A fact family is a group of related math facts that use the same three numbers. With 6, 9 and 15 you get 6 + 9 = 15, 9 + 6 = 15, 15 − 6 = 9 and 15 − 9 = 6. Multiplication and division make fact families too, such as 4, 7 and 28. They show that operations undo each other.
Each fact family uses one set of three numbers. The whole (sum or product) goes at the top; the two parts at the bottom.
Formal definition and notation
For numbersa, b and c with a + b = c, the addition fact family is the set of equations a + b = c, b + a = c, c − a = b and c − b = a. For nonzero a and b with a × b = c, the multiplication fact family is a × b = c, b × a = c, c ÷ a = b and c ÷ b = a. Each family rests on the commutative property and on inverse operations.
Symbol
Read it as
Meaning
a + b = c
a plus b equals c
the starting fact; the family adds b + a = c, c − a = b, c − b = a
a × b = c
a times b equals c
a multiplication family adds b × a = c, c ÷ a = b, c ÷ b = a
Key facts
A family with two different parts has 4 facts. If the two parts are equal, as in 5, 5 and 10, it has only 2: 5 + 5 = 10 and 10 − 5 = 5.
The biggest number (the sum or the product) always starts the subtraction or division facts.
Fact families turn hard facts into known ones: to find 14 − 8, think “8 plus what is 14?” The answer is 6.
Addition and multiplication never mix in one family: 3, 4, 7 is an addition family, 3, 4, 12 is a multiplication family.
A fact triangle puts the whole at the top corner and the two parts at the bottom corners.
0 does not make a full division family: 0 × 5 = 0 and 0 ÷ 5 = 0, but 0 ÷ 0 is undefined.
Fact Family examples, step by step
Example 1: Write an addition family
Warm-up
Write the fact family for 8, 5 and 13.
13 is the whole, since 8 + 5 = 13.
Two addition facts: 8 + 5 = 13 and 5 + 8 = 13.
Two subtraction facts start with the whole: 13 − 8 = 5 and 13 − 5 = 8.
Inverse operations are the pairs that undo each other (add and subtract, multiply and divide). A fact family is a set of four specific facts built from those pairs, such as 3, 5, 8.
A multiplication table lists many products in a grid. One fact family uses one product and its two factors, like 6, 8, 48.
Quick check: 3 questions
Question 1 of 3
Which equation belongs to the fact family of 7, 9 and 16?
Show answer and explanation
Answer: A.16 − 9 = 7 uses only 7, 9 and 16 and starts with the whole. 7 × 9 is 63, not 16; the other two bring in new numbers (2 and 23).
Question 2 of 3
Use the fact family of 6 × 8 = 48. What is 48 ÷ 6?
Show answer and explanation
Answer: C. In the family 6, 8, 48, dividing the product by one factor gives the other: 48 ÷ 6 = 8. 42 and 54 subtract or add 6, and 6 is the divisor itself.
Question 3 of 3
How many facts are in the fact family of 9, 9 and 81?
Show answer and explanation
Answer: D. The two factors are equal, so there are just 9 × 9 = 81 and 81 ÷ 9 = 9. Swapping the factors repeats a fact, so 4 and 3 overcount.
Fact Family: frequently asked questions
Why do teachers use fact families?
Fact families show that subtraction undoes addition and division undoes multiplication. Once a student knows 7 + 5 = 12, they also know 12 − 5 = 7 and 12 − 7 = 5. That cuts the number of facts to memorize and builds the habit of checking: a division answer is right when the multiplication in its family is true.
Can a fact family have only two facts?
Yes. When the two parts are the same number, swapping them gives the same fact. The family of 4, 4 and 8 is just 4 + 4 = 8 and 8 − 4 = 4, and the family of 6, 6 and 36 is 6 × 6 = 36 and 36 ÷ 6 = 6. Square numbers always give two-fact multiplication families.
NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4Commutative PropertyThe commutative property says that changing the order of the numbers does not change the result of adding or multiplying them: a + b = b + a and a × b = b × a. For example, 4 + 9 = 9 + 4 and 3 × 5 = 5 × 3. Subtraction and division are not commutative.Operations · Grade 1, Grade 3, Grade 6Inverse OperationsInverse operations are operations that undo each other. Addition and subtraction are inverses: 9 + 6 − 6 = 9. Multiplication and division are inverses: 7 × 4 ÷ 4 = 7. You use them to check answers and to solve equations, undoing each step in reverse order until the variable is alone.Operations · Grade 1, Grade 3, Grade 6SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1MinuendThe minuend is the number you subtract from: the first number in a subtraction. In 17 − 6 = 11, the minuend is 17, the subtrahend is 6 and the difference is 11. The minuend is the whole you start with, so it equals the difference plus the subtrahend.Operations · Grade 1, Grade 2, Grade 3Multiplication TableA multiplication table is a grid that lists the products of pairs of numbers, usually 1 to 10 or 1 to 12. To read it, find one factor along the top row and the other down the left column; the box where they meet is the product. Row 6 meets column 7 at 42, since 6 × 7 = 42.Operations · Grade 2, Grade 3, Grade 4Square NumberA square number is the result of multiplying a whole number by itself. The first square numbers are 1, 4, 9, 16, 25 and 36, because 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9 and so on. That many dots fill a square exactly, like 16 dots in a 4 by 4 square.Operations · Grade 3, Grade 4, Grade 5EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8AddendAn addend is any number that is added to another number. In 8 + 5 = 13, the addends are 8 and 5, and 13 is the sum. An addition can have two or more addends, and you can add them in any order. When one addend is missing, subtract the known addend from the sum to find it.Operations · Grade 1, Grade 2, Grade 3