3 comments on “Projects: what they’ll remember in 20 years”

Projects: what they’ll remember in 20 years

This is a post in the ongoing Emergent Math mini-series: Routines, Lessons, Problems, and Projects.

Projects - Card

I graduated high school twenty years ago this year. What’s remarkable is how little I actually remember about my classes. I remember certain feelings I had towards particular teachers or classes, but not the actual classroom action itself. There are three exceptions. There are four distinct activities I remember from my classes and they’re all projects:

  1. In my combined Physics/Calculus course we divined the accelleration due to gravity based on an experiment me and Eric Durbin concocted. And we were pretty close! We’ll call this the Learning Project.
  2. In AP Stats we conducted soil testing and surveyed the neighborhood to determine whether they cared about this issue. We’ll call this Project-Based Learning.
  3. In English we had to recreate various scenes from The Lost Horizon. We’ll call this the Dessert Project, for reasons which will become clear.
  4. In Biology we had to collect a bunch of leaves. I don’t remember why, but we had to do it. We had to get certain kinds press them is a special way. I hated it. We’re not calling this project anything other than The Leaf Project

That’s it. That’s all I remember about my classwork in High School. Don’t get me wrong: I remember other stuff too, like that time the time my friend Ash was talking so the teacher made him get up and teach the class,I recited “Shaft” in English class (“Who’s the black private dic that’s a sex machine to all the chicks?”), and my creepy Algebra 2 teacher making about ten too many jokes about “french curves.”

But by and large I remember the projects. Don’t get me wrong, there’s other stuff in there. I graduated college and everything, partially with some knowledge I acquired in school. But I only remember these actions.

Projects are an opportunity to illustrate how crucial your discipline is to the world or our understanding of it. They’re also an opportunity to waste several days or weeks of class time and force students to jump through imaginary hoops concocted by the teacher. In both cases, students will remember.

Projects apply mathematical knowhow to an in-depth, authentic experience. A project occurs over the course of two to four weeks. Ideally, projects are outward facing, community based, and/or personally relevant to students.

Let’s take a look at three types of projects. As with this entire mini-series, I’m painting with a broad brush and I’d happily concede that what I call one thing, might actually be another in another’s eyes.

The Dessert Project

I’ll withhold why it’s called the “dessert” project for now. These are typically given at the end of a unit intended to sum up the content. These often occur as a retelling of the content, such as my Lost Horizon example above. We read the book, we identify crucial scenes, and then we reenact them. We’re barely doing any analysis, let alone synthesis.

The best Dessert Projects take what a student has learned and unleashes it on an appropriate real-world scenario. Now that we’ve learned the content, we’re going to see how it looks in a different context. Most end-of-chapters offer this kind of project.

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A Dessert Project

The Learning Project

In a Learning Project, we learn something germane to the topic at hand through the use of an in-depth investigation. The structure of a Learning Project is more-or-less dictated by the teacher, but there is enough agency awarded to the students to experiment on their own. The WHY and HOW are often provided and the WHAT is relatively self-contained.

In my gravity example above, we were given the task (calculate the accelleration due to gravity), the materials (a video camera that allowed you to fast forward one frame at a time – this was the 90’s mind you), and the format of the product (a lab report). We had

It was a deeply memorable and engaging task. Unlike Dessert Projects, we are asked to actually find out something new, rather than repackaging information. Despite the fact that Learning Projects may not have a community partner, a public presentation, or a shiny final product (ingredients of Project-Based Learning which we’ll get to in a moment), we construct or deepen our understanding of some new knowledge or knowhow.

I’d suggest these as other examples of Learning Projects:

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A Learning Project

Project-Based Learning (PBL)

PBL has gotten the most headlines lately. Schools across the country want to provide deep, authentic, and motivating experiences for kids in all subjects. And to be sure, the best of PBL absolutely achieves that. Students are given a open ended, authentic challenge and students develop and present a solution. Through this process, students acquire new mathematical knowledge and skills.

