Translating between words and expressions
Five less than a number, is that n minus 5 or 5 minus n? And how do you say 3(n + 2) out loud so that the person listening writes down exactly that?
Where you are 5 in Variables
When you reach for it
Most mathematics reaches us in words before it reaches us in symbols. A price board says “a 12-dollar entry fee, then 5 dollars a route”. A friend says “bring twice as many as last time”. An expression says the same thing in a form you can calculate with, compare and, later, solve. Translating is the bridge, and it runs both ways: words into an expression when you meet a situation, and an expression back into words when you explain it or read it to someone who cannot see the page.
Reach for it whenever a quantity depends on a number that is unknown or can change: the cost of hiring a paddleboard for h hours, the routes climbed in a session, the area of a trampoline mat whose side has not been measured yet. The letter is the one from What a letter stands for: it holds a number, and the expression says what is done to that number.
Two things make this harder than swapping each word for a symbol. Some phrases name the numbers in the opposite order from the symbols: “5 less than n” is n - 5. And some phrases bundle several operations into one thing: “three times the sum of n and 2” is 3(n + 2). Both are about what the whole phrase means, which is why everything below asks what the words mean rather than which symbol a single word stands for.
The procedure
Part 1: words into expressions
Name the letter first. An expression only means something once you know what its letter stands for. “Let h be the number of hours” turns 15h into a cost; without that sentence, 15h is fifteen times a number nobody has named. In a bare phrase such as “the sum of p and 11” the letter is already given.
The four results and their names. Each operation has a name for its answer, and the phrase “the ... of a and b” names the two numbers in the order they are written.
| Words | Expression |
|---|---|
| the sum of a and b | a + b |
| the difference of a and b | a - b |
| the product of a and b | a \cdot b, or ab |
| the quotient of a and b | a \div b, or \displaystyle \frac{a}{b} |
A product of a number and a letter is written with the number first and no sign, so the product of 4 and c is 4c; the number in front is its coefficient (Terms, coefficients and constants). A quotient is usually written as a fraction, since \displaystyle \frac{a}{b} means a \div b (A fraction as division).
Order matters for two of the four. Swapping the two numbers of a sum or a product changes nothing, because addition and multiplication are commutative (Properties of addition, Properties of multiplication). Swapping them in a difference or a quotient changes the answer: taking 4 from 10 leaves 6, while taking 10 from 4 cannot be done with whole numbers at all, and 12 \div 3 = 4 while \displaystyle 3 \div 12 = \frac{1}{4}. So for subtraction and division the symbols must follow the order the meaning asks for.
“More than” and “less than” start from the second number. “Nine more than w” starts at w and goes up 9, so it is w + 9. “Nine less than w” starts at w and comes down 9, so it is w - 9. The number you start from is the one named last, so the symbols come out in the reverse order of the words. “Five less than n” is n - 5, never 5 - n: five less than 20 is 15, and that is 20 - 5. The same picture fits “5 subtracted from n”, “n decreased by 5” and “5 fewer than n”. Each starts at n and takes 5 away.
Doubling and halving. “Twice n” and “double n” are 2n, and “triple n” is 3n. “Half of n” is \displaystyle \frac{n}{2}, the same number as \displaystyle \frac{1}{2}n: cutting n into 2 equal parts gives parts of size \displaystyle \frac{n}{2}.
Sentences about a situation. A rate such as “9 dollars an hour” means 9 dollars for every hour, so h hours cost 9 times h, which is 9h. A fee charged once is added once. It is the situation, not any single word, that decides the operation: “4 dollars more” can mean a total that is 4 dollars higher or a rate that is 4 dollars higher per hour, and those are different expressions.
Part 2: grouping, powers and reading aloud
A phrase that treats a sum as one thing needs parentheses. In “three times the sum of n and 2”, the thing being tripled is the sum, n + 2. The Order of operations multiplies before it adds, so 3n + 2 would triple n alone and add 2 afterwards. To make the sum come first, it goes in parentheses:
Compare “the sum of three times n and 2”. Now the two things added are 3n and 2, and the order of operations already does the multiplying first, so no parentheses are needed:
The picture shows the difference. 3(n + 2) is three copies of a whole n + 2 bar, so it holds three 2s; 3n + 2 holds only one. A quotient groups the same way. “The sum of n and 4, divided by 3” divides the whole sum, and a fraction bar holds its top together like parentheses: \displaystyle \frac{n + 4}{3}. “4 more than the quotient of n and 3” divides n alone: \displaystyle \frac{n}{3} + 4.
