Joseph has a master's degree in literature as well as alternative teaching and ESL educator certifications. He has worked with middle school, high school, and college students in writing and language arts.
The Family That Adds and Subtracts Together
Just like a family starting with two people coming together, fact families in math start with two numbers coming together. Let's say, for example, 1 and 2 meet one day. Then, they come together to make 3. Here's their first family portrait:
1 + 2 = 3
But, in addition, we can switch the numbers we're adding and still get the same sum. So 1, 2, and 3 could also arrange themselves as:
2 + 1 = 3
And just like people in families, numbers can have different relationships with each other at different times. For example, 3, 2, and 1 could come together in a subtraction sentence. Since we're subtracting (or taking away from something), start with the largest number because it will be easiest to take away from.
Using our same three fact family members, we could start by subtracting 2 from 3, giving us this portrait of the three numbers:
3 - 2 = 1
And, of course, we could also subtract 1 from 3:
3 - 1 = 2
Fact Family Relationships with Addition and Subtraction
We can situate our three fact family numbers in four different ways using addition and subtraction. We can create two addition sentences because we can add numbers in any order (1 + 2 or 2 + 1). We can create two subtraction sentences because we can subtract either one of the smaller numbers from the largest number (3 - 1 or 3 - 2).
So, an addition and subtraction fact family consists of the four addition and subtraction sentences created using three numbers. The fact family shows the relationships between the numbers in it. All we need to create a fact family is two numbers that we can then add together and the sum of those two numbers.
Let's create another addition and subtraction fact family example. This time, we'll bring together 5 and 10.
When they come together in addition, they create 15. Here's their first family portrait:
5 + 10 = 15
Just like before, we can switch 5 and 10 in addition without changing the sum. In essence, the numbers are just rearranging themselves in their portrait:
10 + 5 =15
We can also make subtraction sentences with these three numbers. We'll subtract from the largest number in the fact family:
15 - 10 = 5
And we can also subtract the other smaller number from the largest, which in this case means taking 5 away from 15:
15 - 5 = 10
So the four number sentences in this fact family look like this:
5 + 10 = 15 so, 10 + 5 = 15. These are the two addition sentences for this fact family.
15 - 10 = 5 and 15 - 5 = 10. These are the two subtraction sentences for this fact family.
Four different number sentences all using the same three numbers; that's a fact family!
Fact Families for Multiplication and Division
We can also bring numbers together with multiplication. Let's use the numbers 5 and 10 again. When we multiply them together, we get a product of 50. This gives us a new portrait:
5 * 10 = 50
Similar to when we're adding, we can switch the numbers we're multiplying and still get the same product. So, the numbers in this family can rearrange themselves for a second portrait:
10 * 5 = 50
10, 5, and 50 are a fact family when multiplying and dividing. Let's take the same three numbers and make a division sentence with them. Just like in subtraction, we'll start with the largest number and divide it by one of the smaller numbers:
50 / 5 = 10
We can also divide the largest number by the other smaller number for a fourth portrait:
50 / 10 = 5
Fact Family Relationships with Multiplication and Division
We can situate the three fact family numbers in four different ways using multiplication and division. We can create two multiplication sentences because we can multiply numbers in any order (5 * 10 = 50 or 10 * 5 = 50). We can create two division sentences because we can divide the largest number by both of the smaller numbers (50 / 5 = 10 or 50 / 10 = 5).
So, a multiplication and division fact family consists of the four multiplication and division sentences we can create using three numbers. The fact family shows the relationships between the numbers in it. All we need to create them is two numbers that we can then multiply together and the product.
Let's multiply the numbers 8 and 12 for another example. We can then create four number sentences by multiplying, dividing, and rearranging the numbers just like we did in the previous example:
8 * 12 = 96
12 * 8 = 96
These are the two multiplication number sentences for this fact family.
96 / 8 = 12
96 / 12 = 8
These are the two division number sentences for this fact family. Altogether that is four number sentences showing the relationship between three numbers.
Fact families are number sentences that show the relationships between three different numbers. Addition and subtraction fact families make four number sentences that show the relationship between three numbers as you combine them and take them apart. Multiplication and division fact families make four number sentences that show the relationship between three numbers as you group them together and break those groups back apart.
Knowing the numbers within fact families shows understanding of number relationships. This means if you know 4 + 6 = 10, then you know 6 + 4 = 10. And if you remember that 4, 6, and 10 are a fact family, then you know that 10 - 4= 6, and 10 - 6 = 4. Similar relationships hold true for three numbers that you can use when multiplying and dividing. If you know that 6 * 7 = 42, then you can remember that 42 / 6 = 7.
Fact Family Key Terms
- Fact Families: number sentences showing the relationship between three different numbers
- Addition & Subtraction Fact Families: four number sentences that show the relationship between three numbers as you take them apart or combine them
- Multiplication and Division Fact Families: four number sentences that display the relationship between three numbers as you break them apart or group them together
All that you learn about fact families from this lesson can be drawn upon when you subsequently:
- Recognize a fact family
- Build addition and subtraction fact families
- Display knowledge of fact families for multiplication and division
- Create multiplication sentences and division sentences
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