Multiply and divide whole numbers, read a decimal and a fraction, and write a number as a product of its prime factors. Useful earlier lessons: Fractions, Decimals & Percentages; Factors, Multiples, Primes, HCF and LCM; Powers, Roots, Indices, Standard Form & Surds (for the square and cube tables that this lesson uses only briefly).

Each family answers a question the smaller family cannot answer. In the natural numbers, $3-5$ has no answer; once negative numbers are added, it is $-2$. In the integers, $3÷4$ has no answer; once fractions are added, it is $\frac{3}{4}$. In the rational numbers, the side of a square of area $2$ has no answer; once irrational numbers are added, it is $\sqrt{2}$. Seeing the reason for each step makes the families easier to remember than a list of names.

Classify each number in the list $\{-7,0,4,\frac{2}{3},\sqrt{25},\sqrt{8}\}$ into the smallest family that contains it. List every family that each number belongs to.

What a complete solution shows: each number simplified first, then placed in its smallest family, with a sentence saying why.

A number written with a root sign is not automatically irrational. $\sqrt{4}=2$, $\sqrt{9}=3$ and $\sqrt{25}=5$ are all rational and whole. Only a root of a number that is not a perfect square is irrational, and you should simplify the root before deciding.

Answers. 1. Not natural; yes, an integer. 2. $\frac{5}{8}$ and $\sqrt{16}=4$; $\sqrt{3}$ is irrational. 3. For example $5$ is an integer, and $\frac{1}{2}$ is rational but not an integer.