Properties of addition
A child who knows that 8 and 3 make 11, does she also know 3 and 8? And with three numbers to add, may she choose which two to put together first?
Where you are 12 in Addition and subtraction
Why this exists
A child learning addition facts meets a great many of them. Every number from 0 to 9 added to every number from 0 to 9 makes a table of a hundred facts, and that is only the start. Learning each one separately would be slow and dull, and it would miss something a child can notice for herself: many of the facts are the same fact twice.
Take 8 + 3 and 3 + 8. Eight blocks and three blocks, or three blocks and eight blocks: either way the same eleven blocks end up in the pile. A child who knows one of these facts knows the other, and a child who can count on can always start from the bigger number, since 3 + 8 may be worked out as “eight ... nine, ten, eleven”.
The same freedom shows up when a story has three amounts in it. A tower of 7 red, 5 blue and 3 yellow blocks asks for 7 + 5 + 3. Going left to right means 7 + 5 first, which crosses ten. But the 7 and the 3 make ten, and ten and five is fifteen, said at once. A child who may choose which two numbers to put together first can choose the easy pair.
Zero needs a word of its own: a group with nothing in it adds nothing. And subtraction needs a warning, because none of this freedom carries over to taking away. This lesson gathers these facts together. They are called the properties of addition, and they are the reasons behind most of the shortcuts a child uses when adding.
Definition
There are three properties of addition, and one property that subtraction does not have.
Order does not change a sum
Two numbers added in either order give the same total.
Grown-ups write this with letters standing for any two numbers at all: a + b = b + a. It has a long name, the commutative property of addition, from a word meaning “to swap”. A child needs neither the letters nor the name. What she needs is the habit of reading 3 + 6 and 6 + 3 as one fact, sometimes called a turn-around fact.
Grouping does not change a sum
When three numbers are added, putting the first two together first or the last two together first gives the same total.
Brackets show which two go first: the part inside the brackets is worked out before the rest.
On the left, 6 + 4 = 10 and then 10 + 5 = 15. On the right, 4 + 5 = 9 and then 6 + 9 = 15. With letters this reads (a + b) + c = a + (b + c), and it is called the associative property of addition. A young child usually shows the grouping without brackets, by writing the chosen pair first or by drawing a small bond under it. Putting the first and the last number together first uses both properties at once, order and grouping (see Any order and any grouping, below).
Adding zero
Adding zero to a number leaves the number exactly as it was.
With letters, a + 0 = a and 0 + a = a. Because zero leaves every number alone when it is added, it is called the identity for addition, and this is the identity property of addition.
Subtraction has an order
Swapping the two numbers in a subtraction changes it, unless the two numbers are the same.
In 7 - 3 there are seven counters and three are taken away, leaving four. In 3 - 7 there are only three counters, and seven cannot be taken from them. Only two equal numbers survive the swap: 5 - 5 turned around is 5 - 5 again, which is 0. The two numbers in a subtraction do different jobs: the first is the pile, the second is how many leave it. Grouping matters too: 9 - 3 - 2 means take 3 from 9, then take 2 more, and doing 3 - 2 first gives a different answer. Subtraction is not commutative and not associative.
Intuition
Each of these properties can be seen with cubes or counters, and a child should see each one before she is told it.
Turn the train around. Snap three dark cubes and six light cubes into one train. Read from the left, it shows 3 + 6. Turn it around and it shows 6 + 3 (Figure 1). Nothing came off the train and nothing was added, so it is still nine cubes.
Bracket the train two ways. A longer train shows grouping. Figure 2 has 6 dark cubes, 4 gray and 5 light. Read the brackets above it: 6 and 4 first, which is 10, and then 5 more, so 10 + 5 = 15. Read the brackets below: 6, and then 4 and 5 together, which is 9, so 6 + 9 = 15. The brackets only change which cubes we look at together. They cannot change how many cubes there are.
An empty group. Zero is a group with nothing in it, like an empty plate or an empty loop drawn on the table (Figure 3). Pushing it into the pile puts no counters there. When a child counts on zero, she says the first number and then says nothing more.
Taking away has a direction. Put seven counters out and cross three off: four are left. Now put three counters out and try to take seven. All three go, and four more are still wanted that are not there (Figure 4). The question “3 take away 7” has no answer among the numbers a child uses now (0, 1, 2 and on), so swapping the numbers did not even give a subtraction that could be worked out.
