|Long ago, or long, long ago, you learned how to do division arithmetic by using the "goes into" algorithm.|
|For example, for you likely did:||
|I suspect this arithmetic experience was something like the Beatles' Magical Mystery Tour. Why put down the 4 in such a strange place? Why multiply? Why subtract?
If you were self-inspired to ask these questions, it is unlikely you ever received a satisfactory answer. The "goes into" algorithm is a collection of short-cut tricks that are difficult to understand in terms of why you do what you do.
A better algorithm for doing division is the subtractive algorithm. It is much easier to understand because it connects directly to the meaning of division - splitting up an amount into equal groups.
Imagine 20 cookies that are going to be shared equally, 4 cookies to each person. How many people get cookies? (This question concerns "How many groups?") A solution can be found concretely as follows:
|The diagram shows 4 cookies removed (shared) at a time. We can do this 5 times. Thus 5 people get the cookies.|
The action of making and removing groups of 4 can be represented symbolically by the following paper-and-pencil algorithm (method). Please note that the following is one style of writing. It is not the true style of writing. Styles are simply styles, and styles usually change as the years go by.
|Here is another style of writing the algorithm. It also shows a total of 5 groups (the group count) of 4 removed.
Thus 20 ÷ 4 = 5
|You can speed up the process by removing larger amounts at a time. For example:|
|There are many writing styles for showing the work. For example, one writing style looks similar to the one for the 'goes into' algorithm. Each division result (the group count) is written above the horizontal line. The final quotient is obtained by adding those group counts. No matter the writing style, the thinking remains the same: remove groups of the 'divisor' and keep track of how many groups you removed (the group count). This applies to any personal strategy for division.|
|For more detail and examples of the subtractive algorithm refer to:|
|Grade 4 Division Algorithm|
|Try these using the subtractive algorithm by removing place value amounts:|