Three Math Properties That Work Every Single Time

Here is something amazing: 4 times 7 equals 28. Flip it around — 7 times 4 also equals 28. It works every time, for any numbers, with no exceptions. That is not a coincidence. It is a mathematical property — a rule that is ALWAYS true. Properties are like the rules of a game that never change. Once you know them, you can use them to solve harder problems faster, check your work, and understand why math works the way it does. Today you will learn three of the most powerful properties in all of arithmetic.

- The Commutative Property: why order does not matter when you add or multiply - The Associative Property: why grouping does not matter when you add or multiply - The Distributive Property: how to break big multiplication problems into easier ones - When these properties do NOT apply (subtraction and division have different rules!)

The word commute means to travel back and forth. The Commutative Property says you can swap the order of numbers when you add or multiply — and the answer stays the same. Addition examples: 3 + 5 = 8 and 5 + 3 = 8 17 + 46 = 63 and 46 + 17 = 63 Multiplication examples: 6 x 9 = 54 and 9 x 6 = 54 12 x 7 = 84 and 7 x 12 = 84 This is great news! If you are not sure whether 8 x 7 or 7 x 8 is in your multiplication table, they are both 56. You only need to learn one of them. WARNING: This does NOT work for subtraction or division. 10 - 4 = 6 but 4 - 10 = -6 (different!) 20 divided by 5 = 4 but 5 divided by 20 = 0.25 (also different!)

The word associate means to group together. The Associative Property says you can change WHICH numbers you group (with parentheses) when adding or multiplying — and the total stays the same. Addition examples: (2 + 3) + 4 = 5 + 4 = 9 2 + (3 + 4) = 2 + 7 = 9 (same answer!) (15 + 25) + 10 = 40 + 10 = 50 15 + (25 + 10) = 15 + 35 = 50 (same answer!) Multiplication example: (2 x 3) x 4 = 6 x 4 = 24 2 x (3 x 4) = 2 x 12 = 24 (same answer!) Pro tip: when adding a list of numbers, look for pairs that make 10 — (7 + 3) + 8 + 2 = 10 + 10 = 20. Regrouping makes the math easier!

The Distributive Property lets you BREAK a hard multiplication into two easier ones. Rule: a x (b + c) = (a x b) + (a x c) Example: 6 x 14 Break 14 into 10 + 4: 6 x (10 + 4) = (6 x 10) + (6 x 4) = 60 + 24 = 84 Example: 7 x 13 Break 13 into 10 + 3: 7 x (10 + 3) = (7 x 10) + (7 x 3) = 70 + 21 = 91 This is exactly the strategy behind long multiplication! When you learn to multiply two-digit numbers, you are using the Distributive Property every single time.