Download Math Course: PRE - ALGEBRA Grade: 7

Math
Course: PRE - ALGEBRA
Grade: 7
MATH 100 Pre-Algebra
No graduation credit
5 days per week; 1 year
Taught in English
th
This is a required class for all 7 grade students in the Mexican and/or U.S. Diploma program. Course content includes
positive and negative numbers, rational numbers, geometric figures, ratios/proportions/percents, areas and volumes.
Students will learn to evaluate variable expressions, do operations with positive and negative numbers, solve equations,
identify fundamental concepts of geometry, find areas and volumes, and to interpret data from graphs.
Textbook:
Randall, Charles I., et.al. Pre-Algebra. Pearson/Prentice Hall. Upper
Saddle River, NJ (2004 Edition)
Prerequisite:
NONE
Benchmark Code
Subject (M, S, SS, LA).Subject #.Strand#.Standard#. Benchmark#
Example: PA.3.1.4.3 –Pre-Algebra, Strand 1, Standard 4, Benchmark 3
Strand 1:
Strand 2:
Strand 3:
Strand 4:
Strand 5:
Strand 6:
Strand 7:
Strand 8:
Strand 9:
Real Numbers, Exponents, Roots and the Pythagorean Theorem.
Variables and Expressions. Variables and Expressions.
Functions
Equations and Identities
Circles
Ratios, Rates, Scaling, and Similarity
Number Representation and Computation
Probability
Number Systems
Strand 1: Real Numbers, Exponents, Roots and the Pythagorean Theorem.
Standard 1: Categorize real numbers as either rational or irrational and know that, by definition,
these are the only two possibilities; extend the properties of computation with rational numbers to
real number computation.
Benchmark Code
Benchmark
PA.1.1.1
Approximately locate any real number on the number line.
Apply the definition of irrational number to identify examples and
recognize approximations.
PA.1.1.2
Recognize and use 22/7 and 3.14 as approximations for the irrational
number represented by pi.
PA.1.1.3
Approximately locate any real number on the number line.
Standard 2: Apply the Pythagorean theorem to solve problems.
Benchmark Code
Benchmark
Determine distances between points in the Cartesian coordinate plane
and relate the Pythagorean Theorem to this process.
Determine the lengths of sides of right angle triangles.
Strand 2: Variables and Expressions.
Standard 1: Interpret and compare the different uses of variables and describe patterns, properties
of numbers, formulas, and equations using variables.
Benchmark Code
Benchmark
PA.2.1.1
Compare the different uses of variables.
PA.2.1.2
Express patterns, properties, formulas and equation using and defining
variables appropriately for each case.
Standard 2: Analyze and identify characteristics of algebraic expressions; evaluate, interpret, and
construct simple algebraic expressions; identify and transform expressions into equivalent
expressions; determine whether two algebraic expressions are equivalent.
Benchmark Code
Benchmark
PA.2.2.1
Use commutative, associative, and distributive properties of number
operations to transform simple expressions into equivalent forms in
order to collect like terms or to reveal or emphasize a particular
characteristic.
PA.2.2.2
Rewrite linear expressions n the form ax + b for constants a and b.
PA.1.2.1
Strand 3: Functions.
Standard 1: Determine whether a relationship is or is not a function; represent and interpret
functions using graphs, tables, words, and symbols.
Benchmark Code
Benchmark
PA.3.1.1
PA.2.1.2
Identify the independent (input) and dependent (output)
quantities/variables of a function.
Make tables of inputs x and outputs f(x) for a variety of rules that
take numbers as inputs and produce numbers outputs.
Define functions algebraically, e.g. g(x) = 3 + 2(x – x2).
Create the graph of a function f by plotting and connecting a
sufficient number of ordered pairs (x, f(x)) in the coordinate plane.
Construct and interpret functions that describe simple problem
situations using expressions, graphs, tables, and verbal descriptions
and move flexibly among these multiple representations.
Solve one-step inequality, two-step inequality.,
Standard 2: Analyze and identify linear functions of one variable; know the definitions of x and
y- intercepts and slope, know how to find them and use them to solve problems.
Benchmark Code
Benchmark
PA.3.2.1
Explain why any function defined by a linear algebraic expression has
a constant rate of change.
Know that a line with slope equal to zero is horizontal and represents a
function while the slope of a vertical line is undefined and cannot
represent a function.
Standard 3: Express a linear function in several different forms for different purposes.
Benchmark Code
Benchmark
PA.3.3.1
Recognize that in the form f(x) = mx + b, m is the slope, or constant
rate of change of the graph of f, that b is the y-intercept and that in
many applications of linear functions, b defines the initial state of a
situation; express a function in this form when this information is
given or needed.
Recognize that in the form f(x) = m(x – xo) + yo, the graph of f(x)
passes through the point (xo, yo); express a function in this form when
this information is given or needed.
Standard 4: Express a linear situation in terms of a linear function f(x) = mx + b and interpret
the slope (m) and the y-intercept (b) in terms of the original linear context. Recognize, graph,
and use direct proportional relationships.
PA.3.4.1
Common examples of linear phenomena include distance traveled
over time for objects traveling at constant speed; shipping costs
under constant incremental cost per pound; conversion of
measurement units (e.g., pounds to kilograms or degrees Celsius to
degrees Fahrenheit); cost of gas in relations to gallons used; the
height and weight of a stack of identical chairs.
PA.3.4.2
Show that the graph of a direct proportional relationship is a line that
passes through the origin (0, 0) whose slopes is the constant of
proportionality.
PA.3.4.3
Compare and contrast the graphs of x= k, y = k, and y = kx, where k
is a constant.
Strand 4: Equations and Identities.
Standard 1: Solve linear equations and solve and graph the solution of linear inequalities in one
variable.
Benchmark Code
Benchmark
Solve equations using the fact that equals added to equals are equal
and that equals multiplied by equals are equal. More formally, if A
PA.4.1.1
= B and C = D, then A + C = B + D and AC = BD.
Strand 5: Circles
Standard 1: Identify and explain the relationships among the radius, diameter, circumference,
and area of a circle; know and apply formulas for the circumference and area of a circle,
semicircle, and quarter-circle.
Benchmark Code
Benchmark
PA.5.1.1
Identify the relationship between the circumference of a circle and
its radius or diameter as a direct proportion and between the area of a
circle and the square of its radius or the square of its diameter as a
direct proportion.
Identify and describe methods for approximating π.
Show that for any circle, the ratio of the circumference to the
diameter is the same as the ratio of the area to the square of the
radius and that these ratios are the same for different circles; identify
the constant ratio A/r2= 2Cr/r2=C/2r=C/d as the number π and know
that although the rational number 3.14, or 22/7= 31/7 are often used
to approximate
, they are not the actual values of the irrational
number pi.
Strand 6: Ratios, Rates, Scaling, and Similarity.
Standard 1: Use ratios, rates, and derived quantities to solve problems.
PA.6.1.1
Interpret and apply measures of change such as percent change and
rates of growth.
PA.6.1.2
Calculate with quantities that are derived as ratios and products.
Create and interpret scale drawings as a tool for solving problems.
Strand 7: Number Representation and Computation
Standard 1: Extend and apply understanding about rational numbers; translate among different
representations of rational numbers.
Benchmark Code
Benchmark
PA.7.1.1
Use inequalities to compare rational numbers and locate them on the
number line; apply basic rules of inequalities to transform numeric
expressions involving rational numbers.
Standard 2: Apply the properties of computation (e.g., commutative property, associative
property, distributive property) to positive and negative rational number computation; know and
apply effective methods of calculation with rational numbers.
Benchmark Code
Benchmark
PA.7.2.1
Demonstrate understanding of the algorithms for additions,
subtraction, multiplication, and division (non-zero divisor) of numbers
expressed as fractions, terminating decimals, or repeating decimals by
applying the algorithms and explaining why they work.
Add, subtract, multiply, and divide (non-zero divisor) rational numbers
and explain why these operations always produce another rational
number.
Interpret parentheses and employ conventional order of operations in a
numerical expression, recognizing that conventions are universally
agreed upon rules for operating on expressions.
Solve practical problems involving rational numbers.
Strand 8: Probability
Standard 1: Represent probabilities using ratios and percents; use sample spaces to determine
the (theoretical) probabilities of events; compare probabilities of two or more events and
recognize when certain events are equally likely.
Benchmark Code
PA.8.1.1
Benchmark
Calculate theoretical probabilities in simple models (e.g., number
cubes, coins, spinners).
Know and use the relationship between probability and odds.
Analyze and interpret actual data to estimate probabilities and predict
outcomes.
Example: In a sample of 100 randomly selected students, 37 of them
could identify the difference in two brands of soft drink. Based on
these data, what is the best estimate of how many of the 2,352 students
in the school could distinguish between the brands of soft drink?
