Pre-Calculus Standards

Pre-Calculus

Course description:

Precalculus is an advanced mathematics course that uses meaningful problems and appropriate technologies to build upon previously learned mathematical concepts to develop the underpinnings of calculus.

Standard 1.0: Models for Real-World Phenomena

Students will model and analyze real-world phenomena using techniques from algebra and data analysis.

The student will:

1.1 select and use appropriate algebraic functions to model real-world situations;

1.2 select and use appropriate techniques from data analysis to model real-world phenomena.

Student Performance Indicators:

model real-world phenomena using techniques of data analysis;

recognize and apply mathematical models of linear, quadratic, exponential, logarithmic, and trigonometric functions;

use scatterplot residuals, and/or correlation coefficients to determine whether a model is appropriate;

apply equations and graphs of conic sections to model real-world phenomena.

use models when appropriate to draw conclusions or make predictions.

Standard 2.0: Algebraic Functions

Students will extend the concepts of function from earlier courses to a wider variety of functions and their graphs and real-world applications.

The student will:

2.1 represent a variety of functions graphically;

2.2 use a variety of methods to analyze and interpret functions;

2.3 determine the slope and equations of lines tangent to curves;

2.4 apply functions in problem situations.

Student Performance Indicators:

sketch the graphs of the basic functions (linear, quadratic, cubic, square root, absolute value, reciprocal, trigonometric, exponential, logarithmic, and greatest integer);

graph transformations and combinations of transformations for all basic functions;

analyze functions, such as by decomposing into simpler functions;

determine if a function is even, odd, or neither;

use an appropriate technology to solve inequalities;

demonstrate an understanding of the concept of the limit of a function;

apply the limit of a function to find the slope of a line tangent to a curve;

write equations of tangents and normals to conic sections;

apply limits to develop the concept of continuity and identify intervals of increase and decrease;

locate critical points on the graphs of polynomial functions and determine if each critical point is a minimum, a maximum, or a point of inflection;

determine an equation of a rational function from a written description.

define and use the logarithmic function as the inverse of the exponential function;

sketch the graphs of exponential and logarithmic functions;

solve exponential and logarithmic equations modeling real-world problems (e.g. growth and decay).

Standard 3.0: Trigonometric Functions

The student will

3.1 apply trigonometry concepts and applications to model and solve problems;

3.2 use trigonometric concepts to represent, apply, and operate with complex numbers;

3.3 solve trigonometric equations and inequalities algebraically or graphically;

3.4 interpret transformations of trigonometric functions.

Student Performance Indicators:

define six circular functions;

sketch graphs of the six trigonometric functions involving period change, amplitude change, phase shift, and/or vertical shift;

use trigonometric functions to model periodic phenomena;

use graphs to develop and verify trigonometric identities;

find values of inverse trigonometric functions, applying appropriate domain and range restrictions;

solve trigonometric equations and inequalities either algebraically or using graphing technology.

derive the Law of Sines and the Law of Cosines and apply them to solve problems involving triangles and vectors;

derive and apply the formulas for the area of a triangle and the sector of a circle;

understand the relationship between measurements in radians and degrees;

apply radian measures in problems related to linear and angular velocity;

understand and apply vectors to solve real world problems;

represent complex numbers in both rectangular and polar form;

apply the trigonometric form of complex number in calculations;

prove and apply DeMoivre’s Theorem to find roots and powers of complex numbers.

Standard 4.0: Sequences and Series

Students will develop the concept of limit by examining infinite sequences and series.

The student will:

4.1 represent sequences and series;

4.2 determine, when possible, the sums of infinite series.

Student Performance Indicators:

demonstrate an understanding of sequences by representing them recursively and explicitly;

use sigma notation to represent a series;

determine whether a given series converges or diverges;

find the sum of an infinite series that converge;

find the sum of an infinite geometric series;

use the Binomial Theorem to expand binomials.

Geometry Standards

Geometry

Course description:

Geometry uses problem situations, physical models, and appropriate technology to investigate and justify geometric concepts and relationships. Problem-solving situations provide all students an environment that promotes communication, engages student reasoning, and fosters connections within mathematics, to other disciplines, and to the real world. Students will use physical models to represent, explore, develop, and apply abstract concepts. The use of appropriate technology will help students develop mathematics needed in an increasingly technological world. The concepts and topics emphasized in the course include measurement, geometric patterns, coordinate geometry, two- and three-dimensional figures, transformational geometry, congruence, similarity, inductive and deductive reasoning, logic, and proof.

Standard Number 1.0: Number and Operations

Students will recognize, order, represent, and graph rational and irrational numbers, including absolute value notation.

Learning Expectations:

The student will:

1.1 demonstrate an understanding of the relative size of rational and irrational numbers;

1.2 choose and use appropriate notations for rational and irrational numbers, including graphic representations;

1.3 demonstrate an understanding of absolute value.

Performance Indicators State:

As documented through state assessment –

At Level 1, the student is able to

order a set of rational and irrational numbers;

find an integral power of a positive rational number (exponents 1-3).

At Level 2, the student is able to

use absolute value to express the distance between two points on a number line and vice versa;

simplify a radical (radicand less than 1000);

match a given irrational number to the appropriate point on a number line and vice versa (e.g., Ö2, Ö30, pi).

