Understanding operations with negative numbers [1]. Introduction.

By the end of year 6 pupils are required to be able to solve problems with negative numbers.

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Negative numbers, conceptually are more difficult to visualise than their positive cousins or have fewer possibilities available to them should I say. For example it is impossible to draw a rectangle with dimensions -5cm x -12cm ( and by the way what would its area be?) and even more tricky to label the possible side lengths, x and y, of the following right-triangle (we could label them but would the dimensions make sense?)

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However if we start at linear measurement, which is still not without its conceptual difficulties, we can begin to see the patterns that emerge and help us to become more fluent with calculation using addition and subtraction.

In our ‘right thinking’ world a move to the right is considered visually, a move in the positive direction and a move to the left is seen as a backward move or a negative direction. Interestingly the italian word for left is sinistra which translates not only to left but to sinister or unearthly also, so, very negative we might argue! All this can be put to good use in coming to terms with the mathematics of addition and subtraction of negative values.

Let us begin with that very modern number ‘zero’, a relative newcomer to western mathematics. The Romans for example in their number system did not have zero and to this day calculating with these numbers is not a worthwhile activity, especially in the primary school classroom where recently they have been added to the mathematics curriculum. You can understand why nought was thought to be insignificant and not worthy of recognition, as algebraically it seemed quite redundant. By this we could understand that 10 denarii could be visualised. Your average Roman citizen would recognise 10 denarii but ask them to draw a picture of 0 denarii and you would be considered a fool. So zero in algebraic terms is not a very tangible concept. If we leap forward a few centuries however and we are able too understand negative numbers in  an economic sense as a debt then zero is that place on the number line that divides debts from credits or in modern terminology negative from positive numbers. Adding as mentioned earlier in positive number terms, is all about moving the value to the right. In very basic terms the diagram below shows addition as an aggregation. There are two numbers (3 and +2) which are to be combined essentially. Because of the nature of this type of addition (aggregation)  the result of the addition is 5, but has 2 been added to 3, or 3 been added to 2, or does it really matter as the result is the same? The diagram could be organised as to show both possibilities. We are not going to explore the commutative nature of addition in this blog however but try to lay the foundations which will help us make sense of operating with negative numbers.

 

 

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How the imagery can set up a better understanding of addition is to see that both numbers being combined and set out from a starting point of ‘0’ and head off in the ‘right’ direction. The result of the addition starts at zero and ends at 5 whole places to the right. The following blog will take the thinking further.

Next week’s blog. Understanding operations with negative numbers [2]. Subtracting a positive.

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