A Square and a Cube: common misconceptions

If your child says… “The two 3s count as two different factors.”

Ask together

Do the two equal sides give different numbers?

What’s true

A factor list includes each value only once.

If your child says… “26 ends in 6, so it must be a square.”

Ask together

Which whole-number side could give 26, between square areas 25 and 36?

What’s true

An allowed ending does not guarantee a square.

If your child says… “The difference is seven, so seven whole numbers lie between.”

Ask together

Which numbers remain when you leave out both 9 and 16?

What’s true

Counting strictly between two whole numbers excludes both endpoints.

If your child says… “Only 5 gives 25 when squared.”

Ask together

What does −5 multiplied by −5 give?

What’s true

A positive number has two square roots, but a physical side length must be positive.

If your child says… “I take each different prime once, so the root is 2.”

Ask together

Does multiplying 2 by itself rebuild all four factors shown?

What’s true

Take one factor from every pair, including repeated pairs of the same prime.

If your child says… “I round up when the larger square is nearer.”

Ask together

Can a piece needing more cloth fit inside the cloth available?

What’s true

A square that needs more area cannot fit, however close its area is.

If your child says… “There are four cubes because the layer has four.”

Ask together

Where have you counted the cubes in the upper layer?

What’s true

The whole solid includes every layer, not just the bottom layer.

If your child says… “Matching pairs prove that a number is a cube.”

Ask together

Does a pair supply enough copies for all three identical products?

What’s true

Every prime factor must belong to a matching group of three, with none left over.