Numerical expressions
Someone writes 3 + 4 × 5 on the board. Is that 35 or 23, and how does everyone who reads it get the same number?
Where you are 5 in Order of operations
Why this exists
Caio has 3 tokens in his pocket at the arcade. He buys 4 cups of tokens, and each cup holds 5 tokens. To say how many tokens he has, you can write one short line:
That line is a numerical expression. It uses numbers and operation signs to name a number, and here it names Caio’s total. But a line on paper is only useful if everyone who reads it gets the same number. Work from left to right, adding first, and you get 7 \times 5 = 35. Multiply first and you get 3 + 20 = 23. Caio has 23 tokens, not 35.
So people who write math have agreed on an order for doing the operations. This lesson is about what an expression is, what that agreed order is, and why it is a good choice.
The lesson takes three days. Part 1 is what a numerical expression is, how it differs from an equation, and how two different expressions can name the same number. Part 2 is why an order has to be agreed, and the order itself. Part 3 uses the order on longer expressions, and compares expressions without working them out.
Definition
Part 1: expressions, values and equations
A numerical expression is a group of numbers joined by operation signs: +, -, \times and \div. It has no equal sign. It names one number, and that number is its value. Finding the value is called evaluating the expression.
These are all numerical expressions:
The last one is a single number with no sign at all. That still counts: 40 is an expression whose value is 40. An expression is a name for a number, and a plain number is the simplest name.
The answer to each operation has its own word. Adding gives a sum, subtracting a difference, multiplying a product and dividing a quotient. So 9 \times 6 is a product, and its value is 54.
An equation is different. It joins two expressions with an equal sign, and it makes a claim: “these two name the same number”. A claim can be true or false.
An expression by itself is not true or false, the same way the words “Caio’s tokens” are not true or false. It just names something. The later lesson expressions, equations and inequalities takes this further.
Equal expressions. Many different expressions can name the same number. The expressions 6 \times 4, 20 + 4 and 48 \div 2 all have the value 24. They are three names for one number. When two expressions have the same value, writing an equal sign between them makes a true equation:
This is what the equal sign means: the two sides name the same number. It does not mean “here comes the answer”.
Intuition
Look again at Caio’s tokens in Figure 1. On the left are the 3 loose tokens and the 4 cups of 5. Count them and you get 23.
On the right is what you get if you add 3 + 4 first and then multiply by 5. That counts 7 cups of 5, which is 35 tokens. Caio never had 7 cups. The add-first reading tells a different story from the one the expression was written for.
The 4 \times 5 is a single thing in the story: the tokens in the cups. It should be worked out as one piece, and the loose 3 added after. That is exactly what the agreed order does.
Results
Part 2: the agreed order
When an expression has no grouping symbols, everyone works it out in this order:
First, multiply and divide, from left to right.
Then, add and subtract, from left to right.
So in 3 + 4 \times 5 the multiplication goes first:
Two things in the rule are easy to miss.
- Multiplication and division are partners of the same rank. Neither one goes before the other. When both appear, take them in the order you meet them, reading left to right.
- Addition and subtraction are partners of the same rank too, and they are also taken left to right.
This order is called the order of operations. It is an agreement, the way everyone agrees that a red light means stop. It is not something that was discovered, and the next section explains why it is a sensible agreement.
Part 3: longer expressions
A long expression with no grouping symbols splits into chunks. The plus and minus signs are where the chunks break. Each chunk is a single number or a string of multiplying and dividing. Work out every chunk, then add and subtract the chunk values from left to right.
In Figure 2 the chunks of 8 + 6 \times 3 - 20 \div 4 are 8, 6 \times 3 and 20 \div 4. They are worth 8, 18 and 5, so the whole expression is 8 + 18 - 5 = 21.
Seeing the chunks also lets you compare expressions without working them out. If two expressions are the same except for one chunk, the one with the bigger chunk added is bigger, and the one with the bigger chunk subtracted is smaller.
Why it is true
No one can prove that multiplying comes first. It is a choice. What can be shown is that it is a good choice, and that some choice had to be made.
Some order had to be agreed. An expression is supposed to name one number. If 3 + 4 \times 5 could mean 23 to one person and 35 to another, it would not name anything. A scoreboard, a price list or a recipe written that way would mean different things to different readers. Picking one order, and everyone using it, fixes that.
Multiplying first keeps the short form and the long form equal. Multiplication is a short way to write adding the same number again and again: 4 \times 5 = 5 + 5 + 5 + 5. Write 3 + 4 \times 5 the long way and there is only adding left:
With multiplication first, the short form 3 + 4 \times 5 also gives 23. The two ways of writing it agree. If adding came first, the short form would give 35 and the long form 23, and the shorthand would change the number it stands for. Dividing first works the same way: 20 \div 4 is one number, the size of each of 4 equal shares, so it is treated as one piece.
