Turning a sentence into an equation

Every equation you have solved so far arrived already written. Lesson 3 handed you 4(x - 2) = 2x + 6. Lesson 4 handed you the phone plan. In real use nobody hands you anything: there's a situation, described in words, and somebody has to write the equation before any of the machinery can start.

That step is where the most reliable error in the whole subject lives, and this lesson is about it.

Try it first

The problem was put to engineering freshmen at an American university by John Clement and his colleagues, in this form.

At a university there are six times as many students as professors. Write an equation using S for the number of students and P for the number of professors.

Write your answer down before reading on. This matters more than usual here, because the point of the lesson is what your own hand does.

Why the wrong answer is so attractive

Read the sentence and watch the order in which the pieces arrive.

six times as many students as professors

Six, then students, then professors. Write those down in that order and you get 6S = P. The equation is a transcription of the English word order, and that's the whole mechanism. It is called word-order matching, and it's not stupidity; it's a reasonable strategy that happens to be wrong, because the order of words in a sentence and the structure of an equation are different things that look alike.

There is a second mechanism behind the same wrong answer, and it's worth knowing because people defend the answer with it. It is called static comparison. You picture the room: six students standing next to each professor. The 6 goes with the students because there are six of them in the picture. The trouble is that an equation isn't a caption on a picture. It's a claim that one number is equal to another number, and it is either true or false when you put the numbers in.

The check that takes ten seconds

If you forget everything else in this lesson, keep this habit.

Put numbers in.

Pick a small, easy number for one quantity, work out the other from the sentence in plain English, and then see whether your equation holds.

Suppose there are 6 professors. The sentence says six times as many students, so there are 36 students.

Now test both candidate equations against those two numbers.

That settles it, and it settles it without any argument about what the words feel like. The correct equation is S = 6P.

The picture is the point. The bar for the bigger group is the longer one, and the 6 has to sit on the side of the smaller number to lift it up to the size of the larger. The multiplier goes with the group there are fewer of, which feels backwards until you've said it out loud a few times.

Say what the letter is, in full

There's a habit that prevents most of this trouble before it starts, and it costs one line of writing.

Write down what each letter stands for, as a number, in words.

Not "S is students". That's a label on a thing, which is lesson 1's misconception coming back for a second try. Write:

• S is the number of students.
• P is the number of professors.

Once the letters are numbers, the question "is 216 equal to 6" is obviously answerable, and the question "does the six go with the students" stops being askable, because six times a number of students is a number of students and not a number of professors.

A worked translation, start to finish

The situation. A jacket costs twelve dollars more than three times the price of a shirt.

Step 1. Name the letters as numbers. Let s be the price of the shirt in dollars, and j the price of the jacket in dollars.

Step 2. Find the relationship in plain English. The jacket's price is worked out from the shirt's price: take the shirt price, triple it, add twelve.

Step 3. Write it. j = 3s + 12

Step 4. Test it with numbers. Say the shirt costs $20. Tripling gives 60, plus 12 gives $72. Substituting into the equation: 72 = 3 × 20 + 12, so 72 = 72. It holds.

Step 5. Test a wrong version too, so that you can see the test doing work. A hand that writes the numbers in the order the words arrive might produce j + 12 = 3s, putting the twelve next to the jacket because the sentence mentions the jacket first. With the same numbers: 72 + 12 = 3 × 20, so 84 = 60. False, immediately, with no debate about the English.

Notice that step 4 is doing something step 3 cannot do for itself. Writing the equation uses your reading of the sentence, and it's exactly your reading of the sentence that may be at fault. Substituting brings in a fresh piece of evidence.

The same trap in subtraction

Multiplicative comparisons are where the error was first measured, but the identical thing happens with "more than" and "fewer than", and there it catches people who would never fall for the student-professor problem.

Maya is seven years younger than Ben.

The word order offers M - 7 = B. Test it. If Ben is 30, Maya is 23, and 23 - 7 = 30 says 16 equals 30. False.

The correct equation is M = B - 7, and testing gives 23 = 30 - 7, which holds.