Standard 2: Describe the relationship between probability and relative frequency; use a
probability distribution to assess the likelihood of the occurrence of an event.
Benchmark Code
Benchmark
PA.8.2.1
Recognize and use relative frequency as an estimate for probability.
PA.8.2.2
PA.8.2.3
Recognize and use relative frequency as an estimate for probability.
Analyze and interpret actual data to estimate probabilities and predict
outcomes. Example: In a sample of 100 randomly selected students,
37 of them could identify the difference in two brands of soft drink.
Based on these data, what is the best estimate of how many of the
2,352 students in the school could distinguish between the brands of
soft drink?
Strand 9: Number Systems
Standard 1: Identify the key features of the Ancient numbers systems. SEP
Benchmark Code
Benchmark
PA.9.1.1
Be able to identify the symbols of the five ancient cultures, Mayan,
Egyptian, Aztec, Babylonian and Roman.
Be able to complete basic arithmetic calculations using 1 and 2 digits.
Math
Course: ALGEBRA 1
Grade: 8
MATH 200 Algebra I
1 credit
5 days per week; 1 year
Taught in English
th
This is a required class for all 8 grade students in the Mexican and/or U.S. diploma program. Course content includes
real numbers, polynomials, fractions, linear equations and systems, rational and irrational numbers, and quadratic
functions. Students will learn to simplify and evaluate numerical and algebraic expressions, to do operations with positive
and negative numbers, to solve operations with and factor polynomials, to solve and graph quadratic equations and
equations with two variables. Students will also learn to use negative exponents and to simplify radicals.
Textbook:
Bellman, Allan E., et. Al. Algebra 1. Prentice/Hall. Upper Saddle River,
NJ (2004 Edition)
Prerequisite:
MATH 100
Strand 1 = Application of Numbers.
Strand 2 = Linear equations and inequalities and systems of linear equations and
Inequalities.
Strand 3 = Linear, Proportional And Piecewise-Linear Functions.
Strand 4 = Exponents And Simple Exponential Functions.
Strand 5 = Patterns Of Growth Through Iteration.
Strand 6 = Quadratic functions and equations over the real numbers.
Strand 7 = Data Analysis.
Strand 8 = Probability.
Strand 9 = Classification and operations with polynomials.
Benchmark code:
Example: A1.1.2.3
Subject:
Number of
Number of
Number of
Algebra 1 :
Strand
:
Standard :
Benchmark :
A1
1
2
3
Strand 1 : APPLICATION OF NUMBERS
Standard 1: The Student will use variables to transform Mathematical sentences into an
algebraic expression.
Benchmark Code
Benchmark
A1.1.1.1
The Student will define the concept of variable.
The Student will transform mathematical sentences into algebraic
expressions using Mathematical basic symbols as addition, subtraction,
division, multiplication, exponents and inequalities’ symbols.
Standard 2 :
The Student will apply the properties of computation (commutative,
associative, distributive properties) to positive and negative rational number
computation; Know and apply effective methods of calculation with rational
numbers).
Benchmark Code
Benchmark
The Student will interpret parentheses and employ conventional order in a
numerical expression, recognizing that conventions are universally agreed
A1.1.2.1
upon rules for evaluating expressions at specified values of their variables.
Standard 3: The Student will classify Real Numbers and represent them in different forms;
Compare their magnitude on the number line.
Benchmark Code
Benchmark
A1.1.3.1
The Student will identify finite (terminating) decimals, and repeating
decimals as rational numbers.
The Student will express a percent having a finite number of digits as a
rational number by expressing it as a ratio whose numerator is an integer
and whose denominator is 100 (or, more generally, whose denominator is a
power of 10)
The Student will transform rational numbers from one form (fractions,
decimals, percents and mixed numbers) to another.
Standard 4: Locate rational numbers on the number line and explain the significance of these
locations.
The Student will show that a number and its opposite are mirror
images
A1.1.4.1
with respect to the point 0.
The Student will identify one or more rational numbers that lie between two
given rational numbers and explain how this can be done no matter how
close together the given numbers are.
Standard 5: The Student will graph and analyze data on the Coordinate plane.
Benchmark Code
Benchmark
A1.1.5.1
The Student will identify the independent and dependent variable.
The Student will identify the quadrants of the Coordinate plane.
The Student will graph data and will appreciate the trend of the data
regarding the graphed values. (“Scatter plot”)
A1.1.5.2
The Student will relate non-linear graphs to real world-events through a
description of the graphs by sections according to its inflexion points and
emphasizing how both variables change regarding each other.
Standard 6: The Student will extend and apply understanding about rates, ratios, and unit
rates, percents and percents of change.
Benchmark Code
Benchmark
The Student will identify applications that can be expressed using
rates, ratios and unit rates.
The Student will convert rates to different units.
A1.1.6.1
The Student will solve and write a percent equation for a given real
world-situation.
The Student will calculate the percent of change for the increment
or decrement of a quantity.
Standard 7: The Student will interpret; compare numbers involving significant figures, orders
of magnitude and scientific notation.
Benchmark Code
Benchmark
The Student will use scientific notation with positive and negative
powers of 10, with appropriate treatment of significant digits, to solve
A1.2.7.1
real-world and math problems.
Strand 2: LINEAR EQUATIONS AND INEQUALITIES AND SYSTEMS OF
LINEAR EQUATIONS AND INEQUALITIES
Standard 1: The Student will solve linear equations involving several variables for one
variable in terms of others.
Benchmark Code
Benchmark
A1.2.1.1
The Student will solve one-step equations, two-step equations, and
multi-step equations.
The Student will solve equations with variables on both sides.
The Student will write the corresponding equation solve a given
real world-situation.
A1.2.1.2
The Student will solve one-step inequality, two-step inequality, and
multi-step inequality.
The Student will solve inequalities with variables on both sides.
Standard 2: The Student will solve linear equations’ systems involving two variables using
algebraic procedures.
Benchmark Code
Benchmark
A1.2.2.1
The Student will solve systems of equations by graphing.
A1.2.2.2
The Student will solve systems of equations by graph substitution.
A1.2.2.3
The Student will solve systems of equations by elimination.
Standard 3: The Student will graph the solution of linear inequalities and systems of linear
inequalities in two variables.
Benchmark Code
Benchmark
A1.2.3.1
The Student will Know what it means to be a solution of a linear
inequality in two variables, represent solutions algebraically and
graphically and provide examples of ordered pairs that lie in the
solution set.
A1.2.3.2
The Student will graph a linear inequality in two variables and
explain why the graph is always a half-plane (open or closed).
Standard 4
The Student will create, interpret and apply Mathematical models to solve
problems arising from contextual situations that involve linear relationships.
Benchmark Code
Benchmark
A1.2.4.1
The Student will distinguish relevant from irrelevant information,
identify missing information and find what is needed or make
appropriate estimates.
A1.2.4.2
The Student will recognize and solve problems that can be modeled
using linear inequalities in two variables or a system of linear
equations in two variables; interpret the solution(s) in terms of the
context of the problem.
A1.2.4.3
The Student will represent linear relationships using tables, graphs,
verbal statements; translate among these forms to extract
information about the relationship.
Strand 3 : LINEAR, PROPORTIONAL AND PIECEWISE-LINEAR FUNCTIONS.
Standard 1: The Student will recognize, graph and use direct proportional relationships.
Benchmark Code
Benchmark
A1.3.1.1
The Student will analyze the graph of direct proportional
relationships f(x) = kx and identify its key characteristics.
The Student will compare and contrast the graphs x=k, y=k and
y=kx where k is a constant.
The Student will recognize quantities that are directly proportional
and express their relationship symbolically.
Standard 2: The Student will recognize, graph and use inversely proportional relationships.
Benchmark Code
A1.3.2.1
Benchmark
The Student will recognize quantities that are inversely proportional
and express their relationship symbolically.
The Student will analyze the graph of inversely proportional
relationships f(x) = kx and identify its key characteristics.
Standard 3: The Student will distinguish between direct proportional and inversely
proportional relationships.
Benchmark Code
Benchmark
A1.3.3.1
The Student will identify whether a table, graph, formula or context
suggests a direct or inversely proportional relationship.
The Student will create graphs of direct proportional and inversely
functions.
The Student will distinguish practical situations that can be
represented by direct proportional and inversely proportional
relationships.
The Student will analyze and use the characteristics of these
relationships to answer questions about a given situation.
Standard 4: The Student will explain and illustrate the effect of varying the parameters
m and b in the function f(x) = mx + b
Benchmark Code
Benchmark
A1.3.4.1
The Student will define the concept of slope (m) and connect it to
the concept of constant of variation (k).
The Student will define the concept of y-intercept (b)
The Student will graph the function f(x) = mx + b using a table of
values with a positive slope value.
The Student will graph the function f(x) = mx + b using a table of
values with a negative slope value.