Totals are often sums of products. At the arcade, 2 games at 5 tokens and 3 games at 4 tokens cost 2 \times 5 + 3 \times 4 tokens. With multiplying first, that line means exactly what it says, 10 + 12 = 22, with no extra marks needed. Totals like this come up all the time, so the agreed order makes the most common kind of expression the easiest to write.
Left to right settles the partners. For adding alone, order never matters: 5 + 8 + 2 is 15 whichever pair you add first. That comes from the commutative and associative properties of addition. Subtracting is different. Look at 20 - 6 - 4 in Figure 3.
Taking away 6 and then 4 lands on 10. If someone did 6 - 4 first, they would get 20 - 2 = 18, a different number. Dividing behaves the same way: 48 \div 4 \div 2 is 12 \div 2 = 6 from left to right, but 48 \div 2 = 24 if the second division went first. So the partners need a rule too, and the rule is to read the way you read words, left to right, the way things happen one after another.
Why are multiplying and dividing partners, and not one above the other? Dividing by a number undoes multiplying by it: 6 \times 3 = 18 and 18 \div 3 = 6. The two belong together, the way adding and subtracting do, so they share a rank.
Some simple calculators do each operation the moment you press the next key. Type 3 + 4 \times 5 on one of those and it shows 35. That calculator is not following the agreed order, so a long expression typed into it needs care.
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, and Examples 5 to 7 with Part 3.
Expression or equation
Thao keeps the scoreboard at trivia night. During a break she writes four lines on the board. For each, say whether it is an expression or an equation. Give the value of each expression, and say whether each equation is true or false.
(a) 45 + 17; (b) 9 \times 7 = 63; (c) 40; (d) 50 - 12 = 37
Solution(a) There is no equal sign, so 45 + 17 is an expression. Its value is 62.
(b) There is an equal sign joining 9 \times 7 and 63, so this is an equation. Since 9 \times 7 is 63, the two sides name the same number and the equation is true.
(c) A single number with no equal sign is an expression. Its value is 40.
(d) This is an equation, because an equal sign joins two expressions. But 50 - 12 is 38, not 37. The two sides name different numbers, so the equation is false. It is still an equation. A false claim is still a claim.
Many names for one number
A small prize at the arcade costs 6 \times 4 prize tickets. Which of these expressions name the same number of tickets?
Then write a true equation that uses two of them.
SolutionFirst find the value of 6 \times 4. It is 24. Now find the value of each expression and compare it with 24.
- 20 + 4 = 24. Same number.
- 30 - 7 = 23. Not the same.
- 48 \div 2 = 24. Same number.
- 3 \times 8 = 24. Same number.
- 6 + 4 = 10. Not the same. It uses the same two numbers as 6 \times 4, but a different operation, so it names a different number.
So 20 + 4, 48 \div 2 and 3 \times 8 all name 24. Any two of them make a true equation, for example
Neither side is “the answer”. Both sides are names for 24, and the equal sign says so.
Multiplying and dividing first
Evaluate each expression.
(a) 30 - 4 \times 6; (b) 18 + 12 \div 3; (c) 7 \times 8 - 6
Solution(a) There is a subtraction and a multiplication. Multiplication comes first, so work out 4 \times 6 = 24, then subtract:
Subtracting first would give 26 \times 6 = 156. That is not the value of this expression.
(b) Division comes before addition. 12 \div 3 = 4, so
(c) Here the multiplication is already on the left, but the order does not depend on where it sits. Multiply first, then subtract:
Partners, left to right
Evaluate each expression.
(a) 40 - 15 + 5; (b) 36 \div 6 \times 3; (c) 60 \div 5 \div 2
Solution(a) Subtraction and addition are partners, so neither goes first because of its sign. Take them left to right. 40 - 15 = 25, then 25 + 5 = 30:
Adding 15 + 5 first would give 40 - 20 = 20, which is not the value.
(b) Division and multiplication are partners, so take them left to right. 36 \div 6 = 6, then 6 \times 3 = 18:
Multiplying first would give 36 \div 18 = 2, which is not the value. Multiplication does not outrank division.
(c) Two divisions, left to right. 60 \div 5 = 12, then 12 \div 2 = 6:
A longer expression
Evaluate 8 + 6 \times 3 - 20 \div 4.
SolutionFind the chunks first, as in Figure 2. The plus and minus signs split the expression into 8, 6 \times 3 and 20 \div 4.
