Division:
key ideas and important concept knowledge
Key ideas
The overarching key ideas of estimation, benchmarks, visualisation, equality and equivalence, language and strategies need to be considered when developing units of work in division. The specific key ideas in division are: equal groups, division with a remainder and properties of division. The properties of division include: inverse property, identity property, division of a number by itself and division by zero.
Equal groups
The quantity in each group is the same.
Division with a remainder
A remainder occurs when a collection cannot be partitioned into equal groups.
Treatment of the remainder depends on the context. The remainder can be:
• discarded to give a smaller whole number
• rounded up to the nearest whole number
• represented as a fractional part
• represented as a decimal.
Properties of division
Inverse property
Division and multiplication are related operations that undo each other, therefore multiplication can be used to solve a division problem.
The inverse property is applied to form fact families.
Identity property
Dividing a number by one will not affect the quantity.
Division of a number by itself
Dividing a number by itself will give a quotient (result) of one.
Division by zero
Dividing a number by zero is undefined.
Important concept knowledge
Meaning of the numbers
The number which we divide is called the dividend. The number by which we divide is called the divisor and the result is called the quotient.
Types of division
Division takes two forms: partition division and quotition division.
Partition division
Partition division (or equal sharing division) is used when the total number to be divided is known (the dividend) and the number of parts is known. The number in each part is not known.
Quotition division
Quotition division (also known as measurement division or repeated subtraction) is when the total number to be divided is known and the number in each part is known. The total number of parts is not known.
Division in fraction form
The result of equal sharing or measuring can be represented as both a number and an operation.
Divisional structures
Division is applicable in a range of settings. These settings are sometimes referred to as ‘problem types’. Table 2 provides some examples of problem types.
Table 2: Divisional structures
Division strategies
Division strategies are methods to solve mathematical problems. The strategies may be mental, written, digital or a mix of the three. The strategies rely on modifying the properties of numbers under multiplication to allow for division as the inverse of multiplication.
Mental strategies are calculations worked in one's mind and may involve using one of the following methods:
• partitioning and recombining numbers using the distributive property, usually with place value
• rounding and adjustment strategies
• proportional adjustment strategies, using factors of the divisor
• equal adjustment to dividend and divisor to simplify an equation.
Written strategies are sometimes algorithms, meaning step-by-step methods to get an answer. The most common algorithm for division applies place-value structure and the distributive property and should only be introduced once students have explored a range of strategies and have developed a sound conceptual understanding of division. Written strategies include:
Misconception
A common misconception is that ‘division makes smaller’. Since learning about division usually starts with separating a larger group into equal smaller groups, students often think the quotient is always smaller than the dividend.
The overgeneralisation that ‘division makes smaller’ causes problems with rational numbers.
