Knowledge base · Maths - grade 4Maths - grade 4
Adding natural numbers
Addends and the sum, adding zero, swapping and grouping addends, addition on the number line and making full tens and hundreds - with step-by-step examples for year 4.
You add more often than you think: when you count the money in your wallet, the points in a game or the minutes left until the end of break. In this lesson you will learn the names of the numbers in an addition, see why the order of the addends does not matter, and pick up ways to work out even 36 + 45 + 24 in your head.
What are the numbers in an addition called?
Every number in an addition has its own name. Look at this sum:
The numbers we add are called addends. The result of an addition is called the sum. Here the addends are 23 and 45, and the sum is 68.
There can be more than two addends. In the sum below there are three of them, and still just one sum:
The sum of two numbers is 50. One addend is 18. What is the other addend?
Hint: Think about how much you have to add to 18 to reach 50. You can go through 20.
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Answer
The other addend is 32. From 18 to 20 is 2, and from 20 to 50 another 30, which makes 32 altogether. Check: 18 + 32 = 50.
What happens when you add zero?
Nothing changes. Zero means “nothing”: if you have 7 stickers and nobody gives you any new ones, you still have 7. It works the same way for every number, small or big:
The sum of any number and zero equals that number. It does not matter whether the zero comes first or last.
Does the order of the addends matter?
It does not. See it on the number line. When you add 3 + 5, you stand at 3 and jump 5 to the right. When you add 5 + 3, you stand at 5 and jump 3. Either way you land in the same place - at 8:
You can swap the addends around - the sum stays the same. This property is called the commutative property of addition.
What is that good for? For example, 2 + 97 is easier to work out the other way round: you start from the bigger number and add just 2.
How can you group the addends?
When you add three numbers, you first add two of them and then add the third. Which two you start with is entirely up to you. The result comes out the same:
You can group the addends in any pairs and add them in any order - the sum stays the same. This property is called the associative property of addition.
The second way was easier, because 8 + 2 makes a full ten, and adding on to a full ten is very easy. So before you start counting, look for pairs that make 10:
The same works with bigger numbers. In 26 + 38 + 14 the ones 6 and 4 make 10 together, so you first group 26 with 14:
Work it out the clever way: 17 + 29 + 3 + 1.
Hint: Look for two pairs of addends that make full tens.
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Answer
The sum is 50. 17 with 3 makes 20, and 29 with 1 makes 30. All that is left is to add two full tens: 20 + 30 = 50.
How do you show addition on the number line?
On the number line, bigger numbers lie further to the right. That is why adding is a jump to the right: you stand at the first addend and jump as many ticks as the second addend. This is 4 + 3:
A big jump does not have to be a single jump. You can split it into several smaller ones, as long as together they add up to the same amount. This is how you add 27 + 16. First you jump to a full ten, then a whole ten, and finally whatever is left.
How do you make numbers up to full tens and hundreds?
Full tens are the numbers 10, 20, 30, 40 and so on - they end in a zero. Full hundreds are 100, 200, 300 and so on - they end in two zeros. Making a number up to a full ten means adding just enough to reach the nearest full ten.
To reach a full ten you need as many ones as it takes to make the ones up to 10. 67 has 7 ones, and 7 + 3 = 10, so you need 3 more to get to 70:
When the target is further away, go in two steps: first to the nearest full ten, then in full tens. How much do you need from 37 to 100? From 37 to 40 you need 3, and from 40 to 100 another 60. That is 63 altogether:
What is it all for? Because from a full ten you can count on without any effort. If you have to add 7 to 58, first add 2 - you get to 60 - and then the remaining 5:
How much do you need to add to reach 70?
How do you add two-digit numbers in your head?
Split both numbers into tens and ones. Add the tens and the ones separately, and at the end put the results together:
Sometimes the ones add up to more than 10. That is fine - the sum of the ones is then a full ten and a bit more, and you add the full ten to the tens:
You can also work it out differently - through a full ten. Add 3 to 47 to get to 50, then the rest, that is 38 − 3 = 35. The result is the same, because either way you add exactly 38.
Work it out in your head: 56 + 27.
Hint: Add the tens (50 and 20) and the ones (6 and 7) separately.
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Answer
The sum is 83. The tens make 70, the ones 13, that is a full ten and 3. Together 83.
Worked examples, step by step
Add two one-digit numbers
Work out 8 + 7.
