Skip to main content

Section 1.4 Flexibility with Numbers and Order of Operations

In Section 1.4, you will use your creativity and play with mathematical operations using the Order of Operations. The Order of Operations is an agreed upon order in which operations must be completed when simplifying a numeric or an algebraic expression. When presented as a list of rules, the Order of Operations can seem arbitrary. However, there are mathematical reasons underlying the order in which operations are completed.

Subsection 1.4.1 The Order of Operations

The Order of Operations follows the historical introduction of operations. Addition happened first. Subtraction is the operation that undoes addition. Multiplication naturally occurred as people needed to determine multiple sums of the same number; multiplication is repeated addition. The operation that undoes multiplication is division; division can also be thought of as repeated subtraction. Exponentiation occurred even later signifying repeated multiplication. Taking a whole number root, the inverse operation of raising a number to the same whole number exponent, can also be thought of as repeated division. When you encounter an expression to be simplified, consider the historical introduction of arithmetic operations and perform the most complex operation first, reducing each subsequent expression in the opposite order that the operations arose historically. Of course, if parentheses are included to alter the order of operations further, expressions inside them must be simplified first!

Let's consider some examples. To simplify the expression, \(3 + 2 \times 7\text{,}\) recall that the multiplication sign represents repeated addition. An equivalent expression written entirely in terms of addition is \(3 + 7 + 7 = 17\text{.}\) Try this with your calculator; do you get this result? Some students want to add 3 and 2 then multiply the result by 7 because the expression is written that way. If you add first to get 5 then multiply by 7, you get 35. This is not equivalent to the original expression. Notice that \(35 = (3 + 2) \times 7\text{.}\) In the original expression, only 2 is multiplied by 7. This example helps us understand why multiplication is completed before addition when simplifying \(3 + 2 \times 7\text{,}\) and in general.

Now consider the expression, \(5 \times 3^2\text{.}\) As indicated above, terms with exponents represent repeated multiplication. The expression, \(5 \times 3^2\text{,}\) can be rewritten as \(5 \times 3 \times 3\text{.}\) To multiply \(5 \times 3\) then square the result changes the meaning of the original expression to \((5 \times 3)^2\text{.}\) Try both versions on your calculator. Because no parentheses are included in the original expression, \(5 \times 32 \ne (5 \times 3)^2\text{.}\)

Try to simplify this expression to see why the Order of Operations is helpful. Change all exponentiation to repeated multiplication then change all multiplication to repeated addition before simplifying.

\begin{equation*} 5 + 2 \times 3^2 \end{equation*}

Look at the simplification below. Explain how each expression arises from the one before it:

\begin{align*} 5 + 2 \times 3^2 \amp = 5 + 2 \times 3 \times 3\\ \amp = 5 + (3 \times 3) + (3 \times 3)\\ \amp = 5 + (3 + 3 + 3) + (3 + 3 + 3)\\ \amp = 5 + 9 + 9\\ \amp = 23 \end{align*}

You can see from this simplification and your explanation that simplifying an expression by replacing exponentiation with repeated multiplication and multiplication with repeated addition will quickly get cumbersome.

The rules for simplifying expressions using the Order of Operations serve as a reminder. To recall which operation to complete first it is helpful to think about what each operation represents. There are cautions, however. Sometimes the Order of Operations can lead us to believe that the order in which we perform operations is rigid. We know that expressions like \(85 + 27 + 15\) and \(25 \times 11 \times 4\) can be simplified more easily if we rearrange the terms. We see that \(85 + 15 = 100\) so the sum \(85 + 27 + 15 = 85 + 15 + 27 = 100 + 27 = 127\text{.}\) We also see that \(25 \times 4 = 100\) so \(25 \times 11 \times 4 = 25 \times 4 \times 11 = 100 \times 11 = 1100\text{.}\) Addition and multiplication are commutative (the order can be changed without changing the result). We also know that we can group numbers differently when they are all combined with addition or with multiplication; this is the associative property: \(14 + (16 + 18) = (14 + 16) + 18\text{,}\) \(15 \times (6 \times 17) = (15 \times 6) \times 17\text{.}\) Both of these properties allow us to group and rearrange numbers combined with one of the operations + or x (not both together, though) to put friendlier, easier to combine, numbers together to aid in simplification.

