Math 261 (Summer 2015)

Brendan Dilloughery

Lecture Notes

... or separate links can be found below to each video in the "class videos & lecture notes by section" portion of the page

Class Videos

... or separate links can be found below to each video in the "class videos & lecture notes by section" portion of the page

Homework Assignments

The exact list of problems to be completed in each section can be found at this link:

Homework Assignments for Blitzer's Introductory Algebra (6th edition)

__Due Date --- Homework Number --- Sections Due__

Wed. June 17, 2015 --- Homework #1 --- Sections 1.1-1.7

Mon. June 22, 2015 --- Homework #2 --- Sections 1.8, 2.1-2.4

Wed. June 24, 2015 --- Homework #3 --- Sections 2.5-2.7

Mon. June 29, 2015 --- Homework #4 --- Sections 3.1-3.3

Wed. July 01, 2015 --- Homework #5 --- Sections 3.4-3.5, 4.1

Mon. July 06, 2015 --- Homework #6 --- Sections 4.2-4.4

Wed. July 08, 2015 --- Homework #7 --- Sections 5.1-5.3

Mon. July 13, 2015 --- Homework #8 --- Sections 5.5, 6.1-6.3

Wed. July 15, 2015 --- Homework #9 --- Sections 6.4-6.6, 7.1

Mon. July 20, 2015 --- Homework #10 --- Sections 7.2-7.4, 7.6

Wed. July 22, 2015 --- Homework #11 --- Sections 7.7, 8.1

Mon. July 27, 2015 --- Homework #12 --- Sections 8.2-8.3, 9.1-9.2

Wed. July 29, 2015 --- Homework #13 --- Section 9.3

Course Materials Organized by Section

This course uses the textbook "Introductory Algebra" by Robert Blitzer (6th Edition)**Class 01 - Entire First Class Video**. This covers introductions, a course overview, and sections 1.1 and 1.2. We begin to discuss section 1.3 but I have included a separate video below.

__Chapter 1 (Introduction to Algebra)__

**Section 1.1**-**Introduction to Algebra**-__[Video]__**[Notes]**- Khan Links**[#1] [#2] [#3]**

Focuses on evaluation expressions by plugging in a number in place of a variable and correctly applying the order of operations.**Section 1.2**-**Fractions**-__[Video]__**[Notes]**- Khan Links__[#1]__

Focuses on the four main operations with fractions (addition, subtraction, multiplication and division) as well as how to deal with improper fractions and mixed numbers.**Section 1.3**-**Real Numbers****[Video]**-**[Notes]**- Khan Links__[#1]____[#2]____[#3]____[#4]____[#5]____[#6]__

Categories of Real Numbers (e.g. rational, natural, whole, integers, etc), comparing real numbers with inequalities (greater than, less than or equal to, etc), and evaluating absolute values**Section 1.4**-**Basic Rules of Algebra****[Video]**-**[Notes]**- Khan Links__[#1]____[#2]____[#3]__

Basic terminology of algebra (terms, coefficients, like terms, etc) and how to simplify expressions by combining like terms**Section 1.5**-**Addition of Real Numbers****[Video]**-- Khan Links__[Notes]____[#1]__

How to add positive and negative numbers as well as what negative numbers mean**Section 1.6**-**Subtraction of Real Numbers****[Video]**-- Khan Links__[Notes]____[#1]__

How to subtract positive and negative numbers**Section 1.7**-**Multiplication & Division of Real Numbers****[Video]**-**[Notes]**- Khan Links__[#1]__

How to multiply and divide positive and negative numbers**Section 1.8**-**Exponents and Order of Operations****[Video]**-**[Notes]**- Khan Links__[#1]__

There was no lecture, but instead a quick 5 question quiz where we go over the five problems together on Order of Operations.

