Tuesday

Raven pours beans onto the bench. "Twenty-five beans. Make a square."

Rocket lines them up: five rows of five. "A 5 by 5 square. So 25 is 5 squared."

"Now go backward," Raven says. "A square of 49 beans. How long is its side?"

"Seven, because 7 × 7 = 49." Rocket grins. "The square root of 49 is 7."

Nova hovers over a stack of blocks, her light on the corner. "And a cube of 27 blocks?"

"Three along, three across, three up. 3 × 3 × 3 = 27. The cube root of 27 is 3."

"What do you notice about a square of 2 beans?" Raven asks.

"No whole side works," Rocket says. "The square root of 2 is not a fraction either. It never comes out even."

A square number is a count of beans that makes a perfect square. 25 beans make a 5 by 5 square.

The square root undoes the square. Because 5² = 25, the square root of 25 is 5, written √25 = 5.

The equation x² = 49 asks which number squared is 49. Its positive solution is x = √49 = 7.

A cube number is a count of blocks that makes a perfect cube. 27 blocks make a 3 by 3 by 3 cube.

The cube root undoes the cube. Because 3³ = 27, the cube root of 27 is 3, written ∛27 = 3.

The equation x³ = 64 has the solution x = ∛64 = 4, because 4 × 4 × 4 = 64.

Not every number is a perfect square. Two beans will not make a square with a whole side.

√2 is irrational: it cannot be written as a fraction of whole numbers, and its decimal never ends or repeats.

It is still a real length. A square with area 2 has a side of √2, a little more than 1.4.

Excellent. Tomorrow the Puzzle Lab counts a whole jar of beans without counting every bean.