The English is doing something sly here. "Maya is seven years younger" mentions Maya and seven close together, so the hand wants to put them close together in the equation. But the seven is the size of the gap, and the gap has to be taken off the larger number, not off Maya's.

Translating a whole problem, then solving it

Most real problems don't stop at the translation. They give you a relationship and a total, and want a number out.

The situation. A theatre sold three times as many standard tickets as concessions, and 480 tickets in all. How many of each?

Name the letters. Let c be the number of concession tickets. Then the number of standard tickets is 3c, because there are three times as many, and testing that with 10 concessions gives 30 standard, which is right.

Naming one letter and writing everything else in terms of it is a move worth having. It keeps you to one unknown, which is all lesson 3 can solve.

Write the second fact. The two kinds together make 480, so 3c + c = 480.

Solve. 4c = 480, so c = 120.

Answer the question that was asked. 120 concessions, and standard is 3 × 120, so 360.

Check against the original sentence, not against your equation. Is 360 three times 120? Yes. Do they add to 480? Yes. Both facts hold, so the answer is right.

Checking your answer in your own equation only tells you that your arithmetic was sound. Checking it against the words tells you that your translation was sound, and the translation is the risky part.

An honest note about this error

It would be easy to read the Clement study and conclude that the human mind is built to get this wrong. The evidence says otherwise.

A 2021 study compared 211 Spanish and 79 South African trainee primary teachers on the same kind of problem. Nearly all of the errors the Spanish group made were reversals. The South African group barely made the error at all. The authors put the difference down to how the two groups had been taught, not to anything about the people.²

So this is a trap that particular teaching builds, and other teaching seems not to. If you wrote 6S = P at the top of this lesson, what that tells you is something about the algebra classes you sat in, and nothing whatever about your capacity for the subject. The habit that fixes it's the numerical check.

What people get wrong

Copying the word order. The order words arrive in an English sentence has no reliable relationship to where symbols go in an equation. This is the main event, and the check is numbers.

Letting a letter stand for a thing. "S is students" invites you to think about students. "S is the number of students" invites you to think about a number, and numbers can be tested.

Reading the equation as a picture. Six students clustered around one professor is a fine mental image and a bad reason to write 6S = P. An equation claims that two numbers are equal.

Checking the answer in your own equation only. If your translation is wrong, your equation will happily confirm a wrong answer. Take the numbers back to the sentence.

Forgetting to answer the question. The theatre problem asks for both ticket counts. Solving gives c = 120 and stopping there answers half of it.

Connections

Lesson 1 insisted that a letter stands for a number rather than a thing. This lesson is where that pays: naming S as the number of students is what makes the numerical check possible.

Lesson 2 taught the equals sign as a claim about two sides being the same size, and the check in this lesson is that claim being tested rather than assumed.

Lesson 3 is what happens after the translation, and every solve in this lesson used it.

Lesson 4 wrote situations as lines, which is the same translation skill with a picture attached.

The next lesson takes on two unknowns at once, and translation becomes harder there, because two sentences have to become two equations that are both about the same pair of numbers.

Sources

[1] John Clement, "Algebra word problem solutions: thought processes underlying a common misconception", Journal for Research in Mathematics Education 13(1), 1982, and Clement, Lochhead and Monk (1981). Thirty seven per cent of engineering freshmen wrote the student-professor relationship backwards, and sixty eight per cent of the incorrect responses took the reversed form.

[2] Carlos Soneira, Sarah Bansilal and Reginald Govender, "Insights into the reversal error from a study with South African and Spanish prospective primary teachers", Pythagoras 42(1), 2021. The samples were 211 Spanish and 79 South African second-year Bachelor of Education students. Nearly all the Spanish errors were reversals, the South African group barely made the error, and the authors attribute the gap to differences in curriculum and instruction.

Go deeper

• OpenStax, Elementary Algebra 2e, chapter 3, free online, which is a long set of word problems worked in full.
• Clement, Lochhead and Monk, "Translation difficulties in learning mathematics", the 1981 paper that first measured the reversal error. It's a scan, so the type is old, but it is short.