Skip to content

The equal sign means "the same as"

Algebra · level 1, lesson 3 of 5

A question to start

Here are five number sentences. Which ones are true?

  • 3 + 2 = 5
  • 5 = 4 + 1
  • 5 = 5
  • 3 + 2 = 4 + 1
  • 3 + 2 = 6

Many people say "only the first one!" They think that 5 = 4 + 1 is written backwards, that 5 = 5 is silly, and that 3 + 2 = 4 + 1 is broken because "the answer is missing". But in fact four of them are true, and only one is false. By the end of this lesson you will know which one, and why. (It is in the worked examples.)

The idea

What does = really mean?

The sign = is called the equal sign. Many children learn to read it as "makes" or "the answer is", like a button on a calculator: you put in 3 + 2, you press =, and out comes 5.

But that is not what = means in math. The true meaning is:

= means "is the same amount as".

So 3 + 2 = 5 says: "3 and 2 more is the same amount as 5." The thing on the left of the sign and the thing on the right of the sign are the same amount, just written in two different ways.

The two parts are called the two sides: the left side and the right side. The equal sign sits in the middle, like the post in the middle of a balance scale.

Seeing it with towers

Build a tower with 3 orange cubes and 2 blue cubes. Build another tower with 5 green cubes. Stand them next to each other.

3 + 2 = 5 the towers are the same height

They are the same height. That is what 3 + 2 = 5 means: the two towers match. It does not matter which tower is on the left and which is on the right. The towers are still the same height if you swap them. So we can also write 5 = 3 + 2, and that is just as true.

The equal sign is a level balance

In the last lesson the balance scale was level when both sides weighed the same. The equal sign is a picture of a level scale. Look at it: two flat lines, both the same length, lying level. Whatever is on the left side weighs the same as whatever is on the right side.

A number sentence with = can be:

  • true, if the two sides really are the same amount (a level scale), or
  • false, if they are not (a tipped scale).

So 3 + 2 = 6 is a sentence that makes a claim. The claim is wrong: 3 and 2 make 5, not 6. A tower of 5 cubes is shorter than a tower of 6. The sentence is false.

Both sides can have adding

Since = just means "the same amount", there can be adding on both sides. Look at 3 + 2 = 4 + 1.

3 + 2 = 4 + 1 both towers have 5 cubes: 3 + 2 = 4 + 1

The left tower is 3 and 2 more: 5 cubes. The right tower is 4 and 1 more: also 5 cubes. They are the same height, so 3 + 2 = 4 + 1 is true. There is no "answer" here, and none is needed. The sentence is saying "these two amounts are the same", and they are.

And 5 = 5?

5 = 5 says "5 is the same amount as 5". Of course it is! It is a very simple true sentence. It looks silly, but it reminds us of something important: any amount is the same as itself. A scale with 5 blocks on each side is level.

Many ways to make 5

A number can be made in lots of ways. Here are all the ways to make 5 from two numbers:

0 + 5 1 + 4 2 + 3 3 + 2 4 + 1 5 + 0
is 5 is 5 is 5 is 5 is 5 is 5

Every one of them is the same amount as 5. So every one of them is the same amount as every other one. That means we can put any two of them on the two sides of an equal sign, and the sentence will be true:

1 + 4 = 2 + 3, 0 + 5 = 4 + 1, 2 + 3 = 3 + 2, and so on.

Why does this work? If two towers are each the same height as a 5-tower, then they are the same height as each other. Things that are the same as the same thing are the same as each other.

Making both sides the same

Now a puzzle. Here is a sentence with a gap: 4 + 1 = 2 + ? What number goes in the gap?

Build it. The left tower is 4 and 1: 5 cubes. The right tower starts with 2 cubes. It is too short. How many cubes must we add to make it as tall as the left tower? Add one at a time and count: 3, 4, 5. We added 3 cubes.

4 + 1 = 2 + 3 add 3 cubes to the short tower: 4 + 1 = 2 + 3

So 4 + 1 = 2 + 3. Check: both sides are 5 ✓.

