What is the commutative property?

The commutative property says you can swap the two numbers in an addition or a multiplication and still get the same answer. So 4 + 9 = 9 + 4 = 13, and 4 × 9 = 9 × 4 = 36. Four bags of 9 marbles hold as many marbles as 9 bags of 4. In commutative property practice here, the gap is always the number the right side still lacks.

How do you use the distributive property to multiply in your head?

Split one number into parts that are easy to multiply, multiply each part, then add the results. For 8 × 14, think 8 × 10 = 80 and 8 × 4 = 32, so the product is 80 + 32 = 112. A nearby round number works too: 6 × 19 = 6 × 20 − 6 × 1 = 120 − 6 = 114. The game uses both kinds of split, so expect a minus sign now and then.

Is there a trick for multiplying three numbers in your head?

Yes, the trick is to multiply the friendliest pair first. Thanks to the commutative and associative properties, any order and any grouping give the same product. In 5 × 7 × 2, work out 5 × 2 = 10 first, then 10 × 7 = 70. In 4 × 9 × 25, pair 4 × 25 = 100, and 100 × 9 = 900. That regrouping is exactly what the associative property game level asks for.

Why does my child think 7 − 2 and 2 − 7 are the same?

Many children learn that order does not matter when adding, then carry the rule over to subtraction, where it breaks. 7 − 2 = 5, but 2 − 7 goes below zero. Show it with real cups: from 7 cups you can take 2 away, but from 2 cups you cannot take 7. The swap rule belongs to adding and multiplying only.

Is this game on the properties of operations free, and can I use it in class?

Yes: all three levels cost nothing and open in a browser, with no signup. For distributive property practice, write 6 × 14 on the board and ask pupils for every split they can find. Both 6 × 10 + 6 × 4 and 6 × 20 − 6 × 6 count, since each gives 84. Then every pupil takes a tablet and fights. With the schools plan, paid per student, a teacher dashboard shows which pupils are improving.

Math practice · reimagined

Properties of Operations Game

In this properties of operations fight, every equation hides one number, as in 6 × 8 = ? × 6, and the right law reveals it without any multiplying. Spot the law at a glance, and your fighter swings while your rival is still calculating.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What are the properties of operations?

The properties of operations are rules that let you rearrange a calculation without changing its answer. Swapping, as in 5 + 3 = 3 + 5, is the commutative property. Moving the parentheses when you only add or only multiply is the associative property. Splitting 6 × 13 into 6 × 10 + 6 × 3 is the distributive property.

Every level of Properties

Example: (4 × 5) × 3 = 4 × (5 × ?)

Every sign here is ×, so the associative property lets the parentheses move without changing the result.
The left side multiplies 4, 5 and 3, so the right side must use the same three numbers.
The right side already shows the 4 and the 5, which leaves 3 for the gap.
Check it: 20 × 3 = 60 and 4 × 15 = 60, so you type 3.

Learn it step by step

The commutative law says order doesn't matter for adding or multiplying. 3 × 7 is exactly the same as 7 × 3.

So if you see 7 × 4 = ? × 7, the blank is just the other number — 4 — because swapping the order can't change the answer. Three rows of chairs or chairs in three columns: same number of chairs.

Fill in the number that makes both sides equal.

Tips and common mistakes

Watch what moves. If the numbers trade places, it is commutative. If only the parentheses move, it is associative. If one number multiplies each part, it is distributive.
A frequent slip: 6 × (10 + 3) is not 60 + 3. The 6 multiplies both parts, so it is 60 + 18 = 78.
Put 2, 7 and 8 buttons in three cups. Ask which two cups to pour together first to make the adding easy: 2 and 8 make 10, then 17.

How to play

Every round shows an equation with one gap, like 9 × 6 = ? × 9, and you type the missing number on the keypad. Find it before your rival and Chilli strikes, harder the quicker you are. On the distributive level some splits subtract, as in 7 × 18 = 7 × 20 − 7 × ?.

— Quick answers —

Properties: the short FAQ

What is the commutative property?: The commutative property says you can swap the two numbers in an addition or a multiplication and still get the same answer. So 4 + 9 = 9 + 4 = 13, and 4 × 9 = 9 × 4 = 36. Four bags of 9 marbles hold as many marbles as 9 bags of 4. In commutative property practice here, the gap is always the number the right side still lacks.
How do you use the distributive property to multiply in your head?: Split one number into parts that are easy to multiply, multiply each part, then add the results. For 8 × 14, think 8 × 10 = 80 and 8 × 4 = 32, so the product is 80 + 32 = 112. A nearby round number works too: 6 × 19 = 6 × 20 − 6 × 1 = 120 − 6 = 114. The game uses both kinds of split, so expect a minus sign now and then.
Is there a trick for multiplying three numbers in your head?: Yes, the trick is to multiply the friendliest pair first. Thanks to the commutative and associative properties, any order and any grouping give the same product. In 5 × 7 × 2, work out 5 × 2 = 10 first, then 10 × 7 = 70. In 4 × 9 × 25, pair 4 × 25 = 100, and 100 × 9 = 900. That regrouping is exactly what the associative property game level asks for.
Why does my child think 7 − 2 and 2 − 7 are the same?: Many children learn that order does not matter when adding, then carry the rule over to subtraction, where it breaks. 7 − 2 = 5, but 2 − 7 goes below zero. Show it with real cups: from 7 cups you can take 2 away, but from 2 cups you cannot take 7. The swap rule belongs to adding and multiplying only.
Is this game on the properties of operations free, and can I use it in class?: Yes: all three levels cost nothing and open in a browser, with no signup. For distributive property practice, write 6 × 14 on the board and ask pupils for every split they can find. Both 6 × 10 + 6 × 4 and 6 × 20 − 6 × 6 count, since each gives 84. Then every pupil takes a tablet and fights. With the schools plan, paid per student, a teacher dashboard shows which pupils are improving.

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