# Reason about the location of mixed numbers in the linear number system (5)

## Description

Sometimes pupils get quite far in their maths education only thinking of a fraction as a part of a whole, rather than as a number in its own right. In year 3, pupils learnt about the location in the linear number system of fractions between 0 and 1 (3F–2). For pupils to be able to add and subtract fractions across 1, or those greater than 1 (4F–3 ), they need to understand how mixed numbers fit into the linear number system. Pupils should develop fluency counting in multiples of unit fractions, using number lines for support. Counting draws attention to the equivalence of, for example,  1 and 2, or 2 and 3. Pupils should practise counting both forwards and backwards.

Pupils should then learn to label marked number lines, extending beyond 1. A common mistake that pupils make is to count the number of marks between labelled intervals, rather than the number of parts. For example, on the number line below they may count 3 marks and incorrectly deduce that the number line is marked in thirds.

Language focus “Each interval is divided into 4 equal parts, so we count in quarters.”

Pupils should also be able to estimate the value or position of mixed numbers on number lines which do not have fractional marks. Pupils must understand that it is not the absolute size of the numerator and denominator which determine the value of the fractional part of the mixed number, but the relationship between the numerator and denominator (whether the numerator is a large or small part of the denominator). Pupils need to be able to reason, for example, that 1 is close to 2, but that 1 is closer to 1.

Pupils must also be able to identify the previous and next whole number, and will then be able to round to the nearest whole number, which further supports estimation and approximation.

Language focus

“1 is between 1 and 2.”

“The previous whole number is 1.”

“The next whole number is 2.”