There are no state-assessed performance indicators at Level 3.

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

estimate the value of an irrational number expressed as a radical (radicand less than 1000);

approximate pi given a table of values for the circumference and diameter of circles.

At Level 2, the student is able to

use radicals and decimal approximations of irrational numbers to indicate calculated lengths or distances;

discuss the relative accuracy of radicals and their decimal approximations.

At Level 3, the student is able to

represent irrational numbers as lengths of lines in the coordinate plane (e.g. ,Ö5 is the length of the diagonal of a rectangle with base 1 and height 2).

Sample Task:

Students will compute the hypotenuse of a given right triangles and arrange themselves in order from smallest to largest.

Linkages:

Mathematics: Estimation, Measurement, and Algebra.

Standard 2.0: Algebra

Students will recognize, extend, create, and analyze a variety of geometric, spatial, and numerical patterns; solve real-world problems related to algebra and geometry; and use properties of various geometric figures to analyze and solve problems.

Learning Expectations:

The student will:

2.1 recognize, extend, and create geometric, spatial, and numerical patterns;

2.2 analyze mathematical patterns related to algebra and geometry in real-world problem solving;

2.3 solve problems connecting geometry with number theory, probability and statistics, and measurement and estimation using algebraic thinking and symbolism;

2.4 apply coordinate geometry to analyze and solve problems;

2.5 apply ratio and proportion to problems involving similar figures.

Performance Indicators State:

As documented through state assessment –

At Level 1, the student is able to

extend or find missing element(s) in a geometric pattern;

solve multistep linear equations to find length, width, perimeter, and area of geometric figures;

apply the concept of rate of change to solve a real-world problem given a pattern of data;

determine the slope given a graph of a linear equation and vice versa;

determine the distance, midpoint, or slope when given the coordinates of two points (answers must be given as decimals to the nearest hundredth).

At Level 2, the student is able to

determine the equation of a line parallel or perpendicular to a given line, from given information (e.g., equations of lines, graphs of lines, or two points);

apply ratio and proportion to solve real-world problems involving polygons, (e.g., scale drawings, similar figures);

apply the triangle inequality property to determine which sets of side lengths determine a triangle;

determine the perimeter, area, or volume given the ratio of two similar polygons or rectangular solids;

apply the Triangle Sum Theorem or Exterior Angle Theorem to determine the measures of the angles of a given triangle with the angle measures expressed algebraically.

At Level 3, the student is able to

determine the equation of a circle given coordinates or the graph of the circle (e.g., the center, the endpoints of the diameter).

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

apply the line of best fit given real-world data from geometric figures using technology (e.g., finding the interior angle sum of polygons when given the number of sides; find the circumference of circles when given the diameter).

At Level 2, the student is able to

explore patterns in geometric situations (e.g., Fibonacci sequence and Golden Ratio);

use manipulatives to determine relationships between linear, square, or cubic measures when one of the measures of the object has changed and represent algebraically.

At Level 3, the student is able to

recognize complete and incomplete networks;

graph plane figures on a coordinate plane and solve problems algebraically.

Sample Task:

Students construct designs using basic geometric constructions. Then they transfer the design to a piece of 8″”X 11″”pane of plexiglass and paint the pane to create a “stained glass.”

Linkages:

Mosaic Tiling.

Standard 3.0: Geometry

Students will investigate, model, and apply geometric properties and relationships and use indirect reasoning to make conjectures; deductive reasoning to draw conclusions; and both inductive and deductive reasoning to establish the truth of statements.

Learning Expectations:

The student will:

3.1 analyze relationships among corresponding parts of similar or congruent geometric figures;

3.2 apply geometric properties of solids, polygons, and circles to solve real-world problems;

3.3 justify conclusions and solve problems using deductive reasoning;

3.4 use inductive reasoning to make conjectures and solve problems;

3.5 communicate position using spatial sense with two- and three-dimensional coordinate systems;

3.6 demonstrate an understanding of transformations of geometric figures (i.e., translations, rotations, dilations, and reflections);

3.7 apply right triangle relationships including the Pythagorean Theorem, the distance formula, and trigonometric ratios;

3.8 describe geometric objects and recognize minimal conditions necessary to define the geometric objects;

3.9 apply reflexive, transitive, and symmetric properties when appropriate;

3.10 demonstrate understanding of geometric properties of congruence, similarity, perpendicularity, and parallelism;

3.11 recognize and articulate relationships among families of geometric figures (e.g., quadrilaterals, prisms);

3.12 use logic and proof to establish the validity of conjectures and theorems.

Performance Indicators State:

As documented through state assessment:

At Level 1, the student is able to

identify corresponding parts of similar and congruent geometric figures given a diagram.

determine the length of a missing side in a right triangle when given two sides (answers must be given as simplified radicals).