In PBL (like Problem Based Learning), the task appears first and necessitates the content. Students learn the content in order to achieve their final product. Often – if not always – PBL occurs in groups.

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Project Based Learning

But don’t be fooled: quality PBL entails a lot more than just giving the students and letting go of the process entirely. The teacher/facilitator crafts the daily lessons and activities to support the process. The following graphic is taken from the New Tech Network, my employer. It explains well the various phases of a project and a menu of options for lessons, activities and assessments throughout a project.

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The project launch occurs at the beginning of the unit. It kicks off and drives the unit. In this case the project is the “meal” as opposed to the “dessert” (recall from early Dessert Projects). The project is how students will learn the material.

As an example, here are a few artifacts from one of my PBL Units about the 2000 Election. I’ve blogged about it before (twice, in fact!).

Project Launch: Have students read the Entry Document (the letter) and collect “knows” and “need-to-knows”.

To the students of Akins New Tech High School,

The US presidential election of November 7, 2000, was one of the closest in history. As returns were counted on election night it became clear that the outcome in the state of Florida would determine the next president. When the roughly 6 million Florida votes had been counted, Bush was shown to be leading by only 1,738, and the narrow margin triggered an automatic recount. The recount, completed in the evening of November 9, showed Bush’s lead to be less than 400.

Meanwhile, angry Democratic voters in Palm Beach County complained that a confusing “butterfly” ballot in their county caused them to accidentally vote for the Reform Party candidate Pat Buchanan instead of Gore. See the ballot above.

We have provided you the county-by-county results for Bush and Buchanan. We would like you to assess the validity of these angry voters’ – and therefore Al Gore’s – claims. Based on these data, is the “butterfly” ballot responsible in some part to the outcome of the 2000 election? What other questions do the data drum up for you? And what can we do to ensure this doesn’t happen again?

We look forward to reading your analysis and insight, no later than May 5.

Sincerely,

Your county clerk

Example project pathway – the 2000 Election

Screen Shot 2018-07-18 at 6.29.39 PM.pngScreen Shot 2018-07-18 at 6.29.56 PM.png

In this example, we still retain many of the lessons and workshops that we would typically teach during this unit, but notably they occur as students are working towards various benchmarks and the final product. The lessons inform the student products.

There is enough to write about PBL to merit its own miniseries. Structures and routines are crucially important in a PBL Unit. Assessment must look different. Managing groups becomes an entirely different challenge. How the authenticity of the product and the external audience enhances the quality of student work.

For now, here are some other tasks that adhere to PBL.

How often?

The genesis of this entire mini-series stems from questions I receive about Problems and Projects. Mainly, “how often should I use problems? How often should I use projects?” The unstated part of that question is, “when do I actually, y’know, teach?” (Actually, sometimes that’s stated). I’ll save another post for “putting it all together” or “adjusting the levels” but know for now that’s why I put together this framework of Routines, Lessons, Problems and Projects.

As for the “how many projects?” question, I’ll give a squishy answer and a non-squishy answer.

My squishy answer: design a project whenever (A) the standards uniquely align such that you can create multiple lessons around one scenario and (B) when you can identify a project scenario that will maintain momentum over the course of several weeks.

My non-squishy answer: One or two a year. Most standard clusters don’t lend themselves to multiple investigations around one, single context. But some do! Content clusters around things like Data and Statistics, Area and Perimeter, and Exponential Growth and Decay are ripe for real-world scenarios that can be analyzed through the lens of multiple content standards.

As challenging as it is to design and facilitate projects, and as little time we have as educators to carve out the time for it, we don’t want to deprive students of the real-world insight math can have. We want to provide these experiences that will live on in students’ minds as the power of mathematics, whether or not they go into the field. So be on the lookout. Look for news articles and community opportunities that might embolden students to use math for maximum impact.

Also in this mini-series:

(Editor’s note: The original post had “dessert” written as “desert,” which is a different kind of project altogether, I imagine.)

