Comments on: [LOA] Check For Understanding

By: Bob Hansen

Bob Hansen — Mon, 05 Nov 2012 01:45:08 +0000

By the way, check out the algebra books here…

You can sign up for free to review.

I think they are pretty good and they are on the “active” side.

By: Bob Hansen

Bob Hansen — Mon, 05 Nov 2012 00:56:20 +0000

When you look at it, even the Seattle problem requires that you know how to solve the problem before you know what data to go after, right? You can’t just go hunting for data without a purpose. You need to establish a model that will reach the intended goal. The probable path would be to recognize that the cost of cleaning windows is related to the area of the windows and that you need to inventory the windows in the buildings in Seattle, by size, and tally it all up, in a spreadsheet.

The dog problem is a technical problem, the kind that an engineer might solve, but the Seattle problem is much more prevalent and 9 out of 10 of your students, if they ever apply math after school, will be confronted with the Seattle problem or a version of it regularly. If you were to examine my hard drive at work you would find that the ratio of Seattle problems to dog problems to be 30 to 1. On the flip side, my programming, especially my use of SQL, more resembles the dog problem, but I am a software engineer. Someone that is not a engineer will have all Seattle problems. Someone that is an engineer will have almost all Seattle problems. Don’t tell the kids, it might depress them.:)

From a reality standpoint, the Seattle problem is much richer than the dog problem. I don’t mean from a math perspective, as far as math goes, the Seattle problem only requires arithmetic and a spreadsheet. I am talking about the rest of the model that this (simple) math ties together. You won’t be the one cleaning the windows, you will be the manager of the project, or one of the financial analysts on the project. And you wouldn’t hire a single firm to clean the windows, you will have to hire multiple. First, the task is too large. There isn’t an Acme window cleaning company sitting around with 300,000 cleaners on stand by ready for the next city to ask to have it’s windows cleaned. Secondly, for a project that large with that much ($) at stake, you wouldn’t put all of your eggs in one basket.

Even once you have established the set of primary contractors, they will subcontract a lot of the work out to more contractors. That is just how business works, especially with large projects. A large portion of the cost in a mega project is managing a mega project. The overhead is much higher because mega projects are one-offs.

In any event, the spreadsheet (actually, there will be many) will involve a lot of organization, rates, discounts, interest, and other numerical data, beyond the area of the window times a price. There will probably be an award to the primes if the job is finished ahead of schedule (time is money). There will be bonds (insurance) to purchase in case one of your window cleaners decide to drop their bucket on some poor guy’s head. There will be weekly payments for this job which is called Cash Flow. Another dimension that affects such a project.

Sadly, they don’t teach any of this anymore. It used to be covered in business math. Nothing but arithmetic and a lot of real world details. I’ll go to work tomorrow morning and review a dozen spreadsheets, involving realities having nothing to do with math, but just as deep, and then tomorrow night I will debate algebra and trig with you.:)

Maybe it doesn’t make a difference. Maybe forcing everyone through the algebra filter still gives us the best spreadsheet makers. Something in me doesn’t buy it. As an engineer I hate all that extraneous detail, but others thrive on it. I wonder how many students that thrive on that extraneous detail, give up at algebra’s and calculus’s door.

Bob Hansen

By: Dan Meyer

Dan Meyer — Sun, 04 Nov 2012 22:24:52 +0000

Distinction without a difference.

By: Bob Hansen

Bob Hansen — Sat, 03 Nov 2012 21:35:43 +0000

I was speaking of the path problem here. I agree that your problem is a “research” problem because obviously you haven’t given me enough information.

By: Dan Meyer

Dan Meyer — Sat, 03 Nov 2012 20:26:23 +0000

If I ask you how much it would cost to wash all the windows in Seattle, you’re likely able to sort out some information that’s useful and some that isn’t, even though you’ve never seen a worked example of the same task before.

By: Bob Hansen

Bob Hansen — Sat, 03 Nov 2012 19:45:29 +0000

“Deciding what information is useful and what is not is a big challenge for most students.”

I have seen this statement, or something like it, dozens of times and never gave it much thought, till now. I suppose that the author(s) might be trying to say something else but it comes out this way. On a problem like this, I myself don’t know what information is useful and what is not. Not until I recall or contrive an applicable model that leads to a solution. Until then, all of the information might be useful. It is only at that last instant when a solution pans out that the decision is made for me as to what information is useful and what is not.

By: Cher Yang

Cher Yang — Sat, 03 Nov 2012 18:49:30 +0000

The biggest mistake that I see most students will be making with this type of abstract thinking is trying to understand what the problem is asking them to find. Deciding what information is useful and what is not is a big challenge for most students.

By: dy/dan » Blog Archive » Building A Better Taco Cart

dy/dan » Blog Archive » Building A Better Taco Cart — Tue, 09 Oct 2012 16:04:13 +0000

[…] Students are guessing. They're deciding what information is useful and useless. ¶ Look at the original task and imagine how many more students are included in this re-imagining. ¶ The task is also […]

By: Bob Hansen

Bob Hansen — Mon, 01 Oct 2012 03:15:12 +0000

I guess I should have included the step where I arrived at the minimum, it was simply looking at the speed on the beach as being a vector (Vb) and thus the rate of closing on the ball is Vb * Sin(DBC) where DBC is the angle formed by DBC. We are at the minimum when this rate is equal to the swimming speed (Vw) because running any further would close the distance to the ball slower than swimming directly towards the ball. Thus…

Vb * Sin(DBC) = Vw
Sin(DBC) = Vw/Vb
DBC = ArcSin(Vw/Vb)

Knowing DBC we can then find DC…

DC = BC * Tan(DBC)

And putting it all together…

DC = BC * Tan(ArcSin(Vw/Vb))

Bob Hansen

By: Bob Hansen

Bob Hansen — Sun, 30 Sep 2012 17:54:13 +0000

After reading the article “Do Dogs Know Calculus?” I tried looking at this problem again from the dog’s point of view. So your master has thrown a ball down the beach and into the water and you want to fetch it as fast as you can. To put this another way, you want to close the distance between you and the ball as quickly as possible. Since the ball is down the beach (and out in the water), at the beginning, running on land closes this distance faster. At some point though, as the angle between you, the beach and the ball widens, running along the beach does not close this distance as fast as swimming directly to the ball. Thus, the dog is simply making a judgement call and changing course when swimming directly to the ball closes this distance faster than running at an angle to it (along the beach). People do this as well when navigating crowds. Rather than plotting a direct path through the crowd we move along side it until a direct path is the faster course.

In any event, the distance DC is simply BC * Tan(ArcSin(Vw/Vb)) where Vw is the speed in water (swimming) and Vb is the speed on the beach (running). In this problem, DC = 5 * Tan(ArcSin(1/4)) = 1.29m.

Note: It doesn’t matter how far down the beach the dog starts, assuming of course that it is farther than 1.29m. The point at which it makes more sense to start swimming towards the ball depends only on how far out the ball is and the speed in water versus on land. In fact, if the dog is anywhere within that 1.29m mark, then jumping straight into the water will be the quickest course of action, and probably the one the dog would choose.

Bob Hansen