- 1
8 needs 2 more to make a full ten. Take those 2 from the 7:
- 2
We took 2 away from the 7, so there are 5 left:
- 3
Add the rest to the full ten. On the number line that is two jumps: of 2 and of 5.
Add three one-digit numbers
Work out 6 + 9 + 4.
- 1
Look for a pair that makes 10. That is 6 and 4 - group them first. You are allowed to, because addends can be grouped and swapped:
- 2
Add the last addend to the full ten:
Add two two-digit numbers
Work out 47 + 38.
- 1
Add the tens: 4 tens and 3 tens make 7 tens.
- 2
Add the ones. It comes to more than 10 - that is a full ten and 5 more ones.
- 3
Put the two results together:
Add three two-digit numbers
Work out 36 + 45 + 24.
- 1
Look at the ones: 36 has 6 and 24 has 4. Together they make 10, so it pays to add these two numbers first.
- 2
Add the third addend to the full tens:
Show an addition on the number line
Show the addition 18 + 15 on the number line and read off the result.
- 1
Split 15 into jumps that are easy to make: first 2, to reach a full ten, then 10, and finally whatever is left.
- 2
Stand at 18 and jump to the right: to 20, to 30 and to 33.
- 3
The last jump ends at 33 - that is the sum.
A word problem
The school library lent out 28 books on Monday, 35 on Tuesday and 12 on Wednesday. How many books did it lend out over those three days?
- 1
You want to know how many there are altogether, so you add the numbers of books from all three days:
- 2
The ones 8 and 2 make 10, so first group 28 with 12:
- 3
Add the books from Tuesday:
- 4
Check by adding in the usual order. The result must be the same, because the order of the addends does not matter:
Where is addition useful?
- Shopping. A roll costs 2 zł, juice 5 zł and apples 8 zł. Start with the roll and the apples, because 2 + 8 = 10 is a full ten. Then add the juice and you get 10 + 5 = 15. You pay 15 zł for everything.
- With time. A film lasts 95 minutes, and the adverts before it 15 minutes. Together that is 110 minutes, which is an hour and 50 minutes.
- In games. After every round you add up the points, and at the end you compare the players’ totals.
- On the way. It is 850 m from home to school, and 450 m from school to the swimming pool. The whole route is 1 300 m, that is a kilometre and 300 metres.
- Saving money. If you put aside 15 zł every week, after four weeks you have 15 + 15 + 15 + 15 = 60 zł.
- On the calendar. There are 3 weeks and 4 days left until the holidays. That is 7 + 7 + 7 + 4 = 25 days.
A puzzle to finish
Write out the numbers from 1 to 10 in order, that is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. You want to add them all up, but without tediously counting one after another.
What is the cleverest way to work out this sum, and what is it?
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Solution
The sum is 55. Pair up numbers that together make 10. Those pairs are 1 and 9, 2 and 8, 3 and 7, and 4 and 6. That is four full tens, or 40. What is left is 5 and 10, which make 15. And 40 + 15 = 55. You are allowed to do this, because addends can be swapped and grouped in any pairs.
Remember
The numbers we add are the addends, and the result is the sum. Adding zero changes nothing. Addends can be swapped and grouped in any pairs and the sum stays the same - so before you start counting, look for pairs that make full tens. On the number line, adding is a jump to the right, and you can split a big jump into smaller ones: first to a full ten, then on from there.
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Frequently asked questions
What are the numbers in an addition called?
The numbers we add are the addends, and the result of the addition is the sum. In 23 + 45 = 68 the addends are 23 and 45, and the sum is 68.
What is the sum of a number and zero?
The same as the number itself - adding zero changes nothing. For example, 356 + 0 = 356 and 0 + 7 = 7.
Can you swap the addends around when adding?
Yes. The sum stays the same: 3 + 5 = 5 + 3 = 8. This property is called the commutative property of addition. When there are more addends, you can also group them in any pairs you like - that is the associative property.
How do you add two-digit numbers in your head?
Add the tens and the ones separately, then put the results together: 47 + 38 = 70 + 15 = 85. You can also first make one number up to a full ten: 47 + 3 = 50, and then add the rest, that is 35.
How do you show addition on a number line?
Stand at the first addend and jump to the right by the second addend. Where you land is the sum. You can split a big jump into smaller ones, for example first up to a full ten.
What does making a number up to a full ten mean?
It means adding just enough to reach the nearest full ten, that is a number like 10, 20 or 30. For example, you have to add 3 to 67 to get 70.