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Students will learn about solving higher order polynomial equations in this unit. A major part of this topic is the study of complex numbers and what it means if the root of a polynomial includes an imaginary number.
The first lesson focuses on the basics of complex numbers including the definition of the imaginary unit i, how to rewrite square roots with negative radicands as complex numbers with the imaginary unit, and an explanation of how both the complex numbers — written as a+bi — and imaginary numbers fit in the Venn diagram of number sets. The cyclical nature of the powers of i is explored and applications of it are shown as students learn to perform basic operations with complex numbers. As a bonus, students will have an opportunity to see an introduction to graphing complex numbers on a complex coordinate plane.
In an expansion of previous knowledge, the second lesson will teach students how to recognize if a quadratic function has complex roots. Students will learn how to factor an expression that previously would have been considered prime and how to list the roots of a quadratic that previously would have simply been stated as having no real solutions.
After students have a firm grasp of what it means to have complex roots, students will begin to learn about higher order polynomial equations. In the beginning, the methods of solving these types of equations are an extension of what has been seen in previous units such as factoring out a greatest common factor, the Quadratic Formula, the Zero Product Property, special products and differences, and solving by graphing.
The key features of polynomial functions and their graphs are explored as well as how these features can be found by looking at the related polynomial equation. The end behavior of a function is determined by the degree of the polynomial and the sign of the leading coefficient. Whether there are relative or absolute extrema can also be known by looking at the degree of the polynomial as well as by finding the number of turning points. As a result of knowing the end behavior and types of extrema, students can determine whether a function is even, odd, or neither. Additionally, the Location Principle allows students to confidently know the interval in which a zero of the function will lie.
Finally, students will learn the various methods for dividing polynomials. Polynomial long division follows the same basic procedure as numerical long division. A shortcut for polynomial long division, called synthetic division, is demonstrated as well but can only be used for quotients where the divisor is a first degree binomial. The Remainder Theorem and the Factor Theorem are taught as ways to deduce whether the division will produce a remainder and what that remainder might be.
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As a continuation of the previous unit, students will learn more ways of finding the zeros of higher order polynomial equations. This unit will focus on the various theorems that can be used to identify key characteristics about the zeros — also known as the roots — of a polynomial such as the number of zeros that the polynomial has, the possible signs of the zeros, and the types of numbers that the zeros can be.
The first lesson teaches students about the Fundamental Theorem of Algebra and Descartes' Rule of Signs. The Fundamental Theorem of Algebra shows that the number of zeros of a polynomial can be determined by the degree of the polynomial. In the process of practicing this theorem, students will think about the multiplicity of zeros and how it applies in this case. It is then explained how Descartes' Rule of Signs can be used to narrow down the possible signs of the zeros by examining the number of sign changes between the coefficients of the polynomial when it is written in standard form.
After mastering the rules introduced in the first lesson, students will be shown the Rational Root Theorem, the Irrational Conjugate Root Theorem, and the Complex Conjugate Root Theorem. The Rational Root Theorem gives instructions for finding a list of potential rational roots, including integers and fractions, which can then be verified by substituting the values into the polynomial. The Irrational Conjugate Root Theorem and the Complex Conjugate Root Theorem both demonstrate a relationship between certain types of roots that come in pairs. If a polynomial has a radical root, a+b, or a complex root, a+bi, then the conjugate of that root is also a root.
The following lesson presents the Binomial Theorem as a method of expanding the multiplication of a binomial that is raised to an exponent. Pascal's Triangle is offered as a tool that can be used by students to quickly recall the coefficients of the polynomial terms in an expanded binomial.
The final lesson allows students the opportunity to see how polynomial functions can be applied to model diverse real-world scenarios such as a company's profit trends and carbon dioxide levels over time. Digital tools are demonstrated as an available option for analyzing real-world data and making predictions.
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The intention of this unit is to teach students how to work with and use radicals more complicated than those seen in previous courses to simplify expressions, graph and transform functions, and solve equations and inequalities.
