Topic Name Description
Course Syllabus Course Syllabus
1.1: Commutative Law of Addition and Multiplication Commutative and Associative Properties

Read this section and do the problems to practice using the commutative property. Complete the practice questions and check your answers.

Watch this video for more explanation and examples of the commutative law of addition.

Commutative Law of Multiplication

Watch this video for more explanation and examples of the commutative law of multiplication.

1.2: Associative Law of Addition and Multiplication Commutative and Associative Properties

Read this section on the communitive and associative properties. Complete the practice questions and check your answers.

Watch this video for more examples of the associative law of addition.

Associative Law of Multiplication
Watch this video for more examples of the associative law of multiplication.

Take this quiz to practice your understanding of the associative and commutative laws.

1.3: Identity Property of Addition Identity Property of Zero

Watch this video. You should be able to recognize an equation that shows the identity property of addition.

Read this section on the properties of identity, inverses, and zero. Complete the practice questions and check your answers.

Watch this video for more examples of the inverse property of addition.

1.5: Identity Property of Multiplication Identity Property of 1

Watch this video for more examples of the identity property of multiplication.

1.6: Inverse Property of Multiplication Inverse Property of Multiplication

Read this section to see examples of how to apply the inverse property of addition. Focus on the examples in the boxes. Note that in the first multiplication example box, we can use the inverse property for fractions as well as whole numbers.

Inverse Property of Multiplication Examples

Watch this video for more examples of the inverse property of multiplication.

1.7: Multiplication by Zero Multiplication by Zero

Read this text to see this definition written formally. Complete the practice questions and check your answers.

1.8: Dividing by Zero Is Undefined Why Dividing by Zero Is Undefined

Watch this video and then summarize the argument in your own words so you understand and can explain why it is impossible to divide a number by zero. What problems emerge if we assume dividing by zero is possible? What happens when we divide $-1$ by smaller and smaller numbers, approaching zero? What inconsistency occurs?

1.9: Distributive Property Using the Distributive Property
Read this text, which explains why the distributive property works. Pay close attention to the worked examples in Sample Set A, which show how to use the distributive property. Do the examples in Practice Set A. Complete the practice questions and check your answers.
The Distributive Law of Multiplication

Watch these videos for more examples of the distributive law of multiplication.

2.1: Greatest Common Factor Greatest Common Divisor

Watch this video for examples showing how to determine GCF for a set of numbers. Note that sometimes GCF is called the greatest common divisor.

The Greatest Common Factor

Read this text. Pay close attention to the section, "A Method for Determining the Greatest Common Factor". Complete the practice questions and check your answers.

2.2: Least Common Multiple The Least Common Multiple

Read this text for a description of multiples and the method to determine the least common multiple of a given set of numbers. Complete the practice questions and check your answers.

Least Common Multiple Examples

Watch this video for more examples of determining the least common multiple for a set of three numbers.

LCM and GCF Word Problems

Finally, watch this video to see applied examples of using the least common multiple and greatest common factor.

3.1: Negative Numbers Introduction to Integers

Read this text, which explains the concept of negative numbers using number lines. Pay close attention to the section "Opposite Notation", as this notation will come up frequently in the next few sections of this course. Complete the practice questions and check your answers.

Negative Numbers Introduction

Watch this video for more examples of negative numbers using a number line as a reference.

Watch this video for examples of how to add negative numbers with each other and how to add them with positive numbers. Note that when we add a negative number, we can rewrite it as subtracting the positive number. That is: $5+(-2)=5-2$.

Complete this assessment to practice adding negative numbers and check your answers. If you need more practice, do Problem Sets 2 and 3.

Subtracting Negative Numbers

You also need to be able to subtract negative numbers comfortably. If you consider negative numbers as debts, then you remove a debt when you subtract a negative number. Removing a debt of 4 dollars is like being given 4 dollars. This explains why the equation $5-(-4)=5+4$ holds: subtracting negative four is the same as adding positive four.

The symbols $-(-4)$ also represent the additive inverse OF (the additive inverse of $4$). Our work with additive inverses can help you write this expression in a different and simpler way. After all, the additive inverse of $4$ is $-4$, and now the additive inverse of $-4$ is, by definition, a number that results in zero when added to $-4$. Of course, the additive inverse of $-4$ is $4$ itself. All of this tells us that $-(-4)=4$. In the next section we will discuss multiplying and dividing negative numbers.

Watch this video for more examples on how to subtract negative numbers.

