Driving and deriving in math class

Here is another example of how a student can do math without the use of a textbook or a worksheet.  Many times I have, and will continue to, argue that math is not about repetition but actually deeper understanding.  Also, in math class instead of learning about or for problem solving, we should be teaching students to learn through problem solving.

Problem solving is NOT completing a word problem at the end of a class, or using an algorithm to solve a problem.  In the words of Andrew Wiles "The definition of a good mathematical problem is the mathematics it generates rather than the problem itself".  Below, I will illustrate the 6 steps of implementing a well thought out problem solving activity in a class:

Step 1:             Present an idea which students would want to solve.

Step 2:             Have students ask the question

Step 3:             Create a range of answers (students will feel safe to provide an answer)

Step 4:             Provide any other information that a student may need to solve the problem....HOWEVER....have the students ask for the information.

Step 5:             Allow for true autonomy to solve the problem.

Step 6:             When students finish, or THINK they are finished, extend the thinking by giving a question that is not more of the same work, but entirely different solving.

Recently, in my calculus class, I showed the following video and gave a laptop to each student, with the video uploaded on it, to analyze it.  Here is my lesson plan.

1)  Watch the video.

2)  I then asked the class, “Any questions about what we just watched?”  (REMEMBER: We should get the students to ask the questions; there will then be no extrinsic rewards but only true intrinsic motivation to solve the problem.)

One student asked “How fast is the other car going?”

For which I responded, “Good question”

I then asked for the lowest possible speed of the car, and we determined that to be 109 km/h, while the upper bound for the speed would be 180 km/h.

As students worked they started to ask me for the length of my vehicle. Answering honestly, I informed them that I did not know this.  They then followed my response with a question about the year and make of the vehicle I was driving, which was a 2011 GMC Terrain.

Students started to pull the actual length from the GMC website.

After several conversions, and 8 minutes of work, I took a picture of a student’s work and showed it on my projector.  We then discussed alternate ideas, and where did this student have to estimate and where was he/she exact in calculations.

3) I then asked, “Is there anything else we could solve?”  Again, another student asked, “What is the change of the distance between each driver as the car drives by?” 

**Since I could not take an aerial photo of the situation, I did provide the students with a distance of 12 feet horizontally between the drivers**

Again, I asked the class to solve the problem.  Within minutes, they realized they would need a specific time, and answered with “Solve for the instantaneous change of distance of the drivers after 3 seconds.”

After some solving, I took another picture and discussed the results with the class.

4) I asked the class if there still anything else we could solve.  One student asked about the change in the angle of the arm that swung the camera.  I responded with "Let's figure out the exact change in the angle at the time above".

This task took a little longer for some, while minutes for others.  Those who did finish, or thought they finished early, I asked to calculate again but use a different method.  By using a different method, they are applying a different strategy and also checking their answer at the same time. 

We then did discuss various methods in the class. 








Overall, I believe this lesson was a great success!!

Here are some pointers I have realized about true problem solving. 

Never:
Tell the students they are correct or incorrect, ask if there is a way they could prove their answer another way.

Tell the students HOW to complete the problem, but instead ask them guiding questions.

Tell the students which way to solve the problem, but let them chose a method.

Do:
Scaffold for struggling students.

Answer questions, even if the answer is irrelevant to the problem.

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.