Huffing and Puffing to
Understand Slope
or
Smoke and You Croak
Lenny VerMaas
ESU #6
210 5 Street P.O. Box 748
Milford, NE 68405
402-761-3341
th
Residence
10620 South 213th St
Gretna, NE 68028
402-641-0043
lvermaas@esu6.org
http://manila.esu6.org/lvermaas
Find out how blowing up a balloon can be used to engage students in learning about volume,
graphing, slope and linear equations. Student collect data from a balloon that is blown up, graph
the data, and make some interesting discoveries. Science connections are also created.
Appropriate for students in pre-algebra or above.
Agenda:

How does your lung work, looking at a model?
o Direct and indirect relationships.
 What effects lung volume?
 Measuring lung volume (estimate first).
 Plotting measurements
 How fast does a balloon travel?
References:
“Functions of a Toy Balloon” Mathematics Teacher November 1994 p 619
“More Functions of a Toy Balloon?” Mathematics Teacher April 1997 p290
SMILES activity
American Cancer Society
Lenny VerMaas
lvermaas@esu6.org
page -1http://manila.esu6.org/lvermaas
Smoke and You Croak
Materials: Balloons—one per person (9 or 12 in, 12 inch works better), cloth tape measures,
chart paper, markers, calculators.
Guided Practice




Begin by discussing the volume of a person’s lung. What will affect the volume?
Exercise, smoking, size, male, female, asthma. Estimate the volume of your lung. (Think
of a 2 liter pop bottle)
Scientific terms relating to volume of lungs include; inspiratory reserve volume, vital
volume, expiratory reserve volume, residual volume, vital capacity, total lung capacity.
For exact definitions visit your friendly science teacher. (See page 7)
Make model of the lung using plastic cups or use ones already made. What makes the
lung work? Many students will start by blowing into the straws. Ask them if someone
blows into their lungs when they breathe? Get them to see that the diaphragm is what
causes the lungs to operate.
Pressure and volume is an example of an inverse or indirect relationship. PV=C When
one increase the other decreases. This is what causes air to enter and exit the lungs.
Look for other examples of inverse as well as direct relationships.
Direct
Velocity and distance traveled
Foot on accelerator and speed
Study time and grade
Volume and diameter of balloon


Inverse or indirect
velocity and time to travel certain distance
speed and gas mileage
driving time and gas in tank
watching TV and intelligence
How can the volume of a person’s lung be measured? Accurately using a spirometer,
approximation—water displacement or blowing up a balloon. Once a balloon is blown
up, how do we find the volume? The formula for volume requires that the radius must be
known. The radius is found by measuring the circumference and using the formula for
circumference.
Vocabulary—consider special strategies to help students learn the terms, circle,
circumference, radius, diameter, sphere, locus of points, directly proportional, and
inversely proportional.
C  d  d 
C

r
d
2
4
V  r 3
3
Worked Out Sample
42  d  d 
42

6.7 
13.7
2
4
V   6.73  1251cm3
3
The volume is 1251 cubic cm is also equal to 1251 ml. which is equal to 1.2 liters.
What if the shape of the balloon is not a sphere, maybe oblong? Measure both directions and use
the average for the diameter.
Lenny VerMaas
lvermaas@esu6.org
page -2http://manila.esu6.org/lvermaas
Data Collection:



Measure the circumference of the balloon with one breath
Keeping the air in the balloon add another breath and measure the circumference
Keeping the air in the balloon add another breath and measure the circumference (may
not be possible with 9” balloons
Calculations

Find the radius and volume for 1, 2, and 3 breaths
Graphing



Have 1st, 2nd, and 3rd breaths recorded in different colors or male female
On one graph plot the diameter of the balloon on the x-axis and the circumference on the
y-axis. Diameter usually goes from 0 to 30 cm and circumference go from 0 to 90 cm to
include all three breaths.
On another graph plot the radius of the balloon on the x-axis and the volume in liters on
the y-axis. Radius usually goes from 1 to 15 cm and volume goes from 0 to 8 liters
Discussion and conclusions