Powers in words. “The square of n” is n^2 and “the cube of n” is n^3, with n as the base (Base, exponent and the value of a power). A power acts only on the symbol just below it, and powers come before multiplication and addition (Order of operations with powers). So “the square of the sum of n and 1” needs parentheses, (n + 1)^2, while “the sum of the square of n and 1” does not, n^2 + 1. In the same way “three times the square of n” is 3n^2, where only n is squared, and “the square of three times n” is (3n)^2.
Reading an expression aloud. Going the other way, the danger is in speech, where nobody can hear parentheses. Say “three times n plus two” and a listener may write 3n + 2 or 3(n + 2); the words fit both. What removes the doubt is to name the operation done last first, as the name of the whole thing, and then its two parts. In 3(n + 2) the last step is multiplying, and its parts are 3 and n + 2: “three times the sum of n and two”. In 3n + 2 the last step is adding, and its parts are 3n and 2: “the sum of three n and two”, or “two more than three times n”. Each of those can be written only one way. Another way to mark a group aloud is the word “quantity”: “three times the quantity n plus two”.
When nothing is grouped, the plain reading is already clear: n - 5 is “n minus five” and \displaystyle \frac{n}{4} is “n over four” or “n divided by four”.
Worked examples
Examples 1 to 4 go with Part 1, and Examples 5 to 7 with Part 2.
Naming the result
Write an expression for each phrase: (a) the sum of p and 11; (b) the product of 7 and t; (c) the quotient of m and 6; (d) the difference of 30 and k.
Solution(a) A sum adds, and the two numbers are p and 11, giving p + 11. Addition is commutative, so 11 + p is just as right.
(b) A product multiplies. With the number first and no sign, the product of 7 and t is 7t.
(c) A quotient divides the first-named number by the second: m \div 6, written \displaystyle \frac{m}{6}. Here the order is fixed, because \displaystyle \frac{6}{m} is a different number.
(d) A difference subtracts the second number from the first, giving 30 - k. Writing k - 30 would subtract in the wrong direction.
Order with more than and less than
Write an expression for each phrase: (a) 9 less than w; (b) 9 more than w; (c) w less than 9; (d) 9 subtracted from w; (e) half of w; (f) twice w. Check (a) with a number.
Solution(a) Start at w and come down 9: w - 9. To check, take w = 20. Nine less than 20 is 11, and 20 - 9 = 11, so the order is right.
(b) Start at w and go up 9: w + 9, or 9 + w, since addition is commutative.
(c) The same words with the two numbers swapped. Now the starting number is 9, and w is taken from it: 9 - w. This is a different expression from (a).
(d) 9 is taken away from w, so this is w - 9 again, the same as (a) in other words.
(e) Half of w is w shared into 2 equal parts: \displaystyle \frac{w}{2}.
(f) Twice w is 2 times w: 2w.
Entry fee and routes
A climbing wall charges a 12-dollar entry fee, paid once, and then 5 dollars for each route climbed. Mosi climbs r routes. Write expressions for (a) the cost of Mosi’s routes, (b) Mosi’s total cost, (c) the number of routes Nyla climbs, which is 3 fewer than Mosi, and (d) the number of routes Omid climbs, which is twice as many as Mosi.
Solution(a) Each route costs 5 dollars and there are r of them, so the routes cost 5 times r, which is 5r dollars.
(b) The entry fee is added once to the cost of the routes: 5r + 12 dollars. In words this is “12 more than five times r”, which names the numbers in the opposite order from the symbols. Writing 5(r + 12) would charge the entry fee as if it were 12 more routes, which the price board never said.
(c) Three fewer than r starts at r and takes 3 away: r - 3 routes. This counts routes, not dollars, and only makes sense if Mosi climbed at least 3 routes.
(d) Twice r is 2r routes.
Hiring a paddleboard
A lake hires out paddleboards at 15 dollars an hour. Sela hires one for h hours. Write expressions for (a) the cost, (b) what Sela pays with a voucher for 10 dollars off the total, (c) her time on the water in minutes, and (d) the hours Tycho is out, which is 2 hours fewer than Sela.
Solution(a) 15 dollars for each of h hours is 15h dollars.
(b) The voucher comes off the total once, so Sela pays 10 less than 15h: 15h - 10 dollars. The order matters. The voucher is subtracted from the cost, not the cost from the voucher.
(c) Each hour is 60 minutes, so h hours is 60 times h: 60h minutes. A letter carries no unit by itself; the unit comes from the sentence that names it, which is why h was named in hours.
(d) Two fewer than h is h - 2 hours.
