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

USE THE MEAN AND STANDARD DEVIATION OF A DATA SET TO FIT IT TO A NORMAL DISTRIBUTION AND TO ESTIMATE POPULATION PERCENTAGES. RECOGNIZE THAT THERE ARE DATA SETS FOR WHICH SUCH A PROCEDURE IS NOT APPROPRIATE. USE CALCULATORS, SPREADSHEETS AND TABLES TO ESTIMATE AREAS UNDER THE NORMAL CURVE.

INTERPRET EXPRESSIONS THAT REPRESENT A QUANTITY IN TERMS OF ITS CONTEXT.

• M.3HSTR.12.a Interpret parts of an expression, such as terms, factors, and coefficients.
• M.3HSTR.12.b Interpret complicated expressions by viewing one or more of their parts as a single entity. (e.g., Interpret P(1 + r)n as the product of P and a factor not depending on P.)

USE THE STRUCTURE OF AN EXPRESSION TO IDENTIFY WAYS TO REWRITE IT. FOR EXAMPLE, SEE X^4 – Y^4 AS (X^2)^2 – (Y^2)^2, THUS RECOGNIZING IT AS A DIFFERENCE OF SQUARES THAT CAN BE FACTORED AS (X^2 – Y^2)(X^2 + Y^2).

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.

KNOW AND APPLY THE REMAINDER THEOREM: FOR A POLYNOMIAL P(X) AND A NUMBER A, THE REMAINDER ON DIVISION BY X – A IS P(A), SO P(A) = 0 IF AND ONLY IF (X – A) IS A FACTOR OF P(X).

IDENTIFY ZEROS OF POLYNOMIALS WHEN SUITABLE FACTORIZATIONS ARE AVAILABLE AND USE THE ZEROS TO CONSTRUCT A ROUGH GRAPH OF THE FUNCTION DEFINED BY THE POLYNOMIAL.

UNDERSTAND THAT STATISTICS ALLOWS INFERENCES TO BE MADE ABOUT POPULATION PARAMETERS BASED ON A RANDOM SAMPLE FROM THAT POPULATION.

REWRITE SIMPLE RATIONAL EXPRESSIONS IN DIFFERENT FORMS; WRITE A(X)/B(X) IN THE FORM Q(X) + R(X)/B(X), WHERE A(X), B(X), Q(X), AND R(X) ARE POLYNOMIALS WITH THE DEGREE OF R(X) LESS THAN THE DEGREE OF B(X), USING INSPECTION, LONG DIVISION, OR, FOR THE MORE COMPLICATED EXAMPLES, A COMPUTER ALGEBRA SYSTEM.

SOLVE SIMPLE RATIONAL AND RADICAL EQUATIONS IN ONE VARIABLE AND GIVE EXAMPLES SHOWING HOW EXTRANEOUS SOLUTIONS MAY ARISE.

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.

UNDERSTAND RADIAN MEASURE OF AN ANGLE AS THE LENGTH OF THE ARC ON THE UNIT CIRCLE SUBTENDED BY THE ANGLE.

EXPLAIN HOW THE UNIT CIRCLE IN THE COORDINATE PLANE ENABLES THE EXTENSION OF TRIGONOMETRIC FUNCTIONS TO ALL REAL NUMBERS, INTERPRETED AS RADIAN MEASURES OF ANGLES TRAVERSED COUNTERCLOCKWISE AROUND THE UNIT CIRCLE.

DECIDE IF A SPECIFIED MODEL IS CONSISTENT WITH RESULTS FROM A GIVEN DATA-GENERATING PROCESS, FOR EXAMPLE, USING SIMULATION. (E.G., A MODEL SAYS A SPINNING COIN FALLS HEADS UP WITH PROBABILITY 0.5. WOULD A RESULT OF 5 TAILS IN A ROW CAUSE YOU TO QUESTION THE MODEL?)

CHOOSE TRIGONOMETRIC FUNCTIONS TO MODEL PERIODIC PHENOMENA WITH SPECIFIED AMPLITUDE, FREQUENCY, AND MIDLINE.

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.)

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.

RELATE THE DOMAIN OF A FUNCTION TO ITS GRAPH AND, WHERE APPLICABLE, TO THE QUANTITATIVE RELATIONSHIP IT DESCRIBES. (E.G., IF THE FUNCTION H(N) GIVES THE NUMBER OF PERSON-HOURS IT TAKES TO ASSEMBLE N ENGINES IN A FACTORY, THEN THE POSITIVE INTEGERS WOULD BE AN APPROPRIATE DOMAIN FOR THE FUNCTION.)

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.

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

• M.3HSTR.38.a Graph square root, cube root and piecewise-defined functions, including step functions and absolute value functions.
• M.3HSTR.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.)

WRITE A FUNCTION THAT DESCRIBES A RELATIONSHIP BETWEEN TWO QUANTITIES. COMBINE STANDARD FUNCTION TYPES USING ARITHMETIC OPERATIONS. (E.G., BUILD A FUNCTION THAT MODELS THE TEMPERATURE OF A COOLING BODY BY ADDING A CONSTANT FUNCTION TO A DECAYING EXPONENTIAL, AND RELATE THESE FUNCTIONS TO THE MODEL.)

IDENTIFY THE EFFECT ON THE GRAPH OF REPLACING F(X) BY F(X) + K, K F(X), F(KX), 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 ILLUSTRATE AN EXPLANATION OF THE EFFECTS ON THE GRAPH USING TECHNOLOGY. INCLUDE RECOGNIZING EVEN AND ODD FUNCTIONS FROM THEIR GRAPHS AND ALGEBRAIC EXPRESSIONS FOR THEM.

FIND INVERSE FUNCTIONS. SOLVE AN EQUATION OF THE FORM F(X) = C FOR A SIMPLE FUNCTION F THAT HAS AN INVERSE AND WRITE AN EXPRESSION FOR THE INVERSE. (E.G., F(X) = 2 X^3 OR F(X) = (X+1)/(X-1) FOR X ≠ 1.)

FOR EXPONENTIAL MODELS, EXPRESS AS A LOGARITHM THE SOLUTION TO A B^CT = D WHERE A, C, AND D ARE NUMBERS AND THE BASE B IS 2, 10, OR E; EVALUATE THE LOGARITHM USING TECHNOLOGY.

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.

INFERENCES AND CONCLUSIONS FROM DATA

MATHEMATICAL MODELING

POLYNOMIALS, RATIONAL, AND RADICAL RELATIONSHIPS

TRIGONOMETRY OF GENERAL TRIANGLES AND TRIGONOMETRIC FUNCTIONS