Work out each chunk: 6 \times 3 = 18 and 20 \div 4 = 5. Now only adding and subtracting are left, and they go left to right:
Writing each step on its own line, with the parts already done replaced by their values, keeps track of what is left.
Partners inside a chunk
Evaluate 50 - 2 \times 9 + 36 \div 4 \times 3.
SolutionThe chunks are 50, 2 \times 9 and 36 \div 4 \times 3.
The second chunk is 2 \times 9 = 18.
The third chunk has a division and a multiplication. They are partners, so take them left to right: 36 \div 4 = 9, then 9 \times 3 = 27.
Now add and subtract the chunk values, left to right:
Notice the subtraction comes before the addition here only because it sits farther left. Had the chunks been added first, 18 + 27 = 45 and 50 - 45 = 5, which is wrong.
Comparing without working it out
At trivia night every right answer in the main round scores 9 points, and each team got 12 right. Then come the bonus points. Hyun’s team scored 9 \times 12 + 15, and Thao’s team scored 9 \times 12 + 19. Answer the three questions without working out the whole values.
(a) Which team scored more, Hyun’s or Thao’s, and by how much?
(b) Which is greater, 100 - 6 \times 7 or 100 - 6 \times 8?
(c) Caio says 3 \times 199 + 3 and 3 \times 200 have the same value. Is he right?
Solution(a) Both expressions start with the same chunk, 9 \times 12. After it, Hyun’s team adds 15 and Thao’s team adds 19. Adding more gives more, so Thao’s team scored more, by 19 - 15 = 4 points.
(b) Both start at 100 and take away a chunk. The first takes away 6 sevens and the second takes away 6 eights, which is more. Taking away more leaves less, so 100 - 6 \times 7 is greater. (The chunks differ by 6 \times 8 - 6 \times 7 = 6, so the first is 6 greater.)
(c) Yes. By the commutative property of multiplication, 3 \times 199 is the same as 199 \times 3, which is 199 groups of 3. Adding 3 more is one more group of 3, which makes 200 groups of 3. That is 200 \times 3, which is 3 \times 200. Both have the value 600, but you can see they are equal without finding that.
Boundaries
Only the four operations, and no grouping symbols. The order in this lesson is for +, -, \times and \div in an expression with no parentheses or brackets. To make 3 + 4 happen first, the expression needs Grouping symbols. Powers, written with a small raised number, have their own place in the order, in the order of operations with powers.
An expression has to name a number. 8 \div 0 names no number at all, because no number of zeros makes 8. An expression with a division by zero in it has no value.
Staying with whole numbers. Every expression here has a whole-number value at every step. Some expressions leave the whole numbers. 7 \div 2 is 3\tfrac{1}{2}, not a whole number. And 5 - 3 \times 2 asks for 5 - 6, which is less than zero. Its value is a negative number, which belongs to the order of operations with negative numbers.
An agreement, not a fact about numbers. The order tells you how to read a written expression. It does not change what the numbers are. If a situation needs the addition done first, the expression has to be written differently, not read differently.
Where this goes next
The next step is Grouping symbols: parentheses, brackets and the fraction bar, which let you override the agreed order and say “do this part first”. With them, an expression can say “add 3 and 4, then multiply by 5” in a single line.
Going the other way, a story or a sentence can be turned into an expression, and an expression can be read back as words without working it out. That is writing and reading expressions from words.
Later the same order grows. Powers take a place before multiplying, and negative numbers join in, but the core stays the same: chunks first, then adding and subtracting, partners taken left to right. When letters stand in for numbers, 3 + 4 \times n follows exactly the same order, and that is the start of algebra.
Exercises
Part 1: expression or equation
Say whether each is an expression or an equation. Give the value of each expression, and say whether each equation is true or false.
- 84 \div 7
- 56 \div 8 = 7
- 85
- 90 - 35 = 65
Part 1: equal expressions
- Which of these name the same number as 9 \times 4? \;30 + 6, \;40 - 6, \;72 \div 2, \;3 \times 12, \;9 + 4
- Write four different expressions with the value 50: one that adds, one that subtracts, one that multiplies and one that divides.
- A prize at the arcade costs 75 tickets. Hyun says the prize costs 5 \times 15 tickets. Caio says it costs 100 - 25 tickets. Are both right? Write an equation that says so.
Part 2: using the order
Evaluate each expression.
- 5 + 6 \times 7
- 45 - 27 \div 9
- 9 \times 6 + 4
- 50 - 32 + 7
- 72 \div 9 \times 4
- 96 \div 8 \div 2
- Thao works out 28 - 4 \times 3 and gets 72. What did she do first? What is the value of the expression?