At Level 2, the student is able to

identify properties of plane figures from information given in a diagram;

identify chords, inscribed angles, or central angles of circles given a diagram;

determine congruence or similarity relations between triangles or quadrilaterals given a diagram;

determine whether a plane figure has been translated, dilated, reflected, or rotated given a diagram and vice versa;

solve problems involving complementary, supplementary, congruent, vertical, or adjacent angles given angle measures expressed algebraically;

determine the trigonometric ratio for a right triangle needed to solve a real-world problem given a diagram;

find a missing side length in a 30-60-90 or 45-45-90 degree triangle without rationalizing the denominator

apply properties of quadrilaterals to solve a real-world problem given a diagram (opposite sides and angles, consecutive sides and angles, or diagonals);

solve real-world problems involving measures of interior or exterior angles of regular polygons;

identify the appropriate segment of a triangle given a diagram and vice versa (i.e. median, altitude, angle bisector, perpendicular bisector);

determine which three-dimensional solid is represented by a given net and vice versa (two-dimensional drawing);

determine the area of indicated regions involving circles, squares, rectangles, and/or triangles;

justify triangle congruence given a diagram (i.e., ASA, SSS, AAS, SAS, or Hypotenuse/ Leg);

determine if a triangle is a right triangle given the length of all the sides of a triangle.

At Level 3, the student is able to

solve problems involving the properties of arcs, chords, tangents, or secants;

find the area of a sector of a circle given a diagram.

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

investigate the Pythagorean Theorem by constructing right triangles using a variety of manipulatives and/or technologies;

construct parallelograms, rectangles, rhombi, and squares using physical materials, manipulatives, or technology.

At Level 2, the student is able to

apply reflexive, transitive, or symmetric properties of equality or congruence;

investigate the properties of angles, arcs, chords, tangents, and/or secants using technology or manipulatives;

use inductive and deductive reasoning to make conjectures, draw conclusions, and solve problems;

recognize and articulate relationships among families of geometric figures (e.g., quadrilaterals, prisms);

write and defend indirect and direct proofs;

use logical reasoning to solve problems in the real world;

use manipulatives to explore the geometric mean of similar triangles;

use appropriate tools or technology to develop geometric and spatial concepts;

construct three-dimensional objects using physical materials and manipulatives;

compare and construct quadrilateral properties using a variety of models (e.g., Venn diagrams, family trees, manipulative mobiles).

At Level 3, the student is able to

use coordinates to communicate the location of a three-dimensional figure that has been rotated or reflected.

Sample Task:

Students construct and use a hypsometer to measure several tall structures on the school grounds.

Linkages: Mathematics: Measurement. Surveying and Art.

Standard 4.0: Measurement

Students will apply appropriate units of measurement; develop effective estimation and computation strategies for solving real world problems involving length, area, and volume; and choose appropriate techniques and tools to measure quantities in order to meet specifications for precision, accuracy, and tolerance.

Learning Expectations:

The student will:

4.1 use concepts of length, area, and volume to estimate and solve real-world problems;

4.2 apply measurement concepts and relationships in algebraic and geometric problem-solving situations;

4.3 choose appropriate techniques and tools to measure quantities in order to meet specifications for precision, accuracy, and tolerance.

Performance Indicators State:

As documented through state assessment –

At Level 1, the student is able to

determine the perimeter or area of a triangle or rectangle when the dimensions are given as first degree binomials in one variable;

solve real world problems involving perimeter or area of three or four sided plane figures.

At Level 2, the student is able to

determine the volume or surface area of a rectangular solid or cylinder in a real-world situation.

At Level 3, the student is able to

determine whether a reading falls within an acceptable tolerance range.

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

determine the measure of an angle using a protractor.

At Level 2, the student is able to

construct bisectors of angles and line segments, perpendicular lines, congruent line segments and angles, and perpendicular bisectors using a variety of methods (e.g., patty paper, technology);

solve problems involving volume of 3-dimensional figures, e.g. right prisms, pyramids, cones, cylinders, and spheres;

solve problems involving surface area of prisms and cylinders.

At Level 3, the student is able to

choose appropriate techniques and tools to measure quantities in order to meet specification for precision, accuracy, and tolerance;

locate the irrational numbers Ö2 and Ö3 on a number line by using the Pythagorean relationship and a straightedge and compass, manipulatives, or technology;

solve problems involving surface area of pyramids, cones, and spheres.

Sample Task:

Students construct designs using basic geometric constructions. Then they transfer the design to a piece of 8″ X 11″ pane of plexiglass and paint the pane to create a “stained glass.” Students construct one of the regular 3-dimensional solid and compute the volume and surface area.

Linkages:

Mathematics – Geometry and Number & Operations. Surveying, construction, and architecture. .Mosaic Tiling.

Standard 5.0: Data Analysis and Probability

The student will investigate, explore, and apply geometric representations to calculate theoretical probability; and will use data from geometric figures to investigate relationships.

Learning Expectations:

The student will:

5.1 apply geometric representations to calculate theoretical probability;

5.2 use data analysis to investigate geometric relationships.

Performance Indicators State:

As documented through state assessment –

At Level 1, the student is able to

make a prediction from a geometric representation of a real-world data set;

At Level 2, the student is able to

determine the probability of an event represented as a subset of the area of a two-dimensional geometric figure.

There are no performance indicators for Level 3 of Data Analysis and Probability.

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

explain and justify the given geometric representation of the probability of an event.

At Level 2, the student is able to

use hands-on activities to model geometric representations of probability;

collect and analyze data to make conjectures about geometric relationships.

At Level 3, the student is able to

analyze and debate the validity of claims made based on the given theoretical probability of a real-world situation.

