There are too few words to describe the complexities of our number system.
I am sure there is nothing new about what I am trying to explain here, much of it is to get my own head around some fundamental issues.However the concepts are often poorly understood through a lack of clear and unambiguous language.
The framework I am trying to describe is this..
The operation suggests a before and after, the number sentence (or equation) suggests you start with 5 and there is an operation of 3 added to this to make a new value of 8. This is AUGMENTATION.
The reverse operation is taking away the 3 from the new starting number 8 so the new value is 5. The diagram could be drawn left to right but here I am emphasising the REVERSE operation which is the INVERSE of adding on, that being taking away.
Interesting to note that you can do the same operation (adding on) but with an INVERSE number (in this case negative 3) which also results in getting back to where you started with.This isn’t a point I would necessarily raise with learners early on but is important for comparing the nature of additive and multiplicative systems.
Of course it is also perfectly possible to start with 3 and add on 5 which is another “story” in whatever context the equation is representing but illustrates the COMMUTATIVE nature of addition.
Additive: same amount expressed different ways
Rightly or wrongly I am using a horizontal bar to describe this but this is just for my own benefit and there is nothing mandatory about this.
Consider the amount 8. This can be PARTITIONED into 5 + 3 among many other ways of course. And recombined by AGGREGATING to find the original sum.
There is no before and after, just a value of 8 (or items if we are considering a context) which can be “split” into two or more separate values and then recombined to make the original value.
Again there is no before and after, but two values occurring at the same time. The comparison is not specifically linked to adding on or taking away but depends which value is your initial focus.
If we consider 5 first, then 8 in comparison is three more than 5.
If we focus on 8 first then 5 in comparison is 3 less than 8.
NOTE that we can add 3 on to 5 to result in 8 or take away 3 from 8 to result in 5.
We can also find the difference by taking away 5 from 8 to get 3 but we do not (normally) take 8 from 5 to get a difference of -3, the difference is always positive.
So the difference between 8 & 5 is the same as the difference between 5 & 8, this has implications for interpreting the number sentence.
Here we have a before and after situation where 5 has been multiplied 3 times. In effect we have an ENLARGEMENT of SCALE FACTOR 3 to produce a new number or PRODUCT which in this case is 15
We can do the REVERSE operation with the same number (this being division – whatever that means) but it looks like we are dividing the starting number 15 into 3 equal sections and choosing just one of them. Another way to think of this could be to say 15 (the new starting number) represents 3 parts and I want to know what one part is.
Of course we can (like additive) do the same operation with an INVERSE number (in this case multiply by 1/3) but language helps if we say what is a “third of”.
Multiplicative: same amount expressed different ways
Most easily shown by an array. There is no starting number or new end result, just in this case 15 items but the can be arranged in such a way that shows 5 groups of 3 or three groups of 5 are equivalent to 15. Similar to partitioning and aggregation this form of expressing a value as a product of its factors is called FACTORISING.
Note there is a language issue which can confuse.
Three lots of five in context is not the same as five lots of three, but the product is the same. Five lots of three is the same as 3 multiplied 5 times the latter of these suggests an operation whereas the former leans more to factorising?
For this example I am going to choose the numbers 10 & 15. Like the additive comparison it depends on which is your initial focus.
In terms of 10, the number 15 is 1 ½ times this.
We can see this by comparing equal parts and in this case can conveniently simplify the comparison of 3 lots of 5 with 2 lots of 5 to 3/2 or 1 ½ .
Note that this also finds us the scale factor to turn 10 into 15 using the multiplicative operation.
But if we focus on 15 first, (I have swapped the bars around) then in terms of 15 the value of 10 is only 2/3 this.
Like the additive comparison there are two ways of looking at it (more than, less than), the multiplicative comparison can also be done both ways, although the comparative RATIO should be written in a specific way depending on the context or focus number, unlike the difference which is always given as a positive number.
Dividing into a ratio
Now we are combining both additive and multiplicative structures such that the two parts of an additive model are in a ratio.
This opens up a whole new set of issues including fractions of amounts.
The whole is divided into equal portions where the word part could be used to describe each of these smaller, equal parts or indeed each of the combined section of portions that constitute the part, part whole of the additive model.
If we take a particular example
This example can be described many ways, e.g. three-fifths of the whole 30 is 18
18 as a comparative multiplicative portion of 30 is 3/5 and 3 is to 5 is in the same proportion as 18 is to 30.
Whereas the ratio of the additive parts 2:3 gives us the constituent parts of the whole in an additive sense, being 18 and 12 which are also in the same proportion as the ratio 2:3
So to go back to my original issue for writing this piece….
There aren’t enough words to describe the complexity of our number system, or perhaps there are but they are not all used commonly. Those that are used do not always clearly express the distinctions that need to be made to gain a full and deep understanding.
Generic terms, particularly the four favourites of Addition, Subtraction , Multiplication & Division and perhaps added to these the words Part & Proportion on their own are not sufficient to distinguish the subtleties that at least need to be understood by teachers if not learners and whilst we continue to use them in a general way ambiguous meanings and therefore misunderstandings will prevail.
The use of diagrammatical models certainly allow us to clarify what we mean and can act as a frame of reference to all when clarifying our meaning but perhaps there is also a need for a clarifying language of terms so we can explain our deeper understanding.
Although I doubt we will ever see KS2 questions quite like this…
“Describe the how the multiplicand and the product in a multiplicative relationship can be used in a comparative way to find the multiplier and the significance of this quotient in describing the proportion between the aforesaid multiplicand and product.”
Although if you have read and understood this blog, you may well be able to give it a go.