Multiply a sum by breaking it apart: 3 × (4 + 2) = (3 × 4) + (3 × 2) = 12 + 6 = 18. Arrays split into sections show this clearly.
Draw arrays split into two parts. Compute each part separately, then add.
Not recognizing when to use the property; forgetting to add the products.
You know from multiplication arrays that a product like 3 × 6 can be visualized as a rectangular arrangement — 3 rows and 6 columns, with 18 total dots. The distributive property is what happens when you split that rectangle into two smaller rectangles that together cover the same area. It gives you a way to break harder multiplications into easier ones by using facts you already know.
Here's the core idea: 3 × 6 = 3 × (4 + 2) = (3 × 4) + (3 × 2) = 12 + 6 = 18. Why does this work? Draw a 3-by-6 array, then draw a vertical line separating the first 4 columns from the last 2. You now have two separate rectangles: a 3×4 (which is 12) and a 3×2 (which is 6). Together they still cover 3×6 = 18 total dots. The "distribution" means that the 3 rows apply equally to both parts — each column in both sections still has 3 dots.
This strategy becomes most useful for products you don't yet have memorized. Suppose you're unsure of 7 × 8. Split 8 into 5 + 3: (7 × 5) + (7 × 3) = 35 + 21 = 56. Or split it as 4 + 4: (7 × 4) + (7 × 4) = 28 + 28 = 56. You choose the split that uses facts you know best. The property works no matter how you split the number, which is what makes it flexible.
The distributive property may look like a trick at this stage, but it is one of the most important ideas in all of mathematics. When you later multiply two-digit numbers (24 × 3 = (20 + 4) × 3 = 60 + 12 = 72), you'll use this exact idea. When you study algebra and expand expressions like (x + 5)(x + 2), that's the distributive property again. The rectangle model you learn now — splitting an array into two parts — is the same geometric intuition that underlies multiplication at every level.
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