MathIt BlogProperties of multiplication: commutative, associative and the distributive property
What are the properties of multiplication?
The properties of multiplication are five rules that work for every number: commutative, associative, distributive, identity and zero. Order and grouping never change a product, so 3 × 5 = 5 × 3. The distributive property splits a factor, so 7 × 8 = 7 × 5 + 7 × 3 = 56. Multiplying by 1 keeps a number the same, and multiplying by 0 gives 0.
Key takeaways
- The properties of multiplication are the commutative, associative, distributive, identity and zero properties, and each one holds for every number.
- The commutative property means 3 × 5 = 5 × 3, so a 10 × 10 times table holds only 55 different facts instead of 100.
- The distributive property splits a factor into friendly parts, so 7 × 8 = 7 × 5 + 7 × 3 = 35 + 21 = 56.
- The associative property lets you regroup three factors, so 4 × 7 × 25 becomes 7 × (4 × 25) = 7 × 100 = 700.
- Subtraction and division are neither commutative nor associative, because 12 ÷ 4 = 3 while 4 ÷ 12 = 1/3.
Your child knows 7 × 5 = 35 by heart but freezes at 7 × 8, and the homework sheet says "use the distributive property". The properties of multiplication are the five rules that let a known fact unlock an unknown one: the commutative, associative, distributive, identity and zero properties. Once a child sees that 7 × 8 is 7 × 5 plus 7 × 3, the times table stops being 100 separate facts to memorize.
What are the properties of multiplication?
The properties of multiplication are rules that hold for every number, whatever its size. The commutative property lets you swap the order, the associative property lets you regroup, the distributive property lets you split a factor, and the identity and zero properties say what 1 and 0 do.
US schools start in grade 3. The Common Core standard 3.OA.B.5 asks students to "apply properties of operations as strategies to multiply and divide", and its examples are the ones in this guide: "If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known", and "one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56". A footnote adds that "students need not use formal terms for these properties", so a third grader can use the idea long before the word "distributive".
Why the commutative property cuts the times table to 55 facts
The commutative property means a 10 × 10 times table holds 55 different facts instead of 100. The grid has 100 cells, and 10 of them are squares such as 7 × 7. The other 90 cells come in turnaround pairs such as 3 × 8 and 8 × 3, which gives 45 pairs, and 45 + 10 = 55.
A dot array shows why the order does not matter. Three rows of 5 dots and five rows of 3 dots are the same 15 dots, turned a quarter of a circle.
How to use the distributive property, step by step
The distributive property says that multiplying a number by a sum equals multiplying it by each part and adding the results. The OpenStax Prealgebra section on the distributive property writes it as "If a,b,c are real numbers, then a(b+c)=ab+ac". In practice it turns one hard fact into two easy ones.
- Pick the harder factor. In 7 × 8, the 8 is the factor to split, because the 7s times 5 and 3 are easier than 7 × 8.
- Split it into friendly parts. Choose parts the child already knows: 8 = 5 + 3, or 14 = 10 + 4. Fives and tens are the usual anchors.
- Multiply each part by the other factor. Work out 7 × 5 = 35 and 7 × 3 = 21. Write both products down before adding.
- Add the partial products. 35 + 21 = 56, so 7 × 8 = 56. For a larger number, 6 × 14 = 6 × 10 + 6 × 4 = 60 + 24 = 84.
- Check with a second split. Split the other way: 7 × 8 = 7 × 4 + 7 × 4 = 28 + 28 = 56. Two routes that agree make the fact stick.
The part people get wrong: multiplying only the first term
The most common distributive property mistake is multiplying the first part of the sum and forgetting the second. A child writes 6 × (10 + 4) = 60 + 4 = 64 instead of 60 + 24 = 84. The fix is to draw an arrow from the 6 to every number in the brackets and to count the products: two parts means two multiplications.
The same split is the method behind written multiplication. The partial products in 23 × 4 are 20 × 4 = 80 and 3 × 4 = 12, which make 92, and the guide on how to teach long multiplication builds the whole column method from that step.
