The World of Numbers: common misconceptions

If your child says… “Dividing by zero gives zero.”

Ask together

What happens when you multiply your proposed answer by zero?

What’s true

Division by zero has no unique answer that reverses multiplication.

If your child says… “Two negative factors give a negative answer.”

Ask together

What must cancel negative one to leave zero?

What’s true

Multiplying two negative numbers gives a positive result.

If your child says… “Seven is not rational because it isn't written as a fraction.”

Ask together

What happens to seven when you divide it by one?

What’s true

A number can be rational without already having a fraction bar.

If your child says… “Different top and bottom numbers mean different amounts.”

Ask together

Did making more folds add any shading?

What’s true

Multiplying or dividing both parts by the same nonzero number preserves the fraction’s value.

If your child says… “Absolute value always flips the sign.”

Ask together

How far from zero is positive three?

What’s true

Positive numbers keep their value because their distances from zero are already nonnegative.

If your child says… “Eventually, there won't be any fractions left between them.”

Ask together

What would stop us adding the new endpoints and halving again?

What’s true

Every pair of distinct rational numbers has another rational number halfway between.

If your child says… “Every square root is irrational.”

Ask together

Which whole number multiplied by itself gives 81?

What’s true

A root sign alone cannot decide whether a number is rational.

If your child says… “The number of digits shown tells me how far to shift.”

Ask together

Would writing more threes change the repeating block?

What’s true

The repeating block’s length determines the shift needed to align the endless tails.