Writing and reading numerical expressions from words
Someone says “add 8 and 7, then multiply by 2”. How do you write that as one line that everyone reads the same way, and how do you read a line back as words without working it out?
Where you are 5 in Order of operations
When you reach for it
At the school scavenger hunt, every clue a team finds is worth points. Minh’s team finds 8 clues in the gym and 7 clues in the library. Then the leader says: “Your clues count double today.” So the team adds 8 and 7, then multiplies by 2.
You could write the work as two lines, 8 + 7 = 15 and then 15 \times 2 = 30. But one line is shorter, and it shows the whole plan at once:
The parentheses matter. Without them, 2 \times 8 + 7 multiplies first, as Numerical expressions agreed, and names 16 + 7 = 23. Only the 8 gym clues would count double. With the parentheses from Grouping symbols, the 8 + 7 is one number, and all 15 clues count double.
You reach for this skill in two directions:
- Words into an expression. A sentence or a story says what to do. You write one expression that does exactly that, in that order.
- An expression into words. You read an expression and say what it means, often without working it out. “2 \times (8 + 7) is twice as large as 8 + 7” needs no adding at all.
The lesson takes three days. Part 1 turns a phrase into an expression, and shows when parentheses are needed. Part 2 reads an expression in words and compares two expressions without evaluating them. Part 3 writes expressions for stories with several steps, and finds two different expressions for the same story.
The procedure
Part 1: from a phrase to an expression
Follow what the words ask you to do. A phrase such as “add 8 and 7, then multiply by 2” is a plan. The expression has to carry out the same plan. So the question is never “which word means which sign”. It is “what is done first, and what is done to the result?”
A finished step is one number. The word “then” says the work so far is finished and is now one number. In “add 8 and 7, then multiply by 2”, the thing being multiplied is the whole 8 + 7. If the expression would not do that part first on its own, it needs parentheses around it. The agreed order already does multiplying and dividing before adding and subtracting. So:
- An addition or subtraction that must happen before a multiplication or division needs parentheses: “add 8 and 7, then multiply by 2” is 2 \times (8 + 7) or (8 + 7) \times 2.
- A multiplication or division that happens first needs none: “multiply 8 by 7, then add 2” is 8 \times 7 + 2, because the agreed order multiplies first anyway. Parentheses there, (8 \times 7) + 2, are allowed but change nothing.
Either way of writing 2 \times (8 + 7) or (8 + 7) \times 2 is correct. By the commutative property of multiplication, the two factors may be swapped.
The order of the words is not always the order of the numbers. Read what each phrase means.
- “Subtract 5 from 12” starts with 12 and takes 5 away. It is 12 - 5, even though the 5 is said first. The number you subtract from is written first. In the same way, “5 less than 12” starts at 12 and goes 5 lower, so it is also 12 - 5.
- “Divide 12 by 4” splits 12 into groups of 4, or into 4 equal parts. It is 12 \div 4. The number being divided is written first.
- “Add 8 and 7” and “multiply 8 by 7” may be written in either order, because adding and multiplying are commutative.
Results have names. The answer to adding is a sum, to subtracting a difference, to multiplying a product, and to dividing a quotient. “The sum of 8 and 7” is 8 + 7. “The difference of 12 and 5” is 12 - 5, in that order. “The quotient of 12 and 4” is 12 \div 4. “Twice” a number is 2 times it, and “half of” a number is that number divided by 2. These names are not clues to hunt for. They are ordinary words, and the sentence around them still has to be read for its meaning.
A name for a result can hold a whole group. “Three times the sum of 8 and 7” multiplies the sum by 3, so the sum is one number: 3 \times (8 + 7).
Part 2: reading an expression in words
Read the last step first. Every expression has one step that is done last: the operation that sits outside all the groups and comes last in the agreed order. That last step says what kind of number the expression is.
- In 2 \times (8 + 7) the adding is inside the parentheses, so the multiplying is last. The expression is a product: “two times the sum of 8 and 7”.
- In 2 \times 8 + 7 the multiplying comes first, so the adding is last. The expression is a sum: “7 more than two times 8”.