The five properties of multiplication side by side
The five properties of multiplication fit in one table: each has a rule with letters, a number example and a job it does in times tables. The letter rules follow Table 3 of the Common Core glossary and the OpenStax section on identity, inverses and zero, which states that "the product of any number and 0 is 0".
| Property | Rule | Number example | Where it helps |
|---|---|---|---|
| Commutative | a × b = b × a | 3 × 5 = 5 × 3 = 15 | Cuts the facts to learn from 100 to 55 |
| Associative | (a × b) × c = a × (b × c) | (2 × 3) × 4 = 2 × (3 × 4) = 24 | Regrouping to make 10 or 100 first |
| Distributive | a × (b + c) = a × b + a × c | 7 × 8 = 7 × 5 + 7 × 3 = 56 | Building hard facts from easy ones; long multiplication |
| Identity | a × 1 = 1 × a = a | 9 × 1 = 9 | The whole 1s column, and equivalent fractions |
| Zero | a × 0 = 0 × a = 0 | 0 × 8 = 0 | The whole 0s column, and spotting a zero factor |
Which property of multiplication to teach first
The commutative property is the one to teach first, because it pays off at once. The national curriculum in England asks Year 2 pupils to "show that multiplication of 2 numbers can be done in any order (commutative) and division of 1 number by another cannot" (mathematics programmes of study).
Identity and zero come next, since they clear two whole columns of the table. The same curriculum lists "multiplying by 0 and 1" and "using the distributive law to multiply two-digit numbers by 1 digit" in Year 4, with the example 39 × 7 = 30 × 7 + 9 × 7, which is 210 + 63 = 273. The steps in how to learn times tables fast follow the same order.
Mental math with the properties of multiplication
The properties of multiplication are the engine behind most mental math shortcuts. The associative and commutative properties together let you reorder and regroup any string of factors, so you can hunt for a 10 or a 100.
Take 4 × 7 × 25. Swap and regroup to 7 × (4 × 25) = 7 × 100 = 700. In the same way, 5 × 18 = 5 × (2 × 9) = (5 × 2) × 9 = 10 × 9 = 90.
The Common Core example shows both orders: "3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30".
The distributive property handles prices and decimals too. OpenStax works out three tickets at $9.25 as 3(9 + 0.25) = 27 + 0.75 = 27.75. More shortcuts of this kind are in mental math tricks that work.
A practice rule: draw the area model
An area model makes the distributive property visible, which helps children who lose track of the parts. Draw a rectangle 7 squares tall and 8 squares wide, cut it after the fifth column, and count each part: 7 × 5 = 35 squares and 7 × 3 = 21 squares. Common Core standard 3.MD.C.7c asks students to "use area models to represent the distributive property in mathematical reasoning".
Splitting a number into factors is the same skill as finding a greatest common factor, and the guide to GCF and LCM shows how 36 + 8 becomes 4 × (9 + 2) = 44.
How to check progress with the properties of multiplication
Five short questions show whether a child or an adult uses the properties of multiplication or only recites facts.
- Turnaround: Ask "If 6 × 9 = 54, what is 9 × 6?" A quick "54" without working it out means the commutative property has landed.
- Split: Ask for 8 × 7 using 8 × 5. The answer to listen for is 40 + 16 = 56.
- Regroup: Give 2 × 13 × 5. A child who sees 10 × 13 = 130 is using the associative property.
- Zero and one: Ask for 0 × 12 and 1 × 12. The answers are 0 and 12; mixing them up is common.
- Explain: Ask why 4 × 6 = 4 × 3 + 4 × 3. "Because 6 is 3 + 3" is enough.
A two-minute weekly test
A two-minute check once a week, with the working said out loud, shows more than a page of drills. Mix one question of each kind, and note which property the child reaches for. Children who skip-count to every answer can bridge the gap with the patterns in skip counting explained.
