Understanding Division

UNDERSTANDING DIVISION

Developed by the Association for Conceptual Studies

Presented by: F Richard Singer III Edition date 11/2005

Overview: Section 1 focuses on the concept of division. Section 2 focuses on algorithms for doing what has been called long division. The last part of this section discusses some purposes of learning about both a standard and a modified algorithm for long division. It also relates this to a constructivist perspective on learning mathematics. Section 2 can be read without reading Section 1, as long as concept of division has been understood in relation to subtraction.

SECTION 1 THE CONCEPT OF DIVISION

The focus of this section is on the types of situations to which we apply the concept of division. There are three basic types of these. We refer to these as dividing between, dividing for size, dividing to compare.

‘Dividing between’ means to divide some quantity equally between a specified number of parts.

‘Dividing for size’ means to divide some quantity to obtain parts of a specified size.

‘Dividing to compare’ means to divide quantities of the same type in order to obtain a ratio.

With 8 padlocks to divide between 2 people, give each person 4 padlocks. 8 ¸ 2 = 4

We focus first on dividing for size and dividing between. To divide 8 padlocks so each person gets 4 padlocks; we can share them among 2 person. 8 ¸ 4 = 2

The same picture can be used to show
dividing 8 padlocks between 2 people and dividing 8 padlocks for sizes of 4.

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While the same picture shows both dividing between and dividing for size,
in first learning to divide, these picture would be obtained in different ways.

To physically divide 8 padlocks between 2 people we start with 8 padlocks or tokens for them.

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We then partition them. The most primitive method is one at a time to each pile.

Send one to each person. Ï ï ÏÏÏÏÏÏ ð Ï

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To divide physically for parts of size 4 is a process of taking away 4 padlocks until none are left.

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Send 4 into a pile. This gives 1 pile and some left. ÏÏÏÏ ï those left ÏÏÏÏ

Send 4 more into a pile, giving 2 piles and none left. ÏÏÏÏ ï ÏÏÏÏ ï none left

Unit Types: There is another significant difference in dividing between and dividing for size, When dividing between we divide a number of padlock by the number of people and obtain a number of padlocks. In general, when we divide between, the dividends and the quotient are quantities of the same type but the divisor is a quantity of some other type. In dividing for size, we divide padlocks by padlocks and the result is a number of people. In general, when we divide for size the divisor and dividend are of a similar type but the quotient is a quantity of some other type.

Representing Numbers: Even before people had names for numbers, they could divide 150 sheep equally between 3 shepherds. Merely separate the sheep, giving one to each shepherd, another to each shepherd, etc. However to divide 150 sheep to see how many groups of size 50 can be obtained, they would need some way to represent 50, perhaps the ability to count to fifty either symbolically or with objects such as stones. First remove 50 sheep, then 50 more, etc. This is still the way children first solve problems of dividing between and dividing for size.

Note: Using the same picture for equations 8 ¸ 2 = 4 and 8 ¸ 4 = 2 depended on the fact the first equation was for dividing between and the second equation was for dividing for size. On the other hand, dividing 8 padlocks between 4 people gives 8 ¸ 4 = 2, and this has a different picture than dividing 8 padlocks between 2 people. It has the same picture as dividing 8 padlocks for sizes of 2, which gives the equation 8 ¸ 2 = 4.

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When dividing with whole numbers, we can imagine many situations that involve dividing between and many situations that involve dividing for size. However, they are not always stated this way. For example, if we can seal 500 envelopes in 5 minutes then how many envelopes can we seal in a minute? This is a case of dividing between because we are sharing 500 envelopes between 5 intervals, each of which is a minute long.

Between: 500 envelopes sealed in 5 minutes, do 100 envelopes in each minute: 500 ¸ 5 = 100.

We can also divide envelopes for sizes. Below we have specified the size job for each minute and are looking for how many jobs there are of this size.

For size: 500 envelopes sealed at 100 envelopes each minute, taking 5 minutes: 500 ¸ 100 = 5.

Dividing Into: Using ordinary language, we might describe dividing 500 objects for size 100 as dividing 500 objects into groups of 100. We then talk about the arithmetic of how many 100s go into 500. Since in this language reverses the numbers that come before and after the word ‘into’, using the phrase ‘dividing into’ in these situations can cause confusion for some children. This is why we recommend initial use of the phrase, ‘dividing for size’ instead of ‘dividing into’. Once concepts are well mastered, the duality in the use of ‘into’ should be clear from context.

Dividing To Compare: Dividing to compare is pictured differently than dividing for size or dividing between. We picture both the divisor and dividend. We then look at how they compare. The quotient is a pure number rather than a quantity of any type.