The difference between the two operations is the heart of it. Adding puts groups into one pile, and a pile does not remember which group went in first. Taking away starts from one pile and removes from it, so it matters which number is the pile.
Results
Order does not change a sum. For any two numbers, a + b = b + a. So 3 + 6 = 6 + 3, and a child who knows one turn-around fact knows its partner.
Grouping does not change a sum. For any three numbers, (a + b) + c = a + (b + c). So (6 + 4) + 5 = 6 + (4 + 5): 10 + 5 = 15 and 6 + 9 = 15.
Adding zero changes nothing. For any number, a + 0 = a and 0 + a = a.
Any order and any grouping. Together, the first two properties let any list of numbers be added in any order, putting together any two first. The total is always the same.
Fewer facts to learn. The table of sums of two numbers from 0 to 9 has 100 facts. Turn-around pairs are the same fact, which leaves 55 different ones. Ten of those have a zero in them and need no learning at all, which leaves 45.
Subtraction has an order. Swapping the two numbers changes a subtraction unless they are equal. 7 - 3 = 4, but 3 - 7 cannot be worked out with counters, so 7 - 3 \neq 3 - 7. And 9 - 3 - 2 = 4, while doing 3 - 2 first gives 9 - 1 = 8.
Adding three numbers
The first two properties are what a child is using whenever she chooses which two of three numbers to add first. Before adding, she scans the numbers for an easy pair.
Look for ten first. The pairs that make ten are 1 and 9, 2 and 8, 3 and 7, 4 and 6, and 5 and 5, from the bonds to ten. The pair can be anywhere: the first two, the last two, or the first and the last with another number between them. Once the ten is made, the third number joins it, and ten and some more is a teen number said without counting (Tens and ones). Written down, a child can show the move by writing the pair first:
Figure 5 shows it with ten frames. The 7 dark counters and the 3 light ones fill the first frame exactly. The 5 gray counters sit in the second. A full frame and 5 more is 15, with no counting.
Look for a double. If no pair makes ten, a pair of equal numbers is the next best find. Doubles are the facts most children know first, so 6 + 3 + 6 goes quickly as 6 + 6 = 12 and then 12 + 3 = 15. A pair that is one away from a double, such as 4 + 5, is nearly as good.
When there is neither. Some sets of three have no ten and no double, such as 7 + 5 + 4. The freedom still holds, so the child picks whichever pair she finds easiest and adds the third number after, making ten on the way if the total passes ten. Left to right is never wrong; choosing the pair is only about ease.
Why it is true
Order. Adding 3 and 6 asks one question: how many counters are there when a group of three and a group of six are pushed into one pile? The pile does not remember which group went in first. The same nine counters are in it either way, and a count does not depend on the order in which things are touched (How counting tells how many), so both orders give the same number. Nothing in this argument used the numbers three and six, so it holds for any two numbers.
Grouping. Adding three numbers asks how many counters there are when all three groups are in one pile. Pushing the first two groups together and then the third, or the last two together and then the first, ends with the same counters in the same pile. Nothing is added and nothing is lost, so the total cannot change. Each counter goes into the pile exactly once, whichever pair went first.
Any order and any grouping. With three numbers, the two properties together reach every arrangement. To bring the 3 at the end of 7 + 5 + 3 next to the 7, group the last two, 7 + (5 + 3), turn that pair around, 7 + (3 + 5), and regroup, (7 + 3) + 5. Each step is allowed, so the total never moves. The same steps, repeated, rearrange a list of any length.
Zero. A group of zero has no counters in it. Pushing it into a pile adds none, so the pile is the size it was. Counting on agrees: after saying “five”, counting on zero means saying no more words, so the count still ends on “five”.
Subtraction. To show that a rule does not always hold, one example where it fails is enough. 7 - 3 = 4, while 3 - 7 is not 4 (with counters it cannot be done at all), so swapping fails. And (9 - 3) - 2 = 4 while 9 - (3 - 2) = 8, so regrouping fails. The reason underneath is that the first number in a subtraction is the pile and the second is how many leave it, and those are different jobs. Even zero shows it: 7 - 0 = 7, but 0 - 7 asks to take seven from an empty plate.
Worked examples
Read each problem aloud. Counters in three colors, cubes that snap together, or buttons, coins and pasta shapes let the child build every one.
One fact, two sums
A child knows that 8 + 3 = 11. Find 3 + 8 = \square.