Parentheses from the words
Write an expression for each phrase: (a) four times the sum of q and 7; (b) the sum of four times q and 7; (c) the difference of q and 3, divided by 5; (d) 3 less than the quotient of q and 5; (e) twice the difference of 10 and q.
Solution(a) What is multiplied by four is the sum, q + 7, as one thing. Parentheses make the sum happen first: 4(q + 7).
(b) The two things added are four times q and 7. The multiplication already comes first by the order of operations, so no parentheses are needed: 4q + 7.
(c) The whole difference q - 3 is divided by 5. Written as a fraction, the bar keeps the top together: \displaystyle \frac{q - 3}{5}.
(d) Start from the quotient of q and 5, which is \displaystyle \frac{q}{5}, and take 3 away from it: \displaystyle \frac{q}{5} - 3. Only q is divided.
(e) The difference of 10 and q is 10 - q, and twice that whole difference is 2(10 - q).
Squares on a trampoline
Write expressions for (a) twice the square of y; (b) the square of twice y; (c) the cube of the sum of y and 1. Then: (d) a trampoline park has square mats with sides of s feet. Write the area of one mat, (e) the area covered by a mat and the padded frame, 1 foot wide, that runs all the way round it, and (f) the total mat area of 6 mats.
Solution(a) Only y is squared, and that square is doubled: 2y^2. The power acts on y alone, before the multiplying.
(b) Now twice y, which is 2y, is squared as one thing, so it needs parentheses: (2y)^2.
(c) The sum y + 1 is cubed as one thing: (y + 1)^3.
(d) The area of a square is its side times itself, so one mat covers s^2 square feet.
(e) The frame adds 1 foot at both ends of every side, so the outer square has side s + 2, not s + 1. Its area is the square of that whole side: (s + 2)^2 square feet. Without the parentheses, s + 2^2 would square only the 2, and it is not the area of any square here.
(f) Each mat covers s^2 square feet, and there are 6: 6s^2 square feet. Only s is squared, because the 6 counts mats, not feet.
Saying it so it is written right
A trampoline park charges 8 dollars for each jump session, plus 3 dollars for a pair of grip socks, bought once. Tycho and his sister each do j sessions and each buy socks. Write the total they pay, and say it over the phone so that Omid writes down the same expression. Then say \displaystyle \frac{j + 6}{4} and \displaystyle \frac{j}{4} + 6 so that a listener can tell them apart.
SolutionOne person pays 8j + 3 dollars: 8 dollars for each of j sessions, and the socks once. Two people pay twice that whole amount, so the total is 2(8j + 3) dollars.
To say it, name the last operation first. The last step is doubling, and the thing doubled is the sum 8j + 3: “twice the sum of eight j and three”, or “two times the quantity eight j plus three”. Saying “two times eight j plus three” would not do, because Omid could fairly write 2 \cdot 8j + 3, which counts the socks only once.
In \displaystyle \frac{j + 6}{4} the last step is dividing, and what is divided is the sum: “a quarter of the sum of j and six”. In \displaystyle \frac{j}{4} + 6 the last step is adding, and the parts are \displaystyle \frac{j}{4} and 6: “six more than a quarter of j”. Each sentence can be written down only one way.
Boundaries
Translating keeps the meaning of the words exactly; it cannot add meaning they do not have. A phrase that fits two expressions, like “three times n plus two”, has no single right translation. The fix is to ask, or to say which reading you chose, not to guess silently.
A phrase becomes an expression. A sentence with a verb such as “is” or “equals”, like “five less than n is 12”, makes a claim, which is an equation or an inequality, not an expression (Expression, equation and inequality).
An expression is only as meaningful as the sentence that names its letter. 15h is a cost in dollars only because h was named as a number of hours and 15 as dollars per hour.
Some expressions make sense only for some values of the letter. Without negative numbers, 9 - w needs w to be at most 9, and r - 3 routes needs at least 3 routes. A quotient such as \displaystyle \frac{60}{g} needs g not to be 0, since nothing can be divided by zero. The story usually tells you which values are possible.
Where this goes next
Once a situation is an expression, putting a number in place of the letter gives the value for that case, which is evaluating an expression. Turning a whole sentence into symbols gives an equation, and a one-step equation is the first kind to solve; a condition such as “at most 20 dollars” becomes an inequality. Two phrases that always give the same number, like “three times the sum of n and 2” and “six more than three times n”, are equivalent expressions, and showing why is a later lesson. An expression with one letter is also a rule that turns each input into one output, which is the Functions as input-output rules.
