Download Essential Standard - Big Pine Unified School District

ALGEBRA ESSENTIAL STANDARDS AND BENCHMARKS
IDENTIFY AND USE MATHEMATICAL OPERATIONS AND PROPERTIES OVER REAL NUMBERS, MONOMIALS, AND POLYNOMIALS.
First Semester: Benchmarks
 1.0
Students add, subtract, multiply, and divide real numbers.
 1.0
Students use numerical properties. (Associative and commutative properties of addition and multiplication and the distributive
property)
 2.0
Students use exponents, roots, and operations. (Taking the opposite and finding the reciprocal)
Second Semester: Benchmarks
 2.0
Students apply the rules of exponents, including negative and zero exponents.
 10.0 Students add, subtract, multiply, and divide polynomials.
 11.0 Students apply basic factoring techniques to second- and simple-third degree polynomials. This includes finding a common factor
for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to lowest
terms.
 13.0 Students add, subtract, multiple, and divide rational expressions and functions.
APPLY APPROPRIATE ALGEBRAIC PROCEDURES TO SOLVE REAL-LIFE PROBLEMS.
First Semester: Benchmarks
 4.0
Students simplify expressions before solving equations and inequalities in one variable.
 5.0
Students solve multi-step problems, including word problems, involving linear equations and inequalities in one variable.
Second Semester: Benchmarks
 10.0 Students add, subtract, multiple, and divide monomials and polynomials. Students solve multi-step problems, including word
problems, by using these techniques.
 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
WRITE, SOLVE, AND GRAPH EQUATIONS AND INEQUALITIES, INCLUDING THOSE INVOLVING ABOLSUTE VALUE.
First Semester: Benchmarks
 3.0
Students solve equations and inequalities involving absolute value.
 4.0
Students simplify expressions and solve equations and inequalities in one variable.
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 6.0
Students graph a linear equation and compute the x- and y- intercepts. They are also able to sketch the region defined by linear
inequality.
 7.0
Students verify that a point lies on a line, given the equation of the line. Students derive linear equations using the point-slope
formula.
 8.0
Students recognize the concepts of parallel lines and perpendicular lines and how their slopes are related. Students find the
equation of a line perpendicular to a given line that passes through a given point.
Second Semester: Benchmarks
 9.0
Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically.
Students solve a system of two linear inequalities in two variables and sketch the solution sets.
 14.0 Students solve quadratic equations by factoring or completing the square.
 13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve challenging rational equations by
using these techniques.
 17.0 Students determine the domain and the range of a function defined by a graph, a set of ordered pairs, or a symbolic expression.
 18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function.
 19.0 Students recognize the quadratic formula and are familiar with its proof by completing the square.
 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.
 21.0 Students graph quadratic functions, including inequalities, and know their roots are the x- intercepts. Students determine
whether the graph is a function.
 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will
intersect the x-axis in zero, one, or two points.
GEOMETRY ESSENTIAL STANDARDS AND BENCHMARKS
LS
KNOW AND APPLY THE BASIC DEFINITIONS AND UNDEFINED TERMS OF GEOMETRY.
First Semester: Benchmarks
 1.0
Students identify and give examples of the three undefined terms of geometry.
 L.S.
Students identify and classify acute angles, obtuse angles, right angles, vertical angles, supplementary angles, and
complementary angles.
 L.S.
Students classify triangles as acute, obtuse, right, equilateral, equiangular, scalene, and isosceles.
 L.S.
Students identify the midpoint, bisector, and perpendicular bisector of a given segment.
 16.0 Students construct with a straight edge and compass the mid point of a segment, an angle bisector, and the
perpendicular bisector of a segment.
 L.S.
Given two points, students derive the equation of a line through the two points and derive the equation of a line perpendicular to
the original line through a given point.
 8.0
Students solve problems involving the area and perimeter of basic geometric shapes.
PARALLEL LINES
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7.0
PROVE, USE AND ANALYZE PARALLEL LINES
First Semester: Benchmarks
 7.0
Students prove, use and analyze parallel lines.
 16.0 Students construct a line parallel to a given line through a given point.
TRANSFORMATIONS
22.0
PERFORM BASIC TRANSFORMATIONS ON VARIOUS GEOMETRIC FIGURES.