The first lesson focuses on performing mathematical operations with radical expressions and the rationalization of denominators. The simplification of radicals can often involve several logical steps; students will learn when absolute value symbols are needed during simplification and a decision tree is provided to assist with this thought process. Various properties — such as the Product Property of Radicals and the Quotient Property of Radicals — are also introduced. After a radical can be fully simplified and rationalized, students will practice determining whether or not radicals are like terms that can be added or subtracted.
The various types of radical functions and inequalities are investigated in the following lessons. In the course of this investigation, students will learn to graph radical functions and inequalities, that the inverse of a radical function is a polynomial with a degree equal to the index of the radical, how the domain and range of radical functions are found, and what the index of the radical says about the domain and range. The opportunity to practice solving radical equations and inequalities both algebraically and by graphing is then given. Emphasis is placed on the importance of checking for extraneous solutions when solving a radical equation; students will know what an extraneous solution is, how it happens, and why it cannot be included in a solution set.
After students have a firm understanding of what a radical function looks like and what it means to be a solution of a radical equation, they will be shown the possible transformations that can be done to radical function. Students will be able to identify horizontal and vertical translations by the location of an added or subtracted constant. Similarly, horizontal and vertical stretches and shrinks can be identified by the location of a coefficient; if the coefficient is a negative number then a reflection may be involved as well.
The final lesson demonstrates the real-world applicability of radical functions. Students will learn about the formula for braking speeds using the coefficient of friction, the metabolic rate of people and animals, the period of a pendulum, and more!
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This unit will teach students how to work with both rational expressions — the ratio of two polynomial expressions — and rational functions — a type of function which contains one or more rational expressions. Working with rational expressions requires a deep understanding of content taught in previous courses, most notably fractions.
The first two lessons focus on simplifying and performing operations with rational expressions. Factoring polynomial expressions is a major part of the simplification process; students will learn when factors create excluded values which affect the domain of the expression and any related equations. Multiplying, dividing, adding, and subtracting rational expressions follow the same rules as operating with fractions. Multiplication and division, including complex fractions, can be performed directly whereas addition and subtraction both require common denominators. Students will practice finding least common denominators with algebraic expressions using least common multiples.
The following lessons discuss the graphs of functions in the rational function family, including the most simplistic form which is called a reciprocal function f(x)=x1. Not only is the process of graphing this type of function demonstrated but the steps for finding the domain, range, asymptotes, oblique asymptotes, and points of discontinuity are explained. Transformations of rational functions are covered as well, such as horizontal and vertical translations, horizontal and vertical stretches and shrinks, and reflections. Students will be able to recognize what the placement and value of a constant or coefficient means in terms of a transformation.
After students have a solid foundation in rational functions, they will learn about rational equations and inequalities. In earlier years, students have studied direct variation equations y=mx. Now new types of variation are introduced — joint variations z=kxy, inverse variations y=xk, and combined variations z=ykx. For more generalized rational equations and inequalities, two main methods of solving rational equations and inequalities are considered; students will be shown how and when to use the Cross Products Property to solve, and when it is more appropriate to solve by finding the least common denominator. All solutions are verified to be certain that they are not extraneous.
The final lesson allows students the opportunity to see rational functions in real-world applications. Some of these situations include wind speed, fuel consumption and efficiency, and chemical concentrations.
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Logarithmic and exponential functions are two of the topics taught in higher level mathematics with the most real-world applications. Students will see several of these applications throughout this unit of the course.
To ensure a solid foundation, students will first learn about the definition of a logarithm and how it relates to exponents — logarithmic equations and exponential equations are inverses. Students will have the opportunity to practice rewriting logarithmic equations as exponential equations and vice versa. The Properties of Logarithms are taught so that complex logarithmic expressions can correctly be simplified and expressions can be evaluated; these properties include the Product Property, the Quotient Property, and the Power Property of Logarithms.
