Equals means the same size
35 min
Two hosts talk the lesson through. The voices are synthetic; the script was written from this lesson and checked against it, and asserts nothing the lesson does not.
- Say what the equals sign claims, and test whether an equation is true for a given number
- Keep an equation true by doing the same thing to both sides
- Find a missing number in an equation with operations on both sides
Try this one before you read anything else. What number goes in the box?
8 + 4 = box + 5
Take your time. Then read on.
Why that question is in every algebra course
Researchers put that exact question to school students from first grade to sixth. Roughly what share got it right, and did the older children do better?
Show the answer
Fewer than ten percent in any grade, and the older children did no better than the younger ones.1 If you guessed that sixth graders would beat first graders, that's the natural guess, and it's the one the study overturned.
Being older did not help, which tells you something important: this is not a hard sum, and getting it wrong is not about ability. It is about what people think the equals sign means.
Two wrong answers dominate: 12 and 17.
Both come from the same place. If you have spent years doing arithmetic, you have seen the equals sign thousands of times in one particular shape: 8 + 4 = ?, 6 × 7 = ?, 20 - 3 = ?. In every one of those, the sign sits just before the answer. So it starts to feel like an instruction, meaning and the answer is. Read 8 + 4 = box + 5 that way and the box is obviously where the answer goes, so it's 12. Read the whole line as one long sum and you get 17.
The right answer is 7, and here is the idea that makes it obvious.
The sign is a claim, not an instruction
= means the two sides are the same size. That's all it has ever meant. It is not a button that says compute; it is a statement that what is on the left and what is on the right are the same amount.
So 8 + 4 = box + 5 says: twelve is the same size as something plus five. Something plus five is twelve. The something is 7.
Once you read the sign that way, some things that used to look wrong stop looking wrong.
7 = 3 + 4 is a perfectly good equation. Seven is the same size as three plus four. If that felt slightly uncomfortable to look at, that discomfort is the old reading of the sign still operating, and noticing it is worth more than any rule you could memorise.
12 = 12 is an equation too, and a true one. 5 + 5 = 3 + 7 is an equation, and true. 5 + 5 = 3 + 6 is an equation, and false. Equations can be false. Instructions cannot be false, which is another way of seeing that the sign is not an instruction.
Is 9 + 6 = 15 + 0 true or false? And is 9 + 6 = 15 = 20 + 5 a sensible thing to write?
Show the answer
The first is true: fifteen is the same size as fifteen. The second is not sensible, even though every individual piece looks familiar. It claims that fifteen is the same size as twenty five, because if 9 + 6 = 15 and 15 = 20 + 5 then fifteen equals twenty five. This is what happens when the sign is used to mean "and then I did this next", which is how a great many people write out their working. Each line of working needs its own equation.
The balance
The picture that carries the rest of this course is a pair of scales.
An equation is a balance that is level. Whatever is on the left pan weighs the same as whatever is on the right pan; that is what the sign is claiming. OpenStax's Elementary Algebra turns the same picture into the formal properties of equality, if you want the textbook version.
Take the same amount off both pans and it stays level. Add the same to both and it stays level. That is the whole reason the moves in lesson 3 are allowed, and it is why doing something to one side only is not a move at all.
Now the rule that the picture makes obvious, and that most people are taught as a ritual instead.
Whatever you do to one side, you must do to the other. Not because a teacher said so. Because the equation is a claim that the two sides are the same size, and if you change one side and not the other, the claim you started with is no longer the claim you have. The balance tips.
Take five off both pans of 8 + 4 = box + 5 and you get 7 = box, which is the answer. That is the whole method of lesson 3, arrived at by looking at a picture.
Testing whether an equation is true
An equation with a letter in it is a claim that may be true for some numbers and false for others, and you can always check by substituting.
3n + 1 = 10
Is it true when n = 2? Substitute: 3(2) + 1 is 7, and 7 is not 10. So no.
Is it true when n = 3? Substitute: 3(3) + 1 is 10, and 10 is 10. So yes.
That is what "solving" will mean in lesson 3: finding the values that make the claim true. And notice you already have a way to check any answer you ever produce, which is to put it back in. Checking is not an optional extra step in algebra; it is the thing that makes the rest of it safe.
Is 3n + 1 = 10 true when n = 4? And is there any other number besides 3 that makes it true?
Show the answer
No: 3(4) + 1 is 13, not 10. And no other number works, because if three times something plus one is ten, three times that something is nine, and only one number times three is nine. Most equations you'll meet in this course are like that, true for exactly one number, though lesson 6 meets some that aren't.