The Student will conduct the deduction of relating the slope’s sign
and the position of the graph.
The Student will write, evaluate and graph the function rule
f(x) = mx + b from a real world-situation.
Strand 4: EXPONENTS AND SIMPLE EXPONENTIAL FUNCTIONS.
Standard 1: The Student will build knowledge on the use of the properties of exponents.
Benchmark Code
Benchmark
A1.4.1.1
The Student will simplify expressions with zero and negative
exponents.
A1.4.1.2
The Student will simplify algebraic expressions using the
multiplication’s properties of exponents.
The Student will simplify algebraic expressions using the division’s
properties of exponents.
Standard 2: The Student will apply the properties of exponents to transform variable
expressions involving integral and rational exponents.
Benchmark Code
Benchmark
A1.4.2.1
The Student will translate between rational exponents and notation
involving integral powers.
A1.4.2.2
The Student will factor out common factors with exponents.
Standard 3: The Student will graph and analyze exponential functions and identify their
characteristics.
Benchmark Code
Benchmark
A1.4.3.1
The Student will identify functions having the general form
f(x) = abx + c
A1.4.3.2
The Student will recognize and represent the graphs of exponential
functions.
Standard 4: The Student will recognize problems that can be modeled using exponential
functions; interpret the solution(s) in terms of the context of the problem.
Benchmark Code
Benchmark
A1.4.4.1
The Student will use exponential functions to represent growth and
decay functions such as f(x) = abx
and
f(x) = ab-x
A1.4.1.3
A1.4.4.1
The Student will use the laws of exponents to determine exact
solutions for problems involving exponential functions where possible;
Otherwise approximate the solutions graphically or numerically.
Strand 5: PATTERNS OF GROWTH THROUGH ITERATION.
Standard 1: The Student will generate and describe sequences having specific
characteristics.
Benchmark Code
Benchmark
A1.5.1.1
The Student will generate and describe arithmetic sequences
expressed recursively.
A1.5.1.2
The Student will generate and describe Geometric sequences
expressed recursively.
Standard 2: The Student will demonstrate the effect of compound interest, exponential
decay or Exponential growth using iteration.
Benchmark Code
Benchmark
A1.5.2.1
The Student will identify the diminishing effect of increasing the
number of times per year that interest is compounded and relate
this to the notion of instantaneous compounding.
Strand 6 : QUADRATIC FUNCTIONS AND EQUATIONS OVER THE REAL
NUMBERS.
Standard 1: The Student will identify quadratic functions expressed in multiple forms;
identify the specific information each form clarifies.
Benchmark Code
Benchmark
A1.6.1.1
The Student will express a quadratic function as a polynomial
f(x) = ax2 +bx+c.
.A1.6.1.2
The Student will express a quadratic function having integral roots
in factored form f(x) = (x – r) (x – s) = 0.
Standard 2: The Student will graph quadratic functions and use the graph to help locate
zeros.
Benchmark Code
Benchmark
A1.6.2.1
The Student will sketch graphs of quadratic functions using a table
of values.
A1.6.2.2
The Student will estimate the real zeros of a quadratic function
from its graph and identify quadratic functions that do not have
real zeros by the behavior of their graphs.
Standard 3: The Student will solve quadratic equations with integral solutions;
The Student will use quadratic equations to represent and solve problems
involving quadratic behavior.
Benchmark Code
Benchmark
A1.6.3.1
The Student will solve quadratic equations that can be easily
transformed into the form (x –a) (x – b) = 0 or (x + a)2 = b.
A1.6.3.2
The Student will estimate the roots of a quadratic equation from
the graph of the corresponding function.
Standard 5: The Student will rewrite quadratic functions and interpret their graphical forms.
Benchmark Code
Benchmark
A1.6.5.1
The Student will write a quadratic function in polynomial or
standard form f(x) = ax2 +bx +c to identify the y-intercept of the
function’s parabolic graph or the x-coordinate of its vertex
x = - b 2a
A1.6.5.2
The Student will write a quadratic function into factored form
f(x) = a(x – r) (x – s) to identify its roots.
A1.6.5.3
The Student will write a quadratic function in vertex form
f(x) = a(x – h)2 + k (x – s) to identify the vertex and axis of
symmetry of the function’s parabolic graph.
A1.6.5.4
The Student will determine the axis of symmetry, maximum and
minimum on a parabola.
Standard 6: The Student will graph quadratic equations and solve those with real solutions
using a variety of methods.
Benchmark Code
Benchmark
A1.6.6.1
The Student will solve quadratic equations having real solutions
by factoring, by completing the square and by using the quadratic
formula.
A1.6.6.2
The Student will recognize and solve practical problems that can
be expressed using quadratic equations having real solutions;
Interpret their solutions in terms of the context of the situation.
Standard 7: The Student will make regular fluent use of basic algebraic identities such as
(a + b)2 = a2 + 2ab + b2; (a - b)2 = a2 - 2ab + b2.
Benchmark Code
Benchmark
A1.6.7.1
The Student will use geometric constructions to illustrate these
formulas.
Strand 7: DATA ANALYSIS
Standard 1: The Student will collect and record; Display data using tables, charts or graphs.
Benchmark Code
Benchmark
A1.7.1.1
The Student will understand that data are numbers in context and
identify appropriate units.
A1.7.1.2
The Student will organize written data records using a chart.
A1.7.1.3
The Student will define measurements that are relevant to the
questions posed.
Standard 2: The Student will analyze and interpret categorical and quantitative data.
Benchmark Code
Benchmark
.A1.7.2.1
The Student will make use of frequency and relative frequency
tables and bar graphs.
Strand 8:
Standard 1 :
PROBABILITY.
The Student will represent probabilities using ratios and percents.
The Student will use spaces to determine the (theoretical) probabilities of
Events.
The Student will compare probabilities of two or more events and recognize
when certain events are equally likely.
Benchmark Code
Benchmark
A1.8.1.1
The Student will calculate theoretical probabilities in simple
models (number cubes, coins, spinners).
A1.8.1.2
The Student will know and use the relationship between
probability and odds.
Standard 2 : The Student will use a probability distribution to assess the likelihood of the
occurrence of an event.
Benchmark Code
Benchmark
A1.8.2.1
The Student will analyze and interpret actual data to estimate
probabilities and predict outcomes.
Strand 9 : CLASSIFICATION AND OPERATIONS WITH POLYNOMIALS.
Standard 1: The Student will identify the key characteristics of monomials, binomials and
polynomials.
Benchmark Code
Benchmark
A1.9.1.1
The Student will classify algebraic expressions according to the
number of terms.
The Student will name algebraic expressions based on their
degree.
Standard 2: The Student will solve basic operations with polynomials.
Benchmark Code
Benchmark
A1.9.2.1
The Student will addition and subtraction of polynomials.
.A1.9.2.2
The Student will multiplication and division of polynomials.
Math
Course: Mathematics 300 Plane Geometry
and Introduction to Algebra
Grade 9
MATH 300 Geometry
1 credit
5 days per week; 1 year
Taught in English
th
This is a required class for all 9 grade students in the Mexican and/or U.S. diploma program. Course content includes the
study of: points lines and planes, parallel lines and planes, congruent triangles, similar polygons, right triangles, circles, and
coordinate geometry. Proofs will be introduced but not emphasized.
Textbook:
Bass, Laurie E., et. al. Geometry. Prentice/Hall. Upper Saddle River, NJ
(2004 Edition)
Prerequisite:
MATH 200
Strand 1: Geometric Representations
Strand 2: Reasoning in Geometric Situations
Strand 3: Using Perpendicular and Perpendicular Lines
Strand 4: Similarity, Congruence and Right Triangles Trigonometry
Strand 5: Quadrilaterals
Strand 6: Area
Strand 7: Three-Dimensional Geometry
Strand 8: Circles
Strand 9: Tools of Algebra
Strand 10: Functions, Equations and Graphs
Strand 11: Linear Systems
Benchmark Code
Example: GEO.9.1.4.3 – Geometry, Grade 9th, Strand 1, Standard 4, Benchmark 3
Strand 1: Geometric Representations
Standard 1: Students will review the coordinate plane to help them make logical transition from algebra
to geometry and to identify, describe, and compare points, lines and planes to be used to define other
geometric terms such as angles, segments and rays.
Benchmark Code
Benchmark
GEO.9.1.1.1
GEO.9.1.1.2
GEO.9.1.1.3
GEO.9.1.1.4
GEO.9.1.1.5
GEO.9.1.1.6
GEO.9.1.1.7
Graph ordered pairs on a coordinate plane.
Identify collinear points.
Identify and model points, lines and planes.
Identify coplanar points and intersecting lines and planes.
Solve problems by listing the possibilities and by using formulas.
Calculate maximum area of a rectangle for a given perimeter.
Calculate distance between two points on a number line and between two
points in a coordinate plane.