SolutionThe two sums use the same two groups, a group of eight and a group of three. Only the order is different, and order does not change a sum. So
and \square = 11. A child who does not yet know the fact can still use the property: rather than counting on eight from 3, she starts at the bigger number and counts on three, “eight ... nine, ten, eleven”. Either way the answer is 11, and a train of eight cubes and three cubes, turned around, shows why.
An empty basket
Ravi has two baskets. One holds 7 apples. The other is empty. How many apples does Ravi have?
SolutionRavi’s apples are all the apples in both baskets, so the two groups are joined: 7 + 0. The empty basket adds no apples, so
Ravi has 7 apples. If the empty basket is counted first, the sum is 0 + 7, and it is still 7: an empty basket adds nothing whichever one is looked at first.
Cars in a row
Ben has 2 red cars, 8 blue cars and 5 green cars. How many cars does Ben have?
SolutionAll three groups of cars are Ben’s, so they are joined: 2 + 8 + 5. Look for ten first. The 2 and the 8 make ten, and they are already side by side. So add them first, then the 5:
Ben has 15 cars. Here left to right and making ten are the same road, because the pair that makes ten comes first.
A pair with a number between
4 + 9 + 6 = \square
SolutionThe 4 and the 9 do not make ten, and neither do the 9 and the 6. But the 4 and the 6 do. They are not next to each other, and that does not matter: order and grouping do not change a sum, so any two of the three may be put together first. A child can show the choice with a bond joining the 4 and the 6 underneath (Figure 6).
Now ten and nine make nineteen:
So \square = 19. Going left to right also gives 19, since 4 + 9 = 13 and 13 + 6 = 19, but the first step crosses ten and the second adds on to a teen number. Finding the ten turned two harder steps into one easy one.
Frogs on a log
6 frogs sit on a log. 3 frogs sit on a rock. 6 frogs swim in the pond. How many frogs are there in all?
SolutionAll the frogs are counted together, so this is 6 + 3 + 6. No two of these numbers make ten: 6 + 3 = 9 and 6 + 6 = 12. But the two 6s are a double.
Every frog on the log has a partner in the pond, so those two groups make 6 + 6 = 12 (Figure 7). Then the 3 frogs on the rock join them:
There are 15 frogs. A child who prefers to go left to right gets 6 + 3 = 9 first, then 9 + 6 = 15 by making ten, which is the same answer by a longer road.
Ten or a double?
8 + 2 + 8 = \square
SolutionThis time there are two good pairs to choose from. The 8 and the 2 make ten:
The two 8s are a double, so the child could start there instead:
So \square = 18 either way. Both choices are correct, because both put the same eighteen counters in one pile. The ten is usually quicker, since ten and eight is read straight off as eighteen, but a child who knows double eight well has a perfectly good road too. Asking “which way did you do it?” and hearing both is worth more than choosing one for her.
No ten and no double
7 + 5 + 4 = \square
SolutionCheck each pair. 7 + 5 = 12, 7 + 4 = 11 and 5 + 4 = 9: none of them is ten. No two of the numbers are the same, so there is no double either. The freedom to choose still holds, so pick the pair that is easiest. The 5 and the 4 are one away from the double 4 + 4 = 8, so they make 9. That leaves 9 + 7, and 9 needs just 1 to make ten:
So \square = 16. Another child might add 7 + 4 = 11 first and then 11 + 5 = 16. Different children will pick different pairs here, and every pick gives 16.
A missing group
Sam, Mia and Jo pick 17 apples in all. Sam picks 3 apples and Jo picks 7 apples. How many apples does Mia pick?
SolutionThe three amounts joined make 17, and one amount is missing:
The same freedom helps here. Sam’s 3 and Jo’s 7 make ten, so put them together first. Then the question is “ten and how many more make seventeen?” (a missing part), and seventeen is ten and seven:
Mia picks 7 apples. Check it by adding all three: 3 + 7 + 7 = 10 + 7 = 17.
Does taking away turn around?
Is 8 - 5 the same as 5 - 8? Does 9 - 3 - 2 give the same answer whichever subtraction is done first?
SolutionBuild 8 - 5: eight counters, take five away, and three are left, so 8 - 5 = 3. Now build 5 - 8: five counters, and eight are to be taken away. All five go and three more are still wanted, so it cannot be done with counters. The two are not the same:
For 9 - 3 - 2, working left to right means take 3 from 9, leaving 6, and then take 2 more, leaving 4. Doing 3 - 2 = 1 first instead gives 9 - 1 = 8. Written with brackets,
so grouping changes a subtraction too. The first way is the right one for the question, because it takes away 3 and then 2, five counters in all. Doing 3 - 2 first takes only one counter away from the nine.