Sample Task:

Construct two 1’ X 1’ dart boards and draw circular targets on each that are externally tangent to each adjacent circle and to the edge of the board. Draw two circles on one dartboard and three on the other. Throw randomly and count the throws that hit the board to determine which board yields the highest probability of a dart’s landing in a circle. Calculate the probability for each bard.

Linkages:

Game theory.

Statistics Standards

Statistics

Course description:

Statistics is an advanced mathematics course that uses meaningful problems and appropriate technologies to use statistical concepts developed in previous courses to develop more advanced means of statistical analyses, interpretations, and predictions.

Standard 1.0 Experimental Design

Students will design and conduct statistical experiments.

Learning Expectations:

The student will:

1.1 design studies that can be addressed with data;

1.2 collect data based on an appropriate sample.

Student Performance Indicators:

formulate questions that can be addressed with data;

describe the role of randomization in surveys and experiments;

select and use a method such as a survey or an experiment to collect data;

demonstrate understanding of bias in sampling;

demonstrate an understanding of the Law of Large Numbers;

demonstrate an understanding of the probability of independent events and conditional probability;

using appropriate probability models, design a method for simulating data from a particular situation, and use the generated data to analyze the situation;

design and conduct a statistical experiment to study a problem, and interpret and communicate the outcomes;

test hypotheses using appropriate statistics.

Standard 2.0: Data Analysis

The student will select and use appropriate statistical methods to analyze data and to develop and evaluate inferences and predictions based on the data.

Learning Expectations:

The student will:

2.1 select and use appropriate displays to represent and summarize the data collected in statistical studies or experiments;

2.2 select and use appropriate statistical methods to analyze data;

2.3 develop and evaluate inferences and predictions based on data.

Student Performance Indicators:

construct and interpret charts, tables, and graphs that display univariate and bivariate data;

calculate and apply measures of central tendency and dispersion in order to make inferences about a data set;

analyze the effects of data transformations on measures of central tendency and variability;

calculate and apply the correlation between data sets.

apply the properties of a normal distribution or a Chi-square distribution in appropriate situations in order to make inferences about a data set;

demonstrate an understanding of the Central Limit Theorem;

use curve-fitting with appropriate technology to make regression equations in order to represent a data set algebraically and to make inferences;

demonstrate an understanding of confidence intervals.

Geometry Sections

1.1 Use inductive reasoning to make conjectures

1.2 Understand basic terms of geometry; Understand basic postulates of geometry

1.3 Identify segments and rays

1.4 Find the lengths of segments; Find the measures of angles

Review

TEST Chapter 1 Test Sections 1 – 4

2.5 Identify angle pairs; Prove and apply theorems about angles

2.1 Recognize conditional statements; Write converses, inverses, and contrapostives of conditional statements; Write biconditional statements

2.2 Write biconditionals; Recognize good definitions

2.3 Use the Law of Detachment and Law of Syllogism

TEST Chapter 1 and Chapter 2 Test

2.4 Connect reasoning in algebra and geometry

2.5 Connect reasoning in algebra and geometry; Identify angle pairs; Prove and apply theorems about angles

Review unit on proofs

TEST Chapter 2 Sections 4 and 5 Proofs

Review angle pairs — algebra based problems and vocabulary

TEST Chapter 2 — Angle Pairs

3.1 and 3.2 Identify angles formed by two lines and a transversal; Prove and use properties of parallel lines; Use a transversal in proving lines parallel; Relate parallel and perpendicular lines

3.3 Classify triangles and find the measures of their angles; Use exterior angles of triangles

3.4 Classify polygons; Find the sum of the measures of the interior and exterior angles of polygons

Review Chapter 3 Sections 1 – 4

TEST Chapter 3 Test

3.5 Graph lines given their equations; Write equations of lines

TEST Nine Weeks Exam

3.6 Relate slope to parallel and perpendicular lines

4.1 Recognize congruent figures and their corresponding parts

4.2 and 4.3 Prove two triangles congruent using the SSS and SAS Postulates; Prove two triangles congruent using the ASA Postulate and the AAS Theorem

4.1, 4.2, 4.3 Review the three previous lessons about congruent triangles

4.4 Use triangle congruence and CPCTC to prove that parts of two triangles are congruent

4.5 Use and apply properties of isosceles triangles

Review Chapter 4

TEST Chapter 4 Test

5.3 Identify and use properties of medians and altitudes of triangles

5.2 Identify and use properties of perpendicular bisectors and angle bisectors of triangles

5.1 Use properties of midsegments to solve problems

5.5 Use inequalities involving angles of triangles; Use inequalities involving sides of triangles

5.4 Use indirect reasoning; Review for Chapter 5 Test

TEST Chapter 5 Test

6.2 Parallelograms; Use relationships among sides and among angles of parallelograms; Use relationships involving diagonals of parallelograms or transversals

6.3 Determine whether a quadrilateral is a parallelogram

6.4 Special parallelograms; Use properties of diagonals of rhombuses and rectangles; Determine whether a parallelogram is a rhombus or a rectangle

6.5 Trapezoids and kites; Verify and use properties of trapezoids and kites

6.1 Define and classify special types of quadrilateral

Review Chapter 6

TEST Chapter 6 Test

Review Algebra Skills — Ratios, Proportions, and Simplifying Radical Expressions

Review Algebra Skills — Simplifying Radical Expressions

Review Algebra Skills — The Pythagorean Theorem

Nine Weeks Test

Midterm Exams

8.1 Write ratios and solve proportions

8.1, 8.2, 8.3 Write ratios and solve proportions; Identify similar polygons; Apply similar polygons