__Chapter 2 - Solving Linear Equations with One Variable__

**Section 2.1**-**Addition Property of Equality****[Video]**-**[Notes]**- Khan Links__[#1]__

Solving linear equations by adding (or subtracting) something to both sides of the equation.**Section 2.2**-**Multiplication Property of Equality****[Video]**-**[Notes]**- Khan Links__[#1]__

Solving linear equations by multiplying (or dividing) something to both sides of the equation ... and then combining this with section 2.1 to solve multiple step problems.**Section 2.3**-**Solving Linear Equations****[Video]**-**[Notes]**- Khan Links__[#1]__

Arguably the single most central section in algebra. This section is all about how to solve general linear equations with one variable such as "2(x+3)-5=6x-7"**Section 2.4**-**Formulas****[Video]**-**[Notes]**- Khan Links__[#1]__

How to solve a formula such as "y=mx+b" for a specific variable such as "solve y=mx+b for x"**Section 2.5**-**Introduction to Problem Solving (word problems)****[Video]**-**[Notes]**- Khan Links__[#1]____[#2]____[#3]____[#4]__

How to translate sentences into math equations such as "five more than a number is eight" into "x+5=8"**Section 2.6**-**Problem solving in geometry****[Video]**-**[Notes]**- Khan Links__[#1]__

Using the area and perimeter formulas of rectangles, triangles and circles to solve various problems.**Section 2.7**-**Solving Linear Inequalities****[Video]**-**[Notes]**- Khan Links__[#1]__

Solving inequalities such as "2x+5>6" and then representing these as an inequality, a graph on a number line, and in interval notation.

__Chapter 3 (Graphing Linear Equations with Two Variables)__

**Section 3.1**-**Introduction to Graphing****[Video]**-**[Notes]**- Khan Links__[#1]__

How to graph linear equations with two variables using points and tables of values, such as "graph the line 2x+4y=8". Really focuses on what it means to have an equation with two variables in it as well as what a solution to an equation means.**Section 3.2**-**X- and Y-Intercepts****[Video]**-**[Notes]**- Khan Links__[#1]__

Focuses on x- and y-intercepts. We find the intercepts from graphs first, then find them from the equations of lines, then combine these skills to find the intercepts from an equation and use these points to graph the line.**Section 3.3**-**Slope****[Video]**-**[Notes]**- Khan Links__[#1]__

Focuses on "slope". We discuss slope in a big picture way as steepness of a line and positive vs negative slopes. We then find the slope of lines from graphs given and follow this by finding the slopes between two points. Also introduced are the ideas of lines being parallel or perpendicular.**Section 3.4**-**Slope-Intercept Form (y=mx+b)****[Video]**-**[Notes]**- Khan Links__[#1]__

This section focuses on the equation "y=mx+b" and its significance. We practice finding the slope and y-intercept of lines when given the equation in this format. We then practice rearranging equations to be in this form to get the slope and y-intercept. We finally put these together and use our slope and y-intercept to graph the line.**Section 3.5**-**Point-Slope For****m**(y-y1=m[x-x1])**[Video]**-**[Notes]**- Khan Links__[#1]__

This section focuses on how to write the equation of a line if you are given some information about the line. We discuss how you could only ever really be given two points or a point and a slope. We then practice using this information to write the equation of lines.

__Chapter 4__

**Section 4.1**-**Introduction to Systems of Linear Equations & Solving by Graphing****[Video]**-**[Notes]**- Khan Links__[#1]____[#2]__

Focuses on the basic idea of a system of equations, such as what a solution looks like and why. It also introduces the viewpoint of a system of equations as two lines on a graph and uses this as our first method to solve the system of equations.**Section 4.2**-**Solving Systems of Equations by Substitution****[Video]**- [Notes] - Khan Links__[#1]__

Focuses on solving the same types of problems as section 4.1 but introduces a second method to solve them. The idea here is that you will solve for a variable, then substitute in this equal value in place of that variable to solve the system.**Section 4.3**-**Solving Systems of Equations by Elimination/Addition****[Video]**- [Notes] - Khan Links__[#1]____[#2]__

Focuses on solving the same types of problems as sections 4.1-4.2 but introduces a third method to solve them. The idea here is that you add the two equations together after cleverly rearranging them so that one of the variables cancels out.**Section 4.4**-**Word Problems Involving Systems of Equations****[Video]**- [Notes] - Khan Links__[#1]__

The problems in this section are the same as in sections 4.1-4.3, but the new part is reading a word problem and turning it into a system of equations.

