Skip to content

True or false number sentences

Algebra · level 2, lesson 5 of 6

A question to start

Are these number sentences true or false?

  • 7 + 5 = 6 + 6
  • 10 − 3 = 3 − 10
  • 25 + 38 = 26 + 37

Here is the challenge: decide without working out either side. No adding up the 25 and the 38! It sounds impossible, but by the end of this lesson you will be able to do it by looking at how the two sides are built. This is how mathematicians think: instead of calculating, they look for reasons. (The answers are in the worked examples.)

The idea

A number sentence makes a claim

A number sentence with an equal sign says "the left side is the same amount as the right side". That is a claim, and a claim can be true or false. 4 + 4 = 8 is true. 4 + 4 = 9 is false.

One way to decide is to work out both sides and compare. That always works. But sometimes the numbers are big, and sometimes you can see the answer much faster by comparing the two sides. That is what this lesson is about.

Reason 1: adding in either order

34 + 20 = 20 + 34. Both sides add the same two numbers, just in a different order. Putting two piles together gives the same total whichever pile you pour in first. So this is true, and you did not need to know that the total is 54.

Reason 2: one goes up, the other goes down

7 + 5 = 6 + 6. Compare the parts: 6 is one less than 7, and the other 6 is one more than 5. So to get from the left side to the right side, one block moved from the first part to the second part.

7 + 56 + 6 one square changed colour; the row is just as long

The total did not change, because nothing was added and nothing was taken away. One square just changed which part it belongs to. So 7 + 5 = 6 + 6 is true.

This is the "keeping the balance" idea from lesson 2: in an adding sentence, if one part goes up by some amount and the other part goes down by the same amount, the total stays the same.

Reason 3: both go up (or both go down)

9 + 4 = 10 + 5. Here 10 is one more than 9, and 5 is one more than 4. Both parts got bigger, so the right side must be bigger. The sentence is false. (In fact the right side is 2 more.)

If both parts go up, the total goes up. If both go down, the total goes down. Either way, the two sides cannot be the same.

Take-away: the order matters

10 − 3 = 3 − 10. With adding, swapping the order is fine. With taking away, it is not.

  • 10 − 3: start with 10 counters, take away 3, and 7 are left.
  • 3 − 10: start with 3 counters and take away 10. You cannot! You only have 3.

Whatever 3 − 10 means (you will meet numbers below zero later, in Negative numbers), it cannot be 7: if you start with only 3, you can never have 7 left. So this sentence is false. Taking 3 from 10 and taking 10 from 3 are very different stories.

Take-away: both go up keeps the difference

Here is a surprise. Is 15 − 7 = 16 − 8 true?

Think of 15 − 7 as the gap between 7 and 15 on the number line: how far it is from 7 up to 15. And 16 − 8 is the gap between 8 and 16.

567891011121314151617 gap from 7 to 15 gap from 8 to 16

Both numbers moved one step to the right. The gap slid along, but it did not stretch or shrink. So the gaps are the same, and 15 − 7 = 16 − 8 is true. (Both are 8.)

Here is another way to see it. You are 7 years old and your big sister is 15. The difference in your ages is 8 years. Next year you will be 8 and she will be 16. The difference is still 8 years. Both ages go up together, so the difference never changes.

Watch out: the two tricks are different

For adding, one up and one down keeps the total the same.

For taking away, both up (or both down) keeps the difference the same.

What happens if you use the adding trick on a take-away? 15 − 7 = 16 − 6. The first number went up 1 and the second went down 1. On the number line, the top end moved right and the bottom end moved left, so the gap stretched by 2. The sentence is false: 15 − 7 is 8, but 16 − 6 is 10.

Worked examples

1. The starting sentences.

  • 7 + 5 = 6 + 6: 6 is one less than 7, and 6 is one more than 5. One down, one up, in an adding sentence: true.
  • 10 − 3 = 3 − 10: taking 3 from 10 leaves 7, but from 3 you cannot have 7 left. False.
  • 25 + 38 = 26 + 37: 26 is one more than 25, and 37 is one less than 38. One up, one down: true. (Both sides are 63, but we did not need to know that.)

2. 48 + 16 = 50 + 14. 50 is 2 more than 48. 14 is 2 less than 16. Up 2, down 2 in an adding sentence: true. This is a handy way to add 48 + 16 in your head: make the 48 into 50, and take the 2 from the 16. 50 + 14 is 64.

3. 30 − 9 = 31 − 10. Both numbers went up by 1, in a take-away sentence. The gap slides but keeps its size: true. And 31 − 10 is easy: 21. So 30 − 9 is 21 too. This is a trick for taking away 9: take away 10 instead, and add 1 to the first number.

4. 12 + 9 = 9 + 13. Swap the right side round: 13 + 9. Compare with 12 + 9: the 9s match, but 13 is one more than 12. So the right side is bigger: false.

5. 20 − 5 = 19 − 4. Both numbers went down by 1, in a take-away sentence. The gap slides left but stays the same size: true. (Both are 15.)

Try it

Decide true or false without working out the totals. Then give your reason.

  1. 9 + 6 = 6 + 9
  2. 8 + 7 = 9 + 6
  3. 10 + 4 = 11 + 5
  4. 14 − 6 = 6 − 14
  5. 17 − 9 = 18 − 10
  6. 25 − 10 = 26 − 9
  7. 39 + 47 = 40 + 46
  8. Fill the box to make the sentence true, without adding up: 56 + 29 = 55 + ?
Answers
  1. True. The same two numbers added in a different order.
  2. True. 9 is one more than 8, and 6 is one less than 7. One up, one down, in an adding sentence. (Both are 15.)
  3. False. 11 is one more than 10 and 5 is one more than 4. Both parts went up, so the right side is bigger. (14 and 16.)
  4. False. 14 − 6 leaves 8. But from 6 you cannot take away 14, and you could never have 8 left from 6. Order matters for taking away.
  5. True. Both numbers went up by 1 in a take-away sentence, so the gap stays the same. (Both are 8.)
  6. False. The first number went up 1 but the second went down 1. In a take-away, that stretches the gap by
  7. (15 and 17.)
  8. True. 40 is one more than 39, and 46 is one less than 47. One up, one down, in an adding sentence. (Both are 86.)
  9. 30. 55 is one less than 56, so the box must be one more than 29. Check: 56 + 29 = 85 and 55 + 30 = 85 ✓.

Common mistake

Using the adding trick on taking away. In adding, one up and one down keeps the total. In taking away, it does not: 15 − 7 and 16 − 6 are different. For taking away, both numbers must move the same way to keep the difference.

Thinking you can swap the order in a take-away. 9 + 4 = 4 + 9, but 9 − 4 is not the same as 4 − 9.

Where it leads