Classroom Robotics – Mathematics Day 7- Pg. 1
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Classroom
Robotics –
Mathematics
Day 7
Overview
Students will derive the formula for energy and understand
the difference between wheels and treads mathematically.
Learning Objectives:
Student’s will be able to…


Derive the formula for energy
Explain the difference between wheels and treads
mathematically
Suitable Ages: 11+
Time Needed: 1 hr 15 min
In this packet
Teacher’s Guide
Work to Kinetic Energy Explanation Sheet
Traction and Area Worksheet
Traction and Area Worksheet Answers
Day 7 Vocabulary Definitions (for your reference)
Background Notes (for your reference)
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TEACHER’S GUIDE
Activity Goal
Students will derive the formula for energy and understand the difference between wheels and treads
mathematically.
Objectives
Student’s will be able to…


Derive the formula for energy
Explain the difference between wheels and treads mathematically
Materials


Copies of Traction and Area Worksheet
LEGO kits
Set Up
Familiarize yourself with the lesson plan, and copy any handouts. Have students bring their programming
task worksheet from Language Arts.
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Procedure
Time
20 min
15 min
5 min
35 min
<75
minutes
Action
1. Derive the formula for Energy, power, velocity and kinetic
energy with the students using the Work to Kinetic Energy
Explanation Sheet.
2. Have the students work through the Traction and Area
Worksheets in groups.
3. Go through the answers to the worksheets, clearing up any
misconceptions.
4. Allow students to work on their programming tasks from
Language Arts.
Materials Needed
Work to Kinetic Energy
Explanation Sheet
Traction and Area Worksheet
Traction and Area Worksheet
Answers
LEGO kits
We encourage and welcome your feedback!
Please send any feedback about your experience and observations and forward it to
<NETID>@utdallas.edu at the Science and Engineering Education Center.
Classroom Robotics – Mathematics Day 7- Pg. 4
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Work to Kinetic Energy Explanation Sheet
So far we have learned several common principles of algebra. We learned in the first week that if you do something to
one side of an algebraic equation you need to do the same to the other side. We changed the Work formula to the
Power formula by doing this.
F x D= W when divided by time (t) on both sides becomes the Power formula
F
d W

 Power
t
t
To convert Work into Kinetic energy we are going to substitute into the equation equal terms. As long as they are
equal, there should not be any change in the outcome of the formula.

Some of the formulas we have used so far are:
Force x Distance= Work or F x D= W
Force = mass x acceleration, or F=ma
Velocity or speed= Distance/time, or v=
d
t
Power= Force x velocity, or P= F x v, or P= F x

d
t
Acceleration is the change of velocity over time, or velocity/time, or
d
d 1 d
t  acceleration , or x = 2  acceleration
t
t t t
 
When scientists were exploring what energy was, they were getting scientific data through experimentation. Their data
graphs were revealing a curve. When have you seen a curve during this project? (during the discussion of Newton’s
Laws and acceleration)
  
While the scientists were gathering data, the mathematicians were determining energy through the use of
mathematical reasoning.
Scientists realized that Archimedes was actually describing energy when he described work, the ability to do work.
From work, mathematicians reasoned to the definition of Kinetic Energy, a formula like Newton’s Second Law of
Physics. Kinetic Energy is
1 2
mv .
2
Here is how they came up with this:
F x D= W

If F x D= W, then you can replace F with “ma” according to Newton’s Second Law, or mass times acceleration to get
a new formula,
ma x D= W
Since we now know that W or Work is really kinetic energy lets replace W with E to get yet another formula,
ma x D= E
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Now lets change acceleration to its equivalent expression, velocity divided by time or
v
t
v d
d 1 d
  t    2  acceleration . Now for the final substitution, mass times
t t
t t t

acceleration times distance equals kinetic energy,
Therefore, acceleration is
F  d  ma  
dm
d
d2

d

m
 mv 2
t2
t2
The “d”s combine to form d squared, d2. Since

d
d2
is velocity and 2 is velocity squared, the formula is mv2. The
t
t
actual formula is:
Kinetic Energy = ½ mv2


