Today I did some substitute teaching math at a Swedish gymnasium (sort equivalent to highschool in the US) but my duties as prescribed by the lesson plan didn’t involve any substantial teaching but instead I was supposed to just aid them as they repeated some earlier work on their own. As they did this the basic problem of the day was that very few of them (okay, I’ll admit it, none…) very few of them knew how to work with multiplication by parenthesis.
The typical kind of uninspiring but nevertheless important type of problem to rewrite something on the lines of into ‘parenthesis free form’ and so on. There wasn’t time for an actual lesson on this so I mostly just game them ‘the rule’ and ‘pattern’ in most cases and moved on but it’s nevertheless really uncomfortable so as I got home I felt I should try to play with some arguments for why the distributive law ought to be true. You know… why you belive it in the first place.
What I’m talking about here is of course the expression
which in a sense is just an axiom but in most interpretations or visualizations of numbers has some very direct manifestation and I’d like to discuss my 3 main ‘spaces’ for where this law occurs starting with natural numbers and ending up with a geometric proof but in each case the real problem is just finding an interpretation of what multiplication is an play with it.
Multiplication as iterated addition
In the context of multiplication with natural numbers you can interpret to read literally as “three fours” or more precisely the sum of 3 fours: , and more generally as:
interpreting a multiplicative expression as an instruction on how to carry some kind of addition. From this perspective the distributive law simply expresses the rearrangement of terms
where the right hand side is simply the embodiment of the sentence “five threes and five twos” while the left hand side was “five threes and twos” which are naturally the same thing.
Working strictly symbols you might say that a ‘proof’ of distributive law for integers is the same.
(This presentation I suppose is in spirit similar to what you would do to prove the distributive law for natural numbers in the Peano- or a Peano-like construction of the natural numbers.)
Visualizing multiplication as area
The area definition of multiplication
The I suppose most common visualization for the distributive law I find in Swedish school book is using a particularly suggestive pictorial image using rectangular areas and which is nice because it’s very direct but has the problem of representing products of numbers as something different than numbers obscuring the super central property of closure but I get ahead of myself. So you start with the notion that if and $b$ are two lengths represented as lengths then can be said to be the area of the rectangle which has and as its two sides.
Then can be visualized as the are of a rectangle which corresponding sides but that rectangle can be seen as the composition of two smaller rectangles.
Comparing the two areas we arrive at the distributive law again.
Another system which has the distributive property and which algebraicly is exactly the same as the (positive) real number system is working with relative proportions of lengths. That ‘2’ represents a length which is twice as long as the length called ‘1’ and adding lengths (‘1 + 3’) is to put lengths ‘1’ and ‘3’ next to each other to form a new length. In virtually every respects lengths are real numbers but just cutting ahead I simply define the length as the length obtained through the following construction:
This is essentially the was Descartes treated multiplication and working with proportions of similar triangles this construction checks out.
Now we can derive (a) distributive law from this construction by constructing all the lengths $a \cdot b$, $a \cdot c$ and $a \cdot (b + c)$ along a line and noting the existence of a congruent triangle which allows the equality to be obtained.
This is probably my favorite trivial example of a ‘proof’ of the distributive law for real numbers but the formal part of the proof unfortunately relies on the most obscure of the Euclidean congruence theorem. To establish that the two triangles are congruent you would note that they both have the same base and since their bases are parallel they share two angles as the base and from the so called ASA-congruence condition (Prop 26 Book I in Euclid).
I think all of these proofs of the distributive law in their respective settings have different benefits. The counting one is naturally the simplest one and is more obviously connected to numbers but it doesn’t give any indication as to why the same thing should hold for real numbers and students are usually averse to formalizing their approach to natural numbers since they’ve already developed so much intuition about them. The area one is the most instructive for real numbers but frankly I hate it since thinking about as an area screws with closure.
The final one is probably closest to an honest ‘proof’ since we closure is maintained and it’s fairly obvious but it relies on the most machinery and especially machine that is’t taught in Swedish schools anymore.
There some other ‘proofs’ of the distributive law I have in mind but none I think would be convincing to a student…