Subtraction with Negative Numbers Exercises

Complete this assessment to practice subtracting negative numbers and check your answers. If you need more practice, do Problem Sets 2 and 3.

3.3: Multiplying and Dividing Integers with Different Signs Multiplying and Dividing Integers

Read this text. Pay special attention to the sections Multiplication of Signed Numbers and Division of Signed Numbers. These sections give summaries of the rules for multiplying and dividing positive and negative integers. Complete the practice questions and check your answers.

Multiplying/Dividing Positive and Negative Numbers

Watch these videos for more worked examples of multiplying and dividing positive and negative numbers.

3.4: Exponents Understanding Exponents

Watch this video for more practice using this notation.

Exponent Examples

Then watch these videos to see an example of solving a problem using exponents.

Introduction to Exponents

Finally, watch this video to see more examples.

Introduction to Exponents Exercises

Complete this assessment to practice working with exponents and check your answers. If you need more practice, do Problem Sets 2 and 3.

3.5: Order of Operations Use the Language of Algebra

Read this text. Pay special attention to the Order of Operations portion that summarizes the rules for performing multi-step calculations.

Many students use a silly phrase to remember the order of operations: Please Excuse My Dear Aunt Sally, where the first letter of each word corresponds to a mathematical operation: parentheses, exponents, multiplication and division, addition and subtraction.

Do Examples 2.8 – 2.12, and check your answers. If you need more practice, you can try Try It 2.15 – 2.24.

Order of Operations Exercises

4.1: Identifying Parts of Fractions Fractions of Whole Numbers

Read this section to learn more about the history and notation of fractions, how to visualize them in relation to integers, and how to identify the different parts (numerator and denominator). Complete the practice questions and check your answers.

Visualize Fractions

Read this section which offers some useful ways to visualize fractions to help you understand the ideas they can represent. Complete the practice questions and check your answers.

4.2: Equivalent Fractions Demonstrating Equivalent Fractions

Watch this video for practice with equivalent fractions. It uses number lines to show how different fractions are equivalent, or the same value.

4.3: Proper and Improper Fractions Proper and Improper Fractions

Watch this video for practice with equivalent fractions. It uses number lines to show how different fractions are equivalent, or the same value.

Proper Fractions, Improper Fractions, and Mixed Numbers

Read this text to see examples of proper and improper fractions using a number line.

4.4: Mixed Numbers and Improper Fractions Visualize Fractions

Indeed, we can convert every mixed number into an improper fraction. We will discuss this conversion trick in the next section. Read this text, complete the practice problems, and check your answers.

4.5: Converting Between Improper Fractions and Mixed Numbers Changing from an Improper Fraction

Watch this video to see how to convert an improper fraction to a mixed number.

Changing to Improper Fractions
Watch this video to see how to convert a mixed number to an improper fraction.
Converting Mixed Numbers and Improper Fractions Exercises

Complete this assessment to practice converting mixed numbers and improper fractions. Be sure to check your answers.

4.6: Fractions in Lowest Terms Fraction in Lowest Terms

Watch these videos to learn how to reduce fractions to lowest terms.

Reducing Fractions
Watch this second video on reducing fractions.
Reducing Fractions Exercises

Complete this assessment to practice reducing fractions and check your answers. If you need more practice, do Problem Sets 2 and 3.

4.7: Finding Common Denominators Finding Common Denominators

Watch this video. Pause it after the narrator presents the second example to see if you can figure out the common denominator yourself. Then watch the rest of the video to see the solution.

4.8: Adding and Subtracting Fractions Add and Subtract Fractions with Common Denominators

Read this text. Pay special attention to the sections on fraction addition and subtraction. They provide an overview of how to add and subtract fractions with the same denominator. Complete the practice questions and check your answers.

Add and Subtract Fractions with Different Denominators

Read this text for more examples and guidance. It gives a good review of how to find common denominators. Pay attention to the "How To" section at the beginning for an overview of the process. Do Examples 4.67 – 4.72 and check your answers.

More with Fractions

Watch this video for more examples of how to add and subtract fractions with like and unlike denominators.

Read this section which explains the two methods for adding mixed numbers with common denominators. Do Examples 4.83 and 4.84 using the method of adding the whole numbers and fractions separately. Then, do Examples 4.85 using the method of converting mixed numbers to improper fractions. Be sure to check your answers.

Subtract Mixed Numbers

Next, read this section which describes two methods for subtracting mixed numbers with common denominators. Pay attention to the gray "How To" boxes for an overview of the methods. Complete Example 4.90 using the whole numbers and fraction method, and Example 4.91 using the improper fraction method. Be sure to check your answers.

