In the previous post, I introduced vector addition in terms of combining two piles of stuff, while keeping track of what kind of stuff is in our piles, and how many of each of the stuff we’ve got.
In this post, I’m going to take a small excursion from the main path of this series, and show a connection between vector addition and the distributive property. This excursion is mostly intended for people who are already familiar with the distributive property – who may even teach it – but who have noticed how the standard formulation of it: a(b+c) = ab + ac really never comes alive for the students they work with.
Let’s take as our starting point the two trips to the gasoline pump from the previous post, and their sum:
and recall that the first two vectors indicate how much gas was pumped on each trip and how much was paid for it, and the resulting sum vector indicates how much gas was pumped in total and how much was paid in total for these trips. The thing I’d like to emphasize at this point is how general a result this is, and how little it depends on detailed knowledge about each trip. For example, it really doesn’t matter if both trips were to the same gas station – it doesn’t even matter if both trips were done with the same car. The first receipt may come from taking your truck to station A, and the second receipt may come from your spouse taking the minivan to station B. Specifically, I’d like to emphasize that there’s nothing about the way we added the two vectors that in any way depends on the price of gas. The price of gas on the first trip may have been higher, or lower, or the same compared to the price of gas on the second trip.
So now let’s bring to this very general scenario a very special condition, one that itself has nothing to do with vector addition: let’s assume that the price of gas was identical between the two trips. That is, the price of gas per gallon hasn’t budged, neither up nor down. In the example above, this condition is met: the price of gas is $2.90 for each gallon on the first trip, and it is also $2.90 per gallon on the second trip. If that’s the case, can we then say something interesting about the sum vector?
The sum vector represents the total amount of gas pumped, and the total amount of money paid for it. We pumped all that gas in two trips, and paid for it in two trips. How much would we have paid if we had pumped 17.7 gallons of gas in a single trip? On the one hand, at $2.90 a gallon, we would have paid 17.7 times 2.90 dollars. On the other hand, we should have paid neither less nor more than we paid in total for the two trips: it’s the same total amount of gas, and it’s all at the same price. The sum vector, when prices stay constant, represents two things at the same time: the sum of the two trips, and the charge for 17.7 gallons of gas.
This relationship between six quantities, in a 2×3 arrangement, appears to be very natural for seventh and eighth graders: they understand it, use it, bend it to their needs. Most seventh graders I’ve worked with don’t need me to explain this to them in detail, they can invent the critical parts of it on their own. I’ve written about that before, e.g. in this post. Interestingly, these tend to be the same kids for whom a(b+c)=ab+ac is meaningless alphabet soup.