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UNIT I

EQUATIONS AND INEQUALITIES IN ONE VARIABLE

  • Working with Expressions and Formulas: When you have to solve a problem, you will always have to simplify a numeric expression after you plug in values. To do it in the right way and get an accurate answer you will have to apply the order of operations. The lesson goes from order of operations, formulas, area formulas, all the way to present you the quadratic formula, which is widely used in future sections and involves several steps that to get them in the right way demand that you apply the order of operations in the right way.

  • Understanding Real Numbers: In future sections of this webpage as in any algebra 1 or 2 course you will need to be fully familiar with the number sets. As these are referred in multiple problems within the narrative that explains what you have to do to solve the problem. Also, to be able to work with equations, formulas, and arithmetic of polynomials you will need to apply the following: Number line, number sets, properties of numbers (commutative, associative, distributive...), simplifying expressions. You may trust this lesson that after you have gone through it (diligently) you will have mastered all this concepts to apply them throughout the whole algebra 1 or 2 courses.

  • Solving Equations, Including Absolute Value Equations: Your skeleton is holding in place your muscles that in turn allow you to perform your movements as needed. In algebra, the skeleton is the ability to solve equations; from there everything else is "holding or attached"; that is, once you know how to handle equations the door is open to you for all other concepts. The lesson is a starting with one variable linear equations. To properly work them you will need to understand the properties of equality that in turn enables you to get the solution of one variable linear equations, and absolute value equations. Future lessons will add more in the way of solving more complex type of equations.

  • Solving Inequalities, Including Absolute Value Inequalities: To fully understand compound inequalities you need to visualize how a table of true may be gotten from an "and" or "or" module in a water pipe circuit, or in an electric circuit. This lesson dwells on the first one and takes you step by step with a rich visual environment to illustrate the proces of building the tables.

  • Solving Inequalities, Including Absolute Value Inequalities: You will get the expertise to apply properties of inequality in the solution of one variable linear inequalities and absolute value inequalities. Some of these may be compound inequalities where you may have modules "and", or "or" that combine 2 or more inequalities and the solution set is the values in the domain that are common to all of them. These are very useful in multitude of problems that require that you find an unknown quantity that may get different values in the solution. The lesson concludes with a real world application where you have to setup a set of inequalities to program a group of micro processors to control a package managing facility.

 

 

 

 

 

 

UNIT II

INTRODUCTION TO LINEAR EQUATIONS

  • Understanding Relations and Functions and Linear Equations: In general we say that all functions are relations, but not all relations are functions. The same way that you may say: "All pickup trucks are motorized vehicles, but not all motorized vehicles are pickup trucks." To be able to solve many of the problems further into the algebra course, you need an understanding of the following concepts taught throughout the lesson: Definition of relation and function, representation of relations as mapping, table, ordered set, and graph. State domain and range. Identify a function (vertical line test), discrete vs. continuous function, identification of linear equations, standard form for two variable linear equations.

  • Learning About Slope: Have you heard that a lineal behavior revealed by the graph that represents an experiment has positive correlation, or negative correlation. This refers to whether the regression line that may be obtained has an inclination that falls the right or to the left. Also, to determine if a line is perpendicular to other line you need to find if the product of the slops yields negative one, or not. In the development of this lesson you will learn the slope formula (run vs. rise and two points), and its cases where it may be falling to the right, horizontal, vertical, negative and positive, and to identify conditions for the slope of parallel and perpendicular lines.