__Chapter 5__

**Section 5.1**-**Adding and Subtracting Polynomials****[Video]**-**[Notes]**- Khan Links__[#1]____[#2]__

Focuses on combining like terms and remembering to distribute the negative sign. Otherwise the section is just addition and subtraction of individual terms.**Section 5.2**-**Multiplying Polynomials****[Video (Part 1)]**--__[Video (Part 2)]__**[Notes]**- Khan Links__[#1]____[#2]____[#3]__

Focuses on the multiplication of monomials and polynomials as well as exponents. For example, you would see problems like (2x)(7x) and (x+2)(5x-4).**Section 5.3**-**Special Products**-__[Video]__**[Notes]**- Khan Links__[#1]__

Focuses on identifying two common forms that products come in. Specifically it focuses on the difference of squares "(a+b)(a-b)" and a perfect square of binomials (a+b) squared.**Section 5.5**- Dividing polynomials by monomials (getting ready to take out the GCF)-__[Video]__**[Notes]**- Khan Links__[#1]____[#2]__

Focuses on ...

Chapter 6

**Section 6.1**-**Greatest Common Factor and Factor by Grouping**-__[Video]__**[Notes]**- Khan Links__[#1]____[#2]__**[#3]**__[#4]____[#5]____[#6]__

Focuses on finding the greatest common factor of a polynomial. This is always the first step in the factoring process. Once we have figured out the greatest common factor, we factor it out of the polynomial. The second half of this section focuses on the method of "factoring by grouping" which is our general plan to factor any polynomial with four or more terms. This method splits a polynomial up into smaller groups that share common factors. After you take the common factors out of the smaller groups, the remaining binomial is the same in the two groups and can be factored out.**Section 6.2**-**Factoring Trinomials with a Lead Coefficient of 1 ("X-Factor")**-__[Video]__**[Notes]**- Khan Links__[#1]__

Focuses on learning to factor the most common types of polynomials you will see: x^2+5x-6**Section 6.3**-**Factoring Trinomials without a Lead Coefficient of 1 ("Hokey Pokey")**-__[Video]__**[Notes]**- Khan Links- Description of Process__[#1]____[#1]__

Focuses on factoring more challenging trinomials (those where the lead coefficient is not 1). There are many strategies to factor these. Most books suggest a form of guess and check. In this lesson I focus on a method I call the "Hokey Pokey". This method is not common, but I find it more useful than the guess and check method (and more enjoyable and elegant as well).**Section 6.4**-**Factoring Special Forms (Difference of Squares)**-__[Video]__**[Notes]**- Khan Links__[#1]____[#2]__

Focuses on recognizing a few special/common forms. Specifically, you need to be able to recognize a difference of squares. The key hint is that when faced with a factoring problem, if there are only two terms this is your only plan of action (so far).**Section 6.5**-**General Factoring Strategy**-__[Video]__**[Notes]**- Khan Links__[#1]____[#2]__

Focuses on an overall strategy to factor mixed problems. For the remainder of this class you will encounter polynomials in most every lesson, and you will need to decide which factoring method ot use to solve the problem.**Section 6.6**-**Solving Quadratic Equations by Factoring**-__[Video]__**[Notes]**- Khan Links__[#1]__

Focuses on the zero product rule and using this to solve equations by factoring. The basic idea is that you cannot isolate the "x" like you could with a linear equation (chapter 2). Instead you are stuck with this polynomial with so many terms added together. If you factor the polynomial into a product though and the product is equal to zero ... well then you know one of the things being multiplied together MUST be zero. We use this to solve for the values of "x".

