Unit 2: Order of Operations
To avoid making errors, mathematicians follow a series of steps to simplify expressions that use the four basic operations, grouping symbols, and exponents. We call these steps the "order of operations". In this unit, we explore how to use exponents and grouping symbols to perform mathematical operations in the correct order.
For example, we can use exponents to write out a repeat multiplication, such as 2 × 2 × 2 × 2 × 2, or 25. We also examine the concept of the greatest common factor (GCF), the largest positive integer that divides evenly into a given group of numbers with zero remainder. For example, for the set of numbers 18, 30 and 42, the GCF is 6. We also use the least common multiple (LCM) to determine the smallest number that two numbers can both divide into. For example, for the set of numbers 2 and 6, the LCM is 6. Finally, we learn how to perform mathematical operations with negative numbers.
Completing this unit should take you approximately 7 hours.
Upon successful completion of this unit, you will be able to:
- find the greatest common factor and least common multiple of whole numbers;
- calculate problems using negative numbers;
- calculate exponents; and
- apply the order of operations.
2.1: Greatest Common Factor
The greatest common factor (GCF) is the largest positive integer that divides evenly into a given group of numbers. For example, let's consider the numbers 12 and eight. We can divide both of those numbers by two: 12/2 = 6 and 8/2 = 4. But, we can also divide both of those numbers by four: 12/4 = 3 and 8/4 = 2. We cannot divide both numbers by any other factor. Therefore, the GCF for these numbers is four.
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.
Read up to Sample Set A. Pay close attention to how to determine the greatest common factor. Complete the problems in Practice Set A and check your answers. If you feel you need more practice, you can try some of the exercises that follow.
2.2: Least Common Multiple
Multiples are the result of multiplying two whole numbers together. We can write out multiples for any given number. For example, consider some multiples of three: 3 × 1 = 3; 3 × 2 = 6, 3 × 3 = 9. The first three multiples of 3 are 3, 6, and 9. We can see that different numbers sometimes have the same multiples. For example, consider some multiples of 2: 2 × 1 = 2, 2 × 2 = 4; 2 × 3 = 6. Both 3 and 2 have 6 as a multiple.
The least common multiple (LCM) is the smallest (or least) multiple that is the same between two or more whole numbers. In our example above, the least common multiple of 3 and 2 is 6.
Read up to Practice Set C to learn about multiples and how to determine the least common multiple of a given set of numbers. Do Practice Set C and check your answers.
Watch this video for an additional example of determining the least common multiple for a set of three numbers.
Watch this video to see some examples of using the least common multiple and greatest common factor in the real world.
2.3: Negative Numbers
So far, we have only been dealing with positive numbers. However, negative numbers are also very common, and we need to be able to perform operations with them. One common example is temperature.
We could have a temperature of −50.0 degrees or we could have a temperature of +50.0 degrees. We use the concept of negative numbers to calculate the difference between these two temperatures.
Read up to Try It exercise 3.10. This section explains the concept of negative numbers using number lines. Pay close attention to the section on opposite notation, which will come up frequently in the next few sections of this course. Do examples 3.2 through 3.5 and check your answer.
Watch this video for more examples of negative numbers using a number line as a reference.
2.4: Adding and Subtracting Negative Numbers
When adding and subtracting negative numbers, we need to follow different rules than when adding and subtracting positive numbers.
Watch this video for examples of how to add negative numbers. When we add a negative number, we can rewrite it as subtracting the positive number. That is: 5 + (−2) = 5 − 2.
After you watch the video, take this quiz and check your answers.
Watch this video for examples of how to subtract negative numbers. When we subtract negative numbers, we can rewrite it as adding the number. That is: 5 − (−4) = 5 + 4.
After you watch the video, take this quiz and check your answers.
2.5: Multiplying and Dividing Integers with Different Signs
When we multiply and divide integers, we must pay attention to the signs of the integers.
- When we multiply or divide two positive numbers, the result is always a positive number.
- Likewise, when we multiply or divide two negative numbers, the result is always a positive number.
- However, when we multiply or divide numbers with different signs (one positive and one negative), the result is a negative number.
Read up to the "Simplify Expressions with Integers" section. Pay special attention to how to multiply and divide signed numbers (multiplying and dividing positive and negative integers). Do examples 3.47 and 3.49, and check your answers. If you would like more practice, do Try It exercises 3.93, 3.94, 3.97, and 3.98.
Watch these two videos for additional worked examples.
Exponents are a way to simplify writing a number multiplied by itself multiple times. We write exponents as superscripts. For example: 42 = 4 × 4 = 16 and 43 = 4 × 4 × 4 = 64.
Watch this video to learn about how to translate a problem to exponent notation.
Then watch these videos to see examples of solving a problem using exponents.
Watch this video for more examples.
After you watch the videos, complete this quiz and check your answers. If you want more practice, try problem sets 2 and 3.
2.7: Order of Operations
When completing a calculation involving different mathematical operations, how do we decide what to do first? In mathematics, we follow a set of rules called the Order of Operations that determines the order for performing different types of calculations in a large multi-step problem.
Read the section beginning with "Simplify Expressions Using the Order of Operations". Pay special attention to the rules for performing multi-step calculations.
Many students remember the order of operations using the silly phrase Please Excuse My Dear Aunt Sally, where the first letter of each word corresponds to a mathematical operation: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
Do Examples 2.8–2.12 and check your answers. If you need additional practice, you can do Try It exercises 2.15–2.24.
Complete this assessment for more practice. If you are struggling with any of the problems, use the videos or hints for help.