  • Working With Systems of Linear Equations: Slope intercept form and point-slope form problems that involve a point and slope, two points, etc. Solving systems of two variable linear equations by substitution, linear combinations and graphing. Introduction to special functions (step function, constant function, identity function and absolute value function). All the items listed above will help in the real world to solve problems where you have what is called a linear patten or linear behavior. There are some special functions that while they are not the typical linear equation they are piecewise functions of linear equations making it up (absolute value function), or a segmented pattern moving in a linear way along the graph (step function), etc. A brief introduction to them is given in the lesson. Now, if you have two variables involved in the process, then you may represent the situation in the coordinate plane, with a system of two linear equations for which you may have one solution, infinite number of solutions, or not solutions at all. The lesson concludes with a real world hypothetical case.
  • Solving Two Variable Linear Inequalities Including Absolute Value Inequalities. When you work with problems that involve two variables and may yield many or an infinite number of solutions, you end up with having to shade areas in the area above, below, right, or left of a line or lines. So the objective of this lesson is to teach you to learn where to shade above or below the line when the line has a slope not parallel to the y-axis. Now, the solution may or may not include the line itself. This is shown by a solid or a broken line.

 

 

 

 

 

 

UNIT III 

SOLVING SYSTEMS OF LINEAR EQUATIONS

  • Using Determinants to Solve Equations In Two Variables and Solving Equations In Three Variables: When you setup equations in problems sometimes you find out that a particular method is not suitable to the equations in question. For example, when you have equations with decimal numbers, sometimes is better to use determinants to find the solution. In the course of the lesson you will go from second order determinant, Cramer's rule, and conclude with systems of three variable linear equations.
  • Solving Systems of Linear Inequalities. Introduction to Linear Programming and Its Applications. People in the production department of factories needs to know with as much accuracy as possible how much material or materials to request from their suppliers, or to determine the optimal combination of materials to get a given batch of a mix. All this is in the realm of linear programming where you have to take into account a set of constraints (conditions to satisfy) that many times translate into linear equations with a straight line as a graph. When several are graphed in the coordinate plane you get a polygon. The vertices of the polygon are the coordinates that are used to try to find the best answer. Some involve the maximum possible value (maximization) others the minimum possible value (minimization). The lesson covers solution of two variable linear inequalities, including absolute value ones. Solution region and feasible region, maximum and minimum of an optimization function and linear programming problems.
  • Using Matrices to Solve Systems of Equations with inverse and Augmented matrix. Working problems that yield two variable linear equations are relatively easy to do with addition and subtraction, or with substitution. Nevertheless, when you are working with three or more linear variables you get systems of equations that may be impractical to try to solve by any of these methods. It is here where using augmented matrix you expedite finding the solution in a more efficient and effective way. The presentation covers addition and subtraction of matrices, multiplication of matrices and inverse of a matrix, to finalize with augmented matrix in the solution of systems of linear equations with three or more variables. Additionally, you will study some applications where coordinates of points in the vertices of polygons are organized in a matrix and subjected to matrix operations to accomplish rotations, translations or reflections of the same.

 

 

 

 

 

 

 

UNIT IV 

MANIPULATING POLYNOMIALS

  • Working with Polynomials: Arithmetic of polynomials is difficult for some students for all the little rules that need to be taken into account. Also, they imply a level of abstraction that for students with no too much exposure to abstract thinking poses a big challenge. This is addressed using manipultives like algebra tiles, and base 10 blocks to represent operations in polynomials. Adding, Subtracting, and Multiplying. Includes modeling with algebra tiles and blocks. F.O.I.L.
  • Dividing Polynomials and Specia Products. What has been learned about adding, subtracting, and multiplying of polynomials will be applied to long division and to special products when multiplying binomials with binomials/trinomials. The lessons makes a quick coverage of expanding and factoring polynomials. Particulaly Difference and Addition of Cubes, Perfect Square Binomials, and Difference of Squares.
  • Learning to Manipulate Radicals and Rational Exponents. Operations with rational exponents sometimes represents too much of an obstacle for students with difficulties working with fractions. If this is you case you may visit the section for basic math in this website to review how to work with fractions. The lesson itself teaches about rational exponents and how they relate to radicals. Roots of real numbers and radical expressions.
  • Solution for Radical Equations and Inequalities. Radical equations involve to determine the domain in which the particular equation will yield a set of solutions that evaluate the equation as true when any is plugged into it. In this lesson you will learn how to solve equations and inequalities with radicals. A review of the inequalities in one variable at unit I of this website would be very helpful before attempting the lesson.
  • Learning About COMPLEX NUMBERS. In the past, many concepts couldn't have been harnessed shouldn't be for the advent of complex numbers with the real and imaginary parts. Fields like electricity and electromagnetism can't be properly understood unless you know how to add, subtract, multiply, and divide complex numbers and represent them in the coordinate plane for complex numbers. In the following lesson you will practice with imaginary roots, simplify expressions with imaginary numbers, simplify complex numbers and expressions. Iterations.