1 comment on “Counting Idling Cars”

Counting Idling Cars

pickupI’m sitting in my car, waiting to pick up my son from school. It’s too cold to wait outside  this time of year so I keep the heat on, the engine running, and continue listening to the Dunc’d On Basketball Podcast, the nerdiest podcast about basketball out there. I’m also quite anti-social, so I prefer to sit in the car, rather than, like, talk to people.

The driver of the car in front of me is doing the same (presumably, minus the podcast listening), ditto for the car behind of me. Maybe they’re reading “The Pickup Line,” an e-mag specifically for parents who sit in the car, waiting to pick up their kid from school. It occurs to me: boy there are a lot of cars idling in front of the school right now. I’d guess about 40. But y’know, someone should really count these up.

I get typically get to the school about 10 minutes before the release bell rings and I’m sort of in the middle of the pack of idling cars. I’d guess it’s about the average of when most cars arrive, again, most of which are idling. While I don’t conduct this environment-destroying practice all year long – when the weather is nice I’ll get out and check my phone, rather than talk to other parents – I practice it for maybe half the school year. That’s about 80 days or so.

80 days x 10 minutes of idling per day. Boy, 800 minutes of idling seems like a lot doesn’t it? And if there are indeed 40 cars at my son’s school, averaging a similar amount of idling time, we’re looking at 32,000 minutes of car idling. That’s over eight days of just idling.

We have a train that goes through town and we have signs encouraging us to turn off our car, rather than sit there idling, while we wait for the train to pass through. And I sometimes follow that instruction! I should probably follow it more often and more aggressively. But what about idling in the school pickup line? Or along the side of the school for us anti-socialites?

How much gas are we wasting?

How much Carbon Monoxide are we putting in the air?

How much gas waste / CO is the entire town/state/country contributing?

Would it be better to just switch off the car and start it later?

Boy, oh boy, someone oughta do the research on this…

What about at your school? How much gas is wasted in a day, week, or school year? Could students do the research? Could they create an awareness campaign for reducing gas waste (and presumably promoting cleaner air at their school)? Seems like something a bunch of go-getter students could handle.

 

8 comments on “How does one provide the complex data of global warming to students?”

How does one provide the complex data of global warming to students?

Update (3/12/2013): An atmospheric scientist friend of mine, Katie, suggested a few edits to this post, primarily to clear up a few of the tools listed here. The edits are in bold.

My initial thesis on this post was originally going to be “why don’t teachers let students investigate global warming very often?” While this may not answer it here’s a terrifying google search for any teacher who is interested in having their students do some independent research on climate change. Google: “global warming raw data“.

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So the first result is a good one. A legit one. There are lots of links to reputable sites maintained by reputable scientists. Then the second result is a yahoo! answers post. The the third (third!) google result for a simple query on raw data turns up World Net Daily, a website for conspiracy theorists and people that think they’re going to be put in FEMA camps any day now. That is not a reputable site. They provide the opposite of “raw data”.

This is not a post about the messy politics and confusion-campaigns around climate change. But this does point to a particular difficulty that you’d hope would be much simpler: where can we find raw temperature data that we can actually use? For the record, a google search of “raw temperature data” yields much more acceptable initial results. But still, many of those results can be extremely difficult for a secondary math or science teacher to pick up and use, let alone students. For one, climate data is often presented in a file format that requires heavy coding knowledge or special programs to process (such as NetCDF). Second, it’s hard to know where to start with temperature data. Do you start by geographic location? Do you take the annual mean across the globe? How would one do that, exactly?

So this is the problem, and maybe a fundamental problem of teaching science: data are messy. We have to rely on others to package it for us. Scientists are interested in providing the raw data because they want people to have access to true observations, but that raw data is so vast and difficult to process (but not that difficult to interpret!) you have to get at least a Master’s degree before you can even start to decipher it. And often, scientists aren’t interested in culling the data to make it more digestible for the public. They’d prefer to show you the graph. This is great for communication, but not great for independent research. And worse, they’re now fighting on the same plane as disingenuous charlatans who are paid to be as such. So let’s provide students of science the raw data in a way that anyone with Microsoft Excel and a genuine curiosity can begin to explore the very real phenomenon of climate change.

