Worked solutions: Multiplying Fractions

1. 1Take the ColumnsFirst take 3/4 of the whole: split the rectangle into 4 columns and shade 3 of them solid; the columns we leave fall into shadow.3 of 4 columns
2. 2Take the Rows of What Is LeftNow take 2/3 of that shaded part: split the height into 3 rows and shade 2 of them. Where the two shadings cross, both fractions are taken at once.2 of 3 rows
3. 3Count the OverlapCount the overlap cells one by one as they turn gold: 6 cells out of 3 × 4 = 12 in the whole grid. That counted part is the product: 6/12 of the whole.2 × 3 = 6 of 12 cells
4. 4Why the Product Shrinks — and the Written Method"Of" a part smaller than 1 takes only a piece of it — that is why the product shrinks. On paper the counted cells become a formula: tops 2 × 3 = 6, bottoms 3 × 4 = 12, and 6/12 simplifies to 1/2.2/3 × 3/4 = 6/12
5. 5Check: Recount by ColumnsCount the overlap by columns instead: 3 columns of 2 cells give the same 6 cells. And the answer 1/2 is smaller than both 2/3 and 3/4 — exactly what "a fraction of a fraction" must be.3 × 2 = 6 cells ✓
6. 6The Size TableauLine the three numbers up by size on three bars: the two factors and the product 1/2 — the product is the shortest bar of all. 2/3 × 3/4 = 1/2: multiplying by a fraction smaller than 1 makes the result smaller.Multiply the tops, multiply the bottoms

Change the numbers and watch the diagram redraw itself: