A quantity is an amount of something which is determined using a number and a unit.

The term ‘manyness’ applies to discreet quantities.

When a set of discrete items is counted, the result assumes that the unit is one collection.

The term ‘muchness’ applies to continuous quantities.

In the abstraction principle, different sized or unrelated objects can be counted and treated the same numerically. 

Items that cannot be seen can also be counted – for example ideas, characters in a story, sounds, etc.

In the one-to-one principle, words in the forward or backward counting sequence are mapped onto the objects being counted; that is, there is one word to one object.

In the stable-order principle, there is a fixed order of words in the sequence when objects are counted.

In the cardinality principle, the last number indicates the total number of items; that is, it is a cumulative count.

In the order-irrelevance principle, the order in which objects are counted does not change the quantity.

In the ordinal principle, numbers are used to indicate the position of an object in a numerical sequence or order.

In conservation of number, the number of objects in a collection does not change as the spatial arrangement of the collection changes.

Subitising is an instant recognition of a small quantity without counting.

A number pattern is a regularity in a sequence of numbers. 

Number patterns are a type of growing pattern.

The number word sequence, forwards or backwards, is the fixed order of number names.

There is a difference between reciting and counting a number word sequence.

Reciting a sequence of number words is by rote, whereas counting is the allocation of each spoken number word with an item.

Students begin to learn the number word sequence and then use this sequence to count collections.

A quantity is an amount of something which is determined using a number and a unit.

A number triad represents the relationship between words, symbols and materials or diagrams of a quantity.

The place of a digit in a number determines its value.

There are three ways to interpret a single digit within a whole number: face value, place value and total value.

Zero can be a placeholder and a digit representing a quantity.

The base-10 system is a number system that is based on grouping and equally partitioning quantities by tens.

Each place has a value that is 10 times greater than the place to its right, and one-tenth of the value of the place to its left.

Place-value partitioning is breaking a whole number into place-value units.

The relative size of two quantities can be compared, either to each other or a benchmark.

Rounding is a context-driven approximation used to make numbers easier to calculate. It is useful for estimation.

The convention for rounding is that numbers are rounded up when the digits to the right of the nominated ‘place’ are equal to or greater than 5, and numbers are rounded down when these digits are less than 5.

Combining is the operation that represents the joining of two sets or quantities.

A quantity can be separated into parts while maintaining a sense of the whole.

A relationship exists between the parts and the whole. This relationship assists in finding the unknown quantity.

Addition has several properties, which are outlined below.

• Identity property

Adding zero to a number will not affect the quantity. Zero is called the ‘identity element’ because it leaves the number unchanged.

• Commutative property

The order in which two numbers are added does not affect the sum.

• Associative property

The order in which three or more addends are added does not affect the sum. Numbers can be arranged in different ways to make them easier to add.

• Inverse property

Addition and subtraction are related operations that undo each other, therefore subtraction can be used to solve an addition problem.

The inverse property is applied to form fact families.

Addition equations have at least two addends (numbers being added) and a sum (total).

Addition strategies are methods to solve mathematical problems. 

With addition, the strategies may be mental, written, digital or a mix of the three.

Mental strategies are calculations worked in one’s mind and may involve using one of the following methods:

• partitioning and recombining numbers, usually using place-value structure (split strategy)

• jumping forward from a given number (jump strategy)

• jumping strategies can be represented on an empty number line

• rounding and adjustment strategies

• transformation strategies, involving the shifting of a quantity from one addend to another

• compensation strategies, involving the adjusting of one of the addends to make an equation easier to solve.

Written strategies are often algorithms, meaning step-by-step methods to find an answer. 

The most common algorithm for addition applies place-value structure and should only be introduced once students have explored a range of other strategies and have developed a sound conceptual understanding of addition.

The unknown in result, change and start.

Result unknown, change unknown and start unknown refer to different locations of the unknown in arithmetic problems.

Result-unknown problems have the answer as the result of the action.

Change-unknown (missing addend) problems have an initial quantity and a result quantity, but ask for the change quantity.

Start-unknown (missing addend) problems ask for the beginning quantity.

Separation is subtracting or 'taking away' a quantity from a given collection.

The relative size of two quantities can be compared and expressed as a difference.

A relationship exists between the parts and the whole. 

This relationship assists in finding the unknown quantity.

A quantity can be separated into parts while maintaining a sense of the whole.

• Identity property

Subtracting zero from the minuend (initial quantity) has no effect on the difference.

• Inverse property

Subtraction and addition are related operations that undo each other, therefore addition can be used to solve a subtraction problem.

