Area Models

Algebra 1 · Axiom Academy

Multiplication is the area of a rectangle. Split the sides, and a product breaks into pieces you can see — all the way up to multiplying polynomials. A product is an area — and an area splits into pieces Multiplying two numbers has a picture: it is the area of a rectangle whose sides are those two numbers. The quiet power of that picture is what happens when you chop each side into parts — the one big multiplication breaks into a handful of smaller ones that add back to the same total. Hold onto that idea: it is the whole reason multiplying messy numbers, and even polynomials, works. Watch a 7-by-7 rectangle. Its width is split into 4 + 3, its height into 5 + 2, so the rectangle is carved into four smaller rectangles. Each one fills in with its own area, and the running total climbs — landing on 7 × 7 = 49 once all four pieces are counted. The four pieces — 20, 8, 15, 6 — add up to the same 49 you would have gotten multiplying 7 × 7 straight across. Why the split makes hard multiplication easy Slide the two numbers. Each side splits at its tens and ones, so a rectangle like 23 × 14 becomes four friendly pieces — 20·10, 20·4, 3·10, and 3·4. The four areas always add back to the full product. That decomposition is the standard multiplication algorithm, drawn. However you split the sides, the four areas always rebuild the exact product — that is the distributive property at work. Multiplying polynomials: (x + 3)(x + 2)

This is the written version of the interactive lesson above. See the full Algebra 1 course.