GEO.9.1.1.8
GEO.9.1.1.9
GEO.9.1.1.10
GEO.9.1.1.11
GEO.9.1.1.12
Use the Pythagorean theorem to find the length of the hypotenuse.
Identify and classify angles.
Use the angle addition postulate to find the measures of angles.
Identify and use congruent angles and the bisector of an angle.
Identify and use adjacent, vertical, complementary, supplementary and linear
pairs of angles and perpendicular lines.
GEO.9.1.1.13
Make inferences to determine what information can and cannot be assumed
from a diagram.
GEO.9.1.1.14
GEO.9.1.1.15
Learn basic constructions.
Find and calculate Perimeter, Circumference and Area of polygons and circles.
Strand 2: Reasoning in Geometric Situations
Standard 1: Explore inductive and deductive reasoning strategies leading to a study of if-then
statements and their logic and conditional statements, their inverses, their converses, and contrapositives.
Benchmark Code
Benchmark
GEO.9.2.1.1
GEO.9.2.1.2
Make conjectures based on inductive reasoning.
Write a statement in "if-then" form with the converse, inverse, and contrapositive of the same if-then statement.
GEO.9.2.1.3
Identify and use basic postulates about points, lines, and planes.
GEO.9.2.1.4
Prove that angles are congruent.
Strand 3: Using Perpendicular and Parallel Lines
Standard 1: State the relationship between points, lines, angles, planes, spheres, parallel lines, and
slopes as well as the angle measures formed when lines intersect.
Benchmark Code
Benchmark
GEO.9.3.1.1
Solve problems by drawing a diagram and identifying the relationships
between two lines or two planes and naming angles formed by a pair of lines
and a transversal.
GEO.9.3.1.2
Find the measures of the angles formed by two parallel lines and a transversal.
GEO.9.3.1.3
GEO.9.3.1.4
GEO.9.3.1.5
Use the properties of parallel lines to determine angle measures.
Find the slopes of lines and use it to identify parallel and perpendicular lines.
Recognize angle conditions that produce parallel lines and prove two lines are
parallel based on given angle relationships.
GEO.9.3.1.6
Recognize and use distance relationships among points, lines, and planes.
GEO.9.3.1.7
Find a distance between a point and a line.
GEO.9.3.1.8
Use the polygon angle-sum theorem to find internal and external angles.
GEO.9.3.1.9
Construct parallel and perpendicular lines.
Strand 4: Similarity, Congruence and Right Triangle Trigonometry
Standard 1: Students will use Algebra skills to solve proportions and identify similar figures and solve
problems using proportions.
Benchmark Code
Benchmark
GEO.9.4.1.1
Recognize and use ratios and proportions and apply their properties.
GEO.9.4.1.2
GEO.9.4.1.3
GEO.9.4.1.4
Identify similar figures and use them to solve problems.
Identify and use similar triangles to solve problems.
Use the properties of similarity in right triangles to find missing sides or angles.
GEO.9.4.1.5
Use proportional parts of triangles to solve problems and divide a segment into
congruent parts.
GEO.9.4.1.6
Recognize and use the proportional relationships of corresponding perimeters,
altitudes, angle bisectors, and medians of similar triangles.
Standard 2: Students will identify, classify and apply theorems relating to triangles. Also apply
congruence to different types and parts of triangles while identifying and using medians, altitudes, angle
bisectors, perpendicular bisectors. Indirect reasoning and indirect proofs will be used to reach
conclusions and solve problems.
Benchmark Code
Benchmark
GEO.9.4.2.1
Identify congruent figures.
GEO.9.4.2.2
Prove triangle congruence by Side-Side-Side , Side-Angle-Side, Angle-SideAngle and Angle-Angle-Side theorems.
GEO.9.4.2.3
Use congruent triangles to prove the CPCTC (corresponding parts of
congruent triangles are congruent).
GEO.9.4.2.4
Identify isosceles and equilateral triangles.
GEO.9.4.2.5
Prove congruence in right triangles.
GEO.9.4.2.6
Use corresponding parts of congruent triangles.
GEO.9.4.2.7
Identify and use medians, altitudes, angle bisectors, and perpendicular
bisectors in a triangle.
GEO.9.4.2.8
Use indirect reasoning and indirect proof to draw conclusions.
GEO.9.4.2.9
Recognize and apply properties of inequalities to the measures of segments
and angles.
GEO.9.4.2.10
Apply the Triangle Inequality Theorem to verify possible sides of triangles.
Standard 3: Investigate and use the Pythagorean theorem and find geometric mean which will be used
to solve triangles using the altitude to the hypotenuse.
Benchmark Code
Benchmark
GEO.9.4.3.1
Use paper folding to develop the Pythagorean Theorem.
GEO.9.4.3.2
Use tangent, sine and cosine ratios to find missing sides or angles.
GEO.9.4.3.3
Use trigonometry to solve problems involving angles of elevation or
depression.
GEO.9.4.3.4
Apply properties of a right triangle to solve magnitude and direction of
vectors.
GEO.9.4.3.5
Use trigonometric functions to calculate area in figures.
Strand 5: Quadrilaterals
Standard 1: Students will be introduce to various quadrilaterals and use technology to investigate some
while employing the properties of parallelograms, rhombi, rectangles, squares and trapezoids to solve
problems.
Benchmark Code
Benchmark
GEO.9.5.1.1
Classify quadrilaterals.
GEO.9.5.1.2
Identify properties of parallelograms.
GEO.9.5.1.3
Prove that a quadrilateral is a parallelogram by analyzing their unique
properties.
GEO.9.5.1.4
Recognize and apply the properties of rectangles and squares.
GEO.9.5.1.5
Recognize and apply the properties of rhombi and trapezoids.
GEO.9.5.1.6
Place figures on a coordinate plane.
GEO.9.5.1.7
Write proofs using coordinate geometry.
Strand 6: Area
Standard 1: Identify and name polygons and investigate interior and exterior angle measures of convex
and regular polygons. Also identify types of tessellations.
Benchmark Code
Benchmark
GEO.9.6.1.1
Find area of parallelograms and triangles.
GEO.9.6.1.2
Apply Pythagorean theorem and its converse to find area.
GEO.9.6.1.3
Apply the properties of special right triangles to find area and missing sides.
GEO.9.6.1.4
Calculate area of trapezoids, rhombi and kites.
GEO.9.6.1.5
Calculate area of regular polygons.
GEO.9.6.1.6
Measure circles and arcs.
GEO.9.6.1.7
Find areas of circles and sectors.
GEO.9.6.1.8
Apply geometric probability to situational problems .
Strand 7: Three-Dimensional Geometry
Standard 1: Recognize and draw many three-dimensional figures and to flatten these figure to look at
nets and surface area; to determine surface area and volume of prisms, cylinders, pyramids, cones and
spheres.
Benchmark Code
Benchmark
GEO.9.7.1.1
Create cross sections and other slices of solids.
GEO.9.7.1.2
Use top, front, side, and corner views of three-dimensional solids to make
models and describe and draw cross sections and other slices of threedimensional figures.
GEO.9.7.1.3
Construct a tetrahedron kite.
GEO.9.7.1.4
Draw three-dimensional figures on isometric dot paper and make twodimensional nets.
GEO.9.7.1.5
GEO.9.7.1.6
GEO.9.7.1.7
GEO.9.7.1.8
GEO.9.7.1.9
GEO.9.7.1.10
GEO.9.7.1.11
GEO.9.7.1.12
GEO.9.7.1.13
Find surface area of three-dimensional solids .
Find the lateral area and surface area of a right prism and a right cylinder.
Find the lateral area and surface area of a regular pyramid
and a right circular cone.
Find the volume of a right prism and a right cylinder.
Compare the volumes of prisms and pyramids and the volumes of cylinders
and cones.
Find the volume of a pyramid and circular cone.
Recognize and define basic properties of spheres.
Find the surface area and the volume of a sphere.
Identify congruent or similar solids and state the properties of congruent
solids.
Strand 8: Circles
Standard 1: Define radius, diameter and circumference; use major and minor arcs and their relationships
to central angles; and develop theorems involving inscribed angles and intercepted arcs using chords,
secants, tangents and their properties.
Benchmark Code
Benchmark
GEO.9.8.1.1
Identify and use parts of circles.
GEO.9.8.1.2
Solve problems involving the circumference of a circle.
GEO.9.8.1.3
Recognize major arcs, minor arc, semicircles, and central angles.
GEO.9.8.1.4
Find measures of arcs and central angles and solve problems by making circle
graphs.
GEO.9.8.1.5
Recognize and use relationships among arcs, chords, and diameters.
GEO.9.8.1.6
Recognize and find measures of inscribed angles and apply properties of
inscribed figures.
Recognize tangents and use their properties.
