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The equal sign as a balance

Algebra · level 2, lesson 2 of 6

A question to start

What number goes in the box?

8 + 4 = ? + 5

Many children quickly write 12. They see 8 + 4, they know it makes 12, and they write 12 straight after the equal sign. It feels right. But it is wrong. In this lesson you will see why, find the right answer two different ways, and learn a trick that finds it without adding up at all.

The idea

What the sentence really says

Remember from level 1: the equal sign means "is the same amount as". It is a picture of a level balance scale. Whatever is on the left side must weigh the same as whatever is on the right side.

So 8 + 4 = ? + 5 says:

"8 and 4 is the same amount as some number and 5."

The box is not "the answer to 8 + 4". The box is a number that sits with the 5 on the right side. The whole right side, box and 5 together, has to match the whole left side.

Why 12 is wrong

Put 12 in the box and check both sides.

  • Left side: 8 + 4 = 12.
  • Right side: 12 + 5 = 17.

12 is not the same as 17. The right side is too heavy, and the scale tips. Here are the two sides as bars: the right bar is longer than the left bar.

left 8 4 right 12 5 with 12 in the box, the right side is too long

So 12 makes the sentence false. The mistake comes from reading = as "the answer is". When there is something after the box, like + 5, the box cannot be the answer to the left side.

Way 1: work out the side you can

The left side has no box, so we can work it out: 8 + 4 = 12. Now the sentence says

12 = ? + 5

This is a missing-number puzzle from lesson 1. The whole is 12, one part is 5, and the box is the other part. 12 − 5 = 7.

Check: left side 8 + 4 = 12. Right side 7 + 5 = 12. Both 12 ✓. The scale is level.

left 8 4 right 7 5 with 7 in the box, both bars are 12 long

Way 2: compare, without adding up

Look at the bars again. Both sides have two parts. Compare them part by part.

  • The second parts: 4 on the left and 5 on the right. The right one is 1 more.
  • So, to keep the sides the same, the first part on the right must be 1 less than 8. One less than 8 is 7.

Why does this work? Imagine two towers of blocks. The left tower is 8 blue blocks and 4 orange blocks. To build the right tower, take the left tower and move one block from the blue part to the orange part. The tower is just as tall (no block was added or taken away), but now the parts are 7 and 5. So 8 + 4 = 7 + 5.

one block changes sides 8 + 4 7 + 5 =

This idea is called keeping the balance: if one part goes up by some amount, the other part must go down by the same amount, and the total stays the same. It is very useful, because you do not need to work out the totals at all. It also works with big numbers that would be hard to add, like 38 + 25 = 40 + ?: 40 is 2 more than 38, so the box is 2 less than 25, which is 23.

The box can be on either side

The box can be on the left: ? + 6 = 9 + 3. Way 1: the right side is 12, so ? + 6 = 12 and the box is 6. Way 2: 6 is 3 more than 3, so the box is 3 less than 9: 6. Same answer. Check: 6 + 6 = 12 and 9 + 3 = 12 ✓.

More than two numbers

Sometimes one side has three numbers: 5 + 5 + 2 = 6 + ?. The left side is 10 + 2 = 12. So 6 + ? = 12 and the box is 6. Check: 6 + 6 = 12 ✓.

Taking away on both sides

The equal sign works the same way with take-away. 10 − 4 = 9 − ?. The left side is 6. So 9 − ? must also be 6. 9 take away what leaves 6? Count down from 9 to 6: three steps. The box is 3. Check: 9 − 3 = 6 ✓. (In lesson 5 you will learn a no-adding trick for take-away too, and find out that it is a little different from the adding trick.)

Worked examples

1. The starting puzzle. 8 + 4 = ? + 5.

  • Way 1: left side 12. So 12 = ? + 5 and the box is 12 − 5 = 7.
  • Way 2: 5 is 1 more than 4, so the box is 1 less than 8: 7.
  • Check: 8 + 4 = 12 and 7 + 5 = 12 ✓.

2. 9 + 6 = 10 + ?. Way 2 is quick: 10 is 1 more than 9, so the box is 1 less than 6: 5. Check: 9 + 6 = 15 and 10 + 5 = 15 ✓. This is how many people add 9 + 6 in their heads: make the 9 into a 10, and take 1 from the 6.

3. 7 + 7 = 5 + ?. 5 is 2 less than 7, so the box is 2 more than 7: 9. Check: 7 + 7 = 14 and 5 + 9 = 14 ✓.

4. ? + 20 = 15 + 25. Way 1: the right side is 40, so ? + 20 = 40 and the box is 20. Way 2: 20 is 5 less than 25, so the box is 5 more than 15: 20. Check: 20 + 20 = 40 and 15 + 25 = 40 ✓.

5. Spotting the trap. 6 + 3 = ? + 1. Someone writes 9. Check: 9 + 1 = 10, not 9. Wrong! The right answer: 1 is 2 less than 3, so the box is 2 more than 6: 8. Check: 8 + 1 = 9 ✓.

Try it

  1. 5 + 4 = ? + 3
  2. 10 + 2 = 6 + ?
  3. ? + 8 = 7 + 9
  4. 9 + 9 = 10 + ?
  5. 3 + 4 + 5 = ? + 2
  6. 12 − 2 = 15 − ?
  7. 48 + 17 = 50 + ? (try Way 2!)
  8. Mia writes 6 + 7 = 13 + 2, because "6 + 7 is 13, and then I added 2". Is her sentence true? How should she write what she did?
Answers
  1. 6. Left side 9, so ? + 3 = 9 and the box is 6. Or: 3 is 1 less than 4, so the box is 1 more than 5. Check: 6 + 3 = 9 ✓.
  2. 6. Left side 12, so 6 + ? = 12 and the box is 6 (a double). Check: 6 + 6 = 12 ✓.
  3. 8. Right side 16. So ? + 8 = 16 and the box is 8 (a double). Or: 8 is 1 less than 9, so the box is 1 more than 7. Check: 8 + 8 = 16 and 7 + 9 = 16 ✓.
  4. 8. 10 is 1 more than 9, so the box is 1 less than 9. Check: 9 + 9 = 18 and 10 + 8 = 18 ✓.
  5. 10. Left side 3 + 4 + 5 = 12. So ? + 2 = 12 and the box is 10. Check: 10 + 2 = 12 ✓.
  6. 5. Left side 10. So 15 − ? = 10, and 15 take away 5 leaves 10. Check: 15 − 5 = 10 ✓.
  7. 15. 50 is 2 more than 48, so the box is 2 less than 17. Check: 48 + 17 = 65 and 50 + 15 = 65 ✓.
  8. False. The left side is 13 and the right side is 15. They are not the same. She used = to mean "and then". She should write two sentences: 6 + 7 = 13, then 13 + 2 = 15. Or one true sentence: 6 + 7 + 2 = 15.

Common mistake

Writing the answer to the left side in the box. In 8 + 4 = ? + 5, writing 12 ignores the "+ 5". The box must make the two sides match. Check every answer by working out both sides.

Moving the wrong way in Way 2. If one part goes up, the other part must go down. In 9 + 6 = 10 + ?, the 9 went up to 10, so the 6 must go down to 5, not up to 7. A quick check with the totals will show you if you moved the wrong way.

Where it leads