My favorite site that does that is this NASA’s GISS Surface Temperature Analysis. In terms of accurate, raw, commentary-free, accessible, customizable, and processable data, I haven’t found a better place to start. Bookmark that site. Tell your students to go to that site. Start locally.

To find specific historic local weather stations, Katie recommends using the map rather than the search function. The map appears to have better functionality. So click on your favorite vacation spot and go find that precious, precious raw data.

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Once you have the ASCII data (shown here), it’s simply a matter of copying and pasting it into Excel, or if you’re incredibly ambitious (or teaching a Stats class perhaps), having students import it into R, one of the industry standards.

For the uninitiated, let me translate a few things: 

D-J-F= December-January-February average

M-A-M=March-April-May average

J-J-A, S-O-N = I think you get the idea….

The last column, metANN = annual mean temperature. This actually might be the best first place to start. 

Berkley also has a nice data set organized by country. However, the accessible to-layperson data is a bit more hidden.

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If you’re not careful, you’ll end up downloading intense, non-accessible-to-the-layperson, NetCDF data. Which, again, is fantastic data, but difficult to work with yourself.

But now we’ve got two sites with data that can be tossed into Excel, R, or even those statistics packages designed for secondary students. Now that we have that data, we can do a lot with it.

Suggested Activities

  • Have students investigate the temperature trend in their area.
  • Create a linear model that predicts temperature as a function of year locally.
  • Assign each group or student a different region of the world to investigate and develop a linear model for.
  • Or what about this: develop a sinusoidal equation that describes monthly temperature. Get some trig in there.
  • Ask the question: is our town/state/country/planet heating up or not? Or is it too uncertain to tell?
  • Can you find local stations that DON’T show a warming trend? Katie suggests looking at weather stations closer to the poles to consider the potential impact of polar temperature trends. This might be a bit science-y, but it’s something I’d happily let students explore in a math class.

Once you have actual data, you can start to test it to assess that last, fundamental question (which then spurs thousands of other questions, like “should I have children?”). Is ß>0 under the general linear model? Once we have that answer, even if it’s just locally, we can start to talk about the implications.

1 comment on “Attention Math Teachers: Slate has graciously discovered your next project”

Attention Math Teachers: Slate has graciously discovered your next project

Slate.com’s always entertaining “The Explainer” segment runs an always-even-more entertaining year-end segment on the Unanswered Questions of the Year in which readers are prompted to vote on the question to be answered (aside: say, that’s a pretty awesome activity for a classroom. Students in the middle of a Problem vote on the question the teacher answers.)

This years’ edition contains burning questions such as “When you urinate in a toilet and there’s splashback, is that urine or toilet water?” and “Why do dogs like having their bellies rubbed?”

The one that I’m interested in is the following.

Artifact

13. When parking in a nearly full parking lot, is it quicker to a) park in the first open space you see and walk, or b) drive a few laps around the lot and grab the closest possible spot? In my experience the two ways are about even, since the extra time spent driving for “b)” means a quicker exit when you leave. Please settle this using statistics as my wife has refused to argue anymore regarding this issue.

Tell me this isn’t a dilemma you play out in your head on a near daily basis.

I suppose you could use statistics, but this would make a super modeling project as well. Moreover, I have no idea what “level” of math a problem such as this requires and that’s a good thing. This could be a middle school project or an Algebra 2 project. This could be attempted by your high-flying AP Calculus-bound students as well as your remediated Algebra 1B students who probably hate the subject you’re teaching them.

So how do we pose this problem to students? A video of you driving around looking for a parking spot with your friend/spouse imploring you to “just park and we’ll walk!” might work. A few simple diagrams might work:

Does your school have parking issues? Particularly in the afternoon? Grab some footage. Or shoot, even the simple question posed in The Explainer may be interesting enough as is.

