Multiplication & Division- the array Part 1

sliced white round cookies on brown wood

Snowy River Farm Eggs_edited_edited_edit

What do you notice? What do you wonder?

How could we work out how many eggs are in the box?

In combination these are nice open-ended question prompts to draw on children's previous experiences and focus their awareness on different strategies for solving problems.

They also lend themself to conversations about what contributed to that efficiency - in this case the organising structure of the egg box which is 2 rows of 5 eggs.

This is called an array. It is a helpful tool that adds structure to quantities.

The larger the quantity, the more helpful it can be.

Children who have used the ten frame when learning to count collections using early arithmetic strategies will be familiar with this structure.

The array is most often associated with representing and solving multiplication and division problems so an early understanding of the concept of grouping is important.

What is an array?

An array is a rectangle divided into rows and columns.

Rows are horizontal - like sitting in rows at the movies to watch the screen.

Columns are vertical - think of columns that hold up a building.

An array can be made up of separate (discrete) items like the eggs above.

It can also be a region (or area) model divided into smaller rectangles (think a block of chocolate).

We generally describe arrays using the phrase 'rows of', telling how many rows and how many are in each row.

For example, in the red stars array there are 2 rows of 5 which is 10 stars.

In the pink rectangles array there are 2 rows of 5 which is 10 rectangles.

If we read the composite units vertically, the above array also tells us that 5 columns of 2 equals 10.

The flexibility of the array and how we 'read' it means we can see that 2 rows of 5 or 2 x 5 , and 5 columns of 2 or 5 x 2 both equal 10.

Arrays all around

Once you start looking for arrays, you notice more and more: as tiles in a pool or a bathroom floor, paint pots, stadium seating, chocolate, windows, seed planting...

Multiplication and division are related. They are inverse operations - one 'undoes' the other. This is helpful to know as it means we can use the array to work out division problems as well.

Building on children's lived experiences and the informal language associated with division of a whole can start with conversations about sharing fairly. It might occur when sharing with one other person, which is dividing by 2 or halving. It might be occur when there is a given quantity of food (say biscuits or pieces of fruit) and there is a discussion about how to divide this whole into equal shares, and what might be done with any left over.

An array that is realistically associated with this situation may be muffins in a baking tray.

Giving children many opportunities to notice and describe arrays, and create their own assists understanding of the structure and how it can be used to promote flexible problem solving strategies. It is important in these early experiences that children are not rushed to use formal expressions with the multiplication or division symbol (Siemon et al., 2017) and that there is an emphasis on the relationship between multiplication and division (Downton, 2008). This is because want to promote conceptual understanding.

Commutative property

The commutative property is helpful when learning to think multiplicatively. The structure of the array represents this property clearly. When an array is rotated (a 1/4 turn or 90 degrees) the commutative property of multiplication is demonstrated, that is: a rows of b is the same quantity as b rows of a.

Let's look at the eggs again to assist.

This first image shows 2 rows of 6 eggs, 12 eggs altogether.

If we rotate (turn) this egg box 90 degrees we see the same quantity differently.

Now we see 6 rows of 2. There are still 12 eggs.

So we can say 2 rows of 6 is the same quantity as 6 rows of 2.

If we were writing this more formally it would say 2 x 6 = 6 x 2.

References

Downton, A. P. (2013). Making connections between multiplication and division. In V. Steinle, L. Ball, & C. Bardini (Eds.), Mathematics Education: Yesterday, Today and Tomorrow. Proceedings of the 36th annual conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 242 - 249). Mathematics Education Research Group of Australasia (MERGA).

Siemon, D., Beswick, K., Brady, K., Clarke, J., Faragher, R. & Warren, E. (2017). Teaching Mathematics: Foundations to Middle Years. Oxford