Find the measures of angles formed by intersecting secants and tangents in
relation to intercepted arcs
GEO.9.8.1.7
GEO.9.8.1.8
GEO.9.8.1.9
Use the properties of chords, secants and tangents to solve segment measure
problems.
GEO.9.8.1.10
Write and use the equation of a circle in the coordinate plane.
Strand 9: Tools of Algebra
Standard 1: Students will evaluate and simplify expressions, solve equations, solve and graph
inequalities
Benchmark Code
Benchmark
ALG.9.9.1.1
Use the order of operations to evaluate expressions and use formulas.
ALG.9.9.1.2
Determine the sets of numbers to which a number belongs.
ALG.9.9.1.3
Translate verbal expressions and sentences into algebraic expressions and
equations.
ALG.9.9.1.4
Solve equations by using the properties of equalities.
ALG.9.9.1.5
Solve equations for a specific variable
ALG.9.9.1.6
Solve equations containing absolute value and problems by using a table.
ALG.9.9.1.7
Solve inequalities and graph the solution sets.
ALG.9.9.1.8
Solve compound inequalities using the properties of “and/or”.
ALG.9.9.1.9
Solve inequalities involving absolute value and graph the solution.
Strand 10: Functions, Equations and Graphs
Standard 1: Students will identify different types of relations and functions, graph relations and
functions on the coordinate plane, look for patterns to solve problems, model real-world data using
scatter plots, and graph inequalities on the coordinate plane.
Benchmark Code
Benchmark
ALG.9.10.1.1
Graph relations, state its domain and range, and determine if it’s a function.
ALG.9.10.1.2
Identify and graph linear equations.
ALG.9.10.1.3
Write linear equations in standard form.
ALG.9.10.1.4
Determine the intercepts of a line and use them to graph an equation.
ALG.9.10.1.5
Determine the slope of a line and use slope and a point to graph an equation.
ALG.9.10.1.6
Identify and use patterns points or linear equations to make conclusions.
ALG.9.10.1.7
Write an equation of a line in slope-intercept form given the slope and/or two
points.
ALG.9.10.1.8
Write an equation of a line that is parallel or perpendicular to the graph of a
given equation.
ALG.9.10.1.9
Draw scatter plots to find and use prediction equations.
ALG.9.10.1.10
Draw graphs of inequalities with two variables.
Strand 11: Linear Systems
Standard 1: Solve systems of equations in two or three variables, solve system of inequalities, use linear
programming to find maximum and minimum values of functions, and solve problems by solving
simpler problems.
Benchmark Code
Benchmark
ALG.9.11.1.1
Solve systems of equations by graphing.
ALG.9.11.1.2
Use the substitution and elimination methods to solve system of equations.
ALG.9.11.1.3
Find the values of second-order determinants.
ALG.9.11.1.4
Solve system of equations by using Cramer’s rule.
ALG.9.11.1.5
Solve system of inequalities by graphing.
ALG.9.11.1.6
Find the maximum and minimum values of a function over a region using
linear programming techniques.
ALG.9.11.1.7
Apply algebraic rules to solve word problem.
ALG.9.11.1.8
Solve problems involving maximum and minimum values by using linear
programming techniques.
ALG.9.11.1.9
Solve a system of three equations with three variables.
ALG.9.11.1.10
ALG.9.11.1.11
Determine the octant in which a point in space is located.
Graph linear equations in space and determine the intercepts.
Math
Course: Algebra II
Grade 10
MATH 401 Algebra II
1/2 credit
5 times per week (1st Semester)
Taught in English
th
This is a required class for all 10 grade students in the Mexican and/or U.S. Diploma program. Major content areas in the
class include properties of real numbers, solving equations in one variable, solving inequalities and absolute value
problems, graphs of linear equations, slope, solving systems of equations and linear inequalities, properties of polynomials
and operations involving polynomials, and factoring polynomials, word and application problems are emphasized
nd
throughout the course. In the 2 Semester the course will cover laws of exponents, operations using rational expressions,
solving fractional equations, properties of radicals, simplifying radicals, real and complex numbers, solving quadratic
equations, graphing quadratic equations, direct and inverse variation, operations involving polynomial equations, and
solving polynomial equations.
Textbook:
Bellman, Allan E., Algebra 2. Prentice/Hall: Upper Saddle River, NJ
(2004 Edition)
Prerequisite:
MATH 300
Benchmark Code – Subject: Algebra II = ALG2
Strand 1: Matrices
Strand 2: Quadratic Equations and Functions
Strand 3: Polynomials and Polynomial Functions
Strand 4: Radical Functions and Rational Exponents
Strand 5: Exponential and Logarithmic Functions
Strand 6: Rational Functions
Strand 7: Sequences and Series
Subject.Grade.Strand#.Standard#. Benchmark#
Example: ALG2.10.1.1.3 – Algebra 2, Grade 10, Strand 1, Standard 1, Benchmark 3
Strand 1: Matrices
Standard 1: Students will create matrices to represent data, solve problems by using matrix logic,
perform operations with matrices, use matrices to achieve transformations of geometric figures, and
use matrices to solve systems of equations.
Benchmark Code
Benchmark
Use a graphing calculator to perform operations with matrices and find
ALG2.10.1.1.1
variables.
Perform scalar multiplication on a matrix and solve matrices for
ALG2.10.1.1.2
determinants and inverses.
Solve problems using matrix logic.
ALG2.10.1.1.3
Add, subtract, and multiply matrices.
ALG2.10.1.1.4
Evaluate the determinant of a 3 x 3 matrix.
ALG2.10.1.1.5
Find the inverse of a 2 x 2 matrix.
ALG2.10.1.1.6
Solve systems of linear equations by using inverse matrices.
ALG2.10.1.1.7
Solve systems of linear equations by using augmented matrices.
ALG2.10.1.1.8
ALG2.10.1.1.9
Use a graphing calculator to solve systems of linear equations.
ALG2.10.1.1.10
Preparation for PSAT and SAT.
Strand 2: Quadratic Equations and Functions
Standard 1: Students will graph quadratic functions, solve quadratic equations, solve problems using
the guess-and-check strategy, and analyze graphs of quadratic functions and inequalities.
Benchmark Code
Benchmark
ALG2.10.2.1.1
Use a graphing calculator to graph and solve quadratic equations.
ALG2.10.2.1.2
Write functions in quadratic form, graph, and solve by graphing.
ALG2.10.2.1.3
Solve problems by using the guess-and-check strategy.
ALG2.10.2.1.4
Solve quadratic equations by factoring, completing the square, and by
using the quadratic formula.
ALG2.10.2.1.5
Use discriminants to determine the nature of the roots of quadratic
equations.
ALG2.10.2.1.6
Find the sum and product of the roots of quadratic equations.
ALG2.10.2.1.7
Find a quadratic equation to fit a given condition.
ALG2.10.2.1.8
Use a graphing calculator to graph and explore similarities between
parabolas.
ALG2.10.2.1.9
Determine the equation of a parabola by using points on its graph.
ALG2.10.2.1.10
Use a graphing calculator to graph and solve quadratic inequalities.
ALG2.10.2.1.11
Graph quadratic inequalities and solve inequalities in one variable.
ALG2.10.2.1.12
Solve non-quadratic equations by using quadratic techniques.
ALG2.10.2.1.13
Preparation for PSAT and SAT.
Strand 3: Polynomials and Polynomial Functions
Standard 1: Students will simplify expressions containing polynomials, factor polynomials; and
solve problems by identifying and achieving sub-goals, evaluate polynomial functions, and identify
general shapes of the graphs of polynomial functions.
Benchmark Code
Benchmark
ALG2.10.3.1.1
Multiply and divide monomials.
ALG2.10.3.1.2
Represent numbers in scientific notation and multiply and divide
expressions written in scientific notation.
ALG2.10.3.1.3
Add, subtract, and multiply polynomials; divide using long division.
ALG2.10.3.1.4
Divide polynomials by binomials using synthetic division.
ALG2.10.3.1.5
Factor polynomials and simplify polynomial quotients by factoring.
ALG2.10.3.1.6
Solve problems by identifying and achieving sub-goals.
ALG2.10.3.1.7
Evaluate polynomial functions and identify general shapes of the graphs
of polynomial functions.
ALG2.10.3.1.8
Find factors of polynomials by using the factor theorem and synthetic
division.
ALG2.10.3.1.9
Use a graphing calculator to graph polynomial functions and approximate
the real zeros of the functions.
ALG2.10.3.1.10
Find the number and type of zeros of a polynomial function.
ALG2.10.3.1.11
Identify all possible rational zeros of a polynomial function by using the
rational zero theorem.
ALG2.10.3.1.12
Find zeros of polynomial functions.
ALG2.10.3.1.13
Solve non-quadratic equations.
ALG2.10.3.1.14
Find the composition of functions.
ALG2.10.3.1.15
Graph the iterations of a function.
