUAGE.

EXPLAIN WHY THE CONDITIONAL PROBABILITY OF A GIVEN B IS THE FRACTION OF B'S OUTCOMES THAT ALSO BELONG TO A, AND INTERPRET THE ANSWER IN CONTEXT.

USE THE STRUCTURE OF AN EXPRESSION TO IDENTIFY WAYS TO REWRITE IT.

CHOOSE AND PRODUCE AN EQUIVALENT FORM OF AN EXPRESSION TO REVEAL AND EXPLAIN PROPERTIES OF THE QUANTITY REPRESENTED BY THE EXPRESSION.

• A1.6.a Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.
• A1.6.b Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.
• A1.6.c Use the properties of exponents to transform expressions for exponential functions.

ADD, SUBTRACT, AND MULTIPLY POLYNOMIALS, SHOWING THAT POLYNOMIALS FORM A SYSTEM ANALOGOUS TO THE INTEGERS, NAMELY, THEY ARE CLOSED UNDER THE OPERATIONS OF ADDITION, SUBTRACTION, AND MULTIPLICATION.

EXPLAIN WHY EXTRANEOUS SOLUTIONS TO AN EQUATION INVOLVING ABSOLUTE VALUES MAY ARISE AND HOW TO CHECK TO BE SURE THAT A CANDIDATE SOLUTION SATISFIES AN EQUATION.

SELECT AN APPROPRIATE METHOD TO SOLVE A QUADRATIC EQUATION IN ONE VARIABLE.

• A1.9.a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)^2 = q that has the same solutions. Explain how the quadratic formula is derived from this form.
• A1.9.b Solve quadratic equations by inspection (such as x^2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.