What are ‘multiplicative relations‘?
For those of you that read the blog on additive relations, you will notice some similar ideas, but for those coming in fresh…
Multiplicative relations refer to the connections between numbers and the processes of multiplication and division.
But it’s more than that.
Multiplication and division are inverse operations. Inverse operations are those where one ‘undoes’ the other, or as the NSW syllabus defines it, “the operation that reverses the effect of another operation.”
Let me give you an example.
If we have 5 groups of 4 items, we have a total of 20 items, and if we have 20, we can share these into groups of 4, making 5 groups in total.
Multiplication fact families, covered formally in Stage 2, comprise all of the combinations that can be found.
For example:
5 groups of 4 = 20
4 groups of 5 = 20
20 divided by 4 = 5
20 divided by 5 = 4
More about learning our ‘tables’ in a future blog.
This concept is dealt with differently in each stage of your child’s learning. Let’s explore…
Early Stage 1
In Kindergarten, your child won’t talk about ‘multiplication’ or' ‘division’, they will talk about forming groups. They will copy, continue and create patterns - making 4 groups of 2, for example, and they’ll begin thinking about what we call division, by sharing a certain number into equal groups. For example, 8 items shared into 4 groups gives us 2 in each group.
Stage 1
In Stage 1, the topic is still called Forming Groups, and your child will be introduced to counting in multiples as well as using equal groups to represent multiplication. They begin to recognise, model and represent multiplication and division using concrete materials (objects) and investigate arrays. They’ll also learn about ‘left overs’ (remainders). Fractions are also introduced through division and they will learn to model and explain doubling and halving.
Stage 2
“When students generate multiplication fact families in Stage 2, they draw on their knowledge of multiples and doubling from Stage 1.” (NSW Maths syllabus)
Your child will generate and describe patterns, using arrays as one way to represent and explain multiplication fact families and begin to use the area model as a strategy to solve multiplication and division problems.
They’ll also investigate and learn about the commutative and associative properties, as well as flexible partitioning which will assist them to solve multiplication and division problems.
Yep, it’s getting serious now!
Stage 3
In Stage 3, your child will work with products and factors, use partitioning and place value to multiply, apply mental strategies to multiply, and begin to work with the algorithm. They’ll represent and solve division problems by selecting and applying the appropriate strategy, which is not necessarily the algorithm.
They’ll work with area models, bar models, multiply decimals, discover the link between division and fractions, and discover and use the order of operations, among many other concepts.
As the syllabus states, “In Stage 3, mental strategies need to be continually reinforced. However, it is important to recognise individual differences in students’ mental strategies … Students may find recording (using jottings) informal mental strategies to be more efficient than using formal written algorithms, particularly in the case of multiplication.
The area model for two-digit by two-digit multiplication in Stage 3 is a precursor to the use of the area model for the expansion of binomial products in Stage 5.”
And this is why it matters that your child understands all of this BEFORE they head off to high school!
Ok, yes … that was a lot!
So, you can see that there is so much going on in this topic, and you can see why trying to ‘fix’ everything in Year 6, before they go to high school, is an almost impossible task.
Your child is going to need your help to be able to get the practice that is crucial to them making this knowledge a permanent fixture in their heads!
And it’s best to start early. Don’t wait until there are problems. Be proactive and help them as they navigate new information year on year.
Not sure how?
That’s ok. Let me show you. It’s what I do.