After the fundamentals are understood, students will expand their knowledge to include two important special cases of logarithms — common logarithms, which have a base of 10, and natural logarithms, which have a base of e. The natural base e is explained as being an irrational constant used throughout many topics in mathematics. Students will learn how to recognize and work with these special types of logarithms as well as how to use the Change of Base Formula to rewrite any logarithm with any base as a quotient of common or natural logarithms so that calculations can be more easily completed with a calculator.
In the next lesson, the graphs of exponential and logarithmic functions will be examined. As part of graphing exponential functions, students will explore the difference between exponential growth and exponential decay. These classifications of natural base exponential functions are determined by observing the sign of the coefficient of the exponent x and have applications such as continuously compounded interest and atmospheric pressures.
With a firm grasp of the necessary definitions and properties, students will continue on to solving more complex exponential and logarithmic equations and inequalities. Different methods and strategies are demonstrated for solving each type of equation or inequality depending on if the bases are the same, and if there is a variable on both sides of the equation or inequality.
The final lesson will focus entirely on the numerous applications of exponential and logarithmic functions. Students will delve into carbon dating, half lives, population growth, and more. As a conclusion to the exploration of real-world topics, students will learn how to utilize graphing calculators to find an appropriate exponential model for any real data set using exponential regression tools.
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Students learned about trigonometric ratios in previous courses. In this unit, they will expand that knowledge to see how these ratios can be used to create trigonometric functions. The first lesson teaches key vocabulary about angles in the coordinate plane which have their vertex at the origin and the unit circle. Students will learn to identify the initial and terminal sides of angles in standard position, what it means for angles in standard position to be coterminal, and how to find a reference angle for standard position angles that are not acute. The unit circle is shown to be constructed from a series of right triangles such that the hypotenuse has a radius of 1 and extends starting from the origin. Trigonometric ratios are used to find the coordinates of the hypotenuse's second endpoint; these points then make up the points that lie on the unit circle.
The next lesson also focuses heavily on vocabulary, this time regarding periodic functions. Understanding the basics of generic periodic functions is essential to understanding trigonometric functions. Students will master how to measure the period of a periodic function, which is the length of a single cycle or the shortest repeating portion of the graph, and how to calculate the amplitude, or total height, and midline of the function.
The following lessons educate students on the various types of graphs of trigonometric functions and how they can be transformed. Drawing graphs of the untransformed sine, cosine, and tangent functions using the unit circle as a method of finding the points along the curve on the coordinate plane is taught. Students will be able to determine the period and amplitude of the functions by examining their equations. This is then broadened to include finding the period and amplitude of functions that have been stretched or shrunk. Students will also see horizontal translations, vertical translations, and reflections of the trigonometric functions.
In addition to learning about the graphs and transformations of the common sine, cosine, and tangent functions, students will be shown the reciprocal trigonometric functions — secant, cosecant, and cotangent. Typically a more challenging topic, the amplitude, period, domain, and range of these functions are thoroughly explained using examples and visuals.
Finally, students will explore several real-world applications of trigonometric and periodic functions such as average wind speed, resting blood pressure, heartbeats measured by electrocardiograms, and sound waves. Real-world examples help students understand the significance of a topic and encourage further research and sustained interest.
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Trigonometric Identities are equations which are always true and that demonstrate relationships between trigonometric functions and ratios. In this unit, students will explore, practice using, and prove the various types of identities for trigonometric functions.
In the first lesson, students are introduced to the most basic of these identities — the Tangent and Cotangent Identities, the Reciprocal Identities, the Pythagorean Identities, the Cofunction Identities, and the Negative Angle Identities. All of these rules can help students rewrite trigonometric functions and ratios as related trigonometric functions or ratios for a variety of reasons such as finding exact values using a unit circle or simplifying an expression without using a calculator. These rules are also used to verify if a trigonometric equation is an identity — if it is always true.