Practice with Unlike Denominators

4.10: Applications of Adding and Subtracting Fractions Adding and Subtracting Mixed Numbers

Watch these two videos for more real-world examples of adding and subtracting mixed numbers.

Practice Adding and Subtracting Mixed Numbers

4.11: Multiplying Fractions and Mixed Numbers Multiplying Fractions and Mixed Numbers

Watch these videos for some real-world examples of multiplying fractions and mixed numbers. Note that you usually need to convert mixed numbers to improper fractions before multiplying.

Practice with Multiplying Fractions

More Practice with Multiplying Fractions

Here are some more practice problems on multiplying fractions. Be sure to check your answers.

4.12: Dividing Fractions and Mixed Numbers Dividing Fractions

Watch these videos to figure out how to divide lots of numbers by fractions. As usual, pay careful attention to the worked examples.

Dividing Mixed Numbers

Watch these videos to learn how to divide mixed numbers and see a few examples.

Dividing Fractions Exercises

Complete this assessment to practice dividing fractions and check your answers. If you need more practice, do Problem Sets 2 and 3.

4.13: Applications of Multiplying and Dividing Fractions Multiplying and Dividing Fractions Word Problems

Watch these videos for more examples on this useful topic.

Multiply and Divide Fractions Word Problems Exercises

Complete this assessment to practice word problems that include multiplying and dividing fractions. Be sure to check your answers.

More Multiply and Divide Fractions Word Problems Exercises

Here are some more practice problems on dividing fractions word problems. Be sure to check your answers.

5.1: Decimal Place Value Introduction to Decimals

Watch this video for more examples with other numbers.

Introduction to Decimals Exercises

Complete this assessment to practice your understanding of decimals and check your answers. If you need more practice, do Problem Sets 2 and 3.

5.2: Rounding Decimals Rounding Decimals

Watch this video for examples of how to round decimals to specific decimal places.

More on Rounding Decimals

Then read this text up to Sample Set A. The second paragraph gives a set of rules for determining how to round a decimal. The first two worked examples under Sample Set A show step-by-step directions for rounding numbers. Complete the practice problems and check your answers.

5.3: Converting Between Decimals and Fractions Decimal Places

Regardless of the flavor you prefer for your non-whole numbers, watch this video to learn how to switch between them. The narrator shows examples of how to convert decimals to fractions.

Decimals and Fractions

Decimal Exercises

Complete these problems to practice converting decimals and fractions and check your answers. If you need more practice, do Problem Sets 2 and 3.

5.4: Converting Repeating Decimals to Fractions Multiplying a Decimal by a Power of 10

Watch this video whose main point is this: every time you multiply a decimal expression by 10, simply move the decimal point one digit to the right.

Converting Repeating Decimals to Fractions

Watch these videos to explore more examples. Many involve repeating decimals.

Repeating Decimals
Now, read this section on repeating decimals. Pay attention to the notation using a line above the repeating portion of the decimal. Complete the practice problems and check your answers.

Watch this video for a worked example of an addition problem with decimal expressions. Pay attention to how to line up the decimal point when adding decimals.

Subtracting Decimals
Watch these videos to learn how to subtract with decimal numbers.
Subtracting Hundredths Exercises
Complete this assessment to practice subtracting hundredths. Be sure to check your answers.
Decimal Operations
Read this text for some more examples. Pay attention to the "How To" box for an outline of how to add and subtract decimals. Complete the practice problems and check your answers.
5.6: Multiplying Decimal Expressions Multiplying Decimals

Watch this video for a worked example of how to multiply decimal numbers. Pay attention to expressions like "move the decimal point".

Multiplying Decimals Exercises

Complete this assessment to practice multiplying decimals. Be sure to check your answers.

More on Multiplying Decimals

Read this text. Pay attention to the "How To" overview of the steps needed to multiply decimals. Complete the practice problems and check your answers.

5.7: Dividing Decimal Expressions Dividing Decimals with Hundredths

Watch these videos for worked examples of how to divide with decimal numbers.

Dividing Decimals

Read this text. Pay attention to the "How To" boxes which give brief step-by-step summaries of how to divide decimals. Complete the practice problems and check your answers.

5.8: Word Problems Using Decimals Applications with Decimals

Decimals also appear in many other places in the real world, especially in problems involving money. Watch this video for more word problems that use decimals.

