The laws of integers are three simple rules that let your child reorder and regroup numbers without changing the answer: the commutative law, the associative law and the distributive law. Once a child owns these three, long sums stop feeling long.
I have taught Grade 6 math for over thirty years, and I introduce the laws exactly as I do below: one law at a time, one worked pair each, then the traps. A free worksheet waits at the end for practice.
What Are the Laws of Integers?
The laws of integers are three rules about order and grouping. The commutative law says order does not change the result. The associative law says grouping does not change the result. The distributive law says multiplication spreads over addition. Together they let a student rewrite a hard sum as an easier one with the same answer.
This is the skill behind Common Core standard 6.EE.A.3, which asks students to apply properties of operations to make equivalent expressions.
What Is the Commutative Law?
The commutative law says you can swap the order of two numbers in addition or multiplication and the answer stays the same. For addition, (-2 + 5) equals (5 + -2), and both give 3. For multiplication, (-4 x 3) equals (3 x -4), and both give -12. Order changes, answer does not.
Think of it like swapping seats: the people in the room stay the same, only their places change. But be careful: the commutative law does not cover subtraction. 5 – 2 = 3, but 2 – 5 = -3.
What Is the Associative Law?
The associative law says you can move the brackets in addition or multiplication and the answer stays the same. For addition, (-3 + 4) + 5 equals -3 + (4 + 5), and both give 6. For multiplication, (-4 x 2) x 3 equals -4 x (2 x 3), and both give -24. Grouping changes, answer does not.
Brackets tell you which pair to work out first. The associative law says that, for addition and multiplication, it does not matter which pair you start with. For example, (-1 + 4) + 2 and -1 + (4 + 2) both give 5.
What Is the Distributive Law?
The distributive law says multiplication spreads over addition inside brackets. The sum 3 x (5 + 2) equals 3 x 5 + 3 x 2, and both give 21. You multiply the outside number by each number inside, then add. The bracket disappears and the answer stays honest.
Try another: 4 x (6 + 3) equals 4 x 6 + 4 x 3, which is 24 + 12 = 36, the same as 4 x 9. For extra practice with this law, try my free Laws of Integers Worksheet for Grade 6.
Where the Laws Do NOT Work
The commutative law fails for subtraction: 5 – 2 = 3, but 2 – 5 = -3. It also fails for division: 12 / 3 = 4, but 3 / 12 = 0.25. The associative law also fails for subtraction and division. For example, 10 – (6 – 2) = 6, but (10 – 6) – 2 = 2. The laws cover addition and multiplication only.
How to Pick the Easy Path
Pick the grouping that makes a round number first. In (-6 + 4) + 6, regroup as -6 + (4 + 6) and the 10 appears at once, giving 4 with far less effort. The laws do not just describe sums, they hand you the simplest version of each sum.
My classroom routine has three steps: first, look for a pair that makes a round number; second, regroup to make that pair; third, finish the easy sum. A shopkeeper shows why this matters: 5 boxes, each holding 4 pens, 6 pencils and 2 erasers, give 20 pens, 30 pencils and 10 erasers, and grouping by item makes the count simple.
For more practice with the easy path, try my free Laws of Integers Worksheet for Grade 6.
Mistakes I See Every Year
Mistake one: calling every law commutative. Ask first: did the order change, or did the grouping change? Order is commutative, grouping is associative.
Mistake two: dropping a sign when regrouping, turning (-6) into 6. Carry the sign with the number, always.
Mistake three: spreading multiplication over only the first number inside the bracket, writing 3 x (5 + 2) as 15 + 2. Multiply by every number inside the bracket.
One habit prevents all three: after rewriting, work both versions and check that the answers match.
Try It Yourself: 3 Quick Questions
Cover the answers, let your child try, then check together.
- Which law is shown by (6 + -2) = (-2 + 6)?
Answer: the commutative law of addition. - Use the distributive law to solve 6 x (5 + 4).
Answer: 6 x 5 + 6 x 4 = 30 + 24 = 54. - Group (3 + -7) + 7 two ways.
Answer: 3. Grouping (-7 + 7) first gives 3 + 0 = 3.
Key Takeaways
- Commutative: order does not change the result, for addition and multiplication.
- Associative: grouping does not change the result, for addition and multiplication.
- Distributive: multiplication spreads over each number inside brackets.
- No law covers subtraction or division.
- Pick the grouping that makes a round number first, then finish the easy sum.
If your child handled those three questions, she is ready for a full page of practice. Print the free Laws of Integers Worksheet for Grade 6 and sit with her for the first two questions only. You will be surprised how quickly she takes over.
For more topics at this level, browse our Grade 6 math worksheets page. Every sheet is free to print, and each one grows out of a lesson I have taught in a real classroom.
Frequently Asked Questions
What are the three laws of integers?
The commutative law is about order, the associative law is about grouping, and the distributive law is about multiplication spreading over addition.
What is the commutative law?
You can swap the order of two numbers in addition or multiplication and the answer stays the same. For example, (-2 + 5) = (5 + -2) = 3.
What is the associative law?
You can move the brackets in addition or multiplication and the answer stays the same. For example, (-3 + 4) + 5 = -3 + (4 + 5) = 6.
What is the distributive law?
Multiplication spreads over addition inside brackets. For example, 3 x (5 + 2) = 3 x 5 + 3 x 2 = 21.
Does the commutative law work for subtraction?
No. 5 – 2 = 3, but 2 – 5 = -3. The commutative law holds only for addition and multiplication.