Evaluating energy efficiency claims


This (or other) energy efficient light bulb package(s).

Energy Efficient Bulb 20-75 w

So many opportunities here, depending on how targeted you want to be. Or, if you prefer, what kind of problem you plan to facilitate. There’s a clear nod to systems of linear equations (when one compares the time of payoff). There’s also an opportunity for some simple, linear equation building: evaluate the truth behind the $44 claim.

I’m even thinking of a 101qs video in which a perplexed customer at a hardware store is comparing this light bulb, and, say, one of these, though, these existence of incandescent bulbs is probably not long for this world. And, being Easter, hardware stores are closed today (fun fact: also, retailers really don’t like it when you take photos and videos in their stores). But that brings up a whole other can of worms: how much energy will countries save by switching to energy efficient bulbs? Like I said, lets of ways to go about this, depending on whether you want to be targeted or more exploratory.

Suggested questions

  • Is that $44 claim reasonable or bogus when you compare it against a bulb that uses 75 watts?
  • How does this compare with other energy efficient bulbs at the old hardware store?
  • What would happen if you switched every bulb in your house/school/neighborhood to energy efficient ones?
  • How much does a kilowatt-hour cost in our town? And what exactly is a kilowatt-hour?

Potential Activities

  • Take some predictions: does $44 savings sound about right over 5 years? Is that too high? Too low?
  • Collect some data on how much your lights are actually on in your house.
  • Plot five years of bulb use and see what happens.
  • Go around your house and count the number of bulb outlets you have. That data may be nice to have on hand.
  • Tables, graphs, equations, the usual bit.

Potential Solutions

Not sure what electricity costs in your particular neck of the woods, but Planet Money suggests a US average of $0.12 per KW-hr. These 20 watt bulbs usually cost around $12 per bulb, give or take. So our function looks like:

cost=$12+(20 W)*(1 KW/1000 W)*($0.12/KW-hr)*hours

Incandescent bulbs go for about $2, and comparing with a 75 watt bulb, our graphs look like this.

I actually get a savings over 8000 hours of $42.8:

(2+75/1000×0.12x 8000)-($12+20/1000 x 0.12 x 8000). That doesn’t take into account replacing incandescent bulbs more often. You could potentially get all stepwise functions if you consider the, perhaps 1000-2000 hour lifespan of an incandescent bulb.

(note the slightly different guesstimations of numbers in the planning form)

Final Word. Pretty much anything involving energy efficiency is going to allow for some systems problems. It’s all about tradeoffs, with higher initial costs gradually replaced by energy savings. Water heaters, A/C Units, automobiles, window insulation, you get what you pay for.

Underground parking spots cost $30000-$50000 to build in D.C. This smells like a systems problem.


The Washington City Paper has a (rather lengthy) post on parking in D.C. Fair warning: it’s pretty wonky with zoning rules, ordinances, etc. However, the numbers caught my eye:

An underground parking spot costs between $30,000 and $50,000 to build, and residents pay for it one way or another.

“Let’s say it’s $100 per month. If you built the parking space, and it cost you $40,000, $1,200 per year doesn’t cover it,” says Four Points Development’s Stan Voudrie.”


On the supply side of the parking problem, costs are fixed: You can’t dig a hole and line it with concrete on the cheap. Demand is more dynamic, and to some extent, it responds to price. Unlike in suburban areas, most District landlords don’t pair spaces with the unit, which means that tenants pay between $100 and $300 more per month for their cars.

As these numbers appear, I suppose it’s more of a simple equation problem, but I feel like we could easily transform this into a nice, in-depth linear systems problem. For example you could contrive or find the cost of a garage parking spot and monthly fees, and figure out which one will pay itself off faster.

Basically, I’m just excited to finally find a potential systems problem that doesn’t involve cell phone plans.

Guiding Questions

  • How much do underground parking spots cost in our area?
  • How much do parking garage spots cost?
  • And what are the monthly costs of renting a spot?
  • How long would it take to pay off each spot?