ALG2.10.3.1.16
Preparation for PSAT and SAT.
Strand 4: Radical Functions and Rational Exponents
Standard 1: Students will simplify expressions containing radicals, complex numbers, or rational
exponents; and they solve equations containing radicals.
Benchmark Code
Benchmark
ALG2.10.4.1.1
Simplify radicals having various indices.
ALG2.10.4.1.2
Use a calculator to estimate roots of numbers.
ALG2.10.4.1.3
Simplify radical expressions and add, subtract, multiply, and divide
radical expressions
ALG2.10.4.1.4
Rationalize the denominator of a fraction containing a radical expression.
ALG2.10.4.1.5
Write expressions with radical exponents in simplest radical form and
vice versa.
ALG2.10.4.1.6
Evaluate expressions in either exponential or radical form.
ALG2.10.4.1.7
Solve equations containing radicals.
ALG2.10.4.1.8
Simplify square roots containing negative radicands.
ALG2.10.4.1.9
Solve quadratic equations that have pure imaginary solutions
ALG2.10.4.1.10
Add, subtract, and multiply complex numbers.
ALG2.10.4.1.11
Simplify rational expressions containing complex numbers in the
denominator.
ALG2.10.4.1.12
Determine the inverse of a function or relation.
ALG2.10.4.1.13
Graph functions and their inverses.
ALG2.10.4.1.14
Work backward to solve problems.
ALG2.10.4.1.15
Graph and analyze square root functions and graph square root
inequalities.
ALG2.10.4.1.16
Preparation for PSAT and SAT.
Strand 5: Exponential and Logarithmic Functions
Standard 1: Students will simplify expressions and solve equations involving real exponents, write
exponential equations in logarithmic form and vice versa, evaluate expressions and solve equations
involving logarithms, find common and natural logarithms and antilogarithms, and solve equations
with variable exponents by using logarithms.
Benchmark Code
Benchmark
ALG2.10.5.1.1
Use a graphing calculator to draw graphs of exponential functions.
ALG2.10.5.1.2
Simplify expressions and solve equations and inequalities involving real
exponents.
ALG2.10.5.1.3
Use a graphing calculator to fit a curve to a scatter plot of real-world data.
ALG2.10.5.1.4
Write exponential equations in logarithmic form and vice versa.
ALG2.10.5.1.5
Evaluate logarithmic expressions.
ALG2.10.5.1.6
Solve equations and inequalities involving logarithmic functions.
ALG2.10.5.1.7
Simplify and evaluate expressions using properties of logarithms.
ALG2.10.5.1.8
Solve equations involving logarithms.
ALG2.10.5.1.9
Identify the characteristic and the mantissa of a logarithm.
ALG2.10.5.1.10
Find common logarithms and antilogarithms.
ALG2.10.5.1.11
Find natural logarithms of numbers.
ALG2.10.5.1.12
Solve equations with variable exponents by using logarithms.
ALG2.10.5.1.13
Use logarithms to solve problems involving growth and decay.
ALG2.10.5.1.14
Preparation for PSAT and SAT.
Strand 6: Rational Functions
Standard 1: Students will graph rational functions, solve problems involving direct, inverse, and
joint variation, simplify rational expressions, solve rational equations and solve problems by
organizing data.
Benchmark Code
Benchmark
ALG2.10.6.1.1
Use a graphing calculator to explore graphs of rational functions.
ALG2.10.6.1.2
Graph rational functions.
ALG2.10.6.1.3
Solve problems involving direct, inverse, and joint variation.
ALG2.10.6.1.4
Simplify rational expressions. Simplify complex fractions.
ALG2.10.6.1.5
Find the least common denominator of two or more algebraic
expressions.
ALG2.10.6.1.6
Add and subtract rational expressions.
ALG2.10.6.1.7
Solve rational equations inequalities.
ALG2.10.6.1.8
Preparation for PSAT and SAT.
Strand 7: Sequences and Series
Standard 1: Students will be introduced to arithmetic and geometric sequences, and arithmetic and
geometric series.
Benchmark Code
Benchmark
ALG2.10.7.1.1
Identify and generate arithmetic sequences and geometric sequences.
ALG2.10.7.1.2
Evaluate arithmetic series and geometric series.
ALG2.10.7.1.3
Use rectangles to approximate the area under a curve.
ALG2.10.7.1.4
Preparation for PSAT and SAT.
Math
Course: Pre-Calculus (Trigonometry)
Grade 11
MATH 501 Trigonometry
1/2 credit
5 times per week (1st Semester)
Taught in English
th
This is a required class for all 11 grade students in the Mexican and/or U.S. Diploma program. Major content
areas in the class include angles, arcs, and sectors of a circle, trigonometric functions, trigonometric equations and
their applications, and triangle trigonometry [including the study of the Pythagorean theorem and the laws of sine
and cosine]. Students will develop an understanding of the different units of angles. They will also understand the
trigonometric functions and be able to apply them to solve problems. In addition, students will develop the ability
to manipulate trigonometric expressions and equations.
Textbook:
Prerequisite:
Larson, Ron and Hostetler, Robert P. Pre-Calculus. Houghton/Mifflin
Company. Boston, MA (2004 Edition)
MATH 402
Benchmark Code – Subject: Pre-Calculus 1 = PC1
Strand 1: Trigonometry
Strand 2: Analytic Trigonometry
Strand 3: Additional Topics in Trigonometry
Subject.Grade.Strand#.Standard#. Benchmark#
Example: PC1.11.1.1.3 – Pre-Calculus 1, Grade 11, Strand 1, Standard 1, Benchmark 3
Strand 1: Trigonometry
Standard 1: Students will measure angles in radians and degrees, evaluate the trigonometric
functions, graph the trigonometric functions and the inverse trigonometric functions and
solve right triangles.
Benchmark Code
Benchmark
PC1.11.1.1.1
Describe an angle and convert between radian and degree
measure.
PC1.11.1.1.2
Find angles that are coterminal with a given angle.
PC1.11.1.1.3
Find the reference angle for a given angle.
PC1.11.1.1.4
Find the length of an arc given the measure of the central angle.
PC1.11.1.1.5
Find the linear and angular velocities.
PC1.11.1.1.6
Find the area of a sector.
PC1.11.1.1.7
Identify a unit circle and its relationship to real numbers.
PC1.11.1.1.8
Evaluate trigonometric functions of any angle.
PC1.11.1.1.9
Use fundamental trigonometric identities.
PC1.11.1.1.10
Sketch the graphs of trigonometric functions and translations of
graphs of sine and cosine functions.
PC1.11.1.1.11
Find the amplitude, period, and phase shift for a trigonometric
function.
PC1.11.1.1.12
Evaluate the inverse trigonometric functions.
PC1.11.1.1.13
Evaluate the compositions of trigonometric functions and inverse
trigonometric functions.
PC1.11.1.1.14
Solve right triangles.
PC1.11.1.1.15
Solve problems involving simple harmonic motion.
PC1.11.1.1.16
Preparation for SAT.
Strand 2: Analytic Trigonometry
Standard 1: Students will use the fundamental trigonometric identities and will verify them.
Benchmark Code
Benchmark
PC1.11.2.1.1
Use the fundamental trigonometric identities to evaluate
trigonometric functions and simplify trigonometric expressions.
PC1.11.2.1.2
Identify and use reciprocal identities, quotient identities,
Pythagorean identities, and symmetry identities.
PC1.11.2.1.3
Verify trigonometric identities.
PC1.11.2.1.4
Preparation for SAT.
Standard 2: Students will be able to solve trigonometric equations and to recognize and use
the sum and difference, multiple-angle and product-to-sum formulas.
Benchmark Code
Benchmark
PC1.11.2.2.1
Use standard algebraic techniques and inverse trigonometric
functions to solve trigonometric equations
PC1.11.2.2.2
Use the sum and difference formulas, multiple angle formulas,
power reducing formulas, half–angle formulas, and product-tosum formulas to rewrite and evaluate trigonometric functions.
PC1.11.2.2.3
Preparation for SAT.
Strand 3: Additional Topics in Trigonometry
Standard 1: Students will be able to solve oblique triangles using the Law of Sines and Law
of Cosines.
Benchmark Code
Benchmark
PC1.11.3.1.1
Use the Law of Sines and the Law of Cosines to solve oblique
triangles.
PC1.11.3.1.2
Determine whether a triangle has zero, one, or two solutions.
PC1.11.3.1.3
Find the areas of oblique triangles.
PC1.11.3.1.4
Preparation for SAT.
Standard 2: Students will be able to draw vectors in a plane, find their components and
perform vector operations.
Benchmark Code
Benchmark
PC1.11.3.2.1
Represent a vector in its polar form, as a sum of unit vectors and
as an ordered pair or an ordered triple.