The inverse property is applied to form fact families.

Subtraction equations that represent separation situations have a minuend (the whole collection) and a subtrahend (the part being removed) and a difference (the result).

Subtraction strategies are methods to solve mathematical problems. 

The strategies may be mental, written, digital or a mix of the three.

Mental strategies are calculations worked in one's mind and may involve using one of the following methods:

• partitioning and recombining numbers, usually using place-value structure (split strategy)

• jumping backwards from a given number (jump strategy)

• rounding and adjustment strategies (compensation)

• equal differences.

Written strategies are often algorithms, meaning they are step-by-step methods to find an answer.

The two most common algorithms for subtraction are decomposition (of the minuend) and equal addition (to both the minuend and subtrahend).

These two methods apply place-value structure and should only be introduced once students have explored a range of other strategies and have developed a sound conceptual understanding of subtraction.

The unknown in result, change and start. 

Result unknown, change unknown and start unknown refer to different locations of the unknown in arithmetic problems.

Result-unknown problems have the answer as the result of the action.

Change-unknown (missing subtrahend) problems have an initial quantity and a result quantity, but ask for the change quantity.

Start-unknown (missing minuend) problems ask for the beginning quantity.

The quantity in each group is the same.

A composite unit is a collection of single items represented as one group.

Multiplication has several properties, which are outlined below.

• Commutative property

The order in which two numbers are multiplied does not affect the product.

• Associative property

The order in which three or more factors are multiplied does not affect the product.

• Distributive property

Factors can be partitioned, multiplied separately and the partial products are then added.

• Null-factor property

Multiplying a number by zero will always give a product of zero.

• Identity property

Multiplying a number by one will not affect the quantity.

• Inverse property

Multiplication and division are related operations that undo each other, therefore division can be used to solve a multiplication problem.

The inverse property is applied to form fact families.

Multiplication equations have a multiplier (how many groups or sets of equal size), a multiplicand (size of the equal sets) and a product  total number).

The × symbol means ‘of‘ (as in equal 'sets' of) and the = symbol represents the equality (sameness) of 9 × 7 (nine 'sets' of seven objects) and 63 (objects).

Multiplication strategies are methods to solve mathematical problems. 

The strategies may be mental, written, digital or a mix of the three.

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

• rounding and adjustment strategies

• proportional adjustment strategies.

Written strategies are often algorithms, meaning they are step-by-step methods to find an answer. 

The most common algorithm for multiplication 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 multiplication.

Multiplication is applicable in a range of settings.

These settings are sometimes referred to as ‘problem types’. 

Refer to table 1 on page 17 for more information on:

• equal groups

• rate

• times as many

• part-part-whole

• cartesian product (combinations)

• rectangular array.

A factor is a whole number that divides exactly into another number.

A prime number is a whole number greater than one with exactly two factors: itself and one.

A composite number is a whole number that has factors other than one and itself (ACARA 2019).

A multiple is many groups of the same quantity.

The quantity in each group is the same.

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.

• 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.

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.

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.

The result of equal sharing or measuring can be represented as both a number and an operation.

Division is applicable in a range of settings.

These settings are sometimes referred to as ‘problem types’. 

Refer to table 2 on page 19 for more information on:

• equal groups

• rate

• times as many

• part-part-whole

• cartesian product (combinations)

• rectangular array.

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. 

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.

A quantity is an amount of something which is determined using a number and a unit.

A number triad represents the relationship between words, symbols and materials or diagrams of a quantity.

A quantity can be separated into parts while maintaining a sense of the whole.

Fractions are equivalent if they represent the same quantity.

A fraction can also be represented as:

• a decimal 

• a percentage 

• a ratio 

Benchmarks are trusted quantities or numbers used as reference points to estimate, calculate or compare.

In any fraction, the top number is called the numerator and the bottom number is called the denominator. 

The horizontal line that separates the numerator from the denominator is called the vinculum. 

The denominator indicates the size of the parts. 

The numerator indicates the number of parts of that size.

• Proper fraction

The numerator is less than the denominator.

• Unit fraction

A proper fraction with a numerator of 1.

• Improper fraction

The numerator is equal to or greater than the denominator.

• Mixed number

A whole number and a proper fraction.

Iteration is a repeated copying of a unit with no gaps or overlaps to form a quantity.

Iteration is important to the understanding of fractions greater than one.