Notice that the missing number is not 5. If you put 5 in the gap, the right side would be 2 + 5, which is 7 cubes. That tower is taller than the 5 cubes on the left. The gap needs the number that makes the sides match, not the "answer" to the left side.

The sign that says "not the same"

There is also a sign for "is not the same amount as". It is an equal sign with a line through it: ≠. So 3 + 2 ≠ 6 is a true sentence: 3 and 2 is not the same amount as 6. You do not need this sign often, but it is nice to know that = makes a claim, and ≠ makes the opposite claim.

Worked examples

1. The starting sentences.

  • 3 + 2 = 5: left side 5, right side 5. True.
  • 5 = 4 + 1: left side 5, right side 4 and 1, which is 5. True. Written "backwards", but = works both ways.
  • 5 = 5: True. Any amount is the same as itself.
  • 3 + 2 = 4 + 1: left side 5, right side 5. True.
  • 3 + 2 = 6: left side 5, right side 6. False. This is the only false one.

2. True or false: 6 = 3 + 3? Left side: 6. Right side: 3 and 3 more is 6. Same, so true. This is a "double": 6 is double 3.

3. True or false: 2 + 5 = 5 + 2? Left side: 7. Right side: 7. True. In fact you can see it without adding: both sides use the same two numbers, a 2 and a 5. The same cubes in a different order make the same tower height.

4. Fill the gap: 3 + 3 = 4 + ? Left side: 6. The right side starts at 4. Count up from 4 to 6: five, six. That is 2 more. So the gap is 2. Check: 4 + 2 is 6, and 3 + 3 is 6 ✓.

5. Fill the gap: 7 = ? + 4. The left side is 7. Which number with 4 more makes 7? Count up from 4 to 7: five, six, seven. That is 3. So the gap is 3. Check: 3 + 4 = 7 ✓. It does not matter that the 7 is on the left.

Try it

  1. True or false? 4 + 2 = 6
  2. True or false? 6 = 4 + 2
  3. True or false? 4 + 2 = 5 + 2
  4. True or false? 1 + 5 = 3 + 3
  5. Fill the gap: 5 + 1 = 4 + ?
  6. Fill the gap: 8 = ? + 6
  7. Write three different true sentences with 2 + 4 on the left side and something else on the right side.
  8. Zoe writes 2 + 3 = 5 + 4 = 9 to show her working for "2 + 3, then add 4". What is wrong with it? How could she write it so that every = is true?
Answers
  1. True. 4 and 2 more is 6.
  2. True. The same sentence turned around. 6 is the same amount as 4 + 2.
  3. False. Left side: 4 + 2 is 6. Right side: 5 + 2 is 7. 6 is not the same as 7. (You can also see it without adding: both sides add 2, but 5 is more than 4.)
  4. True. Left side: 1 + 5 is 6. Right side: 3 + 3 is 6.
  5. 2. The left side is 6. From 4, count up to 6: five, six. That is 2. Check: 4 + 2 = 6 ✓.
  6. 2. Which number with 6 more makes 8? Count up from 6: seven, eight. That is 2. Check: 2 + 6 = 8 ✓.
  7. 2 + 4 is 6, so the right side must be any way of making 6. For example: 2 + 4 = 6, 2 + 4 = 4 + 2, 2 + 4 = 3 + 3, 2 + 4 = 1 + 5, 2 + 4 = 0 + 6. Any three of these are right.
  8. The middle part says 2 + 3 = 5 + 4. But 2 + 3 is 5, and 5 + 4 is 9. 5 is not the same as 9, so that = is false. Zoe is using = to mean "and then". A better way is to write two separate sentences: 2 + 3 = 5, then 5 + 4 = 9. Each = is now true.

Common mistake

Thinking the answer must come right after the equal sign. In 4 + 1 = 2 + ?, many people write 5 in the gap, because 4 + 1 is 5. But the gap is part of the right side, and 2 + 5 is 7, not 5. Always ask: "What makes the two sides the same?" Build the two towers if you are not sure.

Using = to mean "and then". Writing 2 + 3 = 5 + 4 = 9 looks like a story of what you did, but each = makes a claim, and 2 + 3 is not the same as 5 + 4. Use a new line for each new step.

Where it leads