Division
Number
Definition
Division is a process of separating objects into equal parts or groups. The use of multiplicative strategies in division computations is also beneficial. By using 3 x 4 = 12, we establish that 12 ÷ 3 = 4.
Note: If 12 ÷ 3 = 4, then we say that 12 is the dividend, 3 is the divisor and 4 is the quotient.
In the sharing model of division, the divisor indicates a whole number of equal groups and the quotient, the result of division, is the size of each part. In 12 ÷ 3 = 4, twelve is shared into 3 equal groups and there are 4 in each group. An over-reliance on the sharing model of division can contribute to misconceptions about division with decimals. This model is inadequate when the division has a divisor that is less than one.
In the measurement division model, the divisor indicates the size of the subset (number in each group) and the quotient is the number of equal-sized subsets. For 12 ÷ 3 = 4, 12 is divided into groups of 3, and 4 is the number of groups of 3. The measurement division model is sometimes described as quotitive division. (National Numeracy Learning Progression)
Teaching and learning activities
The resources below provide targeted teaching strategies to support student improvement in this skill.
Each downloadable lesson activity includes:
learning intentions
a list of required resources
a step-by-step lesson sequence
printable classroom materials.
Select the download all icon to download all available activities or select each activity separately.
PLAN2 Areas of focus
An Areas of focus template has been created in PLAN2 to support targeted teaching of Text structure in your learning area.
Search for the DoE template titled ‘DoE HSCMinStd Writing: Text structure’ in the Areas of focus template library tab within the Plan menu, and customise it for your students’ needs.
For more information about using PLAN2 Areas of focus templates with this resource, visit the Using this resource with PLAN2 page.
Relevance to the numeracy test marking
According to the ACSF, the feedback for a Level 3 performance in the HSC minimum standard online numeracy test for division with whole numbers states:
Individuals performing at this level are able to “select appropriate strategies from a variety of everyday mathematical processes in familiar and some less familiar contexts. They are also able to interpret and comprehend mathematical information in written material, diagrams, charts and tables”. They can also use large whole numbers in words and figures and demonstrate their ability to calculate with whole numbers.
Connections with ACSF Level 3 descriptors
The relevant Level 3 ACSF descriptors for numeracy are shown here to demonstrate how division with whole numbers are assessed in the HSC minimum standard online test. The performance features identified show what a student is able to do in order to achieve at this level and are provided to support teachers to understand what is required to achieve a Level 3 in this skill.
Numeracy Indicator 3.10: Selects from and uses a variety of developing mathematical and problem solving strategies in a range of familiar and some less familiar contexts.
Focus area: Problem solving processes including estimating and reflecting
Level 3 performance features:
Draws on a combination of hands-on, in-context materials, personal experience, mathematical and other prior knowledge to:
select appropriate methods of solution from a limited range of mathematical processes
use developing estimation, and other assessment skills, to check and reflect on the outcome and its appropriateness to the context and task.
Numeracy Indicator 3.10: Selects from and uses a variety of developing mathematical and problem solving strategies in a range of familiar and some less familiar contexts.
Focus area: Mathematical methods and use of tools
Level 3 performance features:
uses a blend of personal 'in-the-head' methods and formal pen and paper methods to calculate and uses calculator/technological processes and tools to undertake the problem solving process
selects and uses appropriate tools, hand-held devices, computers and technological processes, e.g. uses a tape measure to measure the dimensions of a window in mm or creates a personal weekly budget in a spreadsheet.
Numeracy Indicator 3.10: Selects from and uses a variety of developing mathematical and problem solving strategies in a range of familiar and some less familiar contexts.
Focus area: Mathematical knowledge skills: number and algebra
Level 3 performance features:
calculates with whole numbers including dividing by small whole numbers only and long division worked out on a calculator.
Connections with Numeracy Learning Progressions
The progressions describe a typical developmental sequence of literacy and numeracy learning. The numeracy progression sub-elements, levels and indicators relevant to multiplication with whole numbers are provided here to assist teachers to identify students’ capabilities and needs to support targeted teaching.
Element: Number Sense and Algebra
Sub-element: Multiplicative Strategies (MuS)
MuS5 — Coordinating composite units
identifies and represents division in various ways such as sharing division or grouping division (e.g. sharing a carton of 12 eggs equally between four people, draws 12 dots and circles three groups of four with 3 in each share)
identifies and represents multiplication and division abstractly using the symbols × and ÷ (e.g. represents 3 groups of 4 as 3 × 4; uses 9 ÷ 3 to represent 9 pieces of fruit being equally shared by 3 people)
MuS6 — Flexible strategies for single digit multiplication and division
interprets a range of multiplicative situations using the context of the problem to form a number sentence (e.g. to calculate the total number of buttons in 2 containers, each with 5 buttons, uses the number sentence 2 × 5 = ?; if a packet of 20 pens is to be shared equally between 4, writes 20 ÷ 4 = ?)
applies mental strategies for multiplication to division and can justify their use (e.g. to divide 64 by 4, halves 64 then halves 32 to get an answer of 16)
explains the idea of a remainder as what is ‘left over’ from the division (e.g. an incomplete group, lot of, next row or multiple)
MuS7 — Flexible strategies for multiplication and division
uses multiplication and division as inverse operations to solve problems or to justify a solution
uses known mental and written strategies such as using the distributive property, decomposition into place value or factors to solve multiplicative problems involving numbers with up to three digits and can justify their use (e.g. 7 × 83 equals 7 × 80 plus 7 × 3; to multiply a number by 72, first multiply by 12 and then multiply the result by 6; 327 × 14 is equal to 4 × 327 plus 10 × 327)
uses estimation and rounding to check the reasonableness of products and quotients (e.g. multiplies 200 by 30 to determine if 6138 is a reasonable answer to 198 × 31)