What observations can be made about the graphs? Both look close to straight lines
Graph of diameter and circumference
Points for a straight line, if continued will go through (0,0) WHY
The line serves as a guide to see if calculations were done correctly
Draw a line through points or use a calculator to find the best fit line
Calculate the slope of the line which will be close to PI, WHY
Look at the graph of radius vs volume
This graph looks straight, but not quite
Does it have to go through (0,0)? YES
This means it must be a curved line
Lenny VerMaas
lvermaas@esu6.org
page -3http://manila.esu6.org/lvermaas
Extensions



Find minutes off of ones life for smoking
Use a stop watch to time how long it takes air to be expelled from the balloon. The flow
of air may be changed by connecting a straw to the mouth of the balloon. Try to vary the
length of the straw. Plot flight time against volume, diameter, or radius.
PICTURES BELOW DEMONSTRATE THIS EXTENSION. Fasten a length of straw to
a balloon that is blown up with 1, 2 or 3 breathes. Have a piece of fishing line. You may
need up to 20 yards. Run the fishing line through the straw. Release the balloon and see
how far it will travel down the line. Use a shop watch to measure the time the balloon
takes to travel the distance. Using the distance and time calculate the speed in several
different units, cm/sec, miles/hour, cm/hour etc.
Constructing a Model of the Lung
Material: clear plastic cups, straws, small balloons, rubber bands, tape, dental dam (from
dentist) or latex gloves
1. Make two holes in the top of each cup. One way to make the holes is to use a hot glue gun
and force the point into the cup. The holes can be made to fit better by using a scissors.
2. The straws represent the trachea, the balloons represent the lungs, and the dental dam
represents the diaphragm. The balloons may be attached with tape, (cut off the lip of the
balloon). The diaphragm is attached with rubber band. Seal the straws with a hot blue gun.
The more air tight the seals, the better the model will work.
Lenny VerMaas
lvermaas@esu6.org
page -4http://manila.esu6.org/lvermaas
Huffing and Puffing to Understand Slope
or
Smoke and You Croak
Take a balloon and with one breath blow as much air as possible into the balloon. Measure and
record the circumference of the balloon. Keeping the air in the balloon add another breath and
measure and record the circumference. Add one more breath, measure and record the
circumference.
I estimate the volume of my lungs to be _______ liters. (Think of 2 liter pop bottle)
Volume in
Circumference
Diameter
Radius
cubic
centimeters
After Breath 1
After Breath 2
After Breath 3
Use the formulas below to calculate the diameter, radius, and volume in cubic centimeters for the
balloon for each breath.
C  d  d 
C

r
d
2
4
V  r 3
3
Graphing
Circumference

Record data from your 1st, 2nd, and 3rd breaths.
On one graph plot the diameter of the balloon on the x-axis and the circumference on the
y-axis
On another graph plot the radius of the balloon on the x-axis and the volume in liters on
the y-axis
Volume in liters


Diameter in cm
Lenny VerMaas
lvermaas@esu6.org
Radius in cm
page -5http://manila.esu6.org/lvermaas
Smoke and You Croak
Name ____________________
1. (2pts) Your lungs work because of an inverse relationship between the volume and pressure.
Explain what it means to be inversely related?
2. (2pts) Give another example of an indirect or inverse relationship and explain why it is an
inverse relationship?
3. (2pts) What does it mean to be directly related?
4. (2pts) Give two examples of a direct relationship. Explain why they represent a direct
relationship.
5. (10pts) A balloon is blown up and has a circumference of 42 cm. Find the volume of the
balloon in liters.
c  d
4
V  r 3
3
6. (2pts) The slope of a line plotting circumference against diameter is

. Explain why.
7. (2pts) Plotting the volume against the radius does not produce a straight line. Explain why.
8. (2pts) Define a sphere. Be specific.
9. (6pts) A balloon travels 45 feet in 2.2 seconds. Find its speed in miles per hour.
Bonus 2pts A balloon has the shape of a sphere and a volume of 2 liters. What is the diameter
of the balloon?
Lenny VerMaas
lvermaas@esu6.org
page -6http://manila.esu6.org/lvermaas
Picture of lung capacity
•
•
•
•
•
•
•
inspiratory reserve volume
vital volume
expiratory reserve volume
residual volume
vital capacity
total lung capacity
For exact definitions visit your friendly science teacher.
Lenny VerMaas
lvermaas@esu6.org
page -7http://manila.esu6.org/lvermaas