Friday, September 24, 2010

Do We Teach Any Math in High School?

I am currently working to install pavers on the back patio of my home.  The previous owner had installed pavers without properly preparing the soil underneath, which caused the pavers to settle and become uneven over time.  To prevent this I have had to rip up the pavers and dig out a couple inches of the Devil's own soil, otherwise known as clay.  Next I will have to fill in the missing clay with coarse gravel (3/4" road base to be specific) and then re-lay the pavers.  Such are the joys of home-ownership.

This morning I called the local sand and gravel company to order the 20 cubic feet of gravel that I need to complete the next step of my project.  What follows is the conversation (as best I can reconstruct it) I had with Jim, the guy that takes phone orders for this company.
Me:  I'd like to order 20 cubic feet of 3/4" road base please.
Jim:  I'm sorry, how many square feet do you want?
Me: I want 20 cubic feet.
Jim: We only sell in square feet.
I'm sorry, I want a volume of gravel, so I think that should be measured in cubic feet.
Well we only do square feet, like I said.
Ok, I'd like a volume of 3/4" road base that's 20 square feet across the top and 1 foot thick.
Our minimum order size is 100 square feet.
Ok, let's try this - how much does 100 square feet of 3/4" road base weigh?
I don't know. Let me ask somebody. (Puts me on hold for a couple minutes). Okay, sorry about that. 200 square feet of road base weighs 1 ton.
And how thick is that layer of road base spread over 200 square feet?
2"
(Pause for some quick math) Ok, then I want 128 square feet of 3/4" road base.
That's right - I ordered a volume of gravel in 2-D.  My 128 square feet of gravel will be delivered between 9 AM and 11 AM tomorrow morning.  I weep for the future.

(Note:  I originally wrote the I ordered 228 square feet in this post.  After several intelligent comments pointing out that I would have ordered far too much gravel that way, I went back and checked my order and I did, in fact, order "128 square feet" of road base.)

1. Nick,

That's funny. A slight aside: I'd be interested in knowing if our increased use of calculators and computers has led to any decline in math skill.

2. Is road base ever spread out in different depths or is it always -- if used as intended -- spread out with a depth of 2"? If it is always spread out with the same depth, then calculating by how much surface you want to cover makes sense, doesn't it?

I mean, think about it this way. If you want to paint a surface, you might buy your paint by volume. But what you really want to know is how much surface area is going to be covered by a given volume of paint. The paint supplier could estimate, on the assumption that you are painting pretty uniformly, and tell you how many square feet of paint you are buying.

I guess what I'm wondering is whether the guy you talked to is just trained to give "road base" in terms of surface area. What would happen if you ordered sand or lime or something else that doesn't have a set use? Would it be sold by weight, volume, or area? I'm guessing by weight.

3. Hey Nick. Quick question, didn't you want 120 square feet? Correct me if I'm wrong:

200 ft^2 ==> 200 ft^2 * 2 in (thick) = 200 ft^2 * .16666 ft = 33.3333 ft^3.

So, 200 ft^2 <==> 33.3333 ft^3
So, 20 ft^3 <==> 20 ft^3 * (200 ft^2/33.3333 ft^3) = 120 ft^2

I guess either there's a typo, or you're going to have way too much gravel tomorrow, or I should be weeping for my mathematical future. Anyways, as always, great post.

4. Also, I agree with Jonathan's comments. If you make certain practical assumptions, ordering a volume of gravel in terms of surface area makes perfect sense. (It's only if you don't know the assumptions that it doesn't make sense.) But it makes a better story otherwise.

5. Bill looks right to me ... 228 ft^2 spread 2" thick is 38 ft^3. (To get the volume in cubic feet for one of their measurements given in square feet, just divide their number by 6.)

6. Nick, it's worse than you think. At BYU the grounds people measure volumes of mulch by the yard.

...ok so they really mean a cubic yard, but when I was working grounds I had to explain that to the rest of my crew. They honestly didn't know that when we asked for a "yard of mulch" that meant 1 cubic yard of dirt.

7. Dang Bill beat me to it. So Nick, if you're correct you have stumped 2 PhD Aerospace engineers (asked a colleague to be sure I wasn't being stupid), Jonathan, and Bill. I come up with 120 ft^2 coming at the problem in at least 2 different ways (assuming, of course, that 1 square foot of 3/4" base is 2" thick).

But like Bill, I'm prepared to weep for my mathematical future if I got the problem wrong.

BTW, this is driving me crazy so Nick, please enlighten us as to how you came up with that number! I simply must know whether or not I'm an idiot!

8. Nick,

This is hard labor! You should have hired a contractor to do it.

Really, most counties and cities have regs how thick the base must be, so they have reduced all this to sq ft, it is very quick for them, Greek to us!

9. Ok, so the math guy can't type. I actually ordered 128 square feet of gravel, not 228, which is 20 square feet plus about 10% extra.

When the gravel was delivered I asked the truck driver about it and he said (a) road base is commonly laid to different depths - residential roads generally use between 4" and 8", depending on the underlying soil - and (b) they measure it in square feet when ordering but tons when loading the truck.

10. Let's try that again - I ordered 128 square feet, which is 21.3 cubic feet. I had rounded to 20 cubic feet, but the actual amount I calculated was 19.3 cubic feet, which I rounded up to 20 cubic feet. When I calculated again in "square feet" I started back from the 19.3 cubic feet, divided by 1/6, and then added 10% just to be safe.

11. Oh phew! I'm so relieved. Seriously, I thought my brain would ooze out onto the floor any moment at the thought that I couldn't do math!

Glad it worked out for you Nick.