To compare 8 padlocks to 4 padlocks picture both but separate the larger into groups of the smaller and count these groups. Comparing 8 to 4 and comparing 8 to 2 have different pictures.

Comparing 8 to 4, we have 8 padlocks is twice as many as 4 padlocks. 8 ¸ 4 = 2

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Comparing 8 to 2, we have 8 padlocks is four times as many as 2 padlocks. 8 ¸ 2 = 4

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SECTION 2 LONG DIVISION

The primitive way of dividing for size relates to subtraction and understanding the algorithm for long division depends on this relationship. The algorithm to the right below shows the usual way of writing a solution to the problem of dividing 8475 by 23. Altho this algorithm is concise, many of the underlying ideas are left implicit. Later we show alternatives that make ideas more explicit. If one goal is to master an algorithm for dividing with multi-digit numbers these alternatives can be used instead of the standard algorithm or as a prelude to understanding it. This is discussed at the end of this section.

The process in the first step is often described
as follows:

Take 23 into 84, 3 times 23 is 69, subtracting 69 from 84 gives 15, now bring down the 7.

The novice may not realize that the 3 is really 300 and that the 69 is really 6900.

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To make the underlying idea more explicit we can include zeros in the standard algorithm. However, the fact that the 3 in the quotient is really 300 may not be apparent until the final step. The modified algorithm on the far right makes this explicit. It also stresses the fact that what we are doing is seeing how many 23’s we can subtract from 8475.	$Description: C:\Users\Singers\Documents\A C S\Web Site Files (From UCT)\Elementary Math\Divide 2.GIF$
The intermediate steps for the modified algorithm can be indicated as follows:	$Description: C:\Users\Singers\Documents\A C S\Web Site Files (From UCT)\Elementary Math\Divide 2a.GIF$

When using the standard algorithm, it is important to estimate the partial quotients accurately. This is not the case when using the modified one. How much to subtract each time is a matter of choice. Some choices involve more steps than others do. However, for anyone who is not comfortable with all the multiplication fact, taking more steps might have an advantage.

We do not need to start with a multiple of 100. For example, we could start by taking away 303. This would leave 1506. Of course, there may be no practical advantage in such a choice.

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Some students have trouble when bringing down a single number give something smaller than the divisor.

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Why Learn About Long Division: Calculators are widely available, and it much easier to use them to find the answer to a division problem than to use an algorithm for long division. This raises the question of whether or not it is useful to master such an algorithm. My answer is that this depends on the reason a person P might have for mastering it.

Suppose a P’s only reason for mastering algorithms is to be able to find numerical answers. Most persons are unlikely to encounter situations in which they have no calculator and need answers to long division problems. In the cases that they do, merely understanding how division relates to subtraction will allow them to find an answer. The work involved is likely to be less than the work involved in learning and remembering the standard algorithm. Furthermore, the modified algorithm will do just about as well and it is easier to master. There is a standard algorithm for extracting square roots, but very few people master it, and those who do seldom use it. From the perspective of finding answers, I can see no more reason to master a long division algorithm than to master this algorithm for extracting square roots.

Utility is relative to purpose, and the preceding comments apply only to a very limited purpose. P’s purpose might include being able to apply mathematics to various situations. It might be also include a desire to understand mathematical ideas primarily because they seem interesting. With either or both of these purposes, merely being able to use a calculator is far from sufficient. To do either, it is necessary to understand the concepts involved well enough to be able to chose which concepts are relevant. Altho algorithms can be learned without understanding concepts, they cannot be appreciated without such an understanding. Thus examining or creating algorithms is one way to gain a greater understanding of concepts. Since one important type mathematical activity involves designing algorithms, mastery of some algorithms is a crucial factor in understanding mathematics. This does not imply that either the standard or the modified algorithm for division is necessarily important. However both algorithms relate division to subtraction, and this is certainly an important conceptual relationship.

By a net for a person P, we mean the networks of concepts that P uses. Central to our constructivist perspective on learning mathematics is that math nets must be grown, rather than acquired ready made. A math net for P is a state of affairs that developed within P, and it does not directly transfer. P may try to copy part of another person’s math net, but this is not a matter of Xeroxing. P can only understand another’s math net as P relates it to P’s own nets. To the extent P accomplishs this, P can draw on others in evolving P’s own math nets. It is useful to get help from others, but there are limits on the utility of this. Without creative effort, P’s math nets only develop to levels of minimal utility. Since a person can learn to use algorithms without understanding the concepts involved, learning algorithms can actually have a deleterious effect on a person’s math nets. However if P designs algorithms based on P’s nets then this is likely to enhance these nets. Examination of algorithm developed by others can also enhance them, at least when they are examined in relation to conceptual relations that P understands. Hopefully, the modified algorithm for long division can be used in this manner.