Boundaries
- The freedom belongs to adding. It comes from pushing groups into one pile. Taking away starts from one pile and removes from it, so in a subtraction neither the order nor the grouping may be changed (taking away).
- Zero is an identity on only one side of a subtraction. 7 - 0 = 7, so taking zero away changes nothing. But 0 - 7 is not 7: it asks to take seven from nothing.
- Every number is used once. Moving a number or joining two first must keep all of them. Each counter goes into the pile exactly once, which is what keeps the total the same.
- Choosing the pair is a choice, not a rule. Left to right always gives the right total. A ten or a double makes the work easier, never more correct.
- Totals up to twenty. This lesson keeps the numbers small enough that the totals are at most 20. The properties do not stop there: 9 + 9 + 9 can be grouped any way too, and 25 + 18 = 18 + 25. But adding past twenty is place value work for later.
- Multiplication has its own versions. This is for the adult, not the child yet: when multiplication arrives, order and grouping still do not change a product, but the number that leaves another unchanged there is 1, not 0. 5 \times 1 = 5, while 5 \times 0 = 0 (Properties of multiplication, Identity and inverse elements).
Where this goes next
The same freedom works for more than three numbers. With 2 + 3 + 8 + 7, a child can join the 2 and the 8, then the 3 and the 7, and see two tens: 10 + 10 = 20. Stories with several amounts in them lead on to problems in words and two-step problems, where choosing a good order is often the difference between an easy problem and a hard one.
Turn-around facts are half of a fact family: 3 + 6 = 9 and 6 + 3 = 9 sit beside 9 - 6 = 3 and 9 - 3 = 6. With bigger numbers the habit of looking for tens carries straight into adding in your head: 25 + 18 + 5 is quickest as 25 + 5 = 30 and then 30 + 18 = 48. The properties also explain making ten, which splits one number and regroups the pieces.
Much later, 3 - 7 gets an answer, -4, once there are numbers below zero, and it is still not the same as 7 - 3. The properties named here return for every kind of number in the commutative and associative properties and the properties of real numbers, where algebra uses them on every line. A child who has pushed counters into one pile already knows why they are true.
Exercises
Turn it around
- 5 + 9 = 14, so 9 + 5 = \square.
- \square + 6 = 6 + 8
Adding zero
- 0 + 13 = \square
- 9 + \square = 9
- 6 + 0 + 4 = \square
Pictures
Use the pictures in Figure 8.
- Row A shows 4 + 5 + 6. Which two groups make ten? How many counters are there in all?
- Row B shows 8 + 7 + 2. Which two groups make ten? How many counters are there in all?
- Row C shows 5 + 4 + 5. Which two groups would you put together first? How many counters are there in all?
Look for ten
- 3 + 7 + 6 = \square
- 9 + 4 + 1 = \square
- 2 + 5 + 8 = \square
- 6 + 9 + 4 = \square
Look for a double
- 4 + 4 + 7 = \square
- 3 + 8 + 3 = \square
Missing numbers
- 6 + \square + 4 = 18
- 5 + 3 + 5 = 10 + \square
Stories
- Ella has 5 red beads, 6 blue beads and 5 yellow beads. How many beads does Ella have?
- On a bus there are 3 children at the front, 9 in the middle and 7 at the back. How many children are on the bus?
- There are 14 birds on three branches. 4 birds are on the first branch and 6 are on the second. How many birds are on the third branch?
Subtraction has an order
- 9 - 4 = \square. Can 4 - 9 be worked out with counters?
- Jo works out 10 - 5 - 3 by doing 5 - 3 first. Is she right?
- Which of these can be turned around without changing the answer: 6 + 2, 6 - 2, 0 + 5?
Explain
- Omar adds 6 + 7 + 4 by putting the 6 and the 4 together first. Kate adds from left to right. Do they get the same answer? How do you know?
- Which two numbers would you add first in 5 + 8 + 5? Why?
- Show with counters that 2 + 9 + 8 is the same as 10 + 9.
- Is there a pair that makes ten in 3 + 4 + 5? How could you add these three numbers?
- Without adding, explain why 6 + 9 and 9 + 6 must give the same total.