First Semester: Benchmarks
 22.0 Students identify and perform translations, reflections, and rotations of line segments and polygons.
TRIANGLES
LS
PROVE, USE, AND ANALYZE TRIANGLES.
First Semester: Benchmarks
 15.0 Students solve for the missing side of a right triangle using the Pythagorean theorem.
 5.0
Students prove triangles congruent using (SSS, SAS, ASA, AAS, and HL). Reference: 2.0, 4.0, 5.0
 L.S.
Students calculate lengths and angles related to the mid-segment of a triangle. Reference: 2.0, 4.0, 5.0
Second Semester: Benchmarks
 4.0
Students prove two triangles are similar by AA, SAS, and SSS.
 6.0
Students apply the triangle inequality theorem.
 14.0 Students prove the Pythagorean theorem.
TRIGONOMETRY
LS
USE TRIGONOMETRIC FUNCTIONS AND APPLY THEM IN PROBLEM SOLVING
Second Semester: Benchmarks
 18.0 Students evaluate the sine, cosine and, tangent of a given angle.
 19.0 Students use trigonometric functions to solve for unknown parts of a right triangle.
Students apply the laws of sines.
Students recognize the identity tan A =sin A
cos A
 20.0 Students use the relationships of the special right triangle, e.g., 30, 60, 90 and 45, 45, 90 to solve for a missing portion of
the triangle.
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Students use the special right triangle and apply these to problems using sine, cosine, and tangent.
Students identify the relationship between the various angles and the sides opposite.
POLYGONS
LS
IDENTIFY AND WORK WITH POLYGONS AND THEIR RELATED GEOMETRIC FEATURES.
Second Semester: Benchmarks
 7.0
Students classify quadrilaterals based on the characteristics of sides, angles, and diagonals.
 8.0 & 10.0
Students calculate perimeter and area of polygons.
 12.0
Students identify and name polygons based on their sides and angles.
 12.0 & 13.0 Students calculate interior and exterior angles, as well as the sums of the angles of polygons.
CIRCLES
LS
IDENTIFY AND WORK WITH CIRCLES AND THEIR RELATED GEOMETRIC FEATURES.
Second Semester: Benchmarks
 8.0
Students find the circumference and area of a circle, the area of a sector and the length of an arc.
 21.0 Students identify chords, secants, tangents, major arcs, minor arcs, central angles and inscribed angles.
Students calculate the measures of central angles, inscribed angles and arcs, as well as, calculate the lengths of radii, chords, and
related segments on a circle.
Students solve problems involving polygons inscribed in a circle.
THREE-DIMENSIONAL FIGURES
LS
SOLVE PROBLEMS INVOLVING THREE-DIMENSIONAL FIGURES.
Second Semester: Benchmarks
 9.0
Students determine volume of three-dimensional figures.
 9.0
Students determine surface area of geometric solids.
 11.0 Students describe and quantify changes in dimension related to perimeter, area, and volume.
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PRE-CALCULUS/TRIGONOMETRY/MATHEMATICAL ANALYSIS - ESSENTIAL STANDARDS AND BENCHMARKS
2.0
(MATHEMATICAL ANALYSIS) DEMONSTRATE SKILL IN THE ARITHMETIC OF COMPLEX NUMBERS. USE THE
TRIGONOMETRIC FORM OF COMPLEX NUMBERS AND UNDERSTAND THAT A FUNCTION OF A COMPLEX VARIABLE CAN BE VIEWED AS
FUNCTION OF TWO REAL VARIABLES.
First Semester Benchmarks:
 Students will add, subtract, multiply and divide complex numbers.
Second Semester Benchmarks:
 Students will use the trigonometric form of complex numbers to multiply or divide.
6.0
(MATHEMATICAL ANALYSIS) FIND THE ROOTS OF A RATIONAL FUNCTION AND CAN GRAPH THE FUNCTION AND
LOCATE ITS ASYMPTOTES.