Applications with Decimals Exercises

Complete this assessment to practice calculating applications with decimals. Be sure to check your answers.

6.1: Introduction to Ratios Introduction to Ratios

Watch this video to explore more about how ratios work. Again, notice that ratios are basically fractions, so you will be applying many of the concepts you learned in the fractions unit of this course here.

Ratios

Watch this video on ratios to reinforce what you just learned.

Ratios Exercises

Complete this assessment to practice working with ratios and check your answers.

6.2: Finding Unit Rates and Prices Solving Unit Rates and Prices
Watch these videos to see worked examples of how to calculate unit prices and rates.
Expressing Ratios as Fractions Exercises

Complete this assessment to practice expressing ratios as fractions and check your answers.

6.3: Understanding Proportions Applications of Proportions

Read this text. Pay attention to the gray box labeled Cross Products of a Proportion, which shows the simple way to solve for an unknown value in a proportion. Complete the practice problems and check your answers.

Solving Proportions
Watch this video for more worked examples.
6.4: Applications of Proportions Solving Applications of Proportions

Read this text which discusses real-world examples of how proportions are used. Complete the practice problems and check your answers. These examples are word problems that use proportions.

More Applications of Proportions
Watch this video for some more worked examples.
Applications of Proportions Exercises

Complete this assessment to practice applications of proportions and check your answers.

7.1: Describing the Meaning of Percent The Meaning of Percents

Watch these videos to better understand how percents relate to fractions with a denominator of 100.

7.2: Converting Between Decimals, Fractions, and Percents Converting Percent to Decimal and Fraction

Watch these videos for worked examples of how to do conversions with percents.

Ratios and Rates as Percents

Read this text. Pay attention to the table on making conversions to a fraction, decimal, and a percent for an overview. Also pay close attention to the worked problems in Sample Set B. Complete the practice problems and check your answers.

7.3: Determine the Percent Given Two Numbers Solving Basic Percent Problems

Watch this video to see some worked examples. Many of these problems will show you how to set up and answer questions like the two examples presented above.

Percent Exercises

Complete this assessment to practice working with percents and check your answers.

7.4: Percent Increase or Decrease and Other Percent Applications Calculating Percent Increase, Decrease, and Discount

Watch this video to see worked problems of these real-world applications and how percentages can help us understand the questions and examples mentioned just above.

Percent Increase, Decrease, and Discount

Complete this assessment to practice calculating percent increases, decreases, and discounts. Be sure to check your answers.

Calculating Sales Tax and Commission

Two more important applications of percents include sales tax and commision. When you purchase an item most states require you to pay a certain percent as sales tax (usually not listed on the item's price tag). How do you determine the real price you need to pay? For example, how much do you need to give the cashier to buy a \$20.00 shirt with an 8% sales tax?

We calculate commissions in the same way. How about if you work at the store and your store manager has agreed to give you an 8% commission for your sale?

Watch this video for worked examples of these applications.

Sales Tax and Commission Exercises

Complete this assessment to practice calculating sales tax and commissions. Be sure to check your answers.

8.1: Mean, Median, Mode, and Range Averages and Probability

Read this text. Pay close attention to the "How To" sections which give summaries of how to do mean, median, and mode calculations. After you read the text, complete the practice problems and check your answers.

Finding Mean, Median, and Mode

Watch these three videos to see more worked examples, and see the definition of range in mathematics.

8.2: Stem-and-Leaf Plots Stem-and-Leaf Plots

The first type of graph we will explore is called a stem and leaf plot. Watch this video to learn about this type of plot.

Reading Stem and Leaf Plot Exercises

8.3: Line Graphs Reading Line Graphs

Watch these videos. The first video explains the structure of a line graph using simple examples. The second shows how some presenters manipulate line graphs to misrepresent data.

Make Line Plots Exercises

Complete this assessment to practice making line plots. Be sure to check your answers.

8.4: Bar Graphs Reading Bar Graphs

Watch this video so you are familiar with other types of bar graphs and how to create them.

Create Bar Graphs Exercises

Complete this assessment to practice creating bar graphs. Be sure to check your answers.

8.5: Box-and-Whisker Plots Box-and-Whisker Plots

These videos show examples of how box-and-whisker plots are used, and examples of how to construct them.

Creating Box and Whisker Plots Exercises

Complete this assessment to practice creating box and whisker plots. Be sure to check your answers.

8.6: Circle or Pie Graphs Reading Pie Graphs (Circle Graphs)

Watch this video to learn how to construct and use pie charts in a variety of settings.