Chapter 7

**Section 7.1**-**Simplifying Polynomial Fractions**-__[Video]__**[Notes]**- Khan Links__[#1]__

Focuses on how to reduce/simplify a fraction when there are polynomials involved. The main idea is that you can never cancel one term of a polynomial from the top of a fraction with one term of a polynomial on the bottom of a fraction EVER. Instead, you factor the polynomials in the numerator (top) and denominator (bottom) completely. This changes something with many terms into one big product of factors. Now since you have a product (one big term), you can cancel any factor from the top with any identical factor from the bottom.**Section 7.2**-**Multiplying & Dividing Polynomial Fractions**-__[Video]__**[Notes]**- Khan Links__[#1]__

Focuses on how to multiply and divide fractions that contain polynomials. There is not much new from 7.1 and should be thought of as a continued practice. The idea is when multiplying two fractions, since you are just multiplying the tops together and the bottoms together, you can just treat it as a single fraction and reduce anything from the top with anything from the bottom. As far as division goes, the only new step is that you first need to flip the second fraction and change the sign to multiplication. Other than this single step, you are really just simplifying fractions like in section 7.1.**Section 7.3**-**Adding & Subtracting Polynomial Fractions (with the same denominator)**-__[Video]__**[Notes]**- Khan Links__[#1]__

Focuses on how to add and subtract polynomial fractions that already have the same denominator (or denominators that might be off by a negative sign). Really we just add them like normal fractions with numbers in them. We add the numerators (tops) together and keep the common denominator (bottom) the same. Then you simplify like in section 7.1.**Section 7.4**-**Adding & Subtracting of Polynomial Fractions (with different denominators)**-__[Video]__- Khan Links__[__Notes]__[#1]__

Focuses on how to add/subtract two fractions with different denominators when there are polynomials in the numerators and denominators. The big idea of this section is how to find the least common denominator (LCD) and then how to modify each fraction to have this denominator. It is not different from fractions with numbers, as you follow the same steps, the the difficulty is in performing those same steps with polynomials.**Section 7.6**-**Solving Rational Equations (Equations with Polynomial Fractions)**-__[Video]__**[Notes]**- Khan Links__[#1]__

Focuses on how to solve equations that involve fractions with polynomials in them. The strategy is the same as in chapter two when we had equations with numerical fractions. The big trick is to multiply the entire equations by the LCD of all of the fractions. This immediately cancels out all of the fractions and gives you a normal fraction-less equation. Then we use one of our two methods to solve the equation, either the classic method from chapter two for linear equations or our new method from section 6.6 to solve all non-linear (polynomial) equations.**Section 7.7**-**Applications (Word Problems)**

[Video] -**[Notes]**- Khan Links__[#1]__

Focuses on how to solve rate, time and distance word problems that involve polynomial fractions. There are also a few problems involving similar triangles.

Chapter 8

**Section 8.1**- Roots

[Video] -**[Notes]**- Khan Links__[#1]__

Focuses on ... (description coming soon)**Section 8.2**- Multiplication and Division of Radicals

[Video] -**[Notes]**- Khan Links (Videos ONLY)__[#1]____[#2]__

Focuses on ... (description coming soon)**Section 8.3**- Operations with Radicals

[Video] -**[Notes]**- Khan Links__[#1]__

Focuses on ... (description coming soon)**Section 8.4**- Rationalizing Denominators

[Video] -**[Notes]**- Khan Links (Video ONLY)__[#1]__

Focuses on ... (description coming soon)**Section 8.5**- Radical Equations

[Video] -**[Notes]**- Khan Links__[#1]__

Focuses on ... (description coming soon)

Chapter 9

**Section 9.1**- Solving Quadratic Equations by the Square Root Property

[Video] -**[Notes]**- Khan Links__[#1]__

Focuses on ... (description coming soon)**Section 9.2**- Solving Quadratic Equations by Completing the Square

[Video] -**[Notes]**- Khan Links__[#1]__

Focuses on ... (description coming soon)**Section 9.3**- The Quadratic Formula

[Video] -**[Notes]**- Khan Links__[#1]__

Focuses on ... (description coming soon)

Course Calendar

Listed below are the book sections assigned each class period. ** Homework is due every Monday and Wednesday**. Instructions on how to complete homework assignments, as well as the grading policy can be found in the class syllabus. The exact problems to complete in each section can be found above in the "Homework Assignments ..." link.