You can also read an expression as a plan in time: “add 8 and 7, then multiply by 2”. Both readings say the same thing.
Comparing without evaluating. Reading the last step lets you compare expressions without working them out. If one expression is a number times another whole expression, it is that many times as large. Take 3 \times (18{,}932 + 921): it is three copies of 18{,}932 + 921, so it is three times as large as 18{,}932 + 921, whatever that sum is. This is the same idea as Times as many, with an expression in place of a single number. In the same way, an expression divided by 2 is half as large as the expression.
When two expressions share most of their parts, compare only the parts that differ. Figure 1, in Example 5, shows why multiplying a whole group makes copies of every part in it.
Part 3: expressions for stories
Name the quantities, then build from the inside out. A story with several steps has some quantities that are found first and used later. Each such quantity is a piece of the expression. If a piece is a sum or a difference and is then multiplied or divided, it goes in parentheses. A piece inside another piece may need brackets, as in Grouping symbols.
Check the expression against the story. Read the expression back in words, using Part 2. If the words match the story, the expression is right. If they tell a different story, something is grouped wrongly.
One story, many expressions. A story can often be counted in more than one way, and each way gives its own expression. They look different, but they count the same thing, so they have the same value. Two expressions that tell different stories usually have different values.
Worked examples
Examples 1 to 3 go with Part 1, Examples 4 and 5 with Part 2, and Examples 6 and 7 with Part 3.
Phrases with and without parentheses
Write each phrase as one expression. Then find its value.
(a) Add 6 and 9, then multiply by 4.
(b) Subtract 5 from 32, then divide by 3.
(c) Multiply 7 by 3, then add 10.
Solution(a) The adding is finished first, so 6 + 9 is one number, and that number is multiplied by 4. Multiplying would normally go first, so the sum needs parentheses:
Without them, 6 + 9 \times 4 = 6 + 36 = 42 multiplies only the 9. That is a different plan.
(b) “Subtract 5 from 32” starts at 32 and takes away 5, so it is 32 - 5. That difference is then divided by 3. Dividing would normally go first, so the difference needs parentheses:
(c) The multiplying is done first, and the agreed order multiplies first anyway. No parentheses are needed:
Writing (7 \times 3) + 10 is also correct. The parentheses change nothing here.
Phrases that name results
Write each phrase as one expression, and find its value.
(a) The product of 5 and the sum of 12 and 8.
(b) The difference of 40 and the quotient of 18 and 6.
(c) Divide the sum of 14 and 21 by 7.
Solution(a) A product multiplies two things. Here they are 5 and “the sum of 12 and 8”. The sum is one number, so it goes in parentheses:
(b) A difference subtracts the second thing from the first. The first is 40. The second is “the quotient of 18 and 6”, which is 18 \div 6. So the expression is 40 minus that quotient:
The agreed order divides first, so no parentheses are needed. Writing 40 - (18 \div 6) shows the quotient as one piece, and it has the same value.
(c) The sum of 14 and 21 is divided, so the whole sum must be one number:
A plan with three steps
At the scavenger hunt, Yadira works out a score with this plan: “Add 3 and 5, multiply by 4, subtract 2, then multiply by 10.” Write the plan as one expression and find its value.
SolutionBuild the expression one step at a time. Each step takes the expression so far as one number.
The first step is 3 + 5. The next step multiplies that sum by 4, so the sum needs parentheses: 4 \times (3 + 5).
Then 2 is subtracted: 4 \times (3 + 5) - 2. No new group is needed, since the agreed order does the multiplying before the subtracting anyway.
Last, the whole thing so far is multiplied by 10. The whole thing is a difference, and multiplying would normally go first, so it needs a group of its own. It already has parentheses inside it, so brackets go around it:
Work it from the inside out:
Without the brackets, 10 \times 4 \times (3 + 5) - 2 would subtract the 2 only after multiplying by 10, giving 320 - 2 = 318. The brackets keep the 2 inside the part that is multiplied.
Reading expressions in words
Read each expression in words, and say whether it is a sum, a difference, a product or a quotient. Do not work it out.
