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!

What Teachers Want: An Education Parable Written By a Math Teacher

Our scene opens with the Teacher and the Student in a moderately decorated classroom. The walls are covered with posters, both inspirational and informative. The Student is seated at one of dozens of neatly aligned desks, facing the Teacher, who stands in front of a whiteboard covered in information, examples, and tasks. The Teacher is just finishing the lesson, which was carefully planned out, aligning with the standards of instruction set by the State. In addition to covering the State-mandated material, the Teacher demonstrated seven examples during the lesson of the Concept being taught today. Each step of each example was carefully explained, with the Teacher showing every action required by the Example on the whiteboard.

Upon completion of the lesson, the Teacher assigns the Student an Activity in order to both allow the Student to practice the Concept presented during the lesson and to assess the Student’ s understanding of the Concept. The Activity could be in the form of a worksheet to be completed, or the construction of a poster, or a presentation to be given to the entire class. Whatever the assignment, the Teacher was careful to include only tasks that were specifically addressed in the lesson presented just moments ago. The tasks are very similar to the ones the Teacher just demonstrated in front of the class. In many instances only a few words or numbers were altered from the tasks the Teacher just performed. Now that we have set up the scenario, let us now briefly examine the motivating forces of the Teacher and the Student.

What the Teacher wants is to produce a generation of critical thinkers, able to work through a difficult problem themselves with minimal “handholding.” The Activity assigned is meant to reinforce the Concepts presented on the whiteboard just moments ago. It is the Teacher’s hope that this Activity will spur the Student to be able to reproduce the actions required by the presented Concept.

What the Student wants is to finish the assignment with as little pain and anguish as possible, preferably being awarded a high score. It is in this context that the Student raises his or her hand, followed by a request of instructions on how to do the tasks required by the Activity.

The Teacher is often surprised, baffled, and frustrated by questions such as “How do I do this?” Sometimes the Teacher will even respond with a simple command: “Just think about it.” Because, after all, the Teacher just spent 30 minutes to an hour explaining in detail, step-by-step, how to do several similar looking problems.

The Student is often surprised when the Teacher won’t just tell them how to do this. Because, after all, the Teacher just spent 30 minutes to an hour explaining in detail, step-by-step, how to do several similar looking problems.

“Why,” the Teacher asks, “won’t the Student just apply critical thinking skills to the problem I just imparted? After all, it’s almost identical to the seven problems I just demonstrated in front of them?”

“Why,” the Student asks, “won’t the Teacher just tell us how to do this particular problem, right after the lesson? After all, it’s almost identical to the seven problems the Teacher just demonstrated in front of us?”

Both Teacher and Student have a gripe. The question is this: Which one has a legitimate grievance?

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!

Can you predict how many followers @charliesheen has right now? ; Exponential growth? (Part 2)

Be sure to check out Part 1 of this post, when the activity was assigned (i.e. if you were absent from E/M yesterday). There I provided my data on the number of Charlie Sheen’s twitter followers and the time. I hypothesized that it would be a more-or-less exponential growth curve.

I’ll break this up into a few different time pieces. Here’s the plot of Charlie Sheen’s followers as I have them, only for 3/1/2011, the day he joined twitter.


Not only is that a linear fit, it’s very linear (R-squared of 0.9995). I’m also shocked at how straight that line is. That is, there are not spikes of activity. I would have thought that in the evening, when everyone’s sitting in front of their TVs with their iPad 2’s there would be a marked increase in @charliesheen twitter followers.

Now let’s look at the same graph, but with all of Day 2 (3/2/2011) data included.

Unfortunately, I didn’t check Sheen’s twitter followers in the middle of the night, but it appears to have leveled off slightly going into Day 2 of the twitter account.

It looks like you could either construct this as a step-wise function or a quadratic-decay function.

Now, I’m not 100% sure when the twitter feed account went live – in fact, that was one of the questions I asked last time – but I do have this:

Zooming in a little on the right:

This is the Google realtime search feature. I would assume there’s a strong correlation between the Google realtime searches for “@charliesheen” and the activation of the twitter account. So it looks like the account may have gone active sometime in the mid-afternoon of 3/1/2011. Let’s call it 3:00 PM. In that case, we’d have the following curve.

The followers-curve really looks like it’s tapering off here.

Let’s add a couple data points. I took two more “measurements“. One yesterday, and one, just now.

Can you predict how many followers @charliesheen has right now? ; Exponential growth? (Part 1)

Sometime on March 1, 2011, Charlie Sheen joined twitter at the suggestion of Piers Morgan, who is apparently some type of person. By the time I was alerted of the existence of a @charliesheen twitter feed, it was 4:04 PM Mountain Standard Time. Sheen had yet to tweet, but already had somehow amassed over 100,000 followers. He hadn’t even put up an avatar of himself yet (and somehow the account was “verified”).

What happened over the next several hours was nothing short of amazing. All you had to do was wait a few seconds or minutes and hit ‘refresh’, and just watch the number of followers climb.

I did this for about 50 minutes and took screen shots along the way.

Is there anything wrong with this graph? ; Looking for a good economist

I’m trying extremely hard to stay out of the comentariat concerning education. My love is instruction, not politics. Students, not funding. Engagement, not class sizes.

So with that in mind, let’s take a look at this graph presented by Bill Gates.

Now, I’m not an economist, so can one of you out there that is redraw this graph to include inflation? Is the slope of the line steeper or less steep than the spending line? This is not a trick question, I have no idea. I don’t even really know how to properly quantify inflation even if I knew what it was over the past four decades.

Addendum. I guess I should clarify this first: have these numbers been adjusted for inflation already? Anyone? Bueller?