The difference between two numbers

"The difference between", that's a funny phrase to use with respect to a number. What's the difference between 6 and 9? Actually they're very similar to each other. Well, they're the same shape but one's upside down. What's the difference between 3 and 7, well they're completely different shapes, one's sort of made up of 2 curves, the other is made up of 2 straight lines. But in maths, "the difference between" means "take away". How on Earth did this phrase come to mean "take away"? If you represent the two numbers either by patterns of dice dots or by number lines, and then compare them, then it is quickly apparent where the phrase has come from. With dice dots some part of the dice dot pattern will be common to both numbers, and some part of the pattern will be present in one number but not in the other. It is this difference in pattern which represents what's left over when one number is taken away from the other. The effect is probably seen even more clearly when the number line is used. The number line for both numbers will start at 0 and extend, counting up, until the particular number is reached. This means that the whole of the number line of the smaller number will be common to both numbers. So the difference between the smaller number and the larger number will be that portion of the larger number's numberline that extends beyond the end of the smaller number's numberline. This is the bit that's left over when the smaller number is taken away from the larger number.
To understand the phrase one has to be able to imagine what numbers represent, the things that the numbers stand for (and notice that the number represents the numberline, not the other way round). It is this mental picture that will help a child succeed at maths and avoid the confusion of apparently meaningless abstract concepts. Puppet Maths is dedicated to helping children visualise numbers in a way that will make aritmetic straightforward.