1. Order of Operations
Topic 1.2.6

2. Order of Operations 1.2.6 Topic California Standard:
1.1 Students use properties of numbers to demonstrate whether assertions are true or false.
What it means for you:
You’ll see why it’s important to have a set of rules about the order in which you have to deal with operations.
Key words:
grouping symbols
parentheses
brackets
braces
exponents

3. Order of Operations 1.2.6 Topic
It’s important that all mathematicians write out expressions in the same way, so that anyone can reach the same solution by following a set of rules called the “order of operations.”

4. Order of Operations 1.2.6 Topic
Grouping Symbols Show You What to Work Out First
If you wanted to write a numerical expression representing
“add 4 and 3, then multiply the answer by 2,”
you might be tempted to write × 2.
But watch out — this expression contains an addition and a multiplication, and you get different answers depending on which you do first.
If you do the addition first, you get the answer 7 × 2 = 14.
If you do the multiplication first, you get the answer = 10.

5. Order of Operations 1.2.6 Topic
You might know the addition has to be done first, but somebody else might not.
To be really clear which parts of a calculation have to be done first, you can use grouping symbols.
Some common grouping symbols are:
parentheses ( ),
brackets [ ],
and braces { }.

6. Order of Operations 1.2.6 Topic
Example 1
Write an expression representing the phrase “add 4 and 3, then multiply the answer by 2.”
Solution
You need to show that the addition should be done first, so put that part inside grouping symbols:
The expression should be (4 + 3) × 2.
Solution follows…

7. Order of Operations 1.2.6 Topic Guided Practice
Write numeric expressions for these phrases:
1. Divide 4 by 8 then add 3.
(4 ÷ 8) + 3
2. Divide 4 by the sum of 8 and 3.
4 ÷ (8 + 3)
3. From 20, subtract the product of 8 and 2.
20 – (8 × 2)
4. From 20, subtract 8 and multiply by 2.
(20 – 8) × 2
Solution follows…

8. Order of Operations 1.2.6 Topic Guided Practice
Evaluate the following sums and differences:
5. (3 – 2) – 5 –4
6. 3 – (2 – 5) 6
7. 6 – (11 + 7)
–12
8. (7 – 8) – (–3 – 8) – 11 –1
9. (5 – 9) – (3 – 10) – 2 1
(5 – 3) – 4 7
Solution follows…

9. Order of Operations 1.2.6 5 – [11 – (7 – 2)] Topic
Nested grouping symbols are when you have grouping symbols inside other grouping symbols.
5 – [11 – (7 – 2)]
When you see nested grouping symbols, you always start from the inside and work outwards.

10. Order of Operations 1.2.6 Topic Evaluate {5 – [11 – (7 – 2)]} + 34.
Example 2
Evaluate {5 – [11 – (7 – 2)]} + 34.
Solution
Start from the inside and work outwards:
{5 – [11 – (7 – 2)]} + 34 = {5 – [11 – 5]} + 34
= {5 – 6} + 34
= –1 + 34
= 33
Solution follows…

11. Order of Operations 1.2.6 Topic Guided Practice
Evaluate the following:
[10 + (6 – 9) + 7]
8 + [10 + (–3) + 7] = = 22
12. 9 – {[(–4) + 10] + 7}
9 – {6 + 7} = 9 – 13 = –4
13. [(13 – 12) + 6] – (4 – 2)
[1 + 6] – (4 – 2) = 7 – 2 = 5
– {8 + [5 – (–2)]} – 6
14 – {8 + 7} – 6 = 14 – 15 – 6 = –7
[10 – (4 + 5)] – (11 + 8)
13 + [10 – 9] – (11 + 8) = – 19 = –5
{[7 – (–2)] – (3 – 1)} + (–14)
10 + {9 – 2} + (–14) = – 14 = 3
Solution follows…

12. Order of Operations 1.2.6 Topic
There are Other Rules About What to Evaluate First
This order of operations is used by all mathematicians, so that every mathematician in the world evaluates expressions in the same way.
1. First calculate expressions within grouping symbols — working from the innermost grouping symbols to the outermost.
2. Then calculate expressions involving exponents.
3. Next do all multiplication and division, working from left to right. Multiplication and division have equal priority, so do them in the order they appear from left to right.
4. Lastly, do any addition or subtraction, again from left to right. Addition and subtraction have the same priority, so do them in the order they appear from left to right too.

13. Order of Operations 1.2.6 Topic Simplify {4(10 – 3) + 32} × 5 – 11.
Example 3
Simplify {4(10 – 3) + 32} × 5 – 11.
Solution
Work through the expression bit by bit: {4(10 – 3) + 32} × 5 – 11
= {4 × } × 5 – 11
Innermost grouping symbols first
Now you have to calculate everything inside the remaining grouping symbols:
= {4 × 7 + 9} × 5 – 11
first work out the exponent
= {28 + 9} × 5 – 11
then do the multiplication
= 37 × 5 – 11
then do the addition
Solution continues…
Solution follows…

14. Order of Operations 1.2.6 Topic Solution (continued)
Example 3
{4(10 – 3) + 32} × 5 – 11
= {4 × } × 5 – 11
= {4 × 7 + 9} × 5 – 11
= {28 + 9} × 5 – 11
= 37 × 5 – 11
Solution (continued)
Now there are no grouping symbols left, so you can do the rest of the calculation:
= 185 – 11
do the multiplication first
= 174
and finally the subtraction

15. Order of Operations 1.2.6 Topic Guided Practice
Evaluate the following:
÷ 8 – 2 1
– 4 × 2 1
21 – {–3 × –11 – 9 × 23} + 17 = 21 – {33 – 72} + 17 = 77
– {–3[–5 × ] – 9 × 23} + 17
20.
–3(32 – 4)
÷ [10 + 3(–4)]
–3(9 – 4)
3 + (12 ÷ [–2])
–3 × 5
3 + (–6) =
= 5
{10 ÷ [11 – 6] × (–4)}
8 + {10 ÷ 5 × (–4)} = 8 + {2 × (–4)} = 0
[17 + {–23 × 8 – 17}] ÷ 8 = [17 + {–184 – 17}] ÷ 8 = –23
22. [17 + {(–33 + 4) × 8 – 17}] ÷ 8
Solution follows…

16. Order of Operations 1.2.6 Topic Independent Practice
Evaluate the following:
1. 14 – [9 – 4 × 2] 13
(9 – 3) – (4 ÷ 2) 15
3. (–1) × (7 – ) –9
× [(1 + 2) – (6 – 14)]
111
5. [(11 – 8) + (7 – 2)] × 3 – 13 11
6. [( ) ÷ 2] – (6 – 12) 18
Solution follows…

17. Order of Operations 1.2.6 Topic Independent Practice
Insert grouping symbols in each of the following statements so that each statement is true:
× 24 – 18 ÷ 3 = 44
× [(24 – 18) ÷ 3] = 44 or × [(24 – 18) ÷ 3] = 44
– 14 – 12 × 6 = 517
(20 + 3)2 – (14 – 12) × 6 = 517
9. 4 × 33 – 2 × 32 – 3 × = 1662
(4 × 3)3 – (2 × 3)2 – 3 × (4 + 6) = 1662
Solution follows…

18. Order of Operations 1.2.6 Topic Round Up
You really need to learn the order of operation rules.
You’ll be using them again and again in Algebra I so you might as well make sure you remember them right now.