8.1, 8.2, 8.3, 8.5 Write ratios and solve proportions; Identify similar polygons; Apply similar polygons; Apply triangle proportionality theorems

Review Similar Triangle

TEST Chapter 8 Sections 1, 2, 3, 5

7.2 The Pythagorean Theorem; The Converse of the Pythagorean Theorem

7.3 Using 45-45-90 and 30-60-90 Triangles

8.4 Find and use relationships in similar right triangles

Review Chapter 7 Sections 2, 3 and Chapter 8 Section 4

TEST Chapter 7 Sections 2, 3 and Chapter 8 Section 4

9.1 Use tangent ratios to determine the length of sides and the measure of angles in right triangles

9.2 Use sine and cosine ratios to determine the length of sides and the measure of angles in right triangles

9.3 Use angles of elevation and depression to solve problems

Review Right Triangle Trigonometry

TEST Chapter 9 Sections 1, 2, 3 — Right Triangle Trigonometry

9.E — Law of Sines

9.E — Law of Cosines

Review Oblique Triangle Trigonometry

TEST Solving Triangles

7.6 and 11.1 Circle basics; The measure of central angles and arcs; The relationship between a radius and a tangent; The relationship between two tangents from one point

End of the 3rd Nine Weeks, Nine Weeks Test

11.2 Use congruent chords, arcs, and central angles; Recognize properties of lines through the center of a circle

11.3 Find the measure of an inscribed angle; Find the measure of an angle formed by a tangent and a chord

11.4 Find the measures of angles formed by chords, secants, and tangents; Find the lengths of segments associated with circles

11.5 Write the equation of a circle; Find the center and radius of a circle

Review

TEST Chapter 11 and Chapter 7 Section 6

7.1 Find the area of a parallelogram; Find the area of a triangle

7.4 Find the area of a trapezoid; Find the area of a rhombus or a kite

7.5 Find the area of a regular polygon

7.6 Find the circumference of a circle; Find the length or an arc

7.7 Find the areas of circles, sectors, and segments or circles

8.6 Find the area and perimeter of similar figures

9.5 Find the area of a regular polygon using trigonometry; Find the area of a triangle using trigonometry

TEST Perimeter and Area of Plane Figures

10.3 Find the surface area of a prism; Find the surface area of a cylinder

10.4 Find the surface area of a pyramid; Find the surface area of a cone

10.5 Find the volume of a prism; Find the volume of a cylinder

10.6 Find the volume of a pyramid; Find the volume of a cone

10.7 Find the surface area and volume of a sphere

10.8 Find relationships between the ratios of the areas and volumes of similar solids

TEST Area and Volume of Solids

Nine Weeks Test

Final Exam

Statistics Sections

1.1 The definition of statistics; How to distinguish between a population and a sample and between a parameter and a statistic; How to distinguish between descriptive and inferential statistics

1.2 How to distinguish between qualitative data and quantitative data; How to classify data with respect to the four levels of measurements: nominal, ordinal, interval, and ratio

1.3 How to design a statistical study; How to collect data by taking a census, using a sampling, using a simulation, or performing an experiment; How to create a sample using random sampling, simple random sampling, stratified sampling, cluster sampling, and systematic sampling, and how to identify a biased sample

2.1 How to construct a frequency distribution including limits, boundaries, midpoints, relative frequencies, and cumulative frequencies; How to construct frequency histograms, frequency polygons, relative frequency histograms, and oglives

2.2 How to graph and interpret quantitative data sets using stem-and-leaf plots and dot plots; How to graph and interpret qualitative data sets using pie charts and Pareto charts; How to graph and interpret paired data sets using time series charts

2.3 Find the mean, median, and mode of a population and a sample; Find a weighted mean and the mean of a frequency distribution; Describe the shape of a distribution as symmetric, uniform, or skewed and compare the mean and median for each

2.4 Find the range of a data set; Find the variance and standard deviation of a population and or a sample; Use the Empirical Rule and Chebychev’s Theorem to interpret standard deviation; Approximate the sample standard deviation for grouped data

2.5 Find the first, second, and third quartiles of a data set; Find the interquartile range of a data set; Represent a data set graphically using a box-and-whisker plot; Interpret other fractiles such as percentiles; Find and interpret the standard score (z-score)

3.1 Identify the sample space of a probability experiment and identify simple events; Distinguish among classical probability, empirical probability, and subjective probability; Identify and use properties of probability

3.2 Find the probability of an event given that another event has occurred; Distinguish between independent and dependent events; Use the Multiplication Rule to find the probability of two events occurring in sequence; Use the Multiplication Rule to find conditional probabilities

3.3 Determine if two events are mutually exclusive; Use the Addition Rule to find the probability of two events

3.4 Use the Fundamental Counting Principle to find the number of ways two or more events can occur; Find the number of ways agroup of objects can be arranged in order; Find the number of ways to choose several objects from a group without regard to order; Use counting principles to find probabilities

4.1 Distinguish between discrete random variables and continuous random variables; Construct a discrete probability distribution and its graph; Determine if a distribution is a probability distribution; Find the mean, variance, and standard deviation of a discrete probability distribution; Find the expected value of a discrete probability distribution