 

 

 

 

 

 

UNIT V

QUADRATIC FUNCTIONS

  • Learning About Quadratic Functions and Its Roots. The approach used here is more in line with an algebra 1 curriculum. It uses the polynomial second degree equation of the parabola in terms of the quadric, and linear coefficients and the constant term. Whereas in the following unit for conics it works with the Standard Equation of the parabola, and the General Equation for Conics. It teaches the solution of quadratic equations by graphing, by graphing features, by factoring, by quadratic formula and by completing the square.

  • Discriminant and Product and Sum of Roots. The Quadratic Formula is very, very, very versatile in terms of getting rid of the problem of having to complete the square, or finding the numbers to do the factoring. In essence it exchanges laboriousness for mastery of the math facts. This in turn is time consuming. So, one simple way to do some predictions for the number of solutions before going through the whole laborious solution is to check the radicand. Depending in whether is less, equal, or more than zero, it reveals how many and type of solutions we get. In working this lesson you will learn how to determine if a quadratic equation has one, two, or none real roots, and how to verify the solution using the product and sum of the roots.

 

 

 

 

 

 

 

 

UNIT VI 

UNDERSTANDING CONIC SECTIONS

  • Working With Distance and Midpoint Formulas. This is a short presentation, but a very important concept. In fact, further in the following lessons you will see that each one of the equations for the conics is obtained by means of using the distance formula and the definition of the geometric space for each of the conics. Lesson that teaches how to use the distance formula, how to find the midpoint with two endpoints, and endpoint of a segment given the midpoint and one endpoint.

  • Parabola: Formula and Graphing. You have the Standard Equation of the parabola, and the General Equation for Conics. You may go from the graph to any of the equations, or from any of the equations to the graph. Learn how to use the vertex and general formulas of horizontal and vertical parabolas, and how to find the equation given the graph. The development of the lesson makes an extensive display of the minor details when you expand the binomials, or when you complete the square to get the actual binomials. These are required depending on whether you are going from the equation to the graph, or from the graph to the equation.
  • Parabola: Deriving Standard Equation Proof. Using the distance formula and the definition for geometric space for a parabola; we obtain the equation in Standard Form for the parabolas. Both vertical and horizontal. You will find very helpful the way the lesson presents the graph with all the relevant parts highlighted to be used in the proof. Then as the proofs moves on you see those changes in the graph. Once the initial equation is obtained, you will see all the intermediate steps in working until the Standard Equation of the parabola centered in the origin is obtained. From there then in a visual and very explicit way you see how the parabola is translated to all the four quadrants using a translation vector with components in "x" and "y", and how it is reflected in both the axis "x" and "y". It finalizes with a summary of the formulas and a video to show how the parameters in the formula affect the shape and orientation of the graph.
  • Circles: Formula and Graphing. For this particular lesson as well as the ones for the ellipse, and hyperbola you would benefit from reviewing in past sections of this website how to expand perfect square binomials, and how to complete the square to get the binomials themselves. Inside it, both processes are work to the detail. Mainly, this lesson is aimed at teaching how to use the standard and general formulas of a circle, and how to use the distance and midpoint formulas to find the equation of a circle.
  • Ellipses: Formula and Graphing. Learn the formal definition of an ellipse, and how to use the general and standard formulas of vertical and horizontal ellipses, and how to determine the equation from the graph. You will be taken from how to graph the equation when the values of the parameters are given, to how to get the parameters when the graph is the one given to you. The lesson takes great pains in showing you the laborious process of expanding binomials, or completing the square to get the binomials themselves depending of whether you are going from the graph to the general equation for conics, or from the general equation for conics to the graph.
  • Ellipse: Deriving Standard Equation Proof. For simplicity the equation is obtained by centering the graph in the origin. Then it is very clearly presented how to translate it to each one of the four quadrants in the coordinate plane by means of a translation vector with components in "x" and "y". This is accomplished using the distance formula and the definition for geometric space for ellipses; we obtain the equation in Standard Form for ellipses. Both vertical and horizontal.
  • Hyperbolas: Formula and Graphing. Equation For Conics: It teaches how to graph the standard and general forms of horizontal and vertical hyperbolas, and how to get the equation of the graph of a hyperbola and its asymptotes using the point-slope formula. It finalizes teaching how to determine from the general equation for conics, which conic a given equation represents.

  • Hyperbola: Deriving Standard Equation Proof. Using the distance formula and the definition for geometric space for a hyperbola; we obtain the equation in Standard Form for hyperbolas. Both vertical and horizontal. The lesson goes through the process of setting up the hyperbola and the parameters to use during the proof properly labeled in the graph. Then it goes to the proof itself. It takes special care in visually highlighting the fact that the equation is first obtained centered in the origin for simplicity of the proof and then how it is possible to get it in any of the four quadrants of the coordinate plane using a translation vector with components in "x" and "y".
  • Solving Systems of Equations and Inequalities That Involve Conics. All the lessons given before presented to you how to work with each one of the individual conics. From the circle all the way to the hyperbola. In this case you will learn how to solve systems that involve circles, parabolas, ellipses and hyperbolas in equations and inequalities. This is accomplished algebraically and graphically verified.

  • Conics Generator Learn how circles, parabolas, ellipses and hyperbolas are generated from transversal sections of a right cone, after this is generated by rotating the generator 360 degrees.

  • General Equation for conics: Videos One of the difficulties that you may experience when learning how to manipulate conics is the laborious process to get each different instance when you change a value in any of the parameters and try to see the effect reflected in the graph. If you don't have a graphing calculator it implies doing the computations, and then plotting the points in graph paper. Even with the calculator, you have to enter the values in the calculator, and most of the ones in the market, while very helpful, still lag in terms of making this “cumbersome-free.” This needs to take place over and over for each change in the parameter that controls the conic equation in question. This is streamlined with the advent of the applets that allow to generate in an almost instantaneous way different instances by playing with sliding rulers, or clicking at check boxes. The changes are reflected on the graph in seconds. A big difference with the described process. Due to technology limitations that don't allow to play the actual applets for you to interact directly with them. You have available for you a set of videos of these applets when they are manipulated to accomplish these same results. Learn about circles, parabolas, ellipses and hyperbolas with those animated applets captured in videos

 

 

 

 

 

 

UNIT VII 

LEARNING ABOUT POLYNOMIAL FUNCTIONS

  • Working With Polynomials To Find Zeros. This lesson guides you through the exploratory process when attempting to find the zeros on a polynomial equation. In absence of this methodologies you might have to spent a lot of time in a non-productive way to find the zeros of a polynomial equation. Of course, a graphing calculator will always take the hassle out of your way, if it is allowed to use it when the evaluation comes. Find out if binomial is a factor of a polynomial by using synthetic division, and find the factors of a polynomial given one factor. Learn to apply the remainder and factor theorems to find zeros, and the Descartes’s rule of signs to find out the number of positive, negative and imaginary zeros. Learn to find the polynomial equation for a given number of zeros.

  • Mathematical Models: An Exponential vs a Polynomial Model. This lesson presents a couple of videos of two different bottles filled up at a constant rate and graphing the height vs the volume and then obtaining competing mathematical models using regression features of a graphing calculator. You may complete the suggested linear regression and compare it with the two given before. The process illustrated in this lesson is the same that takes place to get most of the formulas you use in your different classes in physics or chemistry. Of course, much more complicated in most of the cases. But it gives you an idea of how what your are learning may allow you to accomplish similar results.