The ½ comes from the average velocity over the movement of the object.
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Traction and Area
Name___________________________________________________ Class _______________________________
Traction is important to robots. Traction is the amount of grip a robot has to the ground. It is a function of the
stickiness of the tire or the tread that is in contact with the ground and the amount of weight pushing down on the
ground. When you multiply the two factors they give you a number for traction.
Which has more traction treads or wheels? When you prepare for the team competition this may become a very
important question.
1.
What makes the wheels sticky to the ground?
2.
What happens to the stickiness when you take the tires off of the rims?
3.
Driving on the rims of your robot is good if you want them to allow you to turn. Why?
4.
What are the tires made of?
5.
Which is more sticky?
What are the treads made of?
Below are the footprints of a robot with wheels and a robot with treads. Footprint is the amount of rubber on the road.
Footprint of Wheels
Footprint of Treads
6.
Which has more area on the ground wheels or treads?
7.
What is the definition of area? What are the units?
8.
Assume that each wheel is in contact with the ground 15 cm. length and 5 centimeters width. What area do all the wheels
cover? (show your work, and do not forget units)
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9.
Classroom Robotics – Mathematics Day 7- Pg. 7
Assume each tread above covers 300 centimeters in length and 25 centimeters in width. What is the area that both treads
cover? (Show your work, and do not forget units.)
10. Assume that both the wheels and the treads are holding a 3000 kg chassis on top of them what is the amount of weight
each square centimeter is holding? (Show your work, do not forget units.)
11. Which has more traction a tank on treads or a truck on wheels? Explain.
12. How do tanks turn?
13. Why does it work for them?
14. Why do cars get stuck in sand but tanks do not?
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Classroom Robotics – Mathematics Day 7- Pg. 8
Traction and Area Answer Key
Name___________________________________________________ Class _______________________________
Traction is important to robots. Traction is the amount of grip a robot has to the ground. It is a function of the
stickiness of the tire or the tread that is in contact with the ground and the amount of weight pushing down on the
ground. When you multiply the two factors they give you a number for traction.
Which has more traction treads or wheels? When you prepare for the team competition this may become a very
important question.
15. What makes the wheels sticky to the ground?
The material that they are made of. Hard plastic is more slippery than rubber.
16. What happens to the stickiness when you take the tires off of the rims?
The rims are slick compared to the sticky rubber tires
17. Driving on the rims of your robot is good if you want them to allow you to turn. Why?
They will slide around to allow the robot to turn
18. What are the tires made of?
Rubber What are the treads made of? Rubber
19. Which is more sticky?
LEGO treads and tires are equally sticky. They are both made of the same thing
20. Below are the footprints of a robot with wheels and a robot with treads. Footprint is the amount of rubber on the road.
Footprint of Wheels
Footprint of Treads
21. Which has more area on the ground wheels or treads?
Treads
22. What is the definition of area? What are the units?
Length x width= area, area is in square centimeters
23. Assume that each wheel is in contact with the ground 15 cm. length and 5 centimeters width. What area do all the wheels
cover? (show your work, and do not forget units) Each wheel is L x W or 15 cm. x 5 cm.= 75 cm. 2 There are 4 wheels
however so the total amount connected to the ground is
75 cm2 x 4 =300 cm2
24. Assume each tread above covers 300 centimeters in length and 25 centimeters in width. What is the area that both treads
cover? (Show your work, and do not forget units.)
Each tread is 300 cm. x 25 cm. = 7500 cm 2 Because there are two treads the
total rubber on
the road is 7500 cm 2 x 2 = 15000 cm2
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Classroom Robotics – Mathematics Day 7- Pg. 9
25. Assume that both the wheels and the treads are holding a 3000 kg chassis on top of them what is the amount of weight
each square centimeter is holding? (Show your work, do not forget units.)
Car has
3000kg
kg
3000kg
1 kg
2  10
2 The treads are
2 
300cm
cm
15000cm
5 cm 2
26. Which has more traction a tank on
treads or a truck on wheels? Explain.
Truck on wheels has a lot more traction than a tank. A tank floats over the road whereas a truck sticks to the road.
27. How do tanks turn?
One treads turns one way and the other turns the other way. It can make a very tight turn this way.
28. Why does it work for them?
Because the tank is floating on top of the treads and the treads can slide easily over the ground.
29. Why do cars get stuck in sand but tanks do not?
Cars will get stuck in sand and other loose material because the wheels dig into the sand. Tanks do not because they
float. Tanks are very good in desserts and over loose ground.
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Classroom Robotics – Mathematics Day 7- Pg. 10
Day 7 Vocabulary Definitions
ROY G BIV
Infra Red (IR)
Ultra Violet (UV)
X-ray
Microwave
Radio Wave
Electromagnetic Spectrum
Speed of light
Traction
Coefficient of Friction
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Background Notes
A common practice in algebra is to replace equivalencies in equations. For example, during the first week of the
project the students were introduced to force. They used force in the Work Formula as F x D= W or Energy. They also
learned Newton’s definition of Force as mass x acceleration or F= ma. They also learned that acceleration is the
change of velocity over time. Or velocity divided by time. Scientists in the 1800’s, including Mme. Du Chateulet, used
this understanding to develop a new formula for Energy that combined these equivalences.
In the Language Arts Class the students will read the story of Mme. Du Chateulet. She was one of many scientists
who understood the importance of energy, but were only beginning to discover its properties. It took scientists over
100 years after Newton to move from the understanding of Forces to an understanding of Energy. Your students will
have a couple days
During experiments researchers were finding that when a mass struck an object like a piece of soft clay it was
releasing energy equivalent to its mass times its velocity squared. In other words, an object would increase its energy
a lot faster than they would predict intuitively. Experimentally, they were finding that the data points of energy were
creating a parabola, or a curved line, just like acceleration that we saw last week. By looking at the math formulas that
they already had scientists and mathematicians should have been able to predict this by simply substituting equivalent
expressions. This discovery highlighted the importance of mathematics to science. The rules of mathematics also
appear to follow the rules of nature.
Therefore, if F x D= W, then you can replace F with “ma” or mass times acceleration to get a new formula, ma x D=
W. Since we now know that W or Work is really kinetic energy lets replace W with E to get yet another formula, ma x
D= E. Now lets change acceleration to its equivalent expression, velocity divided by time or
v
. Velocity is simply
t
d
d
d
speed or miles/hr, or distance/time, . Therefore, acceleration is t  2 . Now for the final substitution, mass times
t
t t