4.2 Determine if a probability experiment is a binomial experiment; Find binomial probabilities using the binomial probability formula; Construct a binomial distribution and its graph; Find the mean, variance, and standard deviation of a binomial probability experiment

4.3 Find probabilities using the geometric distribution; Find probabilities using the Poisson distribution

5.1 Interpret graphs of normal probability distributions; Estimate areas under a normal curve and use them to estimate probabilities for random variables with normal distributions

5.2 Find the areas under standard normal curve

5.3 Find probabilities for normally distributed variables using a table and using technology

5.4 Find a z-score given the area under the normal curve; Transform a z-score to an x-value; Find a specific data value of a normal distribution given the probability

5.5 How to find sampling distributions and verify their properties; How to interpret the Central Limit Theorem; How to apply the Central Limit Theorem to find the probability of a sample mean

5.6 How to decide when the normal distribution can approximate the binomial distribution; How to find the correction for continuity; How to use the normal distribution to approximate binomial probabilities

6.1 How to find a point estimate and a maximum error of estimate; How to construct and interpret confidence intervals for the population mean; How to determine the required minimum sample size when estimating the population mean

6.2 How to interpret the t-distribution and use a t-distribution table; How to construct confidence intervals when n < 30 the population is normally distributed, and the standard deviation is unknown

6.3 How to find a point estimate for the population proportion; How to construct a confidence interval for a population proportion; How to determine the required minimum sample size when estimating a population proportion

6.4 How to interpret the chi-square distribution and use a chi-square distribution table; How to use the chi-square distribution to construct a confidence interval for the variance and standard deviation

7.1 A practical introduction to hypothesis tests; How to state a null hypothesis and an alternative hypothesis; How to identify type I and type II errors and interpret the level of significance; How to know whether to use a one-tailed or two-tailed statistical test and finding a P-value; How to make and interpret a decision based on the results of a statistical test; How to write a claim for a hypothesis test

7.2 How to find P-values and use them to test a mean; How to use P-values for a z-test; How to find critical values and rejection regions in a normal distribution; How to use rejection regions for a z-test

7.3 How to find critical values in a t-distribution; How to use the t-test to test a mean; How to use technology to find P-values and use them with a t-test to test a mean

7.4 How to use the z-test to test a population proportion p

7.5 How to find critical values for a chi-squared test; How to use the chi-squared test to test a variance or a standard deviation

8.1 An introduction to two-sample hypothesis testing for the difference between two population parameters; How to perform a two-sample z-test for the difference between two means using large independent samples

8.2 How to perform a t-test for the difference between two population means using small independent smaples

8.3 How to decide whether two samples are independent or dependent; How to perform a t-test to test the mean of the difference for a population of paired data

8.4 How to perform a z-test for the difference between two population proportions

9.1 An introduction to linear correlation, independent and dependent variables, and the types of correlation; How to find a correlation coefficient; How to perform a hypothesis test for a population correlation coefficient

9.2 How to find the equation of a regression line; How to predict y-values using a regression line

9.3 How to interpret the three types of variation about a regression line; How to find and interpret the coefficient of determination; How to find and interpret the standard error of estimate for a regression line; How to construct and interpret a prediction interval for y

9.4 How to use technology to find a multiple regression equation, the standard error of estimate and the coefficient of determination; How to use a multiple regression equation to predict y-values

10.1 How to use the chi-square distribution to test whether a frequency distribution fits a claimed distribution

10.2 How to use a contingency table to find expected frequencies; How to use a chi-square distribution to test whether two variables are independent

10.3 How to interpret the F-distribution and use an F-table to find critical values; How to perform a two-sample F-test to compare two variances

10.4 How to use one-way analysis of variance to test claims involving three or more means; An introduction to two-way analysis of variance

11.1 How to use the sign test to test a population median; How to use the sign test to test the difference between two population medians (dependent samples0

11.2 How to use the Wilcoxon signed-rank test to determine if two dependent samples are selected from populations having the same distribution; How to use the Wilcoxon rank sum test to determine if two independent samples are selected from populations have the same distribution

11.3 How to use the Kruskal-Wallis test to determine whether three or more samples were selected from populations having the same distribution

11.4 How to use the Spearman rank correlation coefficient to determine whether the correlation between two variables is significant

Pre-Calculus Sections

A.1 Evaluate algebraic expressions; Determine the domain of a variable; Graph inequalities; Find distance on the real number line; Use the laws of exponents; Evaluate square roots

A.2 Use the Pythagorean Theorem and its converse; Know geometry formulas

A.3 Add and subtract polynomials; Multiply polynomials; Completely factor polynomials

A.4 Reduce a rational expression to lowest terms; Multiply and divide rational expressions; Add and subtract rational expressions; Use the least common multiple method; Simplify mixed quotients

A.5 Divide polynomials using long division and synthetic division

TEST Chapter A Sections 1 – 5

1.1 Use the distance formula and midpoint formula

1.2 Graph equations by plotting points; Graph equations using a graphing utility; Find intercepts

1.3 Test an equation for symmetry with respect to the x-axis, the y-axis, and the origin; Know how to graph key equations; Write the standard form of the equation of a circle; Graph a circle by hand and by using a graphing utility; Find the center and radius of a circle from an equation in general form

