Tuesday, May 3, 2011

Calculus and the justice system

Recently, I showed how mathematics can influence a court’s decision.
My lesson plan:
1) Introduce the court case.  I edited the real court case, and changed the names to below:
Richard Keaton was 17 when a police officer pulled him over on the morning of July 4, 2007, and wrote him a ticket for going 62 mph in a 45-mph zone.
Keaton was ordered to pay a $190 fine, but his parents appealed the decision, saying data from a GPS system they installed in his car to monitor his driving proved he was not speeding.
What ensued was the longest court battle over a speeding ticket in county history. The case also represented the first time anyone locally has tried to beat a ticket using GPS.
Nationally, such cases remain rare, despite the growing use of such technology in vehicles, primarily for mapping purposes.
In her five-page ruling, Commissioner Carla Bonilla noted the accuracy of the GPS system was not challenged by either side in the dispute, but rather they had different interpretations of the data.
All GPS systems in vehicles calculate speed and location, but the tracking device Keaton’s parents installed in his Toyota Celica downloaded the information to their computer. The system sent out a data signal every 30 seconds that reported the car's speed, location and direction. If Keaton ever hit 70 mph, his parents received an e-mail alert.
Keaton was on his way to Infineon Raceway when Officer Steve Johnson said he clocked Keaton’s car going 62 mph about 400 feet west of South McDowell Boulevard.
The teen's GPS, however, pegged the car at 45 mph in virtually the same location
At issue was the distance from the stoplight at Freitas Road -- site of the first GPS "ping" that showed Keaton stopped -- to the second ping 30 seconds later, when he was going 45 mph.
Bonilla said the distance between those two points was 1,980 feet, and the GPS data confirmed the prosecution's contention that Keaton had to have exceeded the speed limit.
"The mathematics confirm this," she wrote.
The defense also attacked the accuracy of radar, saying Johnson's readings could have been affected by everything from reflections off street signs to him erroneously locking on the wrong vehicle.
But Bonilla sided with the officer, stating he received a clear Doppler tone indicating no interference. Given Johnson's experience, including 15 years in the traffic division, and his observations on the morning in question, "the notion that he may have picked up a different vehicle is speculation," Bonilla wrote.
The case also drew interest because of the time and expense that went into what in essence was a fight over the $190 traffic ticket.
Police said it's a matter of routine to defend such challenges, but in this particular instance, concerns that the case could set a legal precedent that could jeopardize law enforcement's use of radar for speed enforcement factored into their decisions.
That included spending $15,000 on an expert in GPS technology -- including for one court appearance that had to be postponed when Andrew Martinez, the attorney retained by Keaton’s family, asked for a continuance.
"This case ensures that other law enforcement agencies throughout the state aren't going to have to fight a case like this where GPS is used to cast doubt on radar," said Sgt. Ken Savano, who oversees the traffic division

The whole story can be found here: Speeding ticket
2)  Divide the class into groups, and assigned each group as the “defense” or the “prosecution”. 
3)  Provide the “case files” to the group.  These are actual photos of intersections and roads described in the court case.

4) Provide each group with a laptop, or access to a computer to look up any extra needed information.
5) Give the students ample time to research and develop a case to support their side. (40 min).
6) Run a court case where each side must state their case, with defending arguments. 
With the use of the laptops, students researched the acceleration rate of the vehicle, and argued using integration techniques and the average value theorem.  It was one awesome problem solving day!
In the real case the defendant was found guilty, but in my courtroom I found the defendant innocent due to the arguments. 
I have to give thanks to John Scammell for giving me this court case and the inspiration to develop a true problem solving lesson plan.


  1. This is awesome! What a great way to show math in the real world. I will have to figure out the math and see if I think the person is guilty or not. Thanks again for the post!

  2. This is an absolutely excellent example demonstrating math in real life. Well done!

  3. Neil Barembaum said . . .

    I am somewhat confused. It seems that the GPS data confirms by MVT that he had to have exceeded the speed limit. You quote Bonilla that the GPS data confirms the prosecution's contention. This is where the math would be. I haven't worked it out, but then if the students working out the problem should get what Bonilla says they should have gotten, why did your kids get that he was not guilty?

    I understand there are arguments that he was not guilty. The officer cited the perp as speeding at that one place (Instantaneous rate of change). This was disputed by the GPS. Using the GPS data that we was speeding at SOME POINT (MVT) is not the same thing. Indeed, an officer cannot cite you for what he can prove happened. He can only cite you for what he observed happened.

    So did the students work out the math and it proved that he was speeding, but used some other criteria to find him not guilty, or did they work out the math and find that Bonilla was wrong to begin with?

  4. Great work with this. You managed to do what I was unable to - Find Google shots of the intersection. There must be a typo in the article, which explains why I couldn't find Freitas Road in Petaluma.

    I'd be interested in hearing more about how the students did with the arguments. How did they present their cases?

  5. I should clear up when I said "in my class he was innocent"..

    The defendant side argued the math more, and talked about how he could be innocent.

  6. In #3, you say provide "case files" to the group. Was there anything else provided other than the 3 pictures you posted here? I am planning to recreate your project in my class starting at the end of this week, and I wanted to make sure there wasn't anything missing.

    Thanks for your post - This is such a cool activity for the students!

  7. Awesome post it is! Thanks for this meaningful post!
    Fight Speeding Tickets

  8. Good to see this: "Our smart solutions improve safety and prevent accidents at entryways, in facilities and on roadways of all kinds."radar feedback signs, radar trailers, radar speed display signs.