Common mistakes and edge cases
Five mistakes cause most wrong answers with the properties of multiplication, and each has a short fix.
- Turning subtraction or division around: OpenStax shows that 12 ÷ 4 = 3 while 4 ÷ 12 = 1/3, and that 7 − 3 = 4 while 3 − 7 = −4. Only addition and multiplication are commutative.
- Regrouping division: (24 ÷ 4) ÷ 2 = 3, but 24 ÷ (4 ÷ 2) = 12. Division and subtraction are not associative, so keep the brackets where they are.
- Mixing up 0 and 1: 8 × 0 = 0 and 8 × 1 = 8. A child who writes 8 × 0 = 8 is applying the addition rule 8 + 0 = 8.
- Distributing over multiplication: 2 × (3 × 4) is 24, while 2 × 3 × 2 × 4 is 48. The distributive property works over a sum or a difference, never over a product.
- Adding before multiplying: 7 × 5 + 7 × 3 means the two products are found first. The guide to the order of operations explains why: working left to right as (7 × 5 + 7) × 3 gives 126 instead of 56.
Edge case: the distributive property over subtraction
The distributive property works over subtraction too: a × (b − c) = a × b − a × c, one of the "other forms" OpenStax lists. It makes the 9s easy: 8 × 9 = 8 × 10 − 8 × 1 = 80 − 8 = 72, and 6 × 99 = 600 − 6 = 594.
Division distributes only when the sum is divided. (20 + 8) ÷ 4 = 5 + 2 = 7 is correct, but 24 ÷ (4 + 2) is 24 ÷ 6 = 4, while 24 ÷ 4 + 24 ÷ 2 = 6 + 12 = 18.
Your properties of multiplication checklist and next steps
- Commutative: learn one fact, get its turnaround free.
- Identity and zero: the 1s and 0s columns are done.
- Distributive: split the hard factor into 5s or 10s and multiply both parts.
- Associative: regroup three factors to make 10 or 100 first.
- Check: subtraction and division keep their order and their brackets.
Grade 3 and 4 practice is on the multiplication tables game page, where the facts the properties unlock turn into quick recall.
MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and daily practice, in 33 languages. Families can race each other in 2–5 player real-time battles, 22 of the games play free in the browser with no sign-up, and 30 printable worksheets come with answer keys.
Practise it in the MathIt app
About 90 math games with adaptive difficulty, daily practice and 2–5 player battles — free on iOS and Android.
Frequently asked questions
What is the commutative property?
The commutative property says that the order of two numbers does not change the answer: a × b = b × a for multiplication and a + b = b + a for addition. For example, 6 × 4 = 4 × 6 = 24. The property does not hold for subtraction or division, because 7 − 3 = 4 while 3 − 7 = −4.
What is the associative property?
The associative property says that the grouping of three numbers does not change the answer when they are all multiplied or all added: (a × b) × c = a × (b × c). For example, (2 × 3) × 4 = 6 × 4 = 24 and 2 × (3 × 4) = 2 × 12 = 24. You can move the brackets to make a ten or a hundred first.
How do you do the distributive property?
The distributive property is used by splitting one factor into two easier parts, multiplying each part by the other factor, then adding the two products. For 6 × 14, split 14 into 10 + 4, work out 6 × 10 = 60 and 6 × 4 = 24, and add them to get 84. Both parts must be multiplied.
What does the distributive property mean?
The distributive property means that multiplying a number by a sum gives the same answer as multiplying it by each part of the sum and adding the results: a × (b + c) = a × b + a × c. The multiplication is shared out, or distributed, over every term inside the brackets, so 3 × (2 + 5) = 6 + 15 = 21.
What is the commutative property of addition?
The commutative property of addition says that you can add two numbers in either order and get the same sum: a + b = b + a. For example, 3 + 9 = 9 + 3 = 12, so a child can start counting on from the larger number. Subtraction is not commutative, because 9 − 3 = 6 while 3 − 9 = −6.
