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Discovering the common threads across ‘subtraction situations'


This Learning Outcome is one of the critical ones in the book, focused on unearthing the key characteristic of mathematically expressed situations using the subtraction operation. This Learning Outcome’s secondary goal is to delineate such situations into broad, non-overlapping categories of subtraction situations.
This Learning Outcome is entirely anchored around one example to keep the discussion connected across situations and remain simple. To that extent, this is a preliminary exploration of the process of subtraction.
Let us start with what we know about subtraction till now:

  • Seeking to know the difference between quantities of different things is a basic human instinct.
  • Such important is the idea of difference to us that we need a mathematical expression for it, a way to quantify difference as much as possible; subtraction is the name we have accorded to the mathematical equivalent of difference.
  • We have also discussed how in many situations, subtraction is the only way to know the difference; counting may not even be possible, or counting can’t give the difference.

The key characteristic of all situations we mathematically express as subtraction is that the situations are about finding some difference!
We now have to categorise situations seeking to find ‘difference’ into a few distinct kinds, if possible. This is important for developing sound mathematical processes around subtraction. Recall that the very nature of math is precision in quantification. One of the foundational sources of precision in math is that all the primary quantification tools – counting, measurement, and arithmetic operations – work on like things, on similar things.
Naturally, the various situations seeking difference can only be mathematised if they are meticulously clubbed into distinguishable ‘like, or similar’ categories if there is more than one kind of situation. And how can we find distinct categories of difference? One simple and obvious means of getting to that is taking a subtraction expression and exploring all the possible distinct kinds of differences.
However, we need to decide on how we will distinguish/categorise differences into distinct kinds? One easy and the positively distinctive way is to look at what created the difference, the source of the difference. Each unique source of difference will be taken to be a unique kind of difference.
Let us consider a simple subtraction expression, INR 50 – INR 20, and explore the distinct situations expressed as INR 50 – INR 20. We will categorise the distinct situations under the name ‘Category I’, ‘Category II’, and ‘Category III’; the generic names are chosen to allow the natural discovery of these situations through the discussion that follows. 

Category I
The top of the mind association with an expression like INR 50 – INR 20 is the difference of the kind created by consumption, giving away (loan, investment), or wastage/spoilage. This kind of difference is rooted in the actions on the original/initial quantity that tend to reduce the original quantity for good (at least in the short term).
In the INR 50 – INR 20 example, it is about spending away INR 20 out of the initial INR 50 so that the amount of money left is INR 20 less than INR 50.
There is just one quantity of things to start within this category of situations, INR 50 in the example. The quantity to be subtracted is taken away from the quantity at the start. The outcome (the quantity left) is part of the initial quantity. Thus, there is only one quantity, and the difference is part of that quantity.
However, the difference shows the quantitative relationship between the initial quantity and a part of it consumed/loaned/wasted.
Moreover, at the end of it all, there would be three quantities involved – one to start with and one taken away, and the other that is left.
Examples of situations resulting in this category of difference expressed as INR 50 – INR 20:

  • What will you be left of INR 50 with you if you let me borrow INR 20 from you?
  • What is the amount left of INR 50 if INR 20 was spent on buying something?
  • What is the amount left of INR 50 if INR 20 is somehow lost? Assuming INR 50 was on account of multiple notes/coins.
  • What is the amount left for shopping if INR 50 has one INR 20 note spoilt beyond exchange other than a bank?

Category II
Another common source/need of ‘difference’ is rooted in situations where there is a comparison of two distinct quantities. Nothing is reduced. There is no consumption, giveaway, or wastage. Nothing changes in any quantity. And there are two initial quantities, instead of one quantity as in the Category I. This is also a typical situation where subtraction is used to find the difference between the two given quantities.
Like the Category I difference, this difference is also part of the given quantities. However, the difference could be part of any of the two quantities, unlike the Category I difference that is part of the initial quantity.
Moreover, like the Category I, this difference also shows a relationship between two quantities. However, the difference shows the quantitative relationship between the two given, independent quantities, unlike the Category I difference which shows the quantitative relationship between the initial and a part of that quantity.
Examples of situations resulting in this category of difference:

  • How much more expensive is that than this?
  • How much longer is that than this?
  • How much heavier is this than that?
  • How much more is this than that?

Category III
Interestingly, there is yet another source of ‘difference’ which is similar to the Category I difference. This difference shows the quantitative relationship between an initial quantity and a part of it! However, there is a distinguishing feature of this difference – there is no consumption, giving away, or wastage. The quantities don’t change.
In this category of situations, there is just one quantity of things to start with, INR 50 in the example. However, no quantity is taken away from the initial quantity - the quantity to be subtracted and the outcome (the quantity left) remain together (they are just distinctly identifiable within the initial quantity).
Thus, there is just one quantity in the end. And there was only one quantity to start with.
Examples of situations resulting in this category of difference:

  • Two things cost INR 50, and one of the things is priced at INR 20. What is the price of the other thing?
  • There are fruits worth INR 50 at home. Of these fruits worth, INR 20 are spoilt. What is the worth of the fruits left at home?
  • What is the price of the fruit bought along with a packet of juice (which costs INR 20)? The fruit and the juice cost INR 50.
  • There are 67 houses in a compound that are either 2-bedrooms or 3-bedrooms. If there are twenty 3-bedrooms houses, how many 2-bedrooms houses could be counted in the compound?

These three different kinds of subtraction situations are the subject of further exploration in the next three Learning Outcomes.
It may also be registered that there are three distinct quantities in every subtraction situation:

  • The quantity from which we subtract other quantities
  • The quantity which is subtracted
  • The quantity that is the outcome of the subtraction process (every process is carried out with some goal – the outcome)

Each of these three quantities is also a subject of an independent Learning Outcome later in the book. 

Summary
There are the only three possible relationships between INR 50 and INR 20 in the expression INR 50 – INR 20:

  • INR 50 and INR 20 are two separate things
  • INR 50 and INR 20 are not separate (we already know that quantities involved in subtraction can be separate or the same). But this quantity can have two physical manifestations –
    • INR 20 is taken away from INR 50
    • INR 20 remains within INR 50

Additional details
The different situations in which subtraction is applied help us visualize the broadest range of real-world events/activities/situations.  
Two things happen when we subtract a quantity from another:

  • The two (or more) quantities in a subtraction expression precisely record the actions, or change, that is happening, or being planned to happen,

For example, the expression INR 50 – INR 20 may be the record of the following actions/change:

  • INR 20 is to be used to play off a debt
  • INR 20 being committed to be loaned to a friend.  Out of the INR 50 one is carrying; one is left with INR 30
  • INR 20 note has been discovered to be torn/spoilt and can’t be used for any purchase while being out for shopping
  • INR 30 is the amount one is planning to save (i.e., spending only INR 20 out of the INR 50 one is carrying)
  • The amount of money one is seeking to have INR 50 (one already has INR 20)
  • The difference shows a quantitative relationship between the quantities, after the actions, or change.

For example, the expression INR 50 – INR 20 gives a difference of INR 30; it is one quantitative relationship between INR 50 and INR 20. The examples above show more quantitative relationships between INR 50 and INR 20, such as ‘The amount of money one is seeking to have INR 50 (one already has INR 20)



Excerpted from the book ‘Foundations of Subtraction (Mathematics as a language)’ by Sandeep Srivastava and Saloni Srivastava