We would expect that the Order of Operations is a universal convention, agreed upon and used everywhere in the world in the same way. This is not the case, as it happens. Consider the case of multiplication and division. The Order of Operations convention used in USA textbooks tells us to complete multiplication and division in the order they appear in an expression, left to right. In Kenya, students are told to complete division first, in the order division arises, then complete multiplication. Simplify the following expressions with each convention. What do you notice?

\begin{equation*} 15 \times 21 \div 7 \times 5 \qquad 9 \times 24 \div 6 \div 3 \times 12 \qquad 35 \div 7 \times 18 \div 2 \end{equation*}

What do you think is happening here? Can you explain why the result is the same regardless of which version of the Order of Operations you use? Think about how multiplication and division are related then rewrite each expression to use only multiplication. Now what do you notice?

Which Order of Operations convention do you think is easier and more reliable to implement? Use that one! The following reminders in Figure 1.4.1.1 of Order of Operations might be helpful. Use whichever one makes more sense to you. Notice that the order you simplify expressions, in both conventions, is the opposite order that each operation was introduced historically; more complex operations are simplified first.

P Simplify Parentheses (and other grouping symbols)
E Simplify Exponents
M
D
Complete Multiplication and Division in
the order found, left to right.
A
S
Complete Addition and Subtraction in
the order found, left to right.
Figure 1.4.1.1. Two visualizations of the Order of Operations.
‚ÄČ11‚ÄČ

Student Page 1.4.2 Three Threes

Consider the Three Threes Clock Face in Figure 1.4.2.1. Verify that each number on the clock face is represented correctly using three threes. Think about how you are using the Order of Operations to check each expression. Use your creativity to find as many expressions for each number as you can.

Figure 1.4.2.1. Three Threes Clock Face

1.

Verify that each expression is correct. Show your work. (As they arise, write any questions you have about any of the expressions.)

2.

Find other expressions for some of the numbers 1 through 12 using three threes. Show your work.

Share your thoughts on how each number on the clock was represented by three threes. Also share alternative ways you represented some of the numbers using three threes.

Student Page 1.4.3 Four Fours

Now that you've had a chance to verify and play with expressions for the numbers 1 through 12 created using three threes, use your creativity to find expressions for other numbers using four fours. The Four Fours Problem provides you additional opportunities to think about the order of operations and the mathematical operations you have studied in previous courses. Study the problem presented below.

How can you make the numbers 1 through 20, 24, 30, and 100 using 4 fours? Record all of the expressions you find for each number in the table. Work alone for at least 5 minutes before you share your results with your group. If you get stuck, consult Figure 1.4.2.1 for inspiration.

Use the number 4 exactly four times to write expressions equivalent to the numbers from 0 through 20. You may use any of the mathematical symbols below. A few examples are provided.

  • Arithmetic operations: \(+, -, \times, \div\)

  • Parentheses: ( )

  • Decimal points or percents: e.g. \(4.4\text{,}\) \(.44\text{,}\) 44%

  • Exponentiation: ^ (to the power of), e.g. \((4 + 4)^4\)

  • Square root: e.g. \(\sqrt{4}\)

  • Factorial: ! (\(4! = (4)(3)(2)(1) = 24\)

  • Concatenation: e.g. 44, 444

  • Repeating decimal: \(. \overline{4} = \frac{4}{9}\)

1.

Complete the table. Find at least one expression using 4 fours for each number. Use the Order of Operations; include grouping symbols as needed. Check your responses using a graphing calculator. Some examples for \(n = 4\) are provided. For the numbers whose solutions are shown in the table, find at least one more expression that works.

Number Four Fours Equation Number Four Fours Equation
0 12
1 \(1 = \left(\frac{4}{4} \right) ^{44}\) 13
2 14
3 15 \(15 = \frac{44}{4} + 4\)
4 16
5 17
6 18
7 \(7 = 4 + 4 - \frac{4}{4}\) 19
8 20
9 24 \(24 = 4! \cdot \left(\frac{\sqrt{4} \times \sqrt{4}}{4} \right)\)
10 30
11 100

2.

What strategies did you use to find solutions?

3.

Find more than one solution for at least 3 of the numbers. Record them in the table.

4.

Which of your solutions are not dependent on the number 4? For example \(1=\ \left(\frac{4}{4}\right)^{44}\) is also true when each 4 is replaced with 2, 6, or 9. Highlight these solutions in your table.

Share the expressions you found using Four Fours with your group. Compare your work with your group members. Do you agree that each expression is correct? Circle any expressions you question; put a checkmark next to expressions with which you agree. Resolve any differences of opinion. When asked to do so, share some of your group's expressions with the class.