• Fraction as a whole

• Fraction as quotient

• Fraction as measure

• Fraction as operator

• Fraction as rates and ratio

Fraction sub-constructs are the work of Tom Kieren (Kieren 1980, 1988, 1993) and are detailed in Table 3 on page 23.

• Residual thinking

Residual thinking is using the ‘left over’ amount when two or more fractions are compared to one, or another benchmark like one-half.

• Converting to equivalent fractions with the same denominator

Converting to equivalent fractions with the same denominator refers to renaming both fractions so the denominators are the same.

Physical and diagrammatic representations can be discrete or continuous: discrete representations involve collections of objects; and continuous representations can be partitioned anywhere to create fractions, and include lengths, area, volumes or capacities and mass.

Most incorrect ideas students possess are the result of overgeneralising the properties of whole numbers and transferring those properties to rational numbers. 

Below are some examples of common misconceptions.

• Ordering by numerators

Given that the numerator tells how many parts, students may incorrectly order fractions by the numerators.

• Ordering by denominators

Ordering by denominators (or reciprocal thinking) refers to students incorrectly believing that fractions can be ordered by finding the smaller denominator. This misconception arises since the more equal parts a whole is cut into, the smaller the parts become.

• Gap thinking

Gap thinking refers to comparing non-unit fractions (i.e. a fraction with a numerator greater than 1) by considering the number of parts rather than the size of the part.

• Adding numerators and denominators

Given that fractions have whole numbers as numerators and denominators, some students think that addition of fractions works like whole numbers, incorrectly adding numerators together and denominators together.

A percentage is a fraction with a denominator of 100.

The literal meaning of the % sign is 'per hundred' which comes from the vinculum (line) of a fraction combined with the two zeros from 100.

Applications of percentages

• Often percentages refer to a part–whole ratio.

• Percentages can be greater than 100, where they represent a comparison of two quantities.

• Percentages also act as operators.

Strategies

Many strategies are useful for calculating with percentages. Some examples include:

• represent percentages as equivalent part–whole ratios using a dual number line

• use 10% as a benchmark

• convert percentages to simple fractions

• find the unit rate (1%) and multiply (unitary method)

• use common factors to simplify the ratio.

A quantity is an amount of something which is determined using a number and a unit.

A number triad represents the relationship between words, symbols and materials or diagrams of a quantity.

A decimal can be equivalent to a fraction or a percentage; for example, a decimal can be expressed as:

• a fraction – 0.5 is the same as one-half

• a percentage – 0.5 is the same as 50%.

A quantity can be separated into parts while maintaining a sense of the whole.

• renamed decimal fractions

• renaming and regrouping

• compact and expanded decimal forms.

The base-10 system is a number system that is based on grouping quantities in tens or partitioning equally into 10 equal parts.

Each place has a value that is 10 times greater than the place to its right, and one-tenth of the value of the place to its left.

The place of a digit in a number determines its value.

There are three ways to interpret a single digit within a decimal fraction. 

They are face value, place value and total value.

Zero can be a placeholder and a digit representing a quantity.

Benchmarks are trusted quantities or numbers used as reference points to estimate, calculate or compare.

According to Roche (2005), ragged decimal fractions have a varying number of digits to the right of the decimal point.

As occurs with fractions, most misconceptions arise when students overgeneralise the properties of whole numbers and transfer those properties to decimals.

Three possible misconceptions that students can have about decimals include: longer is larger, shorter is larger and those who think in terms of money.

• Longer is larger

Longer is larger occurs in ‘whole-number thinking’, such as 4.63 is larger than 4.8 as 63 > 8, and ‘column overflow thinking’, such as 4.63 is greater than 4.8 as 63 tenths is greater than 8 tenths.

• Shorter is larger

Shorter is larger occurs in ‘denominator-focused thinking’. A student might incorrectly generalise that one-tenth is bigger than one-hundredth, meaning that any number of tenths is bigger than any number of hundredths. For example, 0.4 is bigger than 0.83. 

The shorter-is-larger misconception also occurs in ‘reciprocal thinking’. In this case, a student sees the decimal fraction part as the denominator of a fraction, with larger denominators creating smaller fractions. This misconception is revealed when 0.3 is chosen as the larger of 0.3 and 0.4 (as third is larger than quarter).

In ‘negative thinking’, a student believes 0.3 is larger than 0.4 as -3 is larger than -4.

• Money thinkers

Students who are money thinkers have an understanding of the first two decimal places because amounts of money only exist to hundredths of a dollar (cents). They may view decimals as two whole numbers separated by a dot, the first possibly representing dollars and the second cents. It is important to recognise the limitations of teaching decimals through money.