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## Friday, April 6, 2007

### By the NUMB3RS

Every once in a while, I find myself sitting in front of the TV when CBS's crime drama is on. Though I do not regularly tune in to the show, I have certainly enjoyed it when I did. According to the show's website,
NUMB3RS is a drama about an FBI agent who recruits his mathematical-genius brother to help the Bureau solve a wide range of challenging crimes in Los Angeles. The two brothers take on the most confounding criminal cases from a very distinctive perspective. Inspired by actual events, the series depicts how the confluence of police work and mathematics provides unexpected revelations and answers to the most perplexing criminal questions.

In any event, my respect for the show increased when I found out that Texas Instruments has been pairing with the shows producers to design a down-loadable lesson plan for math teachers, so that they can easily use the show during instruction. Although I would hope some enterprising teachers are already employing techniques like this, it is helpful of CBS to provide it for them voluntarily. Here is an example of a lesson based on the episode "Burn Rate."

NUMB3RS Activity: Energy

In “Burn Rate,” Charlie helps the FBI investigate a series of mail bombings with seemingly
unrelated targets. Charlie explains “explosions are all about physics and math: burn rates,
brisance and pressure waves.” He compares the bomb to a hand hitting a table set for a meal.
“By the way objects are displaced, I can tell you the size of the hand, how tight the grip, or how
much energy was imparted on impact; just like I can analyze what kind of bomb we're looking at
here.” In this activity, students will use radical equations to solve problems about energy.
The equation below gives a relationship between the energy E of a certain bomb, the mass m of
a particle in the bomb, and the distance d the particle travels. Energy is measured in joules
(kg m2/s2), mass is measured in kilograms and distance is measured in meters. The constant of
25,000 arises from physics formulae. The physics principles and a unit analysis are presented in
more detail in the extensions section.

E= 25,000md

1. A piece of shrapnel weighing 50 grams is found 10 meters from the site of the bomb.
a. Determine the value of m, remembering mass is measured in kilograms.
b. Calculate E, the energy of that portion of the bomb.

2. To get a sense of the size of a joule, match each item below with its energy level.
Energy level, in joules Item
a. 80 lightning bolt
b. 1,400 kinetic energy of a car at highway speed
c. 4,000 energy consumed by average automobile in US
in one year
d. 350,000 annual power use of one clothes dryer
e. 4(106) average person swinging baseball bat
f. 1.5(109) 1 gram of TNT
g. 3.2(109) 1 kg of TNT
h. 7.2(1010) eruption of Krakatoa
i. 1.5(1017) bullet traveling at 900 m/s

3. Using the energy level you determined in Question 1b, the equation for the situation
presented earlier is 4,000 = 25,000m d . Recall that a 50 gram piece of shrapnel traveled
10 meters.
a. Should an 80 gram piece of shrapnel travel a longer or a shorter distance than the
50 gram piece?
b. Calculate the expected distance from the bomb of the 80 gram piece.

4. a. Solve the equation E= 25,000md for d. Check your rewritten equation by verifying
that when m = 0.05 kg, d =10 m.
b. Calculate the distance a 25 gram piece will travel and the distance a 100 gram piece will
travel.
c. Use the results of Question 4b to determine a rule about the distance a piece will travel
when its mass is doubled.

The goal of this activity is to give your students a short and simple snapshot into a very extensive math
topic. TI and NCTM encourage you and your students to learn more about this topic using the extensions
provided below and through your own independent research.

Extensions

• The original equation for energy is given below. Using the estimated values given, verify the
equation E= 25000md used in the activity, where a = 10°, g = 9.8 m/s2, l = 0.1 m, and
t = 0.0000107 s.
22cos sin
2
ml dg
E = t a a

• The units of a joule are (kg)(m2)/s2. Verify that the equation below results in the correct units
for energy, measured in joules, where E is energy, C is a constant, m is mass in kg, l is
length in m, d is distance in m, g is acceleration of gravity in m/s2, and t is time in s. This
equation is an expansion of the one used in the activity.
2
E Cml dg
t
=

Additional Resources

• An online unit conversion tool can be found at http://www.onlineunitconversion.com.
• An introduction to solving radical equations is available at
http://www.purplemath.com/modules/solverad.htm.
If only would do this, math and physics teachers would be set for life.