Homework 1.4.4 Homework

Complete the following homework before the next class period:

2.

Determine the value of each expression without the use of an electronic tool. Show your work one step at a time.

(a)

\(4! - \left(4/ \sqrt{4} \right)^{4}\)

(b)

\(4 + 4 / \sqrt{4} \times \sqrt{4}\)

3.

While working on Four Fours, students solved the following expression in these four ways.

\begin{align*} \amp 4 + 4 / \sqrt{4} \times \sqrt{4} \amp \amp 4 + 4 / \sqrt{4} \times \sqrt{4} \amp \amp 4 + 4 / \sqrt{4} \times \sqrt{4} \amp \amp 4 + 4 / \sqrt{4} \times \sqrt{4}\\ \amp = 4 + 4 / 2 \times 2 \amp \amp = 4 + 4 / 2 \times 2 \amp \amp = 4 + 4 / 2 \times 2 \amp \amp = 4 + 4 / 2 \times 2\\ \amp = 4 + 4 / 4 \amp \amp = 8 / 4 \amp \amp = 4 + 2 \times 2 \amp \amp = 4 + 2 \times 2\\ \amp = 4 + 1 \amp \amp = 2 \amp \amp = 4 + 4 \amp \amp = 6 \times 2\\ \amp = 5 \amp \amp \amp \amp = 8 \amp \amp = 12 \end{align*}
(a)

Only one solution can be correct. Which one is it? How do you know?

(b)

For the solutions above that are incorrect, identify the error(s) the student made.

(c)

As you work on Four Fours check the expressions you find to make sure they work for the numbers you think they do.

4.

Solve each of the following. Show how you know your work is correct.

(a)

Insert parentheses so this equation is true: \(4 + 4 \div .4 \div \sqrt{4} = 7\text{.}\)

(b)

Insert parentheses so this equation is true: \(\sqrt{4}^{\sqrt{4}} + \sqrt{4} / 4 = 4\text{.}\)

(c)

Revisit Task 1.4.4.2.a. With no parentheses in the left side of the equation, what would the simplification of the expression be? Write the expression and show the simplification.

5.

Look back at your work on the Four Fours Problem. Indicate where you used the mathematical properties and operations listed:

  • Order of Operations

  • Additive identity (adding zero, \(4 +(4 - 4) = 4 + 0 = 4\text{;}\) \(\frac{4}{4}+(4-4)=1+0=1)\))

  • Multiplicative identity (multiplying by \(1= \frac{4}{4}=\frac{4}{\sqrt4\sqrt4}\) )

  • Fractions as divison, \(\frac{44}{4} = 11\)

  • Square root: \(\sqrt{4} = 2\)

  • Exponentiation: \(4^{\sqrt4}=4^2=4 \cdot 4=16\)

  • Factorials: \(4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24\)

  • Concatenation: 44, 444

  • Decimals: \(4.4, .4, . \overline{4} = .4444444\) ‚Ķ

6.

What strategies have you used to find different expressions? Describe at least three.

7.

Which of the expressions you found will give you the same result if you use four copies of another number such as 1, 2, 3, …? For example, \(\frac{4}{4}+\frac{4}{4}=2\text{;}\) notice that this relationship also works if you replace all 4 fours with four of any of these single digit numbers: 1, 2, 3, 5, 6, 7, 8, or 9.

8.

Two students' work on the same math problem is shown below. Assess their work.

\begin{align*} \amp \text{Student 1:} \amp \amp \text{Student 2:}\\ \amp \sqrt{49} - 4 \div 2 + 3! \amp \amp \sqrt{49} - 4 \div 2 + 3!\\ \amp = 7 - 4 \div 2 + 3! \amp \amp = 7 - 4 \div 2 + 6\\ \amp = 7 - 4 \div 2 + 6 \amp \amp = 3 \div 2 +6\\ \amp = 7 - 4 \div 8 \amp \amp = 1.5 + 6\\ \amp = 3 \div 8 \amp \amp = 7.5\\ \amp = \frac{3}{8} \amp \amp \end{align*}
(a)

What error(s), if any, did each student make?

(b)

If neither solution is correct, find the correct solution and show how to find it step-by-step.

Resource: Bay-Williams, Jennifer M. and Sherri L. Martinie. Order of Operations, The Myth and the Math. Teaching Children Mathematics, Vol. 22, No. 1, August 2015.