TEST Chapter 1 Sections 1 – 3

1.4 Solving equations using a graphing utility; Solve linear equations; Solve quadratic equations by factoring; Solve quadratic equations using the quadratic formula; Solve radical equations; Solve equations involving absolute value

1.5 Use interval notation; Use properties of inequalities; Solve linear inequalities algebraically and graphically; Solve combined inequalities algebraically and graphically; Solve absolute value inequalities algebraically and graphically

1.6 Calculate and interpret the slope of a line; Graph lines given a point and the slope; Find the equation of vertical lines; Use the point-slope form of a line; Identify horizontal lines; Find the equation of a line given two points; Write the equation of a line in slope-intercept form; Identify the slope and y-intercept of a line from its equation; Write the equation of a line in general form; Define parallel lines; Find equations of parallel lines; Define perpendicular lines; Find equations of perpendicular lines

TEST Chapter 1 Sections 4 – 6

2.1 Determine whether a relation represents a function; Find the value of a function; Find the domain of a function; Identify the graph of a function; Obtain information from or about the graph of a function

2.2 Graph linear functions; Draw and interpret scatter diagrams; Distinguish between linear and nonlinear relations; Use a graphing utility to find the line of best fit; Construct a linear model using direct variation

2.3 Find the average rate of change of a function; Use a graph to determine where a function is increasing, is decreasing, or is constant; Use a graph to locate local maxima and minima; Determine even and odd functions from a graph; Identify even and odd functions from the equation

2.4 Graph the functions listed in the library of functions; Graph piecewise-defined functions

2.5 Graph functions using horizontal and vertical shifts; Graph functions using reflections about the x -axis or y -axis; Graph functions using compressions and stretches

2.6 Form the sum, difference, product, and quotient of two functions; Form the composite function and find its domain

2.7 Construct and analyze functions

TEST Chapter 2

TEST Nine Weeks Exams

3.1 Graph a quadratic function by hand and by using a graphing utility; Identify the vertex and axis of symmetry of a quadratic function; Determine the maximum or minimum value of a quadratic function; Use the maximum or minimum value of a quadratic function to solve applied problems

3.3 Identify polynomial functions and their degree; Identify the zeros of a polynomial function and its multiplicity; Analyze the graph of a polynomial function

A.7 Add, subtract, multiply, and divide complex numbers; Solve quadratic equations with a negative discriminant

3.7 Use the Remainder and Factor Theorems; Use the Rational Zeros Theorem to list the potential rational zeros of a polynomial function; Find the real zeros of a polynomial function; Solve polynomial equations

3.8 Utilize the conjugate pairs theorem to find the complex zeros of a polynomial; Find a polynomial function with specified zeros; Find the complex zeros of a polynomial function

TEST Chapter 3 Sections 1, 3, 7, 8 and A.7

3.6 Solve polynomial inequalities and rational inequalities algebraically and graphically

3.4 and 3.5 Find the domain of a rational function; Determine the vertical asymptotes of a rational function; Determine the horizontal or oblique asymptotes of a rational function; Analyze the graph of a rational function; Solve applied problems involving rational functions

4.1 Determine the inverse of a function; Obtain the graph of the inverse function from the graph of the function; Find the inverse function f ^-1

Test Chapters 3, 4 Sections 3.4, 3.5, 3.6, and 4.1

4.2 Evaluate exponential functions; Graph exponential functions; Define the number e ; Solve exponential equations

4.3 Change exponential expressions to logarithmic expressions; Change logarithmic expressions to exponential expressions; Evaluate logarithmic functions; Determine the domain of a logarithmic function; Graph logarithmic functions; Solve logarithmic equations

4.4 Work with the properties of logarithms; Write a logarithmic expression as a sum or difference of logarithms; Write a logarithmic expression as a single logarithm; Evaluate logarithms whose base is neither 10 nor e ; Graph logarithmic functions whose base is neither 10 nor e

4.5 Solve logarithmic equations using the properties of logarithms; Solve exponential equations; Solve logarithmic and exponential equations using a graphing utility

4.6 Determine the future value of a lump sum of money; Calculate effective rates of return; Determine the present value of a lump sum of money; Determine the time required to double a lump sum of money

4.7 Find equations of populations that obey the Law of Uninhibited Growth; Find equations of populations that obey the Law of Decay; Use Newton’s Law of Cooling; Use logistic growth models

4.8 Use a graphing utility to fit an exponential function to data; Use a graphing utility to fit a logarithmic function to data; Use a graphing utility to fit a logistic function to data

TEST Chapter 4

10.1 Solve systems of equations using a graphing utility; Solve systems of equations by substitution; Solve systems of equations by elimination; Identify inconsistent systems; Express the solutions of a system of dependent equations

A.8 Translate verbal descriptions into mathematical expressions; Set up applied problems; Solve mixture problems; Solve interest problems; Solve uniform motion problems; Solve constant rate job problems

10.2 Solve systems of three equations containing three variables; Identify inconsistent systems; Express the solutions of a system of dependent equations

10.3 Write the augmented matrix of a system of linear equations; Write the system from the augmented matrix; Perform row operations on a matrix; Solve systems of linear equations using matrices; Use a graphing utility to solve a system of linear equations

