Subtraction

Introduction to Subtraction

Addition and subtraction are the most basic operations. These concepts are building blocks of understanding and operating on numbers. We add and subtract numbers, amounts and values in our everyday lives. Children are mechanically introduced to these concepts using numbers. However, this may not be fruitful in the long run. Mechanically rote memorizing the procedures increases the chances of silly mistakes. 

Let's try and understand what subtraction is. As you can see in the image given below there are 3 oranges kept in a plate. If we take away one of them how many would be remaining. 2 Oranges would be left in the plate. This act of 'taking away' is known as 'Subtraction'. Subtraction is a basic arithmetic operation that we need every day in our lives.

What is subtraction?

The act of 'taking away' a certain number of objects from a collection of objects is called 'subtraction'. Suppose there are 25 people travelling on a bus, and 3 people get off the bus at a stop. To know how many people are still on the bus you would have to subtract 3 from 25. The subtraction statement given below would show this. 

25 – 3 = 22

You can consider subtraction to be the opposite of addition. In 'addition' you keep on putting together objects to get a larger group of objects, but 'subtraction' involves taking away objects from a larger group of objects.

Subtraction can be visualised as 'taking away'. For example the image below shows, there are 15 squares and we take away 4 of them. 11 would be remaining.

Cuemath students learn to first visualise the idea of addition and subtraction using different visual models such as the number line, ten-frame, 20 frame. Once they are comfortable with visual models they move to numbers. Given below are images of a few visual models. 

Use of real-life objects to show subtraction 

A simple idea used for subtraction of two numbers is the place value method. Let us say you need to subtract 234 from 1568. In order to correctly subtract them, you need to know what are the place values of these two numbers. This is how they would be written

So, in order to subtract 234 from 1568, you write both the numbers one below the other in such a way that the place values are all aligned. So, it would look like this:

\(\begin{align} {\rm{1}}\;\;{\rm{5}}\;\;{\rm{6}}\;\;{\rm{8}}\\   {\rm{ - }}\;\,{\rm{2}}\;\;{\rm{3}}\;\;{\rm{4}}\\   \overline {\underline {1\;\;3\;\;3\;\;4} }   \end{align}\)

Topics closely related to Addition and Subtraction 

The image shown below shows the topics one should be well versed with to understand addition & subtraction. It also shows the topics that these two operations are closely related to. 

Visualising Subtraction of Numbers 

Here is an interesting way to visualise subtraction. Know what happens when you borrow while subtracting numbers. 

How is it important?

The concept of borrowing

When we subtracted 234 from 1568, the corresponding numbers in each place was bigger for the larger number than the smaller number. For the unit’s place, 8 was bigger than 4, for the tens place, 6 was bigger than 3, and for hundreds place 5 was bigger than 2. And hence the subtraction was simpler. 

But what if we had to subtract 268 from 1534? In that case, the numbers would be written in this way

\(\begin{align} {\rm{1}}\;\;{\rm{5}}\;\;{\rm{3}}\;\;{\rm{4}}\\   {\rm{ - }}\;\,{\rm{2}}\;\;{\rm{6}}\;\;{\rm{8}}\\   \overline {\underline {\qquad\quad\;\;\,\\} }   \end{align}\)

Can you take away 8 ones from 4? The answer is no.
Can you take away 6 tens from 3? The answer is no. 

So, what would be the solution? The solution would be to ‘borrow’ from the neighbour. What this means is that when this happens in the unit’s place, you need to borrow from the tens place. And if it happens in the tens place, the borrowing would be from the hundred’s place.

    2 1 4
1 5 3 4 
– 2 6 8
         6

As you can see above, a ten has been borrowed from the 3 tens so now the tens place has 2 tens, and the 4 in the unit’s place becomes 14. Now 8 can be subtracted from 14 to give 6 which is written down as the first step.

Next, we see that in the tens place 6 can’t be subtracted from 2, so we borrow one from the hundreds place, and now this is how it would look like, 

    12
4 2 1 4
1 5 3 4
– 2 6 8
      6 6

Now, we can subtract 6 from 12, and can put the answer 6 in the tens place. Now we see that in the hundred’s place it is possible to subtract 2 from 4 and no more borrowing is needed. Also, in the thousand’s place it is possible to subtract 0 from 1. This gives 1266 as the answer.

    12
 4 2 1 4
1 5 3 4 
 – 2 6 8
 1 2 6 6

How do you teach Addition and Subtraction?

1. Ask children to perform subtraction using real-life objects like pens, pencils, chocolates, marbles. 

2. Use math manipulatives like colour counters, abacus, flats, rods and blocks to help children perform subtraction.

3. Pop quiz: you can ask children 5 rapid-fire questions on subtraction.

4. Based on the pocket money children receive Ask children to maintain a log of their expenses. At the end of the month, you can ask for the remaining amount. 

5. Use flats, rods and blocks to show how the procedure works with larger numbers. 

Common mistakes or misconceptions

• When solving 803 - 65, children first have to subtract 5 ones from 3 ones. Instead of borrowing from higher place value, they simply subtract 3 from 5 and write 2 in the ones' place.  Modelling subtraction using Base-10 blocks helps address the misconception.
• While subtracting, if students have to borrow from a higher place value where the digit is zero, children make mistakes. This is because they have to keep going to higher and higher place values until there is a non-zero digit for them to borrow from. Targeted practice of similar questions helps avoid this mistake.
• When subtracting a two-digit number from a three-digit number, students may place the numbers one below the other incorrectly.

This generally happens once again due to an incomplete understanding of place values. Rather than telling students to simply align numbers from the right, it is better to ask them to model this problem using Base-10 blocks, so they understand why the numbers are aligned from the right.

Tips and Tricks

• Help children associate subtraction with common words used in everyday language like “difference”, “less than”, “more than”, “how much more”, “left with”, “remain”, etc. This association will help them with comprehending word problems.
• When subtracting mentally, sometimes counting up from the smaller number to the larger number can be a useful strategy to find the difference. E.g. to find the difference between 33 and 17, it may be easier to start with 17 and see how much we need to add to get to 33. Doing this in steps allows us to mentally calculate the difference as 16.