(a) 6 \times (15 - 9); (b) 6 \times 15 - 9; (c) (45 - 5) \div 8
Solution(a) The subtracting is inside the parentheses, so it is done first. The multiplying is last, so the expression is a product. In words: “subtract 9 from 15, then multiply by 6”, or “six times the difference of 15 and 9”. It is six times as large as 15 - 9.
(b) No parentheses, so the agreed order multiplies first and subtracts last. The expression is a difference. In words: “multiply 6 by 15, then subtract 9”, or “9 less than the product of 6 and 15”. It is 9 less than 6 \times 15.
(c) The parentheses come first, and the dividing is last. The expression is a quotient. In words: “subtract 5 from 45, then divide by 8”, or “the difference of 45 and 5, divided by 8”. It is one eighth of 45 - 5.
Parts (a) and (b) use the same numbers and the same signs. Only the parentheses differ, and they change which step is last, so they change what the expression is.
Comparing without evaluating
At the end of the hunt, Uche’s team has 2{,}365 + 148 points. Answer each part without working out the sums.
(a) Minh’s team has 4 \times (2{,}365 + 148) points. How does that compare with Uche’s team?
(b) Yadira’s team has (2{,}365 + 148) \div 2 points. How does that compare with Uche’s team?
(c) Which is greater, 5 \times (63 + 28) or 5 \times 63 + 28, and by how much?
Solution(a) The last step of 4 \times (2{,}365 + 148) is multiplying by 4, and what is multiplied is the whole of Uche’s score. Figure 1 shows this as four copies of the whole bar.
So Minh’s team has four times as many points as Uche’s team. No adding was needed to say so.
(b) The last step of (2{,}365 + 148) \div 2 is dividing by 2. What is divided is again the whole of Uche’s score. Splitting it into 2 equal parts gives half of it, so Yadira’s team has half as many points as Uche’s team.
(c) Picture the first expression as 5 copies of the bar 63 + 28. Each copy holds a 63 and a 28, so there are 5 sixty-threes and 5 twenty-eights. The second expression has the same 5 sixty-threes but only 1 twenty-eight. So 5 \times (63 + 28) is greater, by 4 twenty-eights:
As a check, 5 \times 91 = 455 and 315 + 28 = 343, and 455 - 343 = 112.
Seeing that five copies of a sum hold five copies of each part is the distributive property:
The second expression, 5 \times 63 + 28, is missing 4 of those 28s.
A story with a group
Uche’s team finds 7 clues in the park and 5 clues at the library. Each clue is worth 15 points. Then the team spends 40 points to buy a hint. Write one expression for the team’s points now, and find its value.
SolutionThe quantities found first are the clues: 7 + 5 of them. Every one of those clues is worth 15 points, so the whole count is multiplied by 15, and the sum needs parentheses: 15 \times (7 + 5). Then 40 points are taken away from that score:
The team has 140 points.
Read it back to check. The last step is subtracting, so the expression is “40 less than fifteen times the sum of 7 and 5”. That is the story: the clue points, then the hint taken away.
The expression 15 \times 7 + 5 - 40 tells a different story. It gives 15 points to each park clue but only 1 point to the whole library trip, and has the value 105 + 5 - 40 = 70.
Two expressions for one story
Yadira’s team has 4 members. Each member finds 3 gold clues worth 10 points each and 2 silver clues worth 5 points each. Write two different expressions for the team’s total, and show that they have the same value. Then say why 4 \times 3 \times 10 + 2 \times 5 is wrong.
SolutionMember by member. One member scores 3 \times 10 for gold and 2 \times 5 for silver, so 3 \times 10 + 2 \times 5. There are 4 members, and each scores that whole amount, so it goes in parentheses:
Clue by clue. The team finds 4 \times 3 gold clues in all, and 4 \times 2 silver clues. The gold clues are worth (4 \times 3) \times 10 and the silver clues (4 \times 2) \times 5:
Both expressions count the same clues, member by member or color by color, so both name 160 points. The parentheses in the second expression change nothing, since the agreed order multiplies first anyway, but they show the two clue counts.
The expression 4 \times 3 \times 10 + 2 \times 5 multiplies the gold points by 4 and leaves the silver points alone. It counts the silver clues of only one member. Its value is 120 + 10 = 130, which is 30 points short.
