Fact families
A child knows that 3 + 4 = 7. How many other adding and taking-away facts do they already know, without working anything out?
Where you are 12 in Addition and subtraction
Why this exists
Children learning to add and take away meet a great many small facts, and at first each one seems to need its own work: 3 + 4, then 4 + 3, then 7 - 3, then 7 - 4. Counted out separately, that is four jobs.
They are really one job. All four sentences are about the same three numbers, a whole of 7 and its two parts 3 and 4, so a child who knows one of them already knows the other three. That set of sentences is a fact family. It cuts the number of facts a child has to hold in mind to about a quarter, and, more importantly, it gives every taking-away fact a partner addition fact to lean on. A child stuck on 7 - 3 can ask “3 and what make 7?” and use what they know about adding.
Definition
Take three numbers where two of them, the parts, make the third, the whole. The fact family for those numbers is the set of true adding and taking-away sentences that use those three numbers, each one once. For the parts 3 and 4 and the whole 7 there are four:
The two adding sentences put the parts together to make the whole. The two taking-away sentences start from the whole and take one part away, leaving the other part.
A fact family is the part-part-whole picture written as sentences, and the number bond is the quickest way to hold it: the whole in the top circle, the parts below, and all four sentences read off the same three circles.
Each sentence can also be written with its two sides swapped, as in 7 = 3 + 4. That is the same fact read from the other side of the equals sign, not a new member of the family. Some classrooms also write 3 + 4 = 4 + 3, which is true for the same reason the two adding sentences agree.
Intuition
Build a train of 7 cubes, 3 dark and then 4 light, and read it (the first row of the picture below). Dark and light together: 3 + 4 = 7. Now turn the whole train around. It reads light first, 4 + 3 = 7, and it is the very same seven cubes.
Now snap the dark cubes off and slide them away. What is left is the light part: from 7 cubes, taking away 3 leaves 4. Put the dark cubes back and snap off the light ones instead: taking away 4 leaves 3.
The child never counted anything new. One set of cubes was turned round once and broken apart twice, and that is why the four sentences cannot disagree. It helps to say the family aloud as it is built: “three and four make seven, four and three make seven, seven take away three is four, seven take away four is three.”
A small habit makes the family easy to write: find the whole first. It is the biggest of the three numbers. In an adding sentence it comes after the equals sign; in a taking-away sentence it comes first, because we can only take away from what we have.
Results
One whole and two parts give four facts. If two parts make a whole, then
| Sentence | In words |
|---|---|
| 3 + 4 = 7 | one part and the other part make the whole |
| 4 + 3 = 7 | the same two parts, the other way round |
| 7 - 3 = 4 | the whole take away one part leaves the other part |
| 7 - 4 = 3 | the whole take away the other part leaves the first |
and the same four shapes work for any whole and its two parts.
Knowing one fact gives the rest of the family. Whichever sentence a child knows, the three numbers in it are the whole family.
A double has only two facts. When the two parts are the same, turning round changes nothing:
A family with zero still has four facts. For the parts 5 and 0:
Why it is true
Everything here comes from the part-part-whole picture. The whole is made of exactly two parts, with every object in one part or the other and none in both.
Putting the two parts together gives back all the objects, so the parts add to the whole. It does not matter which part is put down first, because the same objects end up together either way (the turned train in the picture). That gives both adding sentences.
Taking one part away from the whole leaves exactly the objects that were in the other part, because each object was in one part or the other. Doing that with each part in turn gives both taking-away sentences. Nothing in this argument used the numbers 3, 4 and 7, so it holds for every whole and every way of splitting it in two.
When the two parts are the same size, the two adding sentences are the same sentence, and so are the two taking-away sentences, which is why a double has two facts instead of four.
Worked examples
Read each problem aloud. Let the child build it with cubes or counters whenever they want to.
A family from a train
“Look at this train. How many dark cubes? How many light cubes? How many cubes in all? Now tell me all the facts the train shows.”
There are 2 dark cubes and 5 light cubes, and 7 cubes in all. So 7 is the whole, and 2 and 5 are its parts.
Reading dark then light: 2 + 5 = 7. Turning the train around and reading light then dark: 5 + 2 = 7. Snapping off the dark cubes leaves the light ones: 7 - 2 = 5. Snapping off the light cubes leaves the dark ones: 7 - 5 = 2. The family is
Each sentence matches something the child can do with their hands, which is the best way to see that all four belong together.
A family from a number bond
“Here is a number bond. Write its fact family.”
