## Types of multiplication and division

Students' experience of multiplication and division at school tends to focus on equal groups and using the array. We need to be aware that other types of multiplication that exist. Students need to engage with these to experience different aspects of multiplication and division.

Multiplication can be understood as:

- equal groups and fair shares.
- multiplicative comparisons, using the idea of ‘times as many’ applied as rates, ratio, or scale.
- Cartesian product to find all possible combinations, using the ‘for each’ idea for multiplication.

#### References

Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2021). Teaching mathematics: Foundations to middle years. Oxford University Press.

Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. and Treacy, K. (2012) *First steps in mathematics – Understand operations, calculate and reason about number patterns*. STEPS Professional Development.

## Partitive and quotative division

Students experience the inverse relationship between the processes of multiplication and division through sharing collections into equal groups and recombining these groups. When they act out sharing problems, students encounter ‘the part left over’, which provides a practical experience upon which factors and divisibility are developed. Understanding equal shares requires experience with both partitive and quotative division.

#### Partitive division

This involves sharing a collection equally into a given number of groups to establish how many in each group. For example, if there are 6 tables and 24 students how many students can sit at each table?

#### Quotative division

This involves sharing a collection by allocating a group of a specified size (quota) to establish how many shares there will be. For example, if a table seats 4 students, and there are 24 students, how many tables are needed?

## Multiplication as equal groups – “how many and how much”

Early understanding of multiplication and division with whole numbers requires students to think about three quantities: the whole (or total) quantity, the number of equal groups, and the amount in each group

WA Department of Education, 2013, p. 52

*How many and how much and the whole*

Multiplication involves students recognising and working with equal-sized groups, which needs them to be able to differentiate between *how many* and *how much* (Siemon et al, 2021).

Students’ initial experiences with numbers focus on counting *how many,* where they count in a unitary way, and only keep track of the count. Multiplication is more complex, as students have to keep track of two things- *how many* groups and *how much* in each group, which involves using a binary count.

The double count can be made explicit for students by the teacher using a numeral when you record *how many *and a word when you record *how much *(Siemon et al, 2021). For example, when you refer to 4 bags of 6 lemons, you would record this as **4 sixes**, to show the distinction between the two counts.

When students record their thinking as a multiplication fact, it can be helpful to understand multiplication as the process which finds ‘how many of how much to give the whole’. This can be written as:

$$\text{how many} \times \text{how much} = \text{the whole}$$

The array is a powerful model for expressing the distinction between *how many* and *how much*, as the count of how many indicates *how many rows*, where the count of how much indicates *how much in each row.*

$$\text{factor} \times \text{factor} = \text{product}$$

The row-and-column structure of the array also facilitates the transition to students using the mathematical language of *factors* and *product* when considering these quantities. Multiplication and division problems are anchored in the *factor-factor-product* relationship as they both involve finding a missing quantity. Indeed, students’ understanding of this relationship in multiplication can help them solve division problems.

When students use multiplication, the product is *unknown*, while both of the factors are known. However, for division, the product and one factor are *known*, but the other factor is not. A known multiplication fact can be used to solve a division problem, because when students know one factor and the product, they can work out the unknown factor.

Students not only need to understand the relationship that exists between the two factors and product, but also the inverse relationship that exists between multiplication and division.

#### Fact families

Students need to have a deep understanding of the relationship between the factors and the product in order to be able to use commutativity and the inverse relationship (Jacob & Willis, 2001). Through their understanding of the *factor-factor-product* relationship they can transform one fact into four, to make a *fact family.* For example, knowing the first multiplication fact generates this fact family:

$$4 \times 6 = 24$$

$$6 \times 4 = 24$$

$$24 ÷ 6 = 4$$

$$24 ÷ 4 = 6$$

Using the commutative property creates a different multiplication fact using the same two factors, where the product remains unchanged, but the factors swap places. This can be modelled using the array, by turning it 90o, to swap the number of rows and the number of columns (factors). Pairs of multiplication facts may be referred to as turn-around facts, and this understanding halves the number of multiplication facts that students need to know by memory!

Using the inverse property derives two complementary division facts from the multiplication fact. Whereas multiplication calculates two factors to find the unknown product, division calculates the missing factor by working out what the known factor needs to be multiplied by to make the known product.

The array structure clearly models the multiplicative bond that exist between related multiplication and division facts. When students understand this relationship, it provides the framework for them to find unknown numbers anywhere in multiplication or division number sentences.

#### Reference

Baroody, A. J. (2006). Why children have difficulties mastering the basic number combinations and how to help them. *Teaching children mathematics,* 13(1), 22-31.

Jacob, L., & Willis, S. (2001). Recognising the difference between additive and multiplicative thinking in young children. In J. Bobis, B. Perry, & M. Mitchelmore (Eds.), *Numeracy and beyond: Proceedings of the 24th annual conference of the Mathematics Education Research Group of Australasia*, Sydney, Vol. 2, pp. 306–313, MERGA.

Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R., & Warren, E. (2021). *Teaching mathematics: Foundations to middle years*. Oxford University Press.

Department of Education Western Australia (2013). *First steps in Mathematics: Number - Book 2 Understand operations; Calculate; Reason about number pattern. *Department of Education Western Australia.

## Multiplication as Cartesian product–“for each”

COMING SOON

## Multiplication as scaling–“times as many”

COMING SOON