Tuesday

Raven sets a jar of blue paint on the Numbers Desk. "We have 3/4 of a cup. Each sign needs 1/8 of a cup."

"Three fourths, so three signs," Rocket says at once.

"Pour and see," says Raven. She fills the 1/8 cup and tips it into an empty jar. Again. Again.

Six scoops later the paint jar is empty. "Six signs," Raven says. "What do you notice?"

"Each fourth held two eighths," Rocket says slowly. "So three fourths held six eighths."

Nova hovers near the jars. "There is a shortcut that always works," she says. "Flip something and multiply."

"Flip what?" asks Rocket. Nova hums. "The strips will show you."

3/4 ÷ 1/8 asks: how many 1/8 cups fit into 3/4 of a cup?

Look at the two strips. The top strip shows 3/4. The bottom strip shows the same amount in eighths.

Each fourth holds 2 eighths, so 3 fourths hold 6 eighths. 3/4 ÷ 1/8 = 6.

Rocket guessed 3 because he saw the 3 on top. The divisor matters just as much as the dividend.

The reciprocal of a fraction is the fraction flipped over. The reciprocal of 1/8 is 8/1, and the reciprocal of 3/4 is 4/3.

To divide by a fraction, multiply by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc.

3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6. The same answer the strips gave.

Why does it work? Dividing by 1/8 means counting eighths, so you multiply by 8.

Dividing by 3/8 means counting groups of 3 eighths. Multiply by 8 to count eighths, then divide by 3 to group them.

Nice flipping. Tomorrow is Numbers Lab: you count scoops and folds and write what you find.