The top circle holds the whole, 9, and the circles below hold the parts, 6 and 3. The two parts together make the whole, in either order:
Taking a part away from the whole leaves the other part, so both taking-away sentences start with 9:
A child can check the bond before writing anything: counting on from 6, “seven, eight, nine”, is three counts, so 6 and 3 really do make 9.
Using an adding fact to take away
“Leo knows that 8 + 4 = 12. His sister asks him, ‘What is 12 - 8?’ How can Leo use what he knows?”
SolutionLeo’s fact has the parts 8 and 4 and the whole 12. The question 12 - 8 starts with 12, the whole, and takes away 8, one of the parts. Whatever is left must be the other part, 4.
Leo did not count back eight steps. He asked himself “8 and what make 12?” and he already knew the answer. That is the real use of a fact family: each taking-away fact is hiding inside an adding fact the child knows. The same fact also gives 12 - 4 = 8. Thinking of subtraction this way has its own lesson, subtraction as a missing part.
A family of only two facts
“Write the fact family for 5, 5 and 10.”
The whole is 10 and both parts are 5. Build a train of 5 dark and 5 light cubes: it reads 5 + 5 = 10. Turn it around and it reads light first, which is still 5 + 5 = 10. The turned train gives no new sentence, because the two parts are the same number.
Snap off either color and 5 cubes are left, so the only taking-away sentence is
The family is just 5 + 5 = 10 and 10 - 5 = 5. Two facts instead of four is not a mistake: it is what happens with every double.
Do these numbers make a family?
“Can you make a fact family with 2, 4 and 5? What about 4, 5 and 9?”
SolutionA family needs a whole and two parts, so the two smaller numbers must make the biggest one. For 2, 4 and 5, the biggest is 5, but 2 + 4 = 6, not 5. These three numbers are not a whole and its parts, so they have no fact family. Any sentence written with them, such as 5 - 2 = 4, would be false: 5 - 2 = 3.
For 4, 5 and 9, the biggest is 9, and 4 + 5 = 9. They do make a family:
Adding the two smaller numbers is the test every time.
One family, four stories
“There are 6 red fish and 3 yellow fish in a tank. Tell a story for each fact in the family.”
SolutionThe red fish and the yellow fish are the two parts, and all the fish together are the whole: 6 + 3 = 9, so there are 9 fish. Each fact in the family asks a different question about the same tank:
- 6 + 3 = 9: “There are 6 red fish and 3 yellow fish. How many fish are there?”
- 3 + 6 = 9: “There are 3 yellow fish and 6 red fish. How many fish are there?”
- 9 - 6 = 3: “There are 9 fish. 6 are red. How many are yellow?”
- 9 - 3 = 6: “There are 9 fish. 3 are yellow. How many are red?”
The first two ask for the whole, the last two ask for a part. Seeing that one tank answers all four is what lets a child choose freely between adding and taking away when a story problem comes along.
Boundaries
The three numbers must be a whole and its two parts. The two smaller numbers have to add up to the biggest. Three numbers picked at random, like 2, 4 and 5, have no fact family.
Taking away starts from the whole. Swapping the order works for adding, 3 + 4 and 4 + 3, but not for taking away. With the whole numbers used here, 3 - 7 has no answer, because we cannot take 7 cubes from 3. So both taking-away sentences start with the whole. That taking away cannot be turned round is a fact of its own.
Not every family has four facts. A double has two, as in 5 + 5 = 10 and 10 - 5 = 5. A family with zero still has four, but two of them look odd to a child: 5 - 0 = 5 and 5 - 5 = 0.
One whole has many families. The whole 7 belongs to the family of 3 and 4, and also to the families of 1 and 6, 2 and 5, and 0 and 7. A fact family fixes both parts, not just the whole.
These are adding and taking-away families only. Multiplying and dividing have families of their own, built the same way from equal groups: fact families for multiplication and division.
Where this goes next
The family is the working tool behind subtracting within ten and the bonds to twenty: a child who knows the adding facts is never far from the taking-away facts. It is also why counting up to subtract works, and why a taking-away answer can always be tested with checking a difference by adding: the answer and the part taken away must add back to the whole.
Other strategies lean on it too. Bridging back through ten and making ten split numbers into parts, and every split is a small fact family. Later the same families hold for tens and hundreds, so 30 + 40 = 70 gives 70 - 40 = 30; the pattern comes back for multiplying and dividing in fact families for multiplication and division, and years later it is the idea of inverse operations: adding and taking away undo each other.
Exercises
Write the family
Use the trains for Exercises 1 and 2. Say how many cubes in all, then write the fact family.