Suggested Activities

  • Depending on how in-depth you want to get into this, you could turn this into some sort of apartment building project, or you could restrict it to a few-day investigation, having student find when each type of spot will pay for itself.
  • Again, if you really want to dig into this, you could ask student to develop a pricing system for their parking spots (i.e. would underground spots be worth more? what about spots closer in?).

Aside: this is like my fourth post on traffic or parking and mathematics. I’m starting to wonder if PBL could stand for “Parking Based Learning”.

A stop-sign vs. a stoplight; when does each make sense? (Part 2, Math modeling)

I know it’s been a while since we visited this, but quick recap:

We want to figure out when it is appropriate to have a stop-sign at an intersection versus a stoplight. In Part 1, we made some assumptions about stop-sign wait time modeling and developed an equation that gave us the total wait time as the dependent variable and the number of cars as the dependent. In this post, we’ll try to model the total wait time based on the number of cars at a stop light.

Now, at a stoplight two of the incoming lanes will be green and the other two red. So any approaching car would have a 50/50 chance of winding up at a red light (ASSUMING NO GAP). Now, if you DO manage to hit a green, you’re golden. The wait time is zero. But if you DO hit a red you’re stuck for 60 seconds (note: could also be 30, or 45, or whatever). And even when you start up again, you’ll essentially be proceeding through like it is a stop sign. Maybe slightly longer if there are more cars. According to our tests in Part 1, it looks like the stop-to-full-speed time is about 15 seconds. Let’s summarize our (my) assumptions:

Assumption #4: This is an equal opportunity intersection. That is, each direction (N,S,E,W) sees 25% of the traffic. So half the traffic will be waiting at any given time, while half will have a wait time of zero.

Assumption #5: Stop-to-full-speed time = 15 seconds per car.

Assumption #6: The stop-light has a stop-time of 60 seconds.

Assumption #7: It’ll take about 2 seconds for each car to get through the intersection. Rather, 2 extra seconds that we wouldn’t have used had there not been any light there.

Total wait time for all cars = 0*1/2 x + (60+15+2)*1/2 x

And if there is in fact a three-second delay in between green lights?

Total wait time for all cars = 0*1/2 x + (60+15+2+3)*1/2 x = 40x

So our equation for the total wait time of all cars at a stop-light (with a 60 second timer) looks like this:


Super. So if we compare this model equation against the model equation from Part 1, we get this graph.

So somewhere between 2 and 3 cars being at an intersection is the point at which it becomes beneficial to have a stop-light instead of a stop sign according to our model.

But there’s a bit of a problem: the units (# of cars). “Number of cars” isn’t very descriptive. If we’re evaluating this via traffic, we’ll need something like “number of cars per minute”. And if that “number of cars per minute” exceeds the individual wait time of the cars at the intersection the total wait time will increase exponentially.

So, let’s switch our independent variable in our models from “number of cars” to “number of cars per minute.” So our axes for our graph will look something like this.

Once again, I’m going to give another intermission at this point. Partially because I want to let you ruminate on this quandary, and partially because I need to think about how to model this. Hopefully the intermission between Part 2 and Part 3 of this post won’t be as long as the one between Part 1 and Part 2. Be sure to give feedback/corrections/suggestions in the comments!

Let’s graph this and/or make it better; What constitutes a “blown game?”

A few minutes ago the other day, NPR’s Mike Pesca tweeted his mathematical rule as to what constitutes a “blown game” in basketball.

That means if at any time a team is winning by (# of minutes left in the game)+4, and loses, that’s a “blown game.” There are 48 minutes in a NBA game (40 in a NCAA game). So let’s graph what that could look like.

A stop-sign vs. a stoplight ; when does each make sense? (Part 1, Math Modeling)


Can we use math modeling and/or equations to answer the question of when it becomes advantageous to install a stoplight vs. a stop-sign? Can we “ambush” students by having them create equations without really knowing it?

What are some of the things we need to know about this?

