Two Shadings, One Overlap

Blue shades columns, orange shades rows. Where both land on the same cell is the product — and it is always a smaller share of the grid than either shading was on its own.

3/4 × 2/3 = 6/12 = 1/2

Columns alone would reach 9 of the 12 cells, and rows alone would reach 8 — but only 6 cells are on BOTH a shaded column and a shaded row, which is the green you can actually count, and it is smaller than either shading by itself.

Columns shaded: 3 of 4
Rows shaded: 2 of 3

Drag the blue handle sideways and the orange handle up and down, or use the arrow keys.

Ready to try some? Multiplying fractions worksheets

Where it usually goes wrong

Multiplying is expected to make things bigger, because it reliably did with whole numbers — so 3/4 × 2/3 gets guessed at something close to 3/4, and "multiply the tops, multiply the bottoms" is applied without any sense of whether 6/12 is a reasonable answer to have landed on. Seeing the two shadings overlap is what makes 1/2 obviously right and something close to 3/4 obviously wrong, before a single number is computed.

Try this

Drag the blue handle to shade all 4 columns. The blue reaches the whole grid, and the green overlap becomes exactly the orange shading — multiplying by a whole 4/4 changes nothing.
Pull it back to shade only 1 column. The green overlap shrinks to a sliver, far smaller than the orange shading alone.
Set both handles to shade fewer than half their columns or rows. The overlap becomes a small corner of the grid — two fractions under a half make a product under a quarter.

Where this goes next

Dividing by a fraction is this same picture read the other way — asking how many copies of one shading fit inside the other — and every algebra rule for multiplying rational expressions is this grid with variables standing in for the counts.

Two Shadings, One Overlap

Where it usually goes wrong

Try this

Where this goes next

More about fractions