__1st Class (Monday June 15, 2015)__

**General introductions****Section 1.1**- Introduction to Algebra: Variables & Mathematical Models

**Section 1.2**- Fractions in Algebra

**Section 1.3**- The Real Numbers

**2nd Class (Tuesday ****June 16****, 2015)**

**Section 1.3**(continued)**Section 1.4**- Basic Rules of Algebra**Section 1.5**- Addition of Real Numbers

**Section 1.6**- Subtraction of Real Numbers

**Section 1.7**- Multiplication and Division of Real Numbers

**3rd Class** (Wednesday June 17__, 2015)__ --- First Homework Collected (Sections 1.1-1.7)

**Section 1.8**- Exponents and Order of Operations**Section 2.1**- Solving One Step Linear Equations with Addition and Subtraction (a.k.a. The Addition Property of Equality)**Section 2.2**- Solving One Step Linear Equations with Multiplication and Division (a.k.a. The Multiplication Property of Equality)

- This is the last week to drop the class without receiving a permanent "W" mark on your transcript, as would happen if you drop the class at a later date.

**4th Class (Thursday ****June 18****, 2015)**

**Section 2.3**- Solving Multiple Step Linear Equations (a.k.a. Solving Linear Equations)**Section 2.4**- Formulas and Percents

**Section 2.5**- An Introduction to Problem Solving (a.k.a. Word Problems)- This is the last week to drop the class without receiving a permanent "W" mark on your transcript, as would happen if you drop the class at a later date.

**5th Class** (Monday **June 22**__, 2015)__ --- Second Homework Collected (Sections 1.8, 2.1-2.5)

**Section 2.6**- Problem Solving in Geometry

**Section 2.7**- Solving Linear Inequalities

- Short Review for Test #1 (Wednesday)

**6th Class (Tuesday June 23****, 2015)**

**Sections 3.1**- Graphing Linear Equations in Two Variables ()__not on test #1__- Review for Test #1 (tommorrow)

**7th Class** (Wednesday June 24__, 2015)__ --- Third Homework Collected (Sections 2.5-2.7)

**Test #1**- Covering all of Chapter 1 and Chapter 2. This is the first of four tests (not including the final which is the fifth overall test). You may use a scientific calculator on the test, but not a graphing calculator or cell phone. The test will be for the first 90 minutes of class and then after this we will have a lesson for the remaining 80 minutes of class.**Section 3.1**(continued)**Section 3.2**- Lines and Intercepts

**8th Class (Thursday ****June 25****, 2015)**

**Section 3.2**(continued)**Section 3.3**- Slope

**9th Class** (Monday **June 29**__, 2015)__ --- Fourth Homework Collected (Section 3.1-3.3)

**Section 3.4**- Slope-Intercept Form

**Section 3.5**- Point-Slope Form

__10th Class (Tuesday June 30, 2015)__

- Section 3.5 (continued if necessary)
**Section 4.1**- Solving Systems of Linear Equations by Graphing

**Section 4.2**- Solving Systems of Linear Equations by Substitution

**11th Class** (Wednesday July 1__, 2015)__ --- Fifth Homework Collected (Sections 3.4-3.5, 4.1)

**Section 4.3**- Solving Systems of Linear Equations by Addition (aka Elimination)**Section 4.4**- Problem Solving Using Systems of Equations (aka Word Problems)

**12th Class (Thursday ****July 2****, 2015)**

**Section 5.1**- Adding and Subtracting Polynomials (not on test #2)

**Section 5.2 (Part 1)**- Multiplying Polynomials (Part 1 - only multiplying monomials by polynomials)

**13th Class** (Monday July 6__, 2015)__ --- Sixth Homework Collected (Sections 4.2-4.4)

**Section 5.2 (Part 2)**- Multiplying Polynomials (Part 2 - multiplying polynomials by polynomials)

**Section 5.3**- Special products - For example, perfect squares or difference of squares

(covering chapters 3 and 4)**Test #2 Review**- Important note - the most recent practice test(s) do not include sections 4.2-4.4. Be sure to practice problems from these sections before the test. These problems are on older versions of the test.

**Important Note**- This is the last week to drop this course with a "W". After this week, you will earn a letter grade whether or not you drop the course.