Guiding Questions

  • Does the size of the parking lot have anything to do with the decision to Park and Walk or to Keep Searching?
  • Do the number of parking spaces have anything to do with our decision?
  • How fast do people walk through parking lots?
  • How often are cars vacating their parking spots?

This is a good example of a problem that A) could potentially be immediately interesting to students (or at least could be posed in an interesting way), B) doesn’t necessarily have a correct solution, C) offers multiple routes through the problem, and D) can be accessed regardless of prior mathematics expertise.

What are some potential “next steps” students could take to engage in the problem? What are some potential mathematical routes to a solution? This is something to think about over the break, and as you’re vigorously searching for a parking space at the mall to buy that last-minute gift you’ve been putting off.

6 comments on “So, what exactly am I supposed to eat? The new MyPlate icon vs. the classic Food Pyramid vs. Geometry.”

So, what exactly am I supposed to eat? The new MyPlate icon vs. the classic Food Pyramid vs. Geometry.

A month ago, I was considering writing a post on the old (now “old, OLD”) food pyramid – you know, the one we all grew up with – and the new (now “old”) food pyramid, unleashed in 2005. It would be about area of triangles and trapezoids and Geometry and possibly graphic design.

See, here’s my tweet about it:

I was all set to contest that the new (“old”) food pyramid, adopted in 2005, was garbage mathematically and visually. And the challenge was for students to come up with a better, more mathematically accurate, food pyramid.

Then the United States government dropped the new MyPlate diagram in my lap.

Visually, and graphic design-ally, I think it’s miles better than the new (“old”) food pyramid, where you had no idea what each of the 137 slivers meant and exactly how much area was each of the 137 slivers. Also, it was unclear why it was a triangle at all? Why is that person climbing that pyramid of food? Are you supposed to eventually eat less and less until you eventually eat nothing?? Not sure if it’s mathematically better. Area of a sector of a circle whose vertex doesn’t meet in the exact center? Now THAT’s a mathematical investigation!

So I think, without having done any calculating, the new MyPlate is better. But you know what? That’s probably something for the students to decide, right?

Artifacts

Food Pyramid #1 (Classic)

Food Pyramid #2 (Post-modern?)

MyPlate (2011)

Guiding questions

  • Which of these government sponsored food diagram is the “best” and why?
  • So, how much of each food group are we supposed to eat again?

Suggested Activities

  • I would start by hosting a class discussion on what would make a diagram the “best.” Have students develop a rubric before you even begin. Potential categories: mathematical accuracy, ease of understanding, etc.
  • Have students find the area of each piece of each diagram and report back what the heck it means. You might want to assign different groups a different diagram, or not.
  • Have students attempt to craft their meals according to the three food diagrams for a week. You might want to assign different groups a different diagram to emulate, or not.
  • Compare the diagrams to each other: are they trending in a healthier or unhealthier direction?
  • The areas are representative of a 2,000 calorie diet. What if you had a 2,500 calorie diet? How would we dilate the pieces of the diagram?
  • Have students use their local and regional produce to create a meal based on the MyPlate (and contrast it with the Food Pyramid?).*

There are lots of different ways I think you could go with this as far as guiding questions and activities, but invariably it will come back to the area of these shapes. And tasks that have several entry points and investigation/exploration opportunities for students that require a mathematical understanding are always worth the time. 

How else might we use the Food Pyramids and MyPlate diagrams to better math understanding?

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* Update: Commenter Sneha suggested this activity. I love the idea. Connection to Geography, anyone?

2 comments on “The Dallas Mavericks are 2-16 in playoff games officiated by Danny Crawford. Is this statistically significant?”

The Dallas Mavericks are 2-16 in playoff games officiated by Danny Crawford. Is this statistically significant?

Artifact.

This shocked me.

The Mavs have a 2-16 record in playoff games officiated by Crawford, including 16 losses in the last 17 games. Dallas is 48-41 in the rest of their playoff games during the ownership tenure of Mark Cuban, who has been fined millions of dollars in the last 11 years for publicly complaining about officiating.