PC1.11.3.2.2
Write the component forms of vectors and perform basic vector
operations.
PC1.11.3.2.3
PC1.11.3.2.4
Find the direction angles of vectors and the angle between two
vectors.
Preparation for SAT.
Math
Course: Pre-Calculus 2 (Analytic Geometry)
Grade 11
MATH 502 Analytical Geometry
1/2 credit
5 days per week (2nd Semester)
Taught in English
th
This is a required class for all 11 grade students in the Mexican and/or U.S. diploma program. Major content
areas in the course include Cartesian coordinates, points and lines, conic sections, and the translation of axes of a
graph. Students will be expected to work extensively with graphs; in addition, students will be able to identify
equations of conic sections.
Textbook:
Prerequisite:
Larson, Ron and Hostetler, Robert P. Pre-Calculus. Houghton/Mifflin
Company. Boston, MA (2004 Edition)
MATH 501
Benchmark Code – Subject: Pre-Calculus 2 = PC2
Strand 1: Functions and Their Graphs
Strand 2: Polynomial and Rational Functions
Strand 3: Exponential and Logarithmic Functions
Strand 4: Topics in Analytic Geometry
Subject.Grade.Strand#.Standard#. Benchmark#
Example: PC2.11.1.1.3 – Pre-Calculus 2, Grade 11, Strand 1, Standard 1, Benchmark 3
Strand 1: Functions and Their Graphs
Standard 1: Students will graph functions, solve equations, solve problems using
functions, and analyze graphs of functions.
Benchmark Code
Benchmark
Solve
linear
and
quadratic
equations
using a variety of methods.
PC2.11.1.1.1
Solve equations involving radicals and absolute values.
PC2.11.1.1.2
Plot points, relations and functions in Cartesian planes.
PC2.11.1.1.3
Find x and y-intercepts of a function.
PC2.11.1.1.4
Sketch graphs of equations; use symmetry to sketch graphs.
PC2.11.1.1.5
Graph linear equations using different methods.
PC2.11.1.1.6
Find the slope of a line. Use the slope to determine parallel and
PC2.11.1.1.7
perpendicular lines.
Determine whether relations are functions or not. Domain and range of
PC2.11.1.1.8
functions. Vertical line test.
PC2.11.1.1.9
Zeros of functions. Increasing or decreasing intervals. Even and
odd functions.
PC2.11.1.1.10
Identify and graph cubic, square root and reciprocal functions.
PC2.11.1.1.11
Combinations of functions. Add, subtract, multiply and divide
functions. Composition of functions.
PC2.11.1.1.12
Inverse functions. Find them and graph them.
PC2.11.1.1.13
Preparation for SAT.
Strand 2: Polynomial and Rational Functions
Standard 1: Students will simplify expressions containing polynomial and rational
functions, factor polynomials, evaluate polynomial functions, and graph polynomial and
rational functions.
Benchmark Code
Benchmark
PC2.11.2.1.1
Sketch graphs of quadratic functions.
PC2.11.2.1.2
Add, subtract, and multiply polynomials; divide using long
division.
PC2.11.2.1.3
Divide polynomials by binomials using synthetic division.
PC2.11.2.1.4
Factor polynomials and simplify polynomial quotients by
factoring.
PC2.11.2.1.5
Evaluate polynomial functions and identify general shapes of the
graphs of polynomial functions.
PC2.11.2.1.6
Find factors of polynomials by using the factor theorem and
synthetic division.
PC2.11.2.1.7
Sketch graphs of polynomial functions of higher degree.
PC2.11.2.1.8
Use a graphing calculator to graph polynomial functions and
approximate the real zeros of the functions.
PC2.11.2.1.9
Find the number and type of zeros of a polynomial function.
PC2.11.2.1.10
Identify all possible rational zeros of a polynomial function by
using the rational zero theorem.
PC2.11.2.1.11
Find zeros of polynomial functions.
PC2.11.2.1.12
Operations with complex numbers.
PC2.11.2.1.13
Sketch graphs of rational functions.
PC2.11.2.1.14
Partial fraction decomposition.
PC2.11.2.1.15
Preparation for SAT.
Strand 3: Exponential and Logarithmic Functions
Standard 1: Students will recognize, graph, evaluate and solve logarithmic functions.
Benchmark Code
Benchmark
PC2.11.3.1.1
Recognize and evaluate exponential and logarithmic functions.
PC2.11.3.1.2
Graph exponential and logarithmic functions.
PC2.11.3.1.3
Use the change-of-base formula to rewrite and evaluate
logarithmic expressions
PC2.11.3.1.4
Solve exponential and logarithmic equations
PC2.11.3.1.5
Use exponential growth models, exponential decay models,
Gaussian models, logistic growth models, and logarithmic models
to solve real-life problems.
PC2.11.3.1.6
Preparation for SAT.
Strand 4: Topics in Analytic Geometry
Standard 1: Students will learn about Conics and conic sections (parabolas, ellipses,
hyperbolas), rotation of conics, parametric equations, polar coordinates.
Benchmark Code
Benchmark
PC2.11.4.1.1
Graph parabolas, ellipses, and hyperbolas.
PC2.11.4.1.2
Identify the equation of a specific conic sections.
PC2.11.4.1.3
Use conic sections to model real-world problems.
PC2.11.4.1.4
Graph polar equations.
PC2.11.4.1.5
Preparation for SAT.
Mathematics
Course: Probability and Statistics
Grade: 12
MATH 601 Probability and Statistics
1/2 credit
5 days per week (1st Semester)
Taught in English
th
This is a required class for all 12 grade students in the Mexican and/or U.S. diploma program. In this course
students will study sequences and series, permutations and combinations, discrete mathematics, and data
analysis. Students will develop the knowledge and skills necessary to be able to organize, analyze and graph
data, as well as make decisions based on the data. Students will also be able to model problems and solve
them. Matrices are introduced.
Textbook:
Brase, Charles Henry and Corrinne Pellillo Brase. Understandable Statistics,
Concepts & Methods, Houghton/Mifflin Company. Boston, MA
(2003 Edition).
Prerequisite: MATH 502
Subject : Probability and Statistics = PS
Strand 1: Introduction to Statistics.
Strand 2: Organizing Data.
Strand 3 : Averages and Variation
Strand 4: Elementary Probability Theory.
Strand 5: The Binomial Probability Distribution and Related Topics.
Strand 6: Normal Distributions.
Strand 7: Introduction to Sample Distributions.
Benchmark Code
Subject (M, S, SS, LA).Grade#.Strand#.Standard#. Benchmark#
Example: PS.1.4.3 – Probability and Statistics, Strand 1, Standard 4, Benchmark 3
Strand: 1 INTRODUCTION TO STATISTICS.
Standard 1: Introduction to statistical data. Students will be able to state the importance of the
study of Statistics, the nature of the statistical data, what a sample is, what the sampling methods
are and how to design ways to collect data.
Benchmark Code
Benchmark
PS.1.1.1
Identify variables in a statistical study.
PS. 1.1.2
Distinguish between quantitative and qualitative variables.
PS. 1.1.3
Identify populations and samples.
PS.1.1.4
Determine the levels of measurement.
PS. 1.1.5
Compare descriptive and inferential statistics.
Standard 2: Random samples. Students will be able to explain why random sampling is important
to the study of Statistics, they will know how to use a their calculators o a random number table to
make a simulation, and describe different sampling strategies and how to use them.
Benchmark Code
PS.1.2.1
PS.1.2.2
PS.1.2.3
Benchmark
Explain the importance of random samples.
Construct a simple random sample using random numbers.
Simulate a random process.
PS.1.2.4
Describe stratified sampling, cluster sampling, systematic sampling
and convenience sampling.
Standard 3: Introduction to Experimental Design. Students will learn the basics for planning a
statistical study understanding the differences between observations and experiments.
Benchmark Code
Benchmark
PS. 1.3.1
Be able to explain the term census.
PS.1.3.2
PS.1.3.3
Describe simulations, observational studies and experiments.
Identify control groups, placebo effects and randomized twotreatment design.
PS.1.3.4
Discuss potential pitfalls that might make the data unreliable.
Strand: 2 ORGANIZING DATA
Standard 1: Graphing statistical data. Students should be able to display effectively information
using a variety graphs.
Benchmark Code
Benchmark
PS.2.1.1
Determine types of graphs appropriate for specific data.
PS. 2.1.2
Construct bar graphs, Pareto charts, circle graphs, and time plots.
PS.2.1.3
Interpret information displayed in graphs.
Standard 2: Frequency distributions and histograms. Students will be able to organize data in a
frequency table and construct a histogram or a frequency polygon.
Benchmark Code
Benchmark
PS.2.2.1
Organize raw data using a frequency table.
PS.2.2.2
Construct histograms, relative-frequency histograms, frequency
polygons and orgies.