d
acceleration times distance equals kinetic energy, m  2  d  E The “d”s now combine to form d squared, d2
t
and the formula for kinetic energy becomes:
m

d
d2
2

E
or since
is velocity m  v  E
2
t
t


Mathematics has become a constant inspiration for scientists and has worked hand in hand with science to develop
new insights about nature. Newton was first considered a talented mathematician before he was considered a

scientist. 
Another misconception about robots can easily be debunked with a little math. People often believe that treads on a
robot or tank give the tank or robot more traction. In reality, the opposite is true. The area of a tire and the area of a
tread touching the ground support all the weight of a robot or tank. The smaller the area touching the ground the more
weight each square centimeter must support. Wheels touch the ground with very little tread. The small tread area
must support all the robot or tanks weight. The wheels are pushing very hard against the ground. On the other hand,
the tread of a tank is very wide and is connected to the ground over a wide area. Therefore, each square centimeter
of tread is pushing down with very little pressure. The pressure is spread out over the wide area of the tread. Because
tanks are not “stuck” to the ground but rather float on the ground they can steer by moving one tread and allowing the
other tread to float. Tanks can turn much more sharply because of this. The robots in this project can work the same
way and turn the same way with treads or wheels. If they use wheels, a pair of the tires must be removed to allow
them to slide or float on the table.
The students will also need much of the period today to finish their programming tasks. Programming is mathematical
in regards to the logic that it requires. One process must follow the next in a logical sequence. Math is everywhere!
Teachers should continue to use as many vocabulary words as possible in discussions with the students. The
students also should be encouraged to use the words in conversations with the teacher and amongst themselves.