  • Mathematical Models: An experiment with a ball and a ramp. We go to a supermarket and see how smooth is the checkout process. We put our items on the integrated conveyor belt of the cashier and the store clerk just scan the items giving us the magic figure of how much we have to pay. For them, it is just a matter of reading the value of the amount to give back to the customer, or even better, they just pull the slip of the credit card and hand it to the customer. Markets were not that tidy. In the past you have a multitude of "moms" yelling to a person who had to be jutting down figures and computing at high speed and to the best of his brain power the totals and then yelling back the quantities, getting the cash and then having to compute the change for each one of the sweeting and desperate moms in front of the counter. Technology, took all the hassle out of the way. This lesson tries to give a glimpse of how the development of a system takes place when technology is available and the part that math takes place in that process. Mathematical Models: A LabPro interface controls a sensor and a calculator to read position vs. time from a ball going up and down a ramp. Data sent to the graphing calculator is used to obtain a quadratic regression model. Detailed instructions are given throughout the experiment.

  • Mathematical Models: Finding the volume of a prism with binomials as width, length, and height. There is a shortcoming in many of the math courses taught in schools. Many times due to time constrains students are taught the concepts in a very effective way, but left short from opportunities to see how they are applied in the real world and thus unlocking for them the economic potential of that knowledge. Students will have the opportunity to go from polynomial arithmetic operations of addition, subtraction, multiplication, and division of polynomials to the geometric concept of finding a volume of a prism that involves those operations.

  • Applying The Rational Zero Theorem to Find The Zeros in a Polynomial Function. When you are given a polynomial equation you may find out the solutions just by taking a graphing calculator with the algebraic module integrated. You enter the polynomial expression as ask for the zeros. The calculator in a "magic" way gives you all of them (most of the time), that was not true when those fancy calculators were not in the market. During those days an effective way of tackling the problem of finding the zeros was applying the Rational Zero theorem that generated a "T" table to perform successive synthetic divisions to look in a systematic and orderly way for the possible zeros in the polynomial equation. In the course of this lesson, that is done by presenting you all the intermediate steps (most textbooks assume you know more that sometimes is true we know, or remember from previous courses). It goes from finding the possible candidates for zeros and then setting up the table and performing the synthetic divisions until the zeros are found.
  • Can Quadratic Techniques Solve Polynomial Equations? There are some polynomial functions that are too "tough" to find their zeros. They are even (highest exponent is even) and higher than second degree. Nevertheless, by a more careful observation it may be determined that their exponents may be expressed as powers of a power where the outermost exponent is 2, effectively reducing the polynomial to a second degree one. This is, they may be reduced to second degree equations and then the Quadratic Formula, completing the square, or factoring using special products may come useful to find the zeros. Learn to use the quadratic equations to solve equations of an even degree.
  • Composition e Inverse In Polynomial Functions. All functions may be deconstructed in a way that you find out how they were "made-up" from other functions. For example you may start with the right side as "x" then you add 1 (adding constant function to identity function = adding 2 functions in the composition), then you raise to the second power this sum (composition of a quadratic function with just quadratic term and the previous composition function) and so on. This process is called composition of functions. Find the composition of a given pair  of functions or a relation. Learn to find the inverse function of a linear equation. Find out the result of the composition of a function and its inverse function.

 

 

 

 

 

 

 

 

 

UNIT VIII 

LEARNING ABOUT RATIONAL EXPRESSIONS 

AND EQUATIONS

  • Solving Rational Equations. Rational equations are interesting because they are made up of fractional algebraic expressions at both sides of the equal sign. This at the same time requires to take some extra precautions that we are not concerned about in other type of equations. They may have values for which the denominator algebraic expression may become zero. In those cases, then the division by zero is not defined, and you end up with asymptotes for those values. Also, once you have gone through the whole solution of the equation, you need necessarily to check the solution for extraneous values. Values that are yielded in the solution, but that don't check as true when you test that solution. This lesson teaches how to simplify rational expressions and how to find the LCD and apply this to find the solution of rational equations.