First of all, is that right? That A) the Dallas Mavericks perform so poorly in Crawford-officiated games, and B) Crawford is still being allowed to referee them? Really? Wow.

And there’s this, which might even be more damming: The Mavs are 4-14 against the Vegas spread. ESPN provides a nice chart of the individual games.

I specifically remember those 2006 Finals games against Miami. By many accounts, those were two of the worst officiated games in NBA history, in which Heat guard Dwyane Wade got what seemed to be every favorable foul call. It pretty much ushered in the era of NBA ref scrutiny.

This has to be tested for statistical significance.

Guiding Questions

  • Really?
  • Is this just coincidence or is there something else going on here?
  • Does Crawford have a vendetta against the Mavericks for some reason?
  • Does Crawford have a suspicious record with any other team?
  • Is there a potential way, other than referee malfeasance, that we could explain away this alleged disparity?
  • Maybe the Mavericks are just playoff chokers?

Suggested activities

  • Obviously if this were a statistics course you could look at statistical significance, which we’ll do in a minute.
  • If students are really up for it, they could delve into the games themselves and look for disparities in “referee stuff” like fouls, technicals, travelling, etc. We’re not going to do that in a minute.
  • Homework: students watch tonight’s Mavs-Trailblazers game closely and look for anything fishy from Crawford (although, this might serve as its own lesson in confirmation bias).

Potential Solution

Let’s start with our null hypothesis:

H(o): Danny Crawford is NOT biased against the Mavericks. The Mavericks’ playoffs woes in games he’s officiated is due to random chance.

I suppose first we have to figure out what the Mavericks’ Crawford-officiated games “should be.” The Mavericks are 48-41 in playoff games not officiated by Crawford, good for a winning percentage of 54%. Although, if you’re like me, you believe more in random chance for sporting events, and the “true percentage” is probably pretty much 50% over the course of a decade. But that could be a fun debate point in your class.

We also need to decide on a significance/confidence level α, usually 0.05 or 0.01.

So what is the probability of a team that “should” win 50% of its games (debatable) ending up winning just 2 of 18 games at random? Or rather, that this team should lose 16 (or more) of 18 games by random chance?

A P-Test would could look like this,

Probability of 16 losses + Probability of 17 losses + Probability of 18 losses =

Remember,

.

So no matter what confidence level we choose, this is, again, pretty damning. If we assign a 50% of the Mavericks winning (less than for their other playoff winning percentage) there is only a six hundredths of a percent chance of this being total flukiness.

Before we go nuts, though, let’s look back at that chart. Now, if you’re not familiar with Vegas lines, the negative sign in front of the “DAL Line” column indicates the Mavericks were favored that game. You’ll note that Dallas was only favored/expected to win 8 of those 18 games, and Vegas is usually pretty dead-on about these sorts of things. If we use that as a “true winning percentage” the Mavericks would only be expected to win a mere 44% (a losing percentage of 56%) of their games, not 50%. Let’s recalculate.

Slightly less suspicious, but still grievously suspicious. It’s well below our 5% or even 1% confidence level.

Still, before I would get Ralph Nader involved, I would ask students, investigators, etc. to look for specific evidence pattern within the games themselves (as 82games.com has done for one specific game). The original ESPN article that led to this investigation suggested Dallas had more fouls and less free throws in Crawford-officiated games than others. A next step would be to look at the beneficiaries of the suspect officiating, i.e. Dallas’ opponents for these games. Did they get an inordinate amount of free throws? Did they tend to overperform, just as Dallas underperformed in these games. Now that we have the statistical basis to be suspicious, we can start the investigation in full.

Couldn’t this work as a real nice Project Based Learning Unit for statistics? The Entry Event could be the ESPN article, or Game 5 of the 2006 NBA Finals. The summative event could be that students could present their findings, host a panel or debate, or write a letter to their congressperson. Or Ralph Nader.

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

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:

y=40x

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!