__14th Class____ (Tuesday ____July 7____, 2015)__

**Test #2**- Covering Chapter 3 and Chapter 4

**Section 5.5**- Dividing polynomials by monomials

**Important Note**- This is the last week to drop this course with a "W". After this week, you will earn a letter grade whether or not you drop the course.

__15th Class (Wednesday July 8____, 2015)__ **--- Seventh Homework Collected (Sections 5.1-5.3)**

**Section 6.1**- Factoring out the Greatest Common Factor and Factor by groupings

**Important Note**- This is the last week to drop this course with a "W". After this week, you will earn a letter grade whether or not you drop the course.

**16th Cla ss (Thursday July 9**

**, 2015)**

**Section 6.2**- Factoring trinomials with a leading coefficient of 1

**Section 6.3**- Factoring trinomials that do not have a leading coefficient of 1 (a.k.a. "The Hokey Pokey" or the "AC Method")- Khan Academy Practice ONLY - Practice factoring these trinomials
- This one is not on Khan Academy or in the book but my notes can be found above. Here is a link to a website with the same method: Hokey Pokey/AC Method

**Important Note**- This is the last week to drop this course with a "W". After this week, you will earn a letter grade whether or not you drop the course.

__17th Class (Monday July 13____, 2015)__ **--- Eighth Homework Collected (Sections 5.5, 6.1-6.3)**

**Section 6.4**- Factoring special forms (perfect square trinomials and the difference of squares)

**Section 6.5**- General factoring strategy

__18th Class____ (Tuesday July 14, 2015)__

**Section 6.6**- Solving quadratic equations by factoring

**Section 7.1**- Simplifying polynomial fractions

**19th Class (****Wednesday July 15**__, 2015)__ **--- Ninth Homework Collected (Sections 6.4-6.6, 7.1)**

**Section 7.2**- Multiplying and dividing polynomial fractions

**Section 7.3**- Adding and subtracting polynomial fractions with the same denominator

**20th Class** (Thursday **July 16****, 2015)**

**Section 7.4**- Adding and subtracting of polynomial fractions with different denominators

**Section 7.6**- Solving Rational Equations (a.k.a. equations with polynomial fractions)

__21st Class (Monday July 20____, 2015)__ **--- Tenth Homework Collected (Sections ****7.2-7.4, 7.6)**

**Section 7.7**- Applications (a.k.a. word problems with equations involving polynomial fractions)

**Test #3 Review**- Covering Chapters 5, 6, and 7

__22nd Class____ (Tuesday __**July 21****, 2015)**

**Test #3**- Covering Chapter 5, Chapter 6 and Chapter 7 (more information below)

**Section 8.1**- Roots

__23rd Class (Wednesday July 22____, 2015)__ **--- Eleventh Homework Collected (Sections 7.7, 8.1)**

**Section 8.2**- Multiplication and Division of Radicals

**Section 8.3**- Operations with Radicals

**24th Class** (Thursday July 23, 2015)

**Section 9.1**- Solving Quadratic Equations by the Square Root Property

**Section 9.2**- Solving Quadratic Equations by Completing the Square

__25th Class (Monday July 27, 2015)__ **--- Twelfth Homework Collected (Sections 8.2-8.3, 9.1-9.2)**

**Section 9.3**- The Quadratic Formula

- Test #4 Review

**26th Class** (Tuesday July 28, 2015)

**Test #4**- Covering Chapter 8 and chapter 9 (__not including sections 8.4 or 8.5__)

- Final Exam Review

__27th Class (Wednesday July 29, 2015)__ **--- Thirteenth Homework Collected (Section 9.3)**

- Test #4 returned to you
- Final Exam Review

**28th Class** (Thursday July 30, 2015)

**Final Exam**

__Sections that had to be cut out due to poor time management:__

**Section 8.4**- Rationalizing Denominators

Test Corrections

Corrections are an important part of the test-taking process. In math, like in most subjects, we learn from making mistakes. Without looking back at our mistakes, we are doomed to repeat them. All of the material from the previous tests will show up again, either to be used in class to build upon or solely on the final exam. It is vital that you review each of the places where you erred, so that the next time you work on a similar problem you won’t make the same mistake!