Grade 3 Maths (IMO) : Division

Let’s talk about what division really is — it is repeated subtraction; much the way multiplication is repeated addition.

Let’s say I have the basic problem 16 ÷  4.  I could start with 16 and then subtract 4, subtract another 4, another 4, and another 4 until I run out and reach zero.  I would have to do this 4 times. If I had 16 cookies that I wanted to share equally among 4 friends, I could do the “one for you, one for you, one for you, and one for you” process and still end up with 4 cookies for each.

But what about 375 ÷ 50? If I don’t know how to divide by double digit numbers, the repeated subtraction process might actually be a good choice . . . at least showing some number sense to know that 375 divided by 50 means “How many 50’s in 375?” I know if I subtract 50 six times, I still have 75 left. I can subtract another 50 and I have 25 left over. So 375 ÷ 50 = 7 with a remainder of 25.

Dividing using the distributive law

Division Possible Split Calculation Answer
69 ÷ 3 60 + 9 (60 ÷ 3) = 20

(9 ÷ 3)  = 3

20 + 3 = 23
391 ÷ 3 390 + 1  (390 ÷ 3) = 130

(1 ÷ 3) = cannot be divided

130 with Remainder 1

 Long Division

Before a child is ready to learn long division, he/she has to know:

  • multiplication tables (fairly well)
  • basic division concept, based on multiplication tables
    (for example 25 ÷ 5 or 72 ÷ 8)
  • basic division with remainders (for example 64 ÷ 7 or 23 ÷ 6)

division problems

Dividend: the number that is being divided.

Divisor: the number by which the dividend is divided.

Quotient: the result of division.

Remainder: the amount that is left over after division.

Long division is a method  that repeats the basic steps of

1) Divide;

2) Multiply;

3) Subtract;

4) Drop down the next digit.

In order to avoid confusion it’s better to divide the process into several steps and teach the kids one step at a time.

  • Step 1: Division without remainder
  • Step 2: A remainder in the ones. Now, students practice the “multiply & subtract” part and connect that with finding the remainder.
  • Step 3: A remainder in any of the place values ; division  now also include “dropping down the next digit”and  using 2-digit dividends.

Step 1: Division without remainder

We divide numbers where each of the hundreds, tens, and ones digits are evenly divisible by the divisor. The GOAL in this first, easy step is to get students used to two things:

  1. To get used to the long division “corner” so that the quotient is written on top.
  2. To get used to asking how many times does the divisor go into the various digits of the dividend.

First we will check if the first digit is less than (or equal to) or greater than the divisor. If greater ; then we will  include the second digit and repeat the step.

Example :

                                           

  •  In this case 6 is greater than 1. You can put zero in the quotient in the hundreds place or omit it. But 6 does go into 18, three times. Put 3 in the quotient.
  • The 1 of 186 is 100 in reality. If you divided 100 by 6, the result would be less than 100, so that is why the quotient won’t have any whole hundreds.

    The ones ie 6 goes in 6 by 1 . So we have 31 as the qoutient with no remainder.

  • Check the final answer: 6 × 31 = 186.

 Always remember to check each division by multiplication.

Division tip:

Let the kid estimate if the answer will be in hundreds, tens or ones. This will help a lot in solving division problems correctly.

In the above example the 1 of 186  is  100 in reality. If you divide 100 by 6, the result would be less than 100, so the quotient will be in tens.

Step 2: A remainder in the ones (units) place.

Now, there is a remainder in the ones (units). Thousands, hundreds, and tens digits still divide evenly by the divisor. Let the kids solve the remainder mentally and write the remainder right after the quotient:

Example :

                         H    T    O       

                0  2  2  R 1

                            _______________

        6 137

 

6 does not go into 1 (hundred). So combine the 1 hundred with the 3 tens (130).

6 goes into 13 two times.

6 goes into 7 once, leaving a remainder of 1.

After the kids are thorough with the concept of division we can  find the remainder using the process of “multiply & subtract. This is a very important step and assumes that students have already learned to find the remainder in easy division problems that are based on the multiplication tables (such as 45 ÷ 5 or 18 ÷ 3).

Step 3: A remainder in any place (Using multiply and subtract)

  1. Take the first digits of the dividend, the same number of digits that the divisor has. If the number taken from the dividend is smaller than the divisor, you need to take the next digit of the dividend.
  2. Divide the first number of the dividend (or the two first numbers if the previous step took another digit) by the first digit of the divisor. Write the result of this division in the space of the quotient.
  3. Multiply the digit of the quotient by the divisor, write the result beneath the dividend and subtract it. If you cannot, because the dividend is smaller, you will have to choose a smaller number in the quotient until it can subtract.
  4. After subtraction, drop the next digit of the dividend and repeat from step 2 until there are no more remaining numbers in the dividend.
  5. Check out the examples below to understand the entire concept.

Example 1 :

Steps to divide :

  • First we will check if the first digit is less than (or equal to) or greater than the divisor. If greater ; then we will  include the second digit and repeat the step. Here 6 is greater than 5 and nearest multiple of 5 less (or equal to) 6 is 5X1 = 5.  So we will put 1 in the quotient place on the top and the 5 below 6.
  • Subtract 5 from 6 and bring down the next digit. Now the nearest multiple of 5 less (or equal to) than 12 is 10 ; which is 5 X 2 = 10. Write 2 in the quotients place and 10 below 12.
  • Subtract 10 from 12 and bring down the next digit 5. Nearest multiple less than or equal to is 5 X 5 = 25. Subtract 25 with 25 which gives 0.

Example 2:

Division tricks   

                                                              
Divide by Three Trick

This is a fun trick. If the sum of the digits in a number can be divided by three, then the number can as well.

Examples:

1) The number 12. The digits 1+2=3 and 12 ÷ 3 = 4.

2) The number 1707. The digits 1+7+0+7=15, which is divisible by 3. It turns out that 1707 ÷ 3 = 569.

Divide by 1 – Anytime you divide by 1, the answer is the same as the dividend.

Divide by 2 – If the last digit in the number is even, then the entire number is divisible by 2. Remember that divide by 2 is the same as cutting something in half.

Divide by 4 – If the last two digits divide by 4, then the entire number is divisible by 4. For example, we know that 14237732 can be divided evenly by 4 because 32 ÷ 4 = 8.

Divide by 5 – If the number ends in a 5 or a 0, it is divisible by 5.

Divide by 6 – If the rules for divide by 2 and divide by 3 above are true, then the number is divisible by 6.

Divide by 9 – Similar to the divide by 3 rule, if the sum of all the digits is divisible by 9, then the entire number is divisible by 9. For example, we know that 18332145 is divisible by 9 because 1+8+3+3+2+1+4+5 = 27 and 27 ÷ 9 = 3.

Divide by 10 – If the number ends in a 0, then it is divisible by 10.

Worksheets :

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Answers

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