The prompt aims to develop knowledge about division in primary school classrooms. In particular, the teacher uses it to deepen pupils' conception of a remainder and to act as a bridge to decimal remainders. 

Ambiguity

Do the remainders mean the same in both calculations? There is a deliberate ambiguity in the prompt. Generally, the contention is false. The quotients are not the same when we consider decimal remainders: 42 ÷ 5 = 8.4 and 26 ÷ 3 = 8.666 recurring.

However, in a concrete context in which items cannot be broken into parts, a pupil could argue that the quotients are the same. For example, if 42 pens are divided among five people and 26 pens are divided among three people, then each person receives eight and the two pens remaining would be identical.

However, this is not the case if we could split the item because the divisors are different. For example, if we imagine each item is a packet containing 15 sweets. The packet can be opened and the sweets distributed. In the first case, each of the five people would receive 6 sweets as their share of the two packets left over (30 sweets in total). In the second case, the divisor is smaller and so each person receives 10 sweets.

Reasoning

In order to show the contention in the prompt is false, the teacher could encourage the class to take another step in the chain of reasoning. 

In the first case, 6 sweets represents two-fifths of a packet; in the second case, 10 sweets represents two-thirds of a packet. The quotients in decimal form are 8.4 and 8.666 recurring respectively and are, therefore, not equal.

Guided inquiry

The slides contain four lines of inquiry that develop from the four regulatory cards below.