Boundaries
Subtracting stays at or above zero. “Subtract 12 from 5” asks for 5 - 12, which is less than zero. With whole numbers there is no such value. That belongs to the order of operations with negative numbers.
No dividing by zero. “Divide 8 by the difference of 6 and 6” is 8 \div (6 - 6), which is 8 \div 0. No number does that, so the expression has no value.
Spoken words can be unclear. “Add 3 and 4 times 5” can be heard two ways. It could mean 3 + 4 \times 5 or (3 + 4) \times 5. A careful phrase removes the doubt with “then”, or with a name for a result: “add 3 and 4, then multiply by 5”, or “the sum of 3 and the product of 4 and 5”. When a phrase is unclear, no expression can fix it. Ask which plan is meant.
Numbers only. Every expression here is made of numbers, so it names one value. Expressions with letters standing for numbers that can change come later.
Equal value does not mean the same story. Two expressions with the same value may still tell different stories: 3 \times 4 and 10 + 2 are both 12. Part 3 finds expressions that count the same thing, which is a stronger link than having the same value.
Where this goes next
The next step puts letters in place of numbers. “Add 8 to a number, then multiply by 2” becomes an expression with a letter in it, and the same reading of “then” decides the parentheses, in writing expressions from words with letters.
Reading the last step of an expression grows into naming every part of it and saying what each part means in a story, in Interpreting the parts of an expression. And the two expressions for one story in Example 7 grow into choosing the form of an expression that shows a fact most clearly, in Rewriting an expression to show what it means.
Exercises
Part 1: phrases into expressions
Write each phrase as one expression, and find its value.
- Add 9 and 4, then multiply by 6.
- Subtract 8 from 30, then divide by 2.
- Multiply 9 by 5, then subtract 7.
- Divide 48 by 6, then add 12.
- Three times the sum of 11 and 14.
- Divide the difference of 50 and 14 by 9.
- Add 2 and 5, multiply by 3, then subtract the result from 40.
- Minh writes “subtract 6 from 20, then multiply by 3” as 20 - 6 \times 3. What number does his expression name? What should he write, and what is its value?
Part 2: expressions into words
Write each expression in words, and say whether it is a sum, a difference, a product or a quotient. Do not work it out.
- 5 \times (13 + 7)
- (36 - 6) \div 3
- 8 \times 6 + 2 and 8 \times (6 + 2)
Part 2: comparing without evaluating
Answer without working out the sums and differences.
- How does 6 \times (3{,}418 + 507) compare with 3{,}418 + 507?
- How does (12{,}630 - 1{,}875) \div 5 compare with 12{,}630 - 1{,}875?
- Which is greater, 9 \times (245 + 38) or 9 \times (245 + 83), and by how much?
- Which is greater, 4 \times (70 + 15) or 4 \times 70 + 15, and by how much?
- Which is greater, (864 + 96) \div 3 or (864 + 96) \div 4? Explain.
Part 3: stories
- Each clue is worth 20 points. Minh’s team finds 6 clues in the morning and 4 in the afternoon. Write one expression for the team’s points, and find its value.
- Yadira’s team has 150 points. It spends 10 points on each of 3 hints, then earns a 25 point bonus for finishing first. Write one expression for its points now, and find its value.
- Three teams find 18, 15 and 21 clues. All the clues are collected and split equally among 6 hiding spots for next year. Write one expression for the number of clues in each spot, and find its value.
- The prize is 96 stickers. They are shared equally among 3 teams, and each team shares its stickers equally among its 4 members. Write two different expressions for the number of stickers each member gets, and show they have the same value.
- Write a scavenger hunt story that 5 \times (12 - 4) fits.
- Minh’s team finds 5 gold clues worth 10 points each and 7 silver clues worth 4 points each, then loses 6 points for coming back late. Uche writes the score as 5 \times 10 + 7 \times 4 - 6. Yadira writes (5 \times 10 + 7 \times 4) - 6. Who is right? Find the score.
Reasoning
- “Subtract 5 from 12” names the 5 first, but the expression is 12 - 5. Explain why the 12 comes first.
