Fact Families & Related Facts within 10 - First Grade Mathematics

Addition and subtraction are not strangers; they are members of the same mathematical family! In fact, addition and subtraction are inverse operations, which means they are exact opposites that undo each other. If addition puts two groups together, subtraction takes them apart. Whenever three numbers fit together in a number bond, they create a cozy group of four related equations known as a fact family. In this chapter, you will discover the deep bond between addition and subtraction, learn how to build complete fact families, and use addition to solve subtraction problems with ease.

The Concept of a Fact Family

A fact family is a group of related addition and subtraction facts that all use the exact same three numbers.

The Fact Family House for 3, 5, and 8:

             /\
            /  \
           / 8  \       <- Whole
          /------\
         / 3    5 \     <- Parts
        +----------+
        | 3 + 5 = 8|
        | 5 + 3 = 8|
        | 8 - 3 = 5|
        | 8 - 5 = 3|
        +----------+

The Three Family Members

In this house, the three numbers are 3, 5, and 8. Notice: No outside numbers are allowed inside the house! Every single equation in the family uses ONLY the numbers 3, 5, and 8!

The Four Equations:

Equation 1 (Addition): Part + Part = Whole (3 + 5 = 8) Equation 2 (Addition Turn-Around): Switch the parts! (5 + 3 = 8) Equation 3 (Subtraction): Whole - Part = Other Part (8 - 3 = 5) Equation 4 (Subtraction): Whole - Other Part = First Part (8 - 5 = 3)

Subtraction Always Starts with the Whole

Look closely at the two subtraction equations in the fact family: 8 - 3 = 5 and 8 - 5 = 3.

The Golden Rule of the Fact Family:
Addition equations END with the Whole (Part + Part = WHOLE)
Subtraction equations START with the Whole (WHOLE - Part = Part)

You Cannot Start Subtraction with a Part!

A common beginner mistake is writing 3 - 5 = 8. But 3 is just a small part; you cannot start with 3 and subtract 5 to get 8! In a fact family, subtraction MUST always start with the biggest number: the whole!

Fact Families for Doubles Facts

What happens to a fact family when the two parts are identical, such as in a doubles fact?

The Fact Family for Double 4 (4, 4, 8):
Whole = 8, Parts = 4 and 4

Addition Fact:     4 + 4 = 8
Subtraction Fact:  8 - 4 = 4

Only Two Equations!

Because both parts are 4, turning around 4 + 4 is still 4 + 4. And subtracting 4 from 8 always leaves 4. So a doubles fact family has only two equations instead of four!

Using Addition to Solve Subtraction

Once you understand fact families, you can use addition to solve any subtraction problem in your head. This is called "thinking addition."

Solving 9 - 4 = ?
Think: "What number plus 4 equals 9?"
You know that 4 + 5 = 9!
Therefore, 9 - 4 MUST equal 5!

Practice Exercises

Exercise 1: Write the four related equations that make up the fact family for the numbers 2, 7, and 9.

Exercise 2: Write the complete fact family for the numbers 4, 6, and 10.

Exercise 3: Write the two equations that make up the fact family for the doubles fact 5 + 5 = 10.

Exercise 4: Use the fact 3 + 6 = 9 to solve this subtraction problem without counting: 9 - 6 = __. Explain your thinking.

Solutions and Explanations

Solution to Exercise 1: The four equations for the fact family 2, 7, 9 are: 2 + 7 = 9 7 + 2 = 9 9 - 2 = 7 9 - 7 = 2

Solution to Exercise 2: The four equations for the fact family 4, 6, 10 are: 4 + 6 = 10 6 + 4 = 10 10 - 4 = 6 10 - 6 = 4

Solution to Exercise 3: The two equations for the doubles fact family 5, 5, 10 are: 5 + 5 = 10 10 - 5 = 5

Solution to Exercise 4: 9 - 6 = 3. Because 3, 6, and 9 belong to the same fact family (3 + 6 = 9), when you start with the whole of 9 and remove the part of 6, the remaining part must be 3.