Mathematics
Grade 7
15 min
Properties of division
Properties of division
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1
Introduction & Learning Objectives
Learning Objectives
Identify the dividend, divisor, and quotient in a division problem.
Explain with counterexamples why division is not commutative.
Explain with counterexamples why division is not associative.
Apply the properties of dividing by 1 and dividing 0 by a non-zero number.
Recognize that division by zero is undefined and explain why.
Correctly apply the distributive property of division over addition and subtraction.
Ever tried to split a pizza 🍕 between zero people? It doesn't make sense, right? Division has some unique rules that other operations don't follow!
In this tutorial, we will explore the fundamental properties of division. You'll learn which familiar properties like commutativity don't apply, and discover the special rules for d...
2
Key Concepts & Vocabulary
TermDefinitionExample
DividendThe number that is being divided. It's the total amount you start with.In `12 ÷ 4 = 3`, the dividend is 12.
DivisorThe number that you are dividing by. It's the number of groups you are splitting the dividend into.In `12 ÷ 4 = 3`, the divisor is 4.
QuotientThe result of a division problem.In `12 ÷ 4 = 3`, the quotient is 3.
Commutative PropertyA property where the order of the numbers does not change the result. This property does NOT apply to division.Addition is commutative: `2 + 3 = 5` and `3 + 2 = 5`. But for division, `10 ÷ 2 = 5` while `2 ÷ 10 = 0.2`.
Associative PropertyA property where the grouping of numbers (usually with parentheses) does not change the result. This property does NOT apply to division.Multiplication is associative: `(2 × 3...
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Core Formulas
Division by 1 (Identity Property)
a / 1 = a
Any number divided by 1 is equal to that number itself. Dividing something into one group doesn't change its value.
Division of Zero
0 / a = 0 (where a ≠ 0)
Zero divided by any non-zero number is always zero. If you have nothing to share, everyone gets nothing.
Division by Zero
a / 0 is undefined
You can never divide a number by zero. The operation has no meaning and is considered undefined in mathematics.
Right-Distributive Property
(a + b) / c = (a / c) + (b / c)
Division can be distributed over addition or subtraction when the sum/difference is in the dividend. This is useful for breaking down complex division problems.
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Challenging
For what value of x will the expression 72 ÷ (3x - 12) be undefined?
A.12
B.3
C.4
D.0
Challenging
A teacher has a prize bag with 180 tokens. They want to divide the tokens equally among a group of students. The number of students in the group is represented by (n-5), where n is the total number of students in the class. For what value of n would the plan be mathematically undefined?
A.n = 0
B.n = 5
C.n = 180
D.n = 1
Challenging
Which of the following statements about the properties of division is ALWAYS true for any non-zero integers a, b, and c?
A.a ÷ (b + c) = (a ÷ b) + (a ÷ c)
B.(a ÷ b) ÷ c = a ÷ (b ÷ c)
C.If a ÷ b = 1, then b ÷ a = 1.
D.(a + b) ÷ c = (a ÷ c) + (b ÷ c)
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