- Uche says “multiply 4 by 5, then add 3” and “add 3 to the product of 4 and 5” give the same expression, even though the words come in a different order. Is he right? Explain.
- Write each phrase as an expression: (a) “divide 20 by 4, then add 1”; (b) “add 4 and 1, then divide 20 by the result”. Explain why only one of them needs parentheses, and find both values.
Answers
- (9 + 4) \times 6 = 13 \times 6 = 78. The sum is multiplied, so it needs parentheses.
- (30 - 8) \div 2 = 22 \div 2 = 11. Subtracting 8 from 30 is 30 - 8, and that difference is divided.
- 9 \times 5 - 7 = 45 - 7 = 38. The agreed order multiplies first, so no parentheses are needed.
- 48 \div 6 + 12 = 8 + 12 = 20. The agreed order divides first.
- 3 \times (11 + 14) = 3 \times 25 = 75. The whole sum is tripled.
- (50 - 14) \div 9 = 36 \div 9 = 4.
- 40 - 3 \times (2 + 5) = 40 - 21 = 19. “Subtract the result from 40” starts at 40. Brackets around 3 \times (2 + 5) are allowed but not needed.
- His expression names 20 - 18 = 2, because it multiplies only the 6. He should write (20 - 6) \times 3 = 14 \times 3 = 42.
- “Add 13 and 7, then multiply by 5”, or “five times the sum of 13 and 7”. A product. It is five times as large as 13 + 7.
- “Subtract 6 from 36, then divide by 3”, or “the difference of 36 and 6, divided by 3”. A quotient. It is one third of 36 - 6.
- 8 \times 6 + 2 is “multiply 8 by 6, then add 2”, a sum, 2 more than 8 \times 6. 8 \times (6 + 2) is “add 6 and 2, then multiply by 8”, a product, eight times 6 + 2.
- It is six times as large: the whole sum is multiplied by 6.
- It is one fifth as large: the whole difference is split into 5 equal parts.
- 9 \times (245 + 83) is greater. Both are 9 copies of a sum that starts with 245, and 83 is 45 more than 38, so the second is greater by 9 \times 45 = 405.
- 4 \times (70 + 15) is greater. It has 4 copies of 15 and the other has 1, so it is greater by 3 \times 15 = 45.
- (864 + 96) \div 3 is greater. The same total is split into 3 equal parts or 4 equal parts, and fewer parts means each part is larger. (Their values are 320 and 240.)
- 20 \times (6 + 4) = 20 \times 10 = 200 points.
- 150 - 10 \times 3 + 25 = 150 - 30 + 25 = 145 points. The hints cost 10 \times 3 in all, and the agreed order multiplies first.
- (18 + 15 + 21) \div 6 = 54 \div 6 = 9 clues. The whole sum is divided, so it needs parentheses.
- Team by team: 96 \div 3 \div 4 = 32 \div 4 = 8. Member by member: there are 3 \times 4 members in all, so 96 \div (3 \times 4) = 96 \div 12 = 8. Both give 8 stickers.
- One story: each of 5 teams has 12 clues to find, and by lunch each team has found 4. How many clues are still hidden in all? Each team has 12 - 4 left, and there are 5 teams. Any story works if a difference of 12 and 4 is found first and then taken 5 times. The value is 5 \times 8 = 40.
- Both are right. The parentheses go around a part the agreed order would do first anyway. The score is 50 + 28 - 6 = 72 points.
- Subtracting 5 from 12 means starting with 12 and taking 5 away. An expression writes the starting amount first and then what is taken away, so it is 12 - 5. Writing 5 - 12 would mean starting with 5 and taking 12 away, a different plan.
- Yes. Both phrases say: find the product of 4 and 5, then add 3 to it. The words name the steps in a different order, but the plan is the same. Both are 4 \times 5 + 3, or 3 + 4 \times 5 by the commutative property of addition, and the agreed order multiplies first.
- (a) 20 \div 4 + 1 = 5 + 1 = 6. (b) 20 \div (4 + 1) = 20 \div 5 = 4. In (a) the dividing is first, and the agreed order divides first anyway. In (b) the adding must be first, and the agreed order would divide first, so the sum needs parentheses.