PS.2.2.3
Recognize basic distribution shapes: uniform, symmetric, bimodal,
and skewed.
PS.2.2.4
Interpret graphs in the context of the data setting.
Standard 3: Stem-and-leaf displays. Students will be able to construct a stem-and-leaf display.
Benchmark Code
Benchmark
PS.2.3.1
Construct a stem-and –leaf display from raw dataPS.2.3.2
Use a stem-and-leaf display to visualize data distribution.
PS.2.3.3
Compare a stem-and leaf display to a histogram.
Strand: 3 AVERAGES AND VARIATION
Standard 1: Measures of central tendency. Mean, Median, Mode. Students will be able to
understand, state and/or compute the measures of central tendency.
Benchmark Code
Benchmark
PS.3.1.1
Compute mean, median and mode of raw data.
PS.3.1.2
PS.3.1.3
Interpret what mean, mode and median tell us.
Explain how mean, median and mode can be affected by extreme data
values.
PS.3.1.4
Compute a trimmed means and explains why is used.
Standard 2: Measures of variation. Students will be able to understand, state and/or compute the
measures of variation.
Benchmark Code
Benchmark
PS.3.2.1
Find the range, variance, and standard deviation.
PS.3.2.2
Compute the coefficient of variation from raw data and understand its
importance.
PS.3.2.3
Apply Chebyshev’s theorem to raw data.
PS.3.2.4
Understand the information given by the Chebyshev’s theorem.
Standard 3: Mean and standard deviation of grouped data. Students will be able to understand,
state and/or compute the mean, variance and standard deviation of grouped data.
Benchmark Code
Benchmark
PS.3.3.1
Estimate the mean, variance and standard deviation from grouped data.
PS.3.3.2
Compute a weighted average.
PS.3.3.3
Understand the applications of weighted averages.
Standard 4: Percentiles and box-and whisker plots. Students will be able to understand, state
and/or compute percentiles, the five number summary and construct a box-and-whiskers plot.
Benchmark Code
Benchmark
PS.3.4.1
Interpret the meaning of percentile scores.
PS.3.4.2
Compute the median, quartiles, and five-number number from raw
data.
PS.3.4.3
Make a box-and-whisker plot and interpret its results.
PS.3.4.4
Describe how a box-and-whisker plot indicates spread of data around
the median.
Strand: 4 ELEMENTARY PROBABILITY THEORY
Standard 1: Introduction to Probability. Basic concepts. Students will be able to understand the
methods to assign probabilities and apply them to basic problems.
Benchmark Code
Benchmark
PS.4.1.1
Assign probabilities to events. Relative frequency. Law of large
numbers. Equally likely outcomes.
PS.4.1.2
PS.4.1.3
Explain how the law of large numbers relates to relative frequencies.
Apply basic rules of probability in everyday life. Sample space.
Complement of an event.
PS.4.1.4
Explain the relationship between statistics and probability.
Standard 2: Some probability rules. Compound events. Students will be able to understand and
apply basic probability rules.
Benchmark Code
Benchmark
PS.4.2.1
Compute probabilities of general compound events. What is and
independent event. What is a compound event?
PS.4.2.2
Compute probabilities involving independent events or mutually
exclusive events. Multiplication rule. Addition rule.
PS.4.2.3
Use results to compute conditional probabilities.
Standard 3: Trees and counting techniques. Students will be able to construct tree diagrams,
organize the outcomes of a series of event and assign probabilities to these outcomes.
Benchmark Code
Benchmark
PS.4.3.1
Organize outcomes in a sample space using tree diagrams.
PS.4.3.2
Compute number of ordered arrangements of outcomes using
permutations.
PS.4.3.4
Compute number of (non ordered) groupings of outcomes using
combinations.
PS.4.3.4
Explain how counting techniques relate to probability in everyday life.
Strand: 5 THE BINOMIAL PROBABILITY DISTRIBUTION AND RELATED TOPICS
Standard 1: Introduction to random variables and probability distributions. Students will learn the
difference between continuous and discrete random variables. They will graph discrete
distributions and compute their parameters.
Benchmark Code
Benchmark
PS.5.1.1
Distinguish between continuous and random variables.
PS.5.1.2
Graph discrete probability distributions.
PS.5.1.3
Compute the mean μ and standard deviation σ for a discrete
probability distribution.
PS.5.1.4
Compute the mean μ and standard deviation σ for a linear function of
a random variable x.
PS.5.1.5
Compute the mean μ and standard deviation σ for a linear
combination of two independent random variables.
Standard 2: Binomial probabilities. Students will learn the characteristics of a Binomial
probability distribution and they will learn to compute Binomial probabilities.
Benchmark Code
Benchmark
PS.5.2.1
List the defining features of a binomial experiment.
PS.5.2.2
Compute binomial probabilities using the formula for this probability
distribution.
PS.5.2.3
Use a binomial table to find the probability of an event P(r).
PS.5.2.4
Use the binomial probability distribution to solve real-world
situations.
Standard 3 : Additional properties of the binomial distribution. Students will graph Binomial
distributions and compute their parameters.
Benchmark Code
Benchmark
PS.5.3.1
Make histograms for binomial distributions.
PS.5.3.2
Compute the mean μ and standard deviation σ for a binomial
distribution.
PS.5.3.3
Compute minimal number of trials n to achieve a given probability of
success P(r)
Standard 4 : The Geometric and Poisson Probability distributions. Students will learn the
characteristics of Geometric and Poisson probability distributions and they will learn to compute
Geometric and Poisson probabilities.
Benchmark Code
Benchmark
PS.5.4.1
Use the Geometric probability distribution to compute the probability
that the nth trial is the first success.
PS.5.4.2
Use the Poisson distribution to compute the probability of the
occurrence of events spread out over time or space.
PS.5.4.3.
Use the Poisson distribution to approximate the binomial distribution
when the number of trials is large and the probability of success is
small.
Strand: 6 NORMAL DISTRIBUTION
Standard 1 : Graphs of Normal Probability Distributions. Students will graph normal distributions
and understand its properties.
Benchmark Code
Benchmark
PS.6.1.1
Graph a normal curve and summarize its important properties.
PS.6.1.2
Apply the empirical rule to solve real-life problems.
PS.6.1.3
Use control limits to construct control charts. Examine the chart for
out of control
Standard 2: Standard units and areas under the standard normal distribution. Students will learn to
compute z scores and find the area under the standard normal curve.
Benchmark Code
Benchmark
PS.6.2.1
Given μ and σ, convert raw data to z scores.
PS.6.2.2
Given μ and σ, convert z scores to raw data.
PS.6.2.3
Graph the standard normal distribution, and find areas under the
standard normal curve.
Standard 3: Areas under any normal curve. Students will learn to compute probabilities of
“standardized events” and they will solve guarantee problems.
Benchmark Code
Benchmark
PS.6.3.1
Compute the probability of “standardized events”
PS.6.3.2
Find a z score from a given normal probability (inverse normal).
PS.6.3.3
Use the inverse normal to solve guarantee problems.
Standard 4: Normal approximation to the Binomial distribution. Students will learn to use the
normal approximation of the Binomial distribution.
Benchmark Code
Benchmark
PS.6.4.1
State the assumptions needed for the normal approximation to the
binomial.
PS.6.4.2
Compute μ and σ for the normal approximation.
PS.6.4.3
Use the continuity correction to convert a range of r values to a
corresponding range of normal x values.
PS.6.4.4.
Convert the x values to a range of standardized z scores and find
desired probabilities.
Strand: 7 INTRODUCTION TO SAMPLING DISTRIBUTIONS
Standard 1: Sampling distributions. Students will review statistical terms and will construct
frequency distributions.
Benchmark Code
Benchmark
PS:7.1.1
Review commonly used terms as random sample, relative frequency,
parameter, statistic, and sampling distribution.
PS.7.1.2
From raw data, construct a relative frequency distribution for mean
values or expected values and compare the result to a theoretical
sampling distribution.
Standard 2: The Central Limit Theorem. Students will learn to use the mean and the standard
deviation to construct sampling distributions.
Benchmark Code
Benchmark
PS.7.2.1
For a normal distribution, use μ and σ to construct the theoretical
sampling distribution for the statistic x-bar
PS.7.2.2
For large samples, use sample estimates to construct a good
approximate sampling distribution for the statistic x-bar.
PS.7.2.3
Learn the statement and underlying meaning of the central limit
theorem well enough to explain it.
Standard 3: Sampling Distributions for Proportions. Students will learn to compute the mean and
standard deviation for a given proportion and they will be able to compute probabilities for
proportions.
Benchmark Code
Benchmark
PS.7.3.1
Compute the mean and standard deviation for the proportion p=r/n
PS.7.3.2
Use the normal approximation to compute the probabilities for
proportions p= r/n.
PS.7.3.3
Construct P-Charts and interpret what they tell us.