Introduction to Division
It’s your child’s birthday and you have ordered Pizza for your child and 3 friends. The pizza arrives and it comes in 8 slices. Now, you know that each of the children will get 2 slices per head as there are 4 kids and 8 slices. Ask your child the same question, how would you, before you were well versed with division, have come to this conclusion? Well, by starting to give out the slices one by one of course! That is essentially what division is, equal distribution!
The Big Idea: What is division?
The process of division can be simply visualized as equal distribution. This can be achieved in two ways, one being dividing it into equal parts and counting the number in each part or subtracting the same number repeatedly and then counting the number of groups formed. The final goal is to see how many number of times the smallest number fits into the larger number.
There are 2 ways to interpret the above example only one has been represented pictorially, explained by the first point;
You subtract 5 repeatedly to end up with 4 groups. This is division via repeated subtraction. This is also called quotitive division.
You can divide 20 into 5 groups, each group consisting of 4 items. This is called partitive division.
The entire equation has specific names;
The dividend: the number you are dividing, that is, 20.
The divisor: the number you are dividing by, that is, 4.
Quotient: the result obtained, that is, 5.
Let’s take a look at another case, 2 does not evenly divide 7. Therefore, you pick a number closest to 7, that would be, 2 x 3 = 6. 2 x 4 = 8, and 8 is larger than 7. So, you use 6. 7 – 6 = 1. With 1 left over, it becomes a reminder ‘R’.
The remainder: not all divisions result in obtaining equal parts. Sometimes you are left with a number, called a remainder.
The number of green eggs represents the divisor, the number of blue squares within each egg is the quotient, and the remaining blue square in the line is the remainder.
Now that we have understood primary division, it’s time to begin with long division.
Importance of division
Division is as important as any life skill. The basics and the foundation being strong, the rules of division apply to many situations of our everyday lives. From dividing the money between friends, sharing items, measuring substances and chemicals in the chemistry lab, the applications are widespread and extremely important.
With a proper grasp on multiplication and the multiplication tables, division should be relatively easy. Multiplication involves addition, similarly division involves subtraction. Before getting the hang of division, using subtraction will allow a firm grasp on the concept of division.
Using the example of, 16 ÷ 4;
Repeated subtraction: subtract the smaller number from the bigger number till either 0 is achieved or a number that can no longer be subtracted. Like so,
16 – 4 = 12
12 – 4 = 8
8 – 4 = 4
4 – 4 = 0
To attain 0, you had to subtract 16 from 4, 4 times. This the result is 4. 16 ÷ 4 = 4.
Equally sharing: 16 ÷ 4, distributing 16 among groups of 4 until you have 0 remainders.
Distributing 1 at a time in each of the 4 groups
Errors in the division:
A lot of children tend to divide by 0. They believe that when a number is divided by 0, the answer is 0. They confuse it with the likes of multiplication, where a number multiplied by 0 is in fact 0. The best way to explain it to the child is that a number divided by 0 has no answer at all. It is impossible. Another important detail is the placement of 0 during dropping down into the dividend or in the quotient.
In this the first step is simple, the division is even. So, continue to multiply normally
4 x 4 = 16
in the next step, 4 does not divide 2, since it is larger. So, place a zero in the quotient. You may continue as per normal or drop the next digit from the dividend. In the next step by dropping another digit from the dividend, we get 020. Or 20. 4 x 5 = 20. So, the next number to go up in the quotient is 5.
Tips for long division
Multiplication is the key. Once you have the multiplications table memorised down to the T, the division is just the reverse. Once you begin division, make sure you write the numbers neatly one below the other and cross off numbers you will not use. Make sure to add a zero in all the right places. And not forget about the remainder. Once you have calculated the quotient, make sure to multiply it with the divisor to make sure that you have the right answer. Create a working column close to the sum you are working on for any rough multiplications so that you do not forget what number you are on.
Common mistakes or misconceptions
- Misconception: Students sometimes think that 12 ÷ 4 = 4 ÷ 12. That is, division is commutative.
This probably arises because they are extending their knowledge of multiplication to division. Helping them do an activity with counters to show the act of division helps them see the difference between the dividend and divisor.
- In long division, placing zeros in the quotient is often missed. This often happens if students skip a step and pick two digits from the dividend to divide.
Tips and Tricks
- After solving a division problem, always quickly verify the answer by multiplying the quotient with the divisor and adding the remainder. You should get the dividend.
(Quotient x Divisor) + Remainder = Dividend
- Help children see the connection between a multiplication statement and the division statements that can be derived from it.
E.g. 3 x 4 = 12 also means 12 ÷ 3 = 4 and 12 ÷ 4 = 3.
Using this they can solve many division problems mentally by converting them to a multiplication statement.
- Help children observe that the remainder is always less than the divisor. This fact can help them quickly verify if their working out is correct or incorrect.