Exercises
Part 1: naming results and order
- Write an expression for the sum of d and 13.
- Write an expression for the product of 12 and x.
- Write an expression for the quotient of y and 8.
- Write an expression for the difference of y and 14.
- Write an expression for 6 less than b.
- Write an expression for b less than 6.
- Write an expression for half of z.
- Write an expression for 1 less than twice z.
Part 1: situations
- A climbing wall charges 7 dollars, once, to hire shoes, and 3 dollars for each route. Write the cost in dollars of climbing r routes.
- A shop hires out paddleboards at p dollars an hour. Write (a) the cost of 4 hours, and (b) the price for an hour after it rises by 2 dollars.
- Each jump session at a trampoline park lasts 45 minutes. Write (a) the minutes spent jumping in s sessions, and (b) the number of sessions Nyla does if she does 2 more than s.
- A group of g people shares a 60-dollar bill equally. Write what each person pays.
Part 2: grouping and powers
- Write an expression for five times the difference of k and 2.
- Write an expression for the difference of five times k and 2.
- Write an expression for the sum of m and 9, divided by 4.
- Write an expression for the square of the sum of a and 6.
- Write an expression for 6 more than the square of a.
- Write an expression for three times the cube of w.
- Tycho and 2 friends each pay a 9-dollar entry fee at a climbing wall, and each climb r routes at 4 dollars a route. Write the total the three of them pay.
- Sela hires a paddleboard at 12 dollars an hour, for 2 hours more than h. Write the cost.
Part 2: reading aloud
- Say 7(x - 1) in words so that a listener writes exactly that.
- Say \displaystyle \frac{x}{3} + 8 in words so that a listener writes exactly that.
- Say 2x^3 and (2x)^3 in words so that a listener can tell them apart.
Reasoning
- A message reads “four times n minus one”. Write the two expressions it could mean, and for each, a way of saying it that has only one meaning.
- Explain why “the quotient of n and 5” and “the quotient of 5 and n” are different expressions, using n = 10. Then explain why “the product of n and 5” and “the product of 5 and n” are the same.
Answers
- d + 13. A sum adds the two numbers named; 13 + d is equally right.
- 12x. A product, with the number written first and no sign.
- \displaystyle \frac{y}{8}. The quotient divides the first-named number by the second, y \div 8.
- y - 14. The difference subtracts the second-named number from the first.
- b - 6. Start at b and take 6 away.
- 6 - b. Now the start is 6 and b is taken away from it.
- \displaystyle \frac{z}{2}. Half of z is z shared into 2 equal parts.
- 2z - 1. Twice z is 2z; one less than that takes 1 away from it.
- 3r + 7 dollars. The routes cost 3 times r, and the shoes are added once.
- (a) 4p dollars, 4 hours at p dollars each. (b) p + 2 dollars, the old price raised by 2.
- (a) 45s minutes, 45 for each of s sessions. (b) s + 2 sessions.
- \displaystyle \frac{60}{g} dollars. The bill is divided by the number of people, so 60 comes first; g cannot be 0.
- 5(k - 2). The whole difference is multiplied by five, so it goes in parentheses.
- 5k - 2. The two parts are 5k and 2; multiplication already comes first.
- \displaystyle \frac{m + 9}{4}. The whole sum is divided, and the fraction bar keeps it together.
- (a + 6)^2. The sum is squared as one thing, so it needs parentheses.
- a^2 + 6. Only a is squared, then 6 is added.
- 3w^3. Only w is cubed, and that cube is tripled.
- 3(4r + 9) dollars. One person pays 4r + 9, and three people pay three times that whole amount.
- 12(h + 2) dollars. The hours are h + 2, and every one of them costs 12 dollars.
- “Seven times the difference of x and one.” The last step is multiplying, and the thing multiplied is the difference x - 1.
- “Eight more than the quotient of x and three”, or “eight more than a third of x”. The last step is adding, and its parts are \displaystyle \frac{x}{3} and 8.
- 2x^3 is “twice the cube of x”: only x is cubed. (2x)^3 is “the cube of twice x”: all of 2x is cubed.
- It could mean 4n - 1, said as “one less than four times n”, or 4(n - 1), said as “four times the difference of n and one”. Each version names the last operation first, so it has only one reading.
- With n = 10, the quotient of n and 5 is \displaystyle \frac{10}{5} = 2, but the quotient of 5 and n is \displaystyle \frac{5}{10} = \frac{1}{2}. One value of n giving different answers is enough to show the expressions differ, because division depends on order. Multiplication is commutative, so 5n and n \cdot 5 are the same number for every n.