Guiding Questions

  • How long does a car usually stop at a stop-sign? And how long does it take to start up again?
  • How long is the red light at a stoplight? Can we adjust this?
  • What’s the time gap between a red-to-green light transition? A couple seconds?
  • How long does it take a car to pass through an intersection?
  • What’s the speed limit here?

Suggested Activities

  • Have students use their iPhones or flip-cams to conduct some tests. For instance, here’s a couple of run-throughs through stop-signs in my neighborhood. Stopwatches at the ready! Improperly mashed-up music & driving video now!

(note: even though I’ve posted the videos, I would totally have students test it out on their own, hopefully with a stop-sign/light in their neighborhood that they find particularly ill-placed.)

Test 1

Test 2

Test 3

Test 4

(note: the speed limit WHICH I WAS DRIVING is 30 mph for this neighborhood, if that helps)

Each test took me about 10 to 15 seconds to slow down, come to a complete stop (as I’m sure all your students do) and then get back up to full speed. So it seems reasonable to assume that a stop sign will cause each vehicle about a 15 second delay, even if there are no cars around.

Potential Solutions and Possibly Foolish Assumptions

Assumption #1: It takes 15 seconds to go from full speed, to fully stopped, to full speed again.

If there were another car in front of us, there would be an additional wait time. Let’s say 10 seconds per car in front of us.

Assumption #2: We wait 10 seconds per each car that is also at the stop-sign.

Assumption #3: No matter the direction the other car is coming from, we wait 10 seconds to advance.

At this point, our equation may look like this:

Wait time for my car = (15 seconds) + (10 seconds) * (# of cars)

What about the total wait time for all cars? We’d have to multiply by the number of cars.

Wait time for all cars = (15 seconds)*(# of cars) + (10 seconds)*(# of cars)^2

Or, if we let y be the wait time for all cars and x be the number of cars,

y=15x + 10x^2

I’m going to cut things off and let you ponder this a bit. I’m sure I’ve made mistakes and false assumptions. I’m also going to let you guys think a bit about how we could represent the wait time for all cars at a stop light.

Couldn’t you see this sprawling into a really interesting Algebra project? Imagine students developing this model and bringing it to a community meeting or city planning gathering of some sort. It could happen.

Stay tuned for Part 2, and be sure to correct my math in the comments below!

The Wacky Algebra of NFL Passer Rating


The following formula calculates NFL Passer rating. (wiki)

(note: each component has a predetermined MAX and MIN value that appeared to be pulled out of thin air.)

Personally, I would simply present this equation to students at the beginning of class and let them stare at it a while and try to figure some things out in their heads. Eventually, the teacher and students should probably come up with some guiding questions.

Guiding Questions

  • Who came up with this thing?
  • Which “component” is most important to QB rating? Why?
  • Why aren’t rushing yards included? Could we include them? How?
  • Why do you divide by 6?
  • Why do you multiply by 100 at the end?
  • Does this convoluted formula correlate at all with being a good QB?
  • Does the formula change with era? Should it?
  • Can we change this formula to make Quarterback X look better? (There’s either a fan question or an agent question. For this region of the country, it would be John Elway.)

Suggested activities

There are a lot of routes you could go with it – comparing passer rating in different eras, creating your own formula that is either less or more complex, regressing passer rating with team offensive production to measure the importance of the quarterback and/or the validity of the formula – but I’m going to go one particular route: comparing the top two current career leaders in passer rating, who, as luck would have it, are still playing today.

Currently, the QB with the highest passer rating in NFL history (minimum 1500 attempts) is Green Bay quarterback (and apparently a regular personality here) Aaron Rodgers at 98.4 (stats can be found here). Phillip Rivers is second with 97.2 (stats found here). How many incompletion in a row would Rodgers have to throw in order to fall behind Rivers? How many interceptions would he have to throw in a row? Conversely, how many, say, touchdowns would Rivers have to throw to surpass Rodgers? Can we graph this mess?