West Virginia: High School Mathematics IV TR (Technical Readiness) Math Standards

DERIVE THE FORMULA FOR THE SUM OF A GEOMETRIC SERIES (WHEN THE COMMON RATIO IS NOT 1), AND USE THE FORMULA TO SOLVE PROBLEMS. (E.G., CALCULATE MORTGAGE PAYMENTS.)

PROVE POLYNOMIAL IDENTITIES AND USE THEM TO DESCRIBE NUMERICAL RELATIONSHIPS. FOR EXAMPLE, THE POLYNOMIAL IDENTITY (X^2 + Y^2)^2 = (X^2 – Y^2)^2 + (2XY)^2 CAN BE USED TO GENERATE PYTHAGOREAN TRIPLES.

KNOW AND APPLY THE BINOMIAL THEOREM FOR THE EXPANSION OF (X + Y)^N IN POWERS OF X AND Y FOR A POSITIVE INTEGER N, WHERE X AND Y ARE ANY NUMBERS, WITH COEFFICIENTS DETERMINED FOR EXAMPLE BY PASCAL’S TRIANGLE.

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); FIND THE SOLUTIONS APPROXIMATELY (E.G., USING TECHNOLOGY TO GRAPH THE FUNCTIONS, MAKE TABLES OF VALUES OR FIND SUCCESSIVE APPROXIMATIONS. INCLUDE CASES WHERE F(X) AND/OR G(X) ARE LINEAR, POLYNOMIAL, RATIONAL, ABSOLUTE VALUE, EXPONENTIAL AND LOGARITHMIC FUNCTIONS.

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.

DERIVE THE FORMULA A = 1/2 AB SIN(C) FOR THE AREA OF A TRIANGLE BY DRAWING AN AUXILIARY LINE FROM A VERTEX PERPENDICULAR TO THE OPPOSITE SIDE.

PROVE THE LAWS OF SINES AND COSINES AND USE THEM TO SOLVE PROBLEMS.

UNDERSTAND AND APPLY THE LAW OF SINES AND THE LAW OF COSINES TO FIND UNKNOWN MEASUREMENTS IN RIGHT AND NON-RIGHT TRIANGLES (E.G., SURVEYING PROBLEMS AND/OR RESULTANT FORCES).

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.

CREATE EQUATIONS IN TWO OR MORE VARIABLES TO REPRESENT RELATIONSHIPS BETWEEN QUANTITIES; GRAPH EQUATIONS ON COORDINATE AXES WITH LABELS AND SCALES.

REPRESENT CONSTRAINTS BY EQUATIONS OR INEQUALITIES AND BY SYSTEMS OF EQUATIONS AND/OR INEQUALITIES AND INTERPRET SOLUTIONS AS VIABLE OR NON-VIABLE OPTIONS IN A MODELING CONTEXT. (E.G., REPRESENT INEQUALITIES DESCRIBING NUTRITIONAL AND COST CONSTRAINTS ON COMBINATIONS OF DIFFERENT FOODS.)

REARRANGE FORMULAS TO HIGHLIGHT A QUANTITY OF INTEREST, USING THE SAME REASONING AS IN SOLVING EQUATIONS. (E.G., REARRANGE OHM’S LAW V = IR TO HIGHLIGHT RESISTANCE R.)

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

• M.4HSTR.38.a Graph square root, cube root and piecewise-defined functions, including step functions and absolute value functions.
• M.4HSTR.38.b Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline and amplitude.

WRITE A FUNCTION DEFINED BY AN EXPRESSION IN DIFFERENT BUT EQUIVALENT FORMS TO REVEAL AND EXPLAIN DIFFERENT PROPERTIES OF THE FUNCTION.

RECOGNIZE THE PURPOSES OF AND DIFFERENCES AMONG SAMPLE SURVEYS, EXPERIMENTS AND OBSERVATIONAL STUDIES; EXPLAIN HOW RANDOMIZATION RELATES TO EACH.

COMPARE PROPERTIES OF TWO FUNCTIONS EACH REPRESENTED IN A DIFFERENT WAY (ALGEBRAICALLY, GRAPHICALLY, NUMERICALLY IN TABLES, OR BY VERBAL DESCRIPTIONS). (E.G., GIVEN A GRAPH OF ONE QUADRATIC FUNCTION AND AN ALGEBRAIC EXPRESSION FOR ANOTHER, SAY WHICH HAS THE LARGER MAXIMUM.)

IDENTIFY THE SHAPES OF TWO-DIMENSIONAL CROSS-SECTIONS OF THREE DIMENSIONAL OBJECTS AND IDENTIFY THREE-DIMENSIONAL OBJECTS GENERATED BY ROTATIONS OF TWO-DIMENSIONAL OBJECTS.

USE GEOMETRIC SHAPES, THEIR MEASURES AND THEIR PROPERTIES TO DESCRIBE OBJECTS (E.G., MODELING A TREE TRUNK OR A HUMAN TORSO AS A CYLINDER).

APPLY CONCEPTS OF DENSITY BASED ON AREA AND VOLUME IN MODELING SITUATIONS (E.G., PERSONS PER SQUARE MILE OR BTUS PER CUBIC FOOT).

APPLY GEOMETRIC METHODS TO SOLVE DESIGN PROBLEMS (E.G., DESIGNING AN OBJECT OR STRUCTURE TO SATISFY PHYSICAL CONSTRAINTS OR MINIMIZE COST AND/OR WORKING WITH TYPOGRAPHIC GRID SYSTEMS BASED ON RATIOS).

USE DATA FROM A SAMPLE SURVEY TO ESTIMATE A POPULATION MEAN OR PROPORTION; DEVELOP A MARGIN OF ERROR THROUGH THE USE OF SIMULATION MODELS FOR RANDOM SAMPLING.

USE DATA FROM A RANDOMIZED EXPERIMENT TO COMPARE TWO TREATMENTS; USE SIMULATIONS TO DECIDE IF DIFFERENCES BETWEEN PARAMETERS ARE SIGNIFICANT.

EVALUATE REPORTS BASED ON DATA.

USE PROBABILITIES TO MAKE FAIR DECISIONS (E.G., DRAWING BY LOTS OR USING A RANDOM NUMBER GENERATOR).

ANALYZE DECISIONS AND STRATEGIES USING PROBABILITY CONCEPTS (E.G., PRODUCT TESTING, MEDICAL TESTING, AND/OR PULLING A HOCKEY GOALIE AT THE END OF A GAME).

INFERENCES AND CONCLUSIONS FROM DATA

MATHEMATICAL MODELING

POLYNOMIALS, RATIONAL, AND RADICAL RELATIONSHIPS

TRIGONOMETRY OF GENERAL TRIANGLES AND TRIGONOMETRIC FUNCTIONS