- Leah says “8 - 2 must be the same as 2 - 8, because 8 + 2 is the same as 2 + 8.” Is she right? Explain.
- Why does adding 0 never change a number?
Answers
- 14. The two sums use the same two numbers in the other order, so 9 + 5 = 5 + 9 = 14.
- 8. The right side has a 6 and an 8, so the left side needs the same two numbers: 8 + 6 = 6 + 8.
- 13. Adding zero changes nothing: 0 + 13 = 13.
- 0. The total is still 9, so nothing was added: 9 + 0 = 9.
- 10. The 0 adds nothing, so 6 + 0 + 4 = 6 + 4 = 10.
- The 4 and the 6 make ten; 15 in all. 4 + 6 = 10, and 10 + 5 = 15.
- The 8 and the 2 make ten; 17 in all. 8 + 2 = 10, and 10 + 7 = 17.
- The two 5s; 14 in all. 5 + 5 = 10 is a double and a ten at once, and 10 + 4 = 14.
- 16. 3 + 7 = 10, so 3 + 7 + 6 = 10 + 6 = 16.
- 14. 9 + 1 = 10, so 9 + 4 + 1 = 9 + 1 + 4 = 10 + 4 = 14.
- 15. 2 + 8 = 10, so 2 + 5 + 8 = 2 + 8 + 5 = 10 + 5 = 15.
- 19. 6 + 4 = 10, so 6 + 9 + 4 = 6 + 4 + 9 = 10 + 9 = 19.
- 15. 4 + 4 = 8, and 8 + 7 = 15 by making ten: 8 + 2 + 5 = 10 + 5 = 15.
- 14. 3 + 3 = 6, so 3 + 8 + 3 = 3 + 3 + 8 = 6 + 8 = 14.
- 8. 6 + 4 = 10, so the sum is 10 + \square = 18, and eighteen is ten and eight.
- 3. 5 + 5 = 10, so 5 + 3 + 5 = 5 + 5 + 3 = 10 + 3.
- 16 beads. 5 + 5 = 10, so 5 + 6 + 5 = 5 + 5 + 6 = 10 + 6 = 16.
- 19 children. 3 + 7 = 10, so 3 + 9 + 7 = 3 + 7 + 9 = 10 + 9 = 19.
- 4 birds. 4 + 6 = 10, so 4 + 6 + \square = 14 is 10 + \square = 14, and fourteen is ten and four.
- 5; and no. Take four from nine counters and five are left, so 9 - 4 = 5. For 4 - 9 there are only four counters, and nine cannot be taken from them: all four go and five more are still wanted.
- No, the answer is 2. Take 5 from 10 to leave 5, then 3 more to leave 2: 10 - 5 - 3 = 2. Jo’s way gives 10 - 2 = 8, because doing 5 - 3 first takes only two counters away instead of eight.
- 6 + 2 and 0 + 5. 6 + 2 = 2 + 6 = 8 and 0 + 5 = 5 + 0 = 5, because order does not change a sum. But 6 - 2 = 4, while 2 - 6 cannot be worked out with counters.
- Yes, both get 17. Omar: 6 + 4 = 10 and 10 + 7 = 17. Kate: 6 + 7 = 13 and 13 + 4 = 17. They join the same counters into one pile, so the total cannot change.
- The two 5s. They are a double and they make ten, so 5 + 8 + 5 = 5 + 5 + 8 = 10 + 8 = 18.
- Make 2, 9 and 8 with counters. Push the 2 and the 8 together: 2 + 8 = 10, a full ten frame, with the 9 still beside it. No counter was added or taken away, so 2 + 9 + 8 = 10 + 9 = 19.
- No: 3 + 4 = 7, 3 + 5 = 8 and 4 + 5 = 9. There is no double either. Add any pair first, for example 4 + 5 = 9 (one more than double four), then 9 + 3 = 12. The total is 12 in every order.
- Snap six cubes and nine cubes into one train. Read from one end it is 6 + 9; turn it around and it is 9 + 6. It is the same train, with no cube added or taken off, so the two totals must be the same, whatever that total is.
- No. Adding puts two groups into one pile, and the pile does not care which went in first. Subtracting starts from a pile and takes some away, so the first number has to be the pile. 8 - 2 = 6, but 2 - 8 asks to take eight from two, which cannot be done with counters.
- Adding 0 means joining a group with nothing in it. No counters go into the pile, so the pile has exactly as many as before. Counting on agrees: after saying the first number, counting on zero means saying no more numbers.