- At trivia night Hyun’s team starts the last round with 10 points and gets 3 questions right at 6 points each. The scorekeeper writes 10 + 3 \times 6. What total does it name? What would adding first give, and how many questions would that mean?
Part 3: longer expressions
Evaluate each expression.
- 4 + 9 \times 5 - 36 \div 6
- 100 - 7 \times 8 + 18 \div 3
- 6 \times 8 \div 4 + 30 - 2 \times 9
- 81 \div 9 \times 2 - 3 \times 4 + 10
Part 3: comparing without working it out
Which is greater, or are they equal? Explain without finding both values.
- 47 \times 6 + 12 or 47 \times 6 + 21
- 200 - 9 \times 15 or 200 - 9 \times 16
- 8 \times 99 + 8 or 8 \times 100
Reasoning
- Caio says that with only adding, as in 5 + 9 + 6, the order never matters. Is he right? Does the same hold for 30 - 9 - 6? Explain.
- Put one of +, -, \times, \div in each box so that 6 \;\square\; 4 \;\square\; 2 has the value 14. Is there more than one way?
- Hyun says the order of operations is a fact about numbers, like 2 + 2 = 4. Thao says it is an agreement. Who is right? Why does it matter that everyone uses the same order?
Answers
- An expression, with value 12. There is no equal sign, and 84 \div 7 = 12.
- An equation, and it is true: 56 \div 8 is 7, so both sides name the same number.
- An expression, with value 85. A single number is the simplest expression.
- An equation, and it is false: 90 - 35 = 55, not 65.
- 30 + 6, 72 \div 2 and 3 \times 12. The value of 9 \times 4 is 36. The others are 40 - 6 = 34 and 9 + 4 = 13.
- Many answers. For example 45 + 5, 60 - 10, 5 \times 10 and 100 \div 2. Each has the value 50.
- Yes. 5 \times 15 = 75 and 100 - 25 = 75, so the equation 5 \times 15 = 100 - 25 is true.
- 47. Multiply first: 5 + 42 = 47.
- 42. Divide first: 45 - 3 = 42.
- 58. 54 + 4 = 58.
- 25. Partners left to right: 50 - 32 = 18, then 18 + 7 = 25.
- 32. Left to right: 72 \div 9 = 8, then 8 \times 4 = 32.
- 6. Left to right: 96 \div 8 = 12, then 12 \div 2 = 6.
- She subtracted first: 28 - 4 = 24, then 24 \times 3 = 72. The value is 16, because the multiplication comes first: 28 - 12 = 16.
- 28 points: 10 + 18 = 28. Adding first would give 13 \times 6 = 78, which would mean 13 questions at 6 points, not 10 points and 3 questions.
- 43. The chunks are 4, 45 and 6, so 4 + 45 - 6 = 43.
- 50. The chunks are 100, 56 and 6, so 100 - 56 + 6 = 44 + 6 = 50.
- 24. The first chunk is 6 \times 8 \div 4 = 48 \div 4 = 12, and the last is 2 \times 9 = 18. Then 12 + 30 - 18 = 24.
- 16. The chunks are 81 \div 9 \times 2 = 9 \times 2 = 18, then 3 \times 4 = 12, then 10. So 18 - 12 + 10 = 6 + 10 = 16.
- 47 \times 6 + 21 is greater, by 9. Both start with the same chunk 47 \times 6, and 21 is 9 more than 12.
- 200 - 9 \times 15 is greater, by 9. Both start at 200, and 9 \times 16 takes away one more 9 than 9 \times 15 does.
- They are equal. By the commutative property, 8 \times 99 is 99 groups of 8. One more group of 8 makes 100 groups of 8, which is 8 \times 100.
- Caio is right about adding: by the commutative and associative properties of addition, 5 + 9 + 6 is 20 in any order. Subtracting is different. Left to right, 30 - 9 - 6 = 21 - 6 = 15, but doing 9 - 6 first would give 30 - 3 = 27. That is why subtraction is taken left to right.
- Only one way: 6 + 4 \times 2 = 6 + 8 = 14. There are sixteen ways to fill the two boxes, and every other one gives a value different from 14. For example 6 \times 4 - 2 = 22 and 6 + 4 + 2 = 12.
- Thao is right. Numbers do not choose an order; people agreed on one so a written expression names one number. If readers used different orders, 3 + 4 \times 5 would be 23 for some and 35 for others, and a written score or price could not be trusted. The order chosen is a sensible one, since it keeps 3 + 4 \times 5 equal to 3 + 5 + 5 + 5 + 5.