# National Numeracy Learning Progression

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## Multiplicative strategies description

This sub-element describes how a student becomes increasingly able to use multiplicative strategies in computation. The coordination of units multiplicatively involves using the values of one unit applied to each of the units of the other, the multiplier. This process of coordinating units is equally relevant to problems of division.

Although multiplication of whole numbers can be achieved by repeated addition, this isn’t necessarily the best way to think of multiplication. To determine how many shoes are in 100 pairs of shoes it is possible (but not practical) to add 100 lots of 2. Coordinating ‘100’ as one unit as well as ‘2’ as a unit leads to appreciating a multiplicative relationship between the quantities. Recognising that 100 lots of 2 is the same as 2 lots of 100 is an important multiplicative strategy. This same understanding relates to seeing the two forms of division as being equivalent.

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.

Multiplicative strategies are used in the sub-elements Operating with decimals, Operating with percentages and Interpreting fractions.

Some students will communicate using augmentative and alternative communication strategies to demonstrate their numeracy skills. This may include digital technologies, sign language, braille, real objects, photographs and pictographs.

 Fischbein, E, Deri, M, Nello, MS & Merino, MS 1985, ‘The role of implicit models in solving verbal problems in multiplication and division’, Journal for Research in Mathematics Education, 16, pp. 3–17.

Each sub-element level has been identified by upper-case initials and in some cases lower-case letters of the sub-element name followed by ascending numbers. The abbreviation for this sub-element is MuS. The listing of indicators within each level is non-hierarchical. Subheadings have been included to group related indicators. Where appropriate, examples have been provided in brackets following an indicator.

### MuS1

Forming equal groups

• shares collections equally by dealing (that is, distributing items one to one until they are exhausted)
• makes equal groups and counts by ones to find the total

### MuS2

Perceptual multiples

### MuS3

Figurative (imagined units)

• relies on perceptual markers to represent each group
• uses equal grouping and counting without individual items visible but needs to represent the groups before determining the total
• counts by twos, fives and tens, matching the count to groups of the corresponding size

### MuS4

Repeated abstract composite units

• uses composite units in repeated addition and subtraction using the unit a specified number of times
• uses skip counting and may use fingers to keep track of the number of groups as the counting occurs
• determines the total or number of equal groups where the individual items cannot be seen

### MuS5

Coordinating composite units

• coordinates two composite units (mentally) as an operation (that is, both the number of groups and the number in each group are treated as composite units)
• represents multiplication in various ways (arrays, factors, ‘for each’)
• represents division as sharing division and measurement or grouping division

### MuS6

Flexible strategies for multiplication

• draws on the structure of multiplication to use known multiples in calculating related multiples (uses multiples of 4 to calculate multiples of 8)
• uses known single-digit multiplication facts (7 boxes of 6 donuts is 42 donuts altogether because 7 x 6 = 42)
• applies known facts and strategies for multiplication to mentally calculate (3 sixes is ‘double 6’ plus 1 more row of 6, 5 x 19 is half of 10 x 19 or 5 x 19 is 5 x 20 take away 5)
• uses commutative properties of numbers (5 x 6 is the same as 6 x 5)

Flexible strategies for division

• applies known multiples and strategies for division to mentally calculate (to find 64 divided by 4, halves 64 then halves 32)
• explains the idea of a remainder as an incomplete next row or multiple, and determines what is ‘left over’ from the division

### MuS7

Flexible number properties

• uses multiplication and division as inverse operations
• uses factors of a number to carry out multiplication and division (to multiply a number by 72, first multiply by 12 and then multiply the result by 6)
• uses knowledge of distributive property of multiplication over addition (7 x 83 equals 7 x 80 plus 7 x 3)
• uses decomposition into hundreds, tens and ones to calculate using partial products with numbers of any size (327 x 14 is equal to 4 x 327 plus 10 x 327)
• uses estimation and rounding to check the reasonableness of products and quotients