Here is 13 + 6 = box + 9, worked to the second to last line. The left pan holds 19. Take nine off both pans and the right pan holds just the box, so 19 - 9 = box. Finish it, then check your answer by substituting it back.
Show the answer
Nineteen minus nine is 10, so the box is 10. Check: 13 + 6 is 19 and 10 + 9 is 19, so the two sides are the same size and the equation is true. If you answered 19, you read the sign as "and the answer is", which is the whole thing this lesson is about, and noticing that you did it is the useful part.
Four more shapes, so the sign stops meaning one thing
The research finding behind this lesson is that people are much more likely to read the sign as a claim when they meet it with operations on both sides, and much more likely to read it as an instruction when they only ever meet it in the shape sum = answer.1 So here are the other shapes, deliberately.
Operations on both sides. 15 - 6 = box + 2. The left is 9, so something plus two is nine, and the box is 7.
The answer first. 20 = box × 4. Twenty is the same size as something times four, so the box is 5.
Nothing to compute at all. box = 8. This is a perfectly good equation, and it is telling you the answer directly. Solving an equation, in lesson 3, is the business of turning a complicated equation into this shape without ever breaking the balance.
Both sides identical. n + 3 = n + 3 is true for every number you could put in. n + 3 = n + 4 is false for every number. Most equations are in between: true for some, false for others.
Work these on paper. Check each answer by substituting it back into the original.
7 + 5 = box + 8box + 6 = 4 + 92 × 8 = box + 924 = box × 3True or false:
6 + 7 = 13 + 0True or false:
10 - 4 = 3 + 2Is
4n - 1 = 11true whenn = 3?Someone writes
5 + 2 = 7 + 3 = 10. What is wrong with it, and how would you write what they meant?Write an equation with operations on both sides that is true, and one that is false.
Answers to the practice set, once you have done all nine
Show the answer
- 4, since 12 is the same size as 4 plus 8. 2. 7, since 13 is the same size as 7 plus 6. 3. 7, since 16 is the same size as 7 plus 9. 4. 8. 5. True. 6. False, because 6 is not 5. 7. Yes:
4(3) - 1is 11. 8. It claims 7 equals 10, because chaining the two statements with equals signs says seven is ten. They meant5 + 2 = 7, and then7 + 3 = 10, on separate lines. 9. Anything of the shape6 + 3 = 4 + 5is true, and6 + 3 = 4 + 6is false; the point is that both are equations.
What people get wrong
"The equals sign means the answer comes next." It means the two sides are the same size. Everything else in this lesson follows from that, and so does most of lesson 3.
Chaining equals signs while working. 5 + 2 = 7 + 3 = 10 is not a shortcut; it is a false claim. Each step gets its own line.
Doing something to one side only. That is not a move. It produces a different equation from the one you had, and the answer you get will not solve the original.
Thinking an equation must have a letter in it. 12 = 12 and 5 + 5 = 3 + 7 are equations, and true ones. The letter is what makes an equation worth solving, not what makes it an equation.
Connections
Lesson 1 built expressions, which are recipes for a number. This lesson added the sign that turns two expressions into a claim.
Lesson 3 solves equations, and every move it makes is the balance from this lesson: the same thing done to both sides, so the claim survives.
Lesson 4 writes relationships as equations with two letters, and the balance still holds.
The lesson on turning a sentence into an equation is where the commonest mistake in all of algebra lives, and the check you learned here, substituting a number to see whether the claim is true, is what catches it.
Sources
[1] Nicole McNeil, Laura Grandau, Eric Knuth, Martha Alibali, Ana Stephens, Shanta Hattikudur and Daniel Krill, "Middle-school students' understanding of the equal sign: the books they read can't help", Cognition and Instruction 24(3), 2006, 367 to 385. The box question and the finding that "fewer than 10% in any grade gave the correct answer and performance did not improve with age" are Carpenter and colleagues' result as that paper reports it. The finding that operations on both sides of the sign elicit the relational reading, and that textbooks almost never present the sign that way, is the paper's own.
Go deeper
- OpenStax, Elementary Algebra 2e, section 2.1, free online, which develops the balance idea into formal properties of equality.
- Khan Academy, Algebra basics, for practice with immediate feedback, which is the one thing a written lesson cannot provide.
Check your understanding
This lesson has a 6-question quiz. Pass it and the questions come back on a schedule in Review, so what you learned stays learned. Your progress is saved in your browser; no account needed.