Executive Summary (aka Grade 3 version)
On a ruler we have marks at whole number positions (and if you know about decimals and other fractions we can also place numbered marks at some – but not all – of the points in between them).
Addition of two numbers gives the length you would reach by putting together two rulers end to end beside a longer one and reading off the mark on the longer ruler, and the same idea of combining lengths gives us something that we could call “addition” even when neither one corresponds to an exact whole (or even fractional) number of units on the ruler.
Eg/Ex for 2+3=5 and for a+b=b+a and a+(b+c)=(a+b)+c
We can also use the idea of adding together a lot of copies to multiply any length by a whole number.
Eg/Ex for 3a, 2a+3a=5a, and 3a+3b=3(a+b)
But what about multiplication of a length by another length?
Next time we’ll discover a way of combining any two lengths that agrees with multiplication when either of them is an exact number of units.
Fuller Adult Discussion (aka tl;dr version)
The issues of length measurement are actually shared by any kind of physical measurement.
In the previous Chapter we reviewed how children are introduced to the natural (counting) numbers. These describe the possible amounts of quantities that can be counted (because they consist of collections of discrete objects like pennies or apples or bricks). But not every quantity can be counted in that way. For example we can try to estimate the amount of water in a glass by counting out the number of times water from the glass can fill a one ounce measure, but there may be a partial measure left at the end and even if we keep using tinier and tinier fractions of the initial measure we won’t necessarily get an exact count unless we go all the way down to individual molecules (which isn’t very useful because then the numbers would be so large). Another approach would be to pour the water into a calibrated tube with markings for each cubic centimetre (aka millilitre) and perhaps finer and finer markings for smaller and smaller units, but we would still run into the same problem of not filling up an exact whole number of units at any realistic scale.
In the above example, given the right kind of measuring tube (of constant cross section), the problem of volume measurement becomes equivalent to measuring a length, and in fact almost every physical measurement amounts in the end to determining the position of a level (eg of mercury in a thermometer) or pointer on a scale (eg in an electrical voltage meter) – and each possible amount of the measured quantity then just corresponds to a length on the appropriate scale. So the problem of assigning amounts to physical quantities of any kind can be considered as similar to the case when the quantity of interest is a geometric length.
So let’s talk about comparing lengths of geometric line segments.
Two line segments have the same length if one can be moved to exactly cover the other and if we move two unequal segments both to run in the same direction from the same starting point we can easily see which is greater and which is less than the other. So the set of possible lengths has an ordering relationship much like that of numbers.
If we combine lengths by placing them end to end it seems natural to call that process addition of lengths because it seems rather like combining of sets of objects that corresponds to adding numbers. It also seems to be a geometric fact (often taken as an “axiom” in deductive geometry) that if we put two lengths together in that way, then it doesn’t matter which is put first, and that if we put more than two together then the order doesn’t matter. So the “addition” of lengths satisfies the same “commutative” and “associative” properties that are familiar to us for addition of numbers.
Now if we can make copies of a line segment, then for any counting number m we can add up that many copies of any length a by placing them end to end, and it seems reasonable to call the resulting length m times a and to refer to the process as the “multiplication” of a length by a counting number.
Eg/Ex Check that ma+na=(m+n)a and m(a+b)=ma+mb
Eg/Ex Check also that m(na)=(mn)a