TEST Chapter 10

Nine Weeks Exam

Midterm Exams

Geometry Review 5.0 Solve right triangles; Solve applied problems using right triangle trigonometry

5.1 Convert between degrees, minutes, seconds, and decimal forms for angles; Find the arc length of a circle; Convert from degrees to radians; Convert from radians to degrees; Find the area of a sector of a circle; Find the linear speed of an object traveling in circular motion

5.2 Find the exact values of the trigonometric functions using a point on the unit circle; Find the exact values of the trigonometric functions of quadrantal angles; Find the exact values of key trigonometric functions and for certain integral multiples of each; Use a calculator to approximate the values of a trigonometric function

5.3 Determine the domain and range of the trigonometric functions; Determine the period of the trigonometric functions; Determine the signs of the trigonometric functions; Find the values of the trigonometric functions utilizing fundamental identities; Find the exact values of the trigonometric functions of an angle given one of the functions and the quadrant of the angle; Use even-odd properties to find the exact values of the trigonometric functions

TEST Chapter 5 Sections 0 – 3

7.1 Find the value of trigonometric functions of acute angles; Use the complementary angle theorem; Solve right triangles; Solve applied problems using right triangle trigonometry

7.2 The Law of Sines; Solve SAA and ASA triangles; Solve SSA triangles; Solve applied problems using the Law of Sines

7.3 The Law of Cosines; Solve SAS and SSS triangles; Solve applied problems using the Law of Cosines

7.4 Find the area of SAS and SSS triangles

TEST Chapter 7

5.4 Graph transformations of the sine and cosine functions; Determine the amplitude and period of sinusoidal functions; Graph sinusoidal functions; Find an equation for a sinusoidal graph

5.5 Graph transformations of the tangent function and cotangent functions; Graph transformations of the secant and cosecant functions

5.6 Determine the phase shift of sinusoidal functions; Graph sinusoidal functions; Find a sinusoidal function from data

TEST Chapter 5 Sections 4, 5, 6

6.3 Establish trigonometric identities

6.4 Use sum and difference formulas to find exact values; Use sum and difference formulas to establish identities; Use sum and difference formulas involving inverse trigonometric functions

6.5 Use double-angle formulas to find exact values; Use double-angle and half-angle formulas to establish identities; Use half-angle formulas to find exact values

6.6 Express products as sums; Express sums as products

6.7 and 6.8 Solve equations involving a single trigonometric function; Solve trigonometric equations quadratic in form; Solve trigonometric equations using identities; Solve trigonometric equations linear in sine and cosine; Solve trigonometric equations using a graphing utility

TEST Chapter 6 Sections 3 – 8

Nine Weeks Exam

8.1 Plot points using polar coordinates; Convert from polar coordinates to rectangular coordinates; Convert from rectangular coordinates to polar coordinates

8.2 Graph and identify polar equations by converting to rectangular equations; Graph polar equations using a graphing utility; Test polar equations for symmetry; Graph polar equations by plotting points

8.3 Convert a complex number from rectangular from to polar form; Plot points in the complex plane; Find products and quotients of complex numbers in polar form; Use De Moivre’s Theorem; Find complex roots

8.4 Graph vectors; Find a position vector; Add and subtract vectors; Find a scalar product and the magnitude of a vector; Find a unit vector; Find a vector from its direction and magnitude; Work with objects in static equilibrium

8.5 Find the dot product of two vectors; Find the angle between two vectors; Determine whether two vectors are parallel; Determine whether two vectors are orthogonal; Decompose a vector into two orthogonal vectors; Compute work

TEST Chapter 8

11.1 Write the first several terms of a sequence; Write the terms of a sequence defined by a recursive formula; Use summation notation; Find the sum of a sequence by hand and by using a graphing utility; Solve annuity and amortization problems

11.2 Determine if a sequence is arithmetic; Find a formula for an arithmetic sequence; Find the sum of an arithmetic sequence

11.3 Determine if a sequence is geometric; Find a formula for a geometric sequence; Find the sum of a geometric sequence; Find the sum of a geometric series

11.5 Evaluate a binomial coefficient; Expand a binomial

TEST Chapter 11

12.1 Find all the subsets of a set; Find the intersection and union of sets; Find the complement of a set; Count the number of elements in a set

12.2 Solve counting problems using the multiplication principle; Solve counting problems using permutations; Solve counting problems using combinations; Solve counting problems using permutations involving n non-distinct objects

12.3 Construct probability models; Compute probabilities of equally likely outcome; Utilize the addition rule to find probabilities; Utilize the complement rule to find probabilities; Compute probabilities using permutations and combinations

12.E Empirical Rule and Standard Deviation

TEST Chapter 12

9.1 Conics, Circles; Find the equation of a circle; Graph circles; Solve applied problems involving circles

9.2 Find the equation of a parabola; Graph parabolas; Discuss the equation of a parabola; Work with parabolas with vertex at (h,k); Solve applied problems involving parabolas

9.3 Find the equation of an ellipse; Graph ellipses; Discuss the equation of an ellipse; Work with ellipses with center at (h,k); Solve applied problems involving ellipses

9.4 Find the equation of a hyperbola; Graph hyperbolas; Discuss the equation of a hyperbola; Find the asymptotes of a hyperbola; Work with hyperbolas with center at (h,k); Solve applied problems involving hyperbolas

TEST Chapter 9