First Semester Benchmarks:
 Given one of the following, students will be able to identify the center, vertices, foci, asymptotes, directrix and other critical values that apply to the
graph of the given equation.
a). (1/(4p))(x-h) 2= (y-k)
b). (x-h)2 + (y-k)2 = r2
c). (x-h)2 + (y-k)2 =1
d). (x-h)2 – (y-k)2 =1
2
2
a
b
a2
b2
 Given a polynomial or rational function, students will find the roots, y-intercept, and asymptotes that apply.
 Given Ax2 +Bxy +Cy2 + Dx + Ey + F = 0, students will transform the equation, identify and graph the conic section.
L.S.
KNOW THE SIX TRIGOMETRIC FUNCTIONS.
Second Semester Benchmarks:
 Students will find the ratios for the sine, cosine and/or tangent for a specified right triangle.
 Students will use the special right triangle relationship to find the missing information for the triangle.
 Students will use the trigonometric functions to find the missing information for a right triangle.
 Students will be able to graph any sine, cosine or tangent trigonometric functions.
 Students will be able to find any of the six trigonometric values given a specific angle.
 Students will find the slope of a line given the angle it forms with the x-axis.
8.0
(TRIGONOMETRY) STUDENTS KNOW THE DEFINITION OF THE INVERSE FUNCTIONS.
First Semester Benchmarks:
 Students will find the inverse of a given algebraic function.
Second Semester Benchmarks:
 Students will graph the inverse of a specified trigonometric function.
4.0
(TRIGONOMETRY) GRAPH THE FUNCTIONS OF THE FORM F(t) = A SIN (B(t + C)) OR F(t) = A COS (B(t + C)) AND
INTERPRET A, B, AND C IN TERMS OF AMPLITUDE, FREQUENCY, PERIOD, AND PHASE SHIFT.
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Second Semester Benchmarks:
 Given an equation of the form y=A sin(B(x-h)) + k, students will tell the amplitude, period, phase shift and graph the equation (cosine and tangent
possibilities).
 Students will convert the angle measure from degrees to radians and vice versa.
 Students will use the trigonometric identities to simplify various trigonometric problems.
 Students will use the laws of sines and/or the law of cosines to solve fore missing information in given triangles.
 Students will determine the area of a triangle given two sides and the included angle.
9.0
(TRIGONOMETRY) WILL COMPUTE, WITHOUT A CALCULATOR, THE VALUES OF THE TRIGONOMETRIC FUNCTIONS
AND THE INVERSE TRIGONOMETRIC FUNCTIONS AT VARIOUS STANDARD POINTS.
Second Semester Benchmarks:
 Students will compute without a calculator the values of the six trigonometric functions.
 Students will compute the angle measure given the inverse trigonometric function.
11.0
(TRIGONOMETRY) STUDENTS DEMONSTRATE UNDERSTANDING OF HALF-ANGLE AND DOUBLE-ANGLE FORMULAS FOR SINES
AND COSINES.
Second Semester Benchmarks:
 Students will calculate the sine, cosine, or tangent of a specified angle using the trigonometric addition or subtraction formulas for special angles.
 Students will calculate the sine, cosine, or tangent of a specified angle using the trigonometric half angle and/or double angle formulas.
16.0
(TRIGONOMETRY) REPRESENT EQUATIONS GIVEN RECTANGULAR COORDINATES.
7.0
(MATHEMATICAL ANALYSIS) DEMONSTRATE EQUATIONS IN TERMS OF POLAR COORDINATES AND RECTANGULAR TO
PARAMETRIC.
Second Semester Benchmarks:
 Students will translate between rectangular and polar coordinates.
 Students will translate between rectangular and polar equations.
 Students will translate between rectangular and parametric equations.
19.0
(TRIGONOMETRY) USE TRIGONOMETRY IN A VARIETY OF APPLICATIONS AND WORD PROBLEMS.
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Second Semester Benchmarks:
 Students will solve a variety of word problems using trigonometry.
L.S.
COMPUTE A 2 X 2 AND 3 X 3 DETERMINANTS AND THE DOT PRODUCT OF TWO VECTORS AND ARE FAMILIAR WITH
THE GEOMETRIC INTERPRETATIONS.
Second Semester Benchmarks:
 Students will compute the determinant of a 2 x 2 and a 3 x 3 matrix.
 Students will compute the dot products of two vector.
 Students will determine whether two vectors are perpendicular.
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