Multiplication and division structures

 Glossary of key terms Repeated aggregation: the process modelled by multiplication related to the idea of ‘so many sets of so many’; also called repeated addition. Scaling: the process modelled by multiplication in which a given quantity is increased by a scale factor, doubling and trebling are examples of scaling by factors of 2 and 3 respectively. Scaling by a factor less than 1 (for example, halving) reduces the size of the quantity. Commutative law of multiplications: the principle that the order of two numbers in a multiplication calculation makes no difference. For example, 5 x 7 = 7 x 5. In symbols, the commutative law of multiplication states that, whatever the numbers a and b, a x b = b x a. Rectangular array: a set of objects or shapes arranged in rows and columns, in the shape of a rectangle, for example, 7 rows of 5 counters, or a 7 by 5 grid of squares. Rectangular arrays are important images to be associated with multiplication. Per: an important word in multiplication and division situations, meaning ‘for each’; used, for example, in problems about cost per unit of measurement. Pro rata increase: an increase applied, for example, to salaries or prices, in which each amount is increased by the same scale factor. Equal sharing: the process modelled by division in which a set of items or a given quantity is shared equally between a number of individuals; the key phrase in the equal sharing structure of division is ‘shared equally between’. Inverse of multiplication: the process modelled by division in which the question is ‘how many groups of a given number there are in a given set?’ For example, ‘how many 4s make 20?’ corresponds to 20 divided by 4. Repeated subtraction (division): one of the ways of experiencing the inverse-of-addition structure of division by repeatedly subtracting a quantity from a given amount, for example, ‘how many times can €6 be taken away €24 until there is nothing left?’ is connected with 24 divided by 6. Repeated addition (division): one of the ways of experiencing the inverse of addition structure of division by repeatedly adding a quantity to reach a given target, for example, ‘how many payments of €6 are required to make a total of €24?’ is also connected with 24 divided by 6. Ratio: the inverse of the scaling structure of multiplication, where division is used to compare two quantities; for example, the ratio of €36 to €12 is 3; this is represented by the division 36 divided by 12 = 3.