 

 

 

 

 

 

 

 

 

 

 

UNIT IX 

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

  • Working with Exponential and Logarithmic Functions. Nowadays we have calculators. In the past, they were nonexistent. When they have to work with large numbers. For example distances among the starts in the sky, then it was very difficult to make operations with such numbers. Way too long to handle, with way too many digits. The way they used to tackle the problem was to work with logarithms. By going through this presentation you will learn how to use the properties of rational exponents and radicals to simplify expressions with radical exponents. Learn how to apply the properties of logarithms to solve logarithmic equations and inequalities. Find the logarithms and antilogarithms of numbers and learn about exponential and natural logarithms.

 

 

 

 

 

 

 

 

 

 

 

UNIT

ARITHMETIC AND GEOMETRIC SERIES

  • Working with Arithmetic and Geometric Sequences and Series. If you start with a given number and then you add the same amount to this number and to each one of the following numbers generated this way, then you have an arithmetic sequence, if rather than adding the number you multiply for this number then you have a geometric sequence. In the first case you talk of a common difference, and in the second you talk of a common ratio. Studying the lesson will teach you to find the terms of arithmetic and geometric series and learn to calculate the sum, including infinite geometric series.

 

 

 

 

 

 

 

 

 

 

 

 

UNIT XI 

LEARNING PROBABILITY

  • The Counting Principle.This lesson teaches how to find out the number of outcomes for a set of events happening in succession one after the other. Students may see the outcomes for each digit or letter position.

  • Permutations and Combinations. Suppose you are told to take 4 letters from the English alphabet and 3 digits from the numbers 0 to 9. Then you are told to makeup passwords with them. Would be the same "abc" than "acb", what is the difference? You will learn that if you consider that the order is important then they are two different outcomes, but if you disregard the order in which they are, then they are no distintive outcomes. In this lesson students will learn the difference between a permutation and a combination, and how repeated elements or circular arrangements will affect the number of outcomes.
  • An Introduction to Probability. This lesson presents the formal and informal definitions of probability. Students learn how to find probabilities using a spinner, a dice and a deck of cards. Students also apply probability to problems involving permutations and combinations. If you want to practice what you just learn with real data, then try to work the following lessons below this one. They have real video from dice, cards, and a wheel of fortune with a "T" table next to them to be fill in with the data generated in the trials, that in turn may be used to calculate probabilities.
  • Probability for a spinner wheel. One thing is to be told imagine a spinner with 8 sectors with green, brown, blue, and yellow evenly distributed, that if they present to you the real thing spinning in full shape and color. This lesson took the care to record several trials from a real wheel of fortune and displays a "T" table next to them for you to complete with the corresponding outcomes after each trial, or spin. You will be able to practice how to calculate probabilties with real data.
  • Dependent and Independent Probability. If you are playing with cards any game where you draw one at a time. There is a difference whether or not you return the card. Also, if you are expecting to win with a combination of cards that have common outcomes, it is different that if they don't have any common outcome. The first example relates to independent vs. dependent events, and the second to inclusive vs. mutually exclusive events. Great lesson to learn about dependent and independent probability and exclusive and inclusive events and how they affect the probability of an event. Students will understand when they add probabilities and when they multiply them.
  • Dependent and Independent Probability. For many students probability translates in something somehow abstract and difficult to understand for the fact that it is taught in the context of board games with cards and dice. If by any chance they didn't have experience with those games, then we are talking of lack of what is called background knowledge. In this lesson that is tackled by presenting the real game. Probability experiments applied to dice and cards. Real videos and the opportunity to work real wold application problems directly on the screen giving the option to complete the adjacent "T" tables to calculate probabilities with that data.
  • Experimental and Geometric Probability. Learn in an easy way how to find experimental probability from data gathered from a survey, and how to find probability using the ratio of lengths, areas and volumes. Lesson targeted to be expanded in the near future. Meanwhile with the use of animations and sequence it presents you the ideas in a very easy to follow way.