Multiplicative models are useful for developing students’ conceptual understanding of multiplication and division situations. They can help students focus on the meaning of a problem and draw out the links between different forms of textual, visual, structural and symbolic representations.
Models used in middle primary situations include equal groups or sets, arrays, number lines and bar models. The equal group model has already been explored. You can revisit it here. In this post we will focus on arrays, number lines and bar models.
We can model the multiplicative situation as arrays using objects in rows or columns or as grids. Arrays possess a number of key advantages.
They help students see the connections between multiplication and division, that they are both representations of a situation with three key quantities:
a number of equal groups or sets
an amount in, or the size of, each set
a total amount
Arrays make it easier for students to focus on these three quantities at the same time, and to link multiplication and division.
This same array can be thought of as multiplication:
4 rows of 4 counters is 16 altogether OR
4 columns of 4 counters is 16 altogether
4 x 4 = 16
or as division:
16 counters placed in rows of 4 gives us 4 rows OR
16 counters placed in columns of 4 gives us 4 columns
16 ÷ 4 = 4
Different array visualisations help students move from additive (equal group) thinking to multiplicative thinking (proportions or scaling).
Arrays are also useful for developing conceptual understanding of the commutative and distributive properties of multiplication.
We will look at these properties in greater depth in the next post.
A basic knowledge of forwards and backwards skip counting is a useful foundation for understanding and working with the equal groups (repeated addition) multiplicative structure. Number lines help students develop their skip counting skills and understandings. Empty or partially empty number lines are particularly useful as students need to mentally construct and test the viability of their own numerical sequences, checking and revising their thinking as they go.
Recently, the Singapore bar model method has come under the spotlight as a useful tool for developing understanding of and fluency in the operations. Like the array model, it also appears to possess some key advantages:
It represents algebraic situations visually, providing a useful ‘missing link’ between concrete and symbolic representations
It makes the structure of the mathematical situation overt
It links key phases of the problem solving process; interpreting the problem and extracting the mathematics, representing the structure of the problem and plugging the data in, and then recontextualising the mathematics and/or representing it procedurally.
Let’s look at an example of this in action. Below is a problem where the the number of parts and the total are known and students need to identify the size of the parts. The information is presented as a word problem and students must extract the mathematical information and decide where each piece fits in the model. The model provides useful organisational clues.