Division Concepts and Strategies

Introduction to division

Division is a mathematical operation where we divide or group a number into equal parts. When we divide, we determine how many times one number, called the divisor, fits into another number, called the dividend. The result is called the quotient, and sometimes there may be a remainder.

Division equation

An example of a division equation is:

36 ÷ 4 = 9

Here, 36 is the dividend, 4 is the divisor, and 9 is the quotient. This means that 4 fits into 36 a total of 9 times without a remainder.

Understanding the terms in division

Before we get into the segmentation strategies, let us understand the key terms used in segmentation:

• Dividend: The number you want to divide by.
• Divisor: The number you are dividing by.
• Quotient: The result of division.
• Remainder: The amount left after division if the dividend is not evenly divided by the divisor.

Basic segmentation concepts

Viewing partitions with grouping

One way to understand division is through grouping. Let's say we want to divide 12 by 3 using grouping:

In the figure, each rectangle represents a single group when 12 is divided by 3, resulting in four such groups.

Long division

Long division is a step-by-step method of dividing one number by another. It is especially useful for large dividends. Here is an example using long division:

Divide 432 by 7.

Phase:

1. Write down the quotient (432) and the divisor (7).
2. Determine how many times 7 will fit into 43. It will fit 6 times because 7 x 6 = 42.
3. Write 6 in the quotient place above 43.
4. Subtract 42 from 43, then bring down the next digit by 2 to make it 12.
5. Determine how many times 7 fits into 12. It fits once. Write 1 in the quotient.
6. Subtract 7 from 12 to get the remainder 5.

6 1 
-----
7 | 432 
   -42 
-----
   12 
   - 7 
----- 
    5

The quotient is 61 and the remainder is 5, so dividing 432 by 7 will give 61 R5.

Steps in long division

Let us summarise the steps involved:

• Division: Determine how many times the divisor fits into the first few digits of the current number or dividend.
• Multiply: Multiply the divisor by the quotient obtained in the division step.
• Subtract: Subtract the result from the current number of dividends.
• Bring down: Bring down the next digit of the dividend and repeat the process.

Partial quotient method

Another strategy for dividing is to use the partial quotient method. This method involves subtracting the parts and then adding the quotients. Here's how you use the partial quotient method:

Let us divide 150 by 12 using the partial quotient method.

Respectively:

1. Estimate how many times 12 fits into 150. Start with an easy number, like 10, to make the calculation easier.
2. 12 x 10 = 120. Subtracting 120 from 150 leaves 30.
3. Next, guess again for 30. You can try 2 because 12 x 2 = 24.
4. If you subtract 24 from 30, the remainder will be 6.
5. Add the quotient parts: 10 + 2 = 12.

10 2 
---
12 (Add to get the final quotient of 12)

On dividing, we get quotient 12 and remainder 6.

Interpretation of the remains

Sometimes there is a remainder after division. How we interpret the remainder depends on the context:

• Ignore remainder: If you don't need accuracy, you can ignore the remainder.
• Round off the quotient: If you need to cover every item in the group, round off.
• Express as a fraction: Represent the remainder as a fraction of the divisor.
• Convert to a decimal: If necessary, continue dividing to convert the remainder to a decimal.

Consider an example: distributing 37 cookies among 5 people.

37 ÷ 5 = 7 R2

This means that each person will get 7 cookies, and there will be 2 cookies left over. Depending on the context, you may want to:

• Keep the remaining 2 as is.
• Have each person get 8 cookies (which will require more cookies).
• Express that each person will get 7 2/5 cookies, or 7.4 as a decimal.

Examples for practice

Practice different division problems for better understanding.

Example 1

Divide 385 by 5 using long division:

Phase:

1. How many times will 5 go into 38? It will go into 7 times
2. 7 x 5 = 35, 38 – 35 = 3
3. Bring 5 down to 35
4. 5 goes into 35 exactly 7 times

7 7 
-----
5 | 385 
   -35 
----- 
   35 
  -35 
-----
    0

So, 385 ÷ 5 = 77.

Example 2

Divide 250 by 6 using partial quotients:

1. Estimate: 6 x 40 = 240, remainder 10 after subtraction
2. Repeat: 6 x 1 = 6, subtract and you get 4 left
3. Add: 40 + 1 = 41
4. The remainder is 4

40 1 
---
41 (Quotient with remainder 4)

Conclusion

Division is an essential mathematical operation that involves grouping, sharing, or distributing a number into equal parts. Using different strategies such as long division and partial quotients helps to solve division problems and better understand the concepts of dividend, divisor, quotient, and remainder. With practice and these strategies, solving division problems can become more manageable and practical.

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