Equal Differences
Equal differences is a mathematical concept primarily used in teaching subtraction, particularly at the elementary level. It offers students an alternative method for solving subtraction problems by transforming them into simpler equations while maintaining the same outcome. For example, instead of directly solving 84 - 27, a student might reframe the problem as 87 - 30, allowing them to avoid borrowing and facilitating easier calculations. This strategy is beneficial for students who struggle with traditional methods of subtraction.
Additionally, equal differences can be explored through exercises that demonstrate how different pairs of numbers can yield the same difference, reinforcing the understanding of subtraction’s foundational principles. The idea also extends into other fields, such as cryptography, highlighting its versatility in various mathematical contexts. Overall, equal differences serves as an important tool in enhancing numerical comprehension and problem-solving skills in a supportive learning environment.
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Subject Terms
Equal Differences
In mathematics, difference usually refers to the amount remaining after a subtrahend has been removed from a minuend in the operation of subtraction. In the statement 10 - 4 = 6, 10 is the minuend, 4 the subtrahend, and 6 the difference. The formal terms are derived from the Latin, as is the word subtraction, which comes from the Latin subtrahere, "to take away from below"; subtrahend therefore means "the thing to be taken away," while minuend derives from minuere, "to diminish," and means "the thing to be diminished."
Some methods of teaching subtraction (particularly involving large numbers) involve the sum of partial differences. For instance, first the difference between the numbers in the thousands position is calculated, then the numbers in the hundreds position, with the results then summed. Partial differences are sometimes called decrements. In set theory, the difference of two sets is another term for the relative complement of set A in B, that is, the elements which are found in set B but not set A.
Overview
The idea of "equal differences" has varying significance in different mathematical contexts. At the elementary school level, it is a method of teaching students subtraction and helping them become comfortable with the operation. As a strategy, equal differences is an approach to subtraction that changes the problem at hand into one that is easier for the student to solve but has the same answer. For instance, 84 −27 is more difficult for some students because, in the traditional American method of subtraction, it involves "borrowing." The student begins to subtract 7 from 4, and borrows a ten from the tens column in order to subtract 7 from 14 instead. Borrowing may be avoided by instead adding 3 to both the subtrahend and the minuend: 87 – 30. This is a quicker problem to solve, since the student need look only at the tens column and subtract 3 from 8. Because 3 was added to both the minuend and the subtrahend, the answer remains the same: 47. The equal differences strategy is practiced by asking the student whether two subtraction equations have the same answer or not.
Exercises may also be constructed to show students that different pairs of numbers can have the same differences between them; that is, 11 − 7 = 4, but 13 − 9 = 4 as well. Sometimes this concept may precede the idea of equal differences as a subtraction strategy, as students can be led to discover that strategy on their own, by noting that 13 is 11 + 2 just as 9 is 7 + 2, and that adding 2 to both the subtrahend and the minuend thus preserves the value of the difference.
Equal difference is also a concept in cryptography, such that if x1, x 2, x 3, x 4 are binary strings of length n with
.
Bibliography
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