- Train A.
- Train B.
- A number bond has 8 in the top circle and 5 and 3 in the part circles. Write its fact family.
- Write the fact family for 4, 6 and 10.
- Write the fact family for 7, 9 and 16.
Use one fact to find another
- You know 6 + 3 = 9. What is 9 - 6?
- You know 7 + 5 = 12. What is 12 - 5?
- You know 10 - 4 = 6. What is 4 + 6?
- You know 9 + 8 = 17. What is 17 - 9?
- You know 30 + 40 = 70. What is 70 - 40?
- Find the number in the box: 5 + \square = 8. Then find 8 - 5.
Is it a family?
- Do 3, 5 and 8 make a fact family?
- Do 2, 3 and 6 make a fact family?
- Write the fact family for 6, 6 and 12. How many facts are in it?
- Write the fact family for 7, 0 and 7.
Explain
- Pat wrote the family for 4, 5 and 9: 4 + 5 = 9, 5 + 4 = 9, 9 - 4 = 5 and 4 - 9 = 5. Which one is wrong? How do you know?
- Use cubes to show why 10 - 3 = 7 is in the same family as 3 + 7 = 10.
- Why does the family for 5, 5 and 10 have only two facts?
- The whole is 7. Find two different fact families that have 7 as the whole.
Answers
- 5 cubes. 1 + 4 = 5, 4 + 1 = 5, 5 - 1 = 4, 5 - 4 = 1. The dark cube and the light cubes are the parts.
- 8 cubes. 6 + 2 = 8, 2 + 6 = 8, 8 - 6 = 2, 8 - 2 = 6. Six dark and two light make eight.
- 5 + 3 = 8, 3 + 5 = 8, 8 - 5 = 3, 8 - 3 = 5. The top circle, 8, is the whole, so both taking-away sentences start with 8.
- 4 + 6 = 10, 6 + 4 = 10, 10 - 4 = 6, 10 - 6 = 4. The biggest number, 10, is the whole.
- 7 + 9 = 16, 9 + 7 = 16, 16 - 7 = 9, 16 - 9 = 7. Check that the parts make the whole: 9 + 7 = 16, since 9 and 1 make 10 and 6 more is 16.
- 3. The whole is 9 and 6 is one part, so what is left is the other part: 9 - 6 = 3.
- 7. The whole is 12; take away the part 5 and the part 7 is left: 12 - 5 = 7.
- 10. The fact 10 - 4 = 6 says 10 is the whole and 4 and 6 are its parts, so 4 + 6 = 10.
- 8. In 9 + 8 = 17 the parts are 9 and 8, so 17 - 9 = 8.
- 30. The parts are 30 and 40 and the whole is 70, so taking away the part 40 leaves the other part: 70 - 40 = 30. It is the family of 3, 4 and 7 counted in tens: 3 tens and 4 tens make 7 tens.
- \square = 3, and 8 - 5 = 3. “5 and what make 8?”: “six, seven, eight”, three counts. The same three numbers give the taking-away fact.
- Yes. 3 + 5 = 8, so 8 is the whole: 3 + 5 = 8, 5 + 3 = 8, 8 - 3 = 5, 8 - 5 = 3.
- No. The biggest number is 6, but 2 + 3 = 5, not 6, so these are not a whole and its parts.
- Two facts: 6 + 6 = 12 and 12 - 6 = 6. Both parts are 6, so turning round gives the same sentences.
- 7 + 0 = 7, 0 + 7 = 7, 7 - 0 = 7, 7 - 7 = 0. Taking away nothing leaves all 7; taking away all 7 leaves nothing.
- 4 - 9 = 5 is wrong. Taking away must start from the whole, and you cannot take 9 from 4. The right fact is 9 - 4 = 5, which she already has, so the missing one is 9 - 5 = 4.
- Make a train of 3 dark and 7 light cubes: 3 + 7 = 10. Snap off the 3 dark cubes and the 7 light ones are left: 10 - 3 = 7. It is the same train, so the two facts are in one family.
- The two parts are both 5. Turning 5 + 5 round gives 5 + 5 again, and taking away either part leaves 5, so there is one adding fact and one taking-away fact.
- Any two splits of 7, for example 2 and 5: 2 + 5 = 7, 5 + 2 = 7, 7 - 2 = 5, 7 - 5 = 2; and 1 and 6: 1 + 6 = 7, 6 + 1 = 7, 7 - 1 = 6, 7 - 6 = 1. The pairs 3 and 4, and 0 and 7, work too.