The next two lessons focus on the Sum and Difference Angle Identities and the Double and Half-Angle Identities. Both types of identities are used when a trigonometric function involves an angle which is not one of the special angles that have commonly memorized measures but that can be rewritten in one of the following ways.
Finally, students are presented with the concept of inverse trigonometric functions; arccosine, arcsine, and arctangent being the inverse functions of cosine, sine, tangent. They are then given the opportunity to use all of these various identities and the inverse functions to solve trigonometric equations.
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This unit will introduce students to the concept of arithmetic and geometric series, which are directly related to what they have learned in previous courses about arithmetic and geometric sequences. In a series, the terms of the sequence are added or subtracted rather than simply listed in order.
In the first lesson, students will come to understand the difference between finite and infinite series and the various pieces of information that can be found by reading a series written in sigma notation. Several opportunities are given to practice rewriting series in sigma notation and vice versa. The formulas for sums of special series – such as the sum of first n positive integers and the sum of the squares of the first n positive integers – are also shown. As a brief history lesson, students will learn about Carl Friedrich Gauss and the Gauss Summation.
The following lessons focus on methods of finding the sums of series. Students will take what they have been taught about sigma notation, common differences, and common ratios to apply the formulas for the sums of finite arithmetic series, finite geometric series, and convergent infinite geometric series; the convergence or divergence of an infinite geometric series is determined by inspecting its common ratio. In later courses, students will learn about the concept of limits. The thought process required to ascertain the number to which a convergent series will converge will help with this future concept.
Throughout the unit, real-world situations that involve series are presented. Some examples of these situations include calculating how much will be paid in total on a loan, the spread of a virus, and the bounce height of a basketball over time.
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Statistics is a subcategory of mathematics which has an extremely high level of real-world applicability. This unit is about how data is gathered and analyzed, as well as how data from different sources can be standardized and compared.
The first lesson focuses on how statistical studies are designed. Students will learn to recognize the difference between a population and a sample, and how to measure them both — measurable characteristics of populations being parameters and of samples being statistics. This lesson will also cover the variety of ways hypotheses can be tested using simulations and statistical studies including experiments, observational studies, and surveys. Potential bias and ways to prevent bias in these testing methods is discussed.
The next lesson expands upon what students have learned in previous courses about the most common types of data distributions, which are shown in the form of frequency distributions and box-and-whisker plots. Students will learn what it means for a distribution to be symmetric, skewed-left, skewed-right, bimodal, or uniform.
With a firm grasp on types of distributions, the following lessons introduce students to probability distributions, with an in depth focus on binomial distributions and normal distributions. Students will understand that probability distributions give the probability of individual outcomes in a sample space while binomial distributions are probability distributions in which the individual data points represent the number of successes out of a defined number of binomial trials in a binomial experiment. The Law of Large Numbers, which dictates that over time the experimental probability of a conducted experiment or simulation will tend towards its theoretical probability, is presented as an explanation of why experimental probabilities and probability distributions are reliable.
Special attention is given to normal distributions. Normal distributions are a type of probability distribution where the mean, median, and mode are all equal to one another; their unique bell-shaped curve follows the Empirical Rule which prescribes what percentages of data fall within a certain number of standard deviations from the mean which allows for probabilities and percentiles to be quickly found for any data point. Standardized normal distributions make it so that any two normally distributed data sets can be directly compared. In the process of learning how to standardize a distribution using z-scores, students will come to understand the significance of normal distributions and the extensive number of use cases for this type of distribution.
In the final lesson, students are taught about confidence intervals, hypothesis testing, and statistical significance. Confidence intervals are introduced as a probable range, or maximum error of estimate, within which a population parameter will lie based on the sample statistics. Students will relate the confidence interval to the standard normal curve by realizing that the confidence level is equal to the area under the standard normal curve around the mean and whose cut off points are given by critical values. A hypothesis test is a test made using sample data to either reject or fail to reject a claim made about a population parameter. Once again, students will use the standard normal curve to set significance levels and appoint critical regions for rejecting a hypothesis.
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