An array is a rectangular arrangement of objects in rows and columns. A 4-by-6 array has 4 rows and 6 objects in each row, representing 4 × 6 = 24. Arrays make multiplication concrete and visible, supporting mental strategies and understanding of commutativity.
You've worked with arrays before as a way to see what multiplication looks like. Now you're using them as a thinking tool — a way to break hard facts into easier ones. The key insight is that you can split any array into two smaller arrays whose answers you already know, add those answers, and get the fact you wanted. This is called the break-apart strategy (or decomposition), and arrays make it visible.
Suppose you want to figure out 4 × 7, and you're not sure of that fact yet. Draw a 4-by-7 array. Now draw a vertical line after the 5th column, splitting it into a 4×5 array and a 4×2 array. You know 4 × 5 = 20 (skip-count by 5s four times) and 4 × 2 = 8. Put them together: 20 + 8 = 28. So 4 × 7 = 28 — and you figured it out from facts you already knew. The array is the visual proof that this split is legal: the 28 tiles in the full array really do equal the 20 tiles plus the 8 tiles.
This strategy also explains why multiplication distributes over addition — the foundation of a rule you'll use throughout algebra. For now, think of arrays as your reasoning tool. Rotating an array 90° proves commutativity (4×6 and 6×4 have the same tiles). Splitting an array proves that you can break facts apart. Eventually, when you study area of rectangles, you'll see that an array of tiles and a rectangle in geometry are describing the same mathematical relationship — length times width. Arrays built that bridge from multiplication facts to geometry, and you're standing on it right now.