Singapore's CPA method teaches multiplication through three stages. First, students use concrete objects they can touch and manipulate. Then they move to pictorial representations with drawings and diagrams. Finally, they work with abstract symbols and equations. This progression helps students deeply understand multiplication properties: commutative, associative, distributive, and identity.
The commutative property shows that order doesn't matter in multiplication. Using CPA method: First, concrete stage uses physical objects - three rows of four dots equals four rows of three dots, both giving twelve. Second, pictorial stage shows area models where a three-by-four rectangle has the same area as a four-by-three rectangle. Finally, abstract stage uses symbols: three times four equals four times three, or generally, a times b equals b times a.
The associative property shows that grouping doesn't change the result. Using CPA: Concrete stage groups objects differently - first multiply two times three, then times four, or multiply three times four first, then times two. Pictorial stage shows rectangles can be regrouped while maintaining the same total area. Abstract stage uses parentheses to show different groupings give the same result: twenty-four.
The distributive property lets us break apart multiplication over addition. Using CPA: Concrete stage shows four groups of three plus four groups of two equals four groups of five total. Pictorial stage uses area models where a rectangle is split into two parts, showing the total area equals the sum of both parts. Abstract stage shows four times three plus two equals four times three plus four times two, giving twenty. This works in reverse too: six times seven plus six times three equals six times ten, or sixty.
The identity property shows that multiplying by one leaves numbers unchanged. Using CPA: Concrete stage shows one group of five objects remains five objects. Pictorial stage shows the same area is maintained. Abstract stage uses the formula a times one equals a. All four properties work together in complex problems. For example: two times three times one plus two times four uses identity, distributive, and commutative properties to simplify calculations and develop flexible problem-solving strategies.