Integers in the Classroom - 1

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Integers are difficult for school students to understand. In this chapter and the next, we will describe strategies using bottle tops which make it easy for students to visualize and understand the nature of integers and the arithmetic operations with them.

We have already seen how bottle tops help separate the two meanings (verb &amp; adverb) in which the operators + &amp; - are used.

Arithmetic Operations

Addition

Addition means physically &ldquo;putting together&rdquo; (bringing together) bottle tops representing 2 numbers.

(+ 3) + (-2) would mean bringing together 3 &ldquo;open&rdquo; and 2 &ldquo;closed&rdquo; bottle tops. What to do next would be dealt with in subsequent paragraph.

Subtraction

Subtraction means physically &ldquo;taking away&rdquo; a certain number of bottle tops representing a number from another set of bottle tops representing another number.

(+3) – (+2) would mean &ldquo;taking away&rdquo; 2 &ldquo;open&rdquo; bottle taps from a set of 3 &ldquo;open&rdquo; bottle tops. What to do next would be dealt with in subsequent paragraph.

Please note that the direction of any bottle top has to remain same, through any operation.

Zero Pairs

Before proceeding further, we need to understand the concept of &ldquo;Null&rdquo; or &ldquo;Zero&rdquo; pairs.

Since a &ldquo;open&rdquo; and a &ldquo;closed&rdquo; pair of bottle tops represent +1 &amp; -1 then a pair of them together represents 0. Similarly, 3 &ldquo;open&rdquo; &amp; 3 &ldquo;closed&rdquo; bottle tops represent +3 &amp; -3 and hence also represent 0.

In general, an equal number of bottle tops with equal number of &ldquo;open&rdquo; and &ldquo;closed&rdquo; bottle tops would be equal to 0.

Introducing In or Removing of Zero pairs

0 added to any number or subtracted from any number does not change its value.

Hence sets of Zero pairs can be &ldquo;put in&rdquo; or &ldquo;removed&rdquo; from a mix of bottle tops without changing the value of the number represented by the bottle tops.

The &ldquo;introducing in&rdquo; or &ldquo;removing&rdquo; operation would need to be done depending on the context. We will explain this in the next paragraph which will illustrate all varieties of addition &amp; subtraction operations.

The Next Step (Normalization)

After the first steps in addition or subtraction, there could be a mix of &ldquo;open&rdquo; and &ldquo;closed&rdquo; bottle tops. The answer to any operation with integers has to be either a positive number or a negative one. Hence we &ldquo;normalise&rdquo; the set by &ldquo;putting in&rdquo; or &ldquo;removing&rdquo; zero pairs, so that what remains is a set of &ldquo;open&rdquo; bottle tops or &ldquo;closed&rdquo; bottle tops.

The final set of bottle tops after normalization would be the answer of the operation.

A few examples would make this process clear.

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