Water Flow
How Does It
Affect Your Life?
Presented by:
Marilyn J. Phillips
Palacios High School
Palacios I.S.D.
Algebra II Advanced/
Pre-Calculus/
AP Calculus
Grade Level: 9-12
E3 Engineering Program for Teachers
June 27th, 2008
Table of Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
Introduction to the lesson
page 3
TEKS for the lesson
page 4
Educational objectives of the lesson
page 5
Vocabulary for the lesson
page 5
Materials needed for the lesson
page 5
Pre-Test of Basic Knowledge
pages 6 – 11
Lab Set-up (Day 2)
page 12
Lab Procedure (Day 3)
pages 13 – 14
Discussion Questions
page 15
Day 4—Power Point Presentation
Counter-current Flow Limitation
page 16
Volume formulas
page 16
Post-Test (Day 5)
pages 17 – 22
Follow-Up Projects
page 23
Algebra II Word Problems
page 24
Pre-Calculus Half-Life Problems
page 25
AP Calculus Related-Rate Problems page 26
Summary
page 27
2
This lesson is designed primarily for Algebra II
students, but it would work well in any math class.
The goal of this lesson is to introduce the students to the
concept of water flow and give them a hands-on activity
that they can do to calculate the water flow rate and also the
velocity of the water. This experience will help them
understand water flow problems for fully and give them the
tools that they need to apply this knowledge to real-life
examples.
I plan to teach this lesson at the end of the unit on linear
functions. This will tie in very well to the unit on quadratic
functions, so the time frame is about the second week in
October. This is the week of October 13th – 17th.
Here is a summary of the schedule for this lesson.
Day One:
Day Two:
Day Three:
Day Four:
Pre-Test on Basic knowledge
Do the Experiment
Make Graphs & Discuss the Data
Have a class discussion about how water flow
relates to the safety of a nuclear plant.
Day Five: Post Test to determine what the students
have learned.
3
These are the Algebra TEKS that will be emphasized in this
lesson:
A.1) Foundations for functions. The student understands that a function
represents a dependence of one quantity on another and can be described in a
variety of ways.
(A) The student is expected to describe independent and dependent
quantities in functional relationships;
(B) The student is expected to gather and record data and use data sets to
determine functional relationships between quantities;
(C) The student is expected to describe functional relationships for given
problem situations and write equations or inequalities to answer questions
arising from the situations;
(D) The student is expected to represent relationships among quantities
using concrete models, tables, graphs, diagrams, verbal descriptions,
equations, and inequalities; and
(E) The student is expected to interpret and make decisions, predictions,
and critical judgments from functional relationships.
These are the Geometry TEKS:
G11 (D) The student is expected to describe the effect on perimeter, area, and
volume when one or more dimensions of a figure are changed and apply this
idea in solving problems.
These are the Pre-Calculus TEKS:
P3 (B) The student is expected to use functions such as logarithmic, exponential,
trigonometric, polynomial, etc. to model real-life data
(D) The student is expected to use properties of functions to analyze and solve
problems and make predictions.
These are the Math Models TEKS:
(M.8) The student uses algebraic and geometric models to describe situations
and solve problems.
4
Educational Objectives:
During this lesson, the students will learn the following
objectives:
(1) How changing dimensions affects the cross-sectional
area and the volume of the water in the pipe.
(2) How to calculate the water flow coming out a pipe by
using the Δ Amount of Water / Δ Time.
(3) The students will also make a table, graph the data
points and draw a conclusion from the graph.
(4) The students will calculate the velocity of the water for
each size of pipe.
The following is a list of vocabulary words that the students
will learn:
Water Flow
Velocity
Cross-sectional Area
Volume
1 Gallon = _128__oz.
1 Gallon = _231__in3
Independent Variable
Dependent Variable
“Best-fit” line
Rate of Change
Counter current
Slope
During this lesson, the students will be using cooperative
learning, peer tutoring, hands-on activities, and whole
group instruction.
The students will need graph paper, and three colored
pencils ( green, blue, and red).
5
Pre-Test of Basic Knowledge
of Water Flow
Vocabulary Terms:
(1) One gallon of water contains _______ oz.
(A) 32 oz
(B)
64 oz.
(C) 128 oz
(D) 256 oz.
(2) How many cubic inches are in one gallon?
(A) 216 in3
(B) 231 in3
(C) 246 in3
(D) 343 in3
(3) How many cubic inches are there in one cubic foot?
(A) 36 in3
(B) 144 in3
(C) 864 in3
(D) 1728 in3
(4) In an algebraic equation, the variable x is the
Dependent / Independent variable.
(Circle the correct term)
(5) In an algebraic equation, the variable y is the
Dependent / Independent variable.
(Circle the correct term)
6
(6) Given the following set of data points:
Time (seconds)
3
5
7
Volume (oz)
32
64
80
Write an equation for the “best-fit” line.
(7) Using the data in the previous problem, what is the
slope of the line.
(8) Using the data from problem # 6, what is the average
rate of change in oz/sec.?
(9)
In a ¾ inch pipe, what is the cross-sectional area of the pipe
in square inches?
(A) 1.767 in2
(B) 1.326 in2
(C) 0.884 in2
(D) 0.442 in2
(10) If water is flowing out of a 100 gal. cylindrical tank at
5 gal/min, how long will it take to empty the tank?
7
(11) How fast will the water level fall in a tank that has a
diameter of 3 feet and a height of 2 feet, if the tank is
draining at 8 gal/min?
(Write your answer to the nearest tenth of an inch/minute)
(12) Calculate the water flow in gal/min, if the pipe size
is ½ inch and the velocity of the water is 6 ft/sec.
(A) 3.672 gal/min (B) 7.344 gal/min (C) 11.016 gal/min (D) 14.688 gal/min
(13) If 8 oz. of water is leaking out of a faucet in 2 hours,
How much water will leak out of the faucet in gal/day?
(A) ½ gal/day (B) ¾ gal/day
(14)
(C) 1 gal/day (D) 1.5 gal/day
If 16 oz of water is leaking out of a faucet in 2 hours,
How much water will leak out in one month.
(Use 1 month = 30 days)
(A) 22.5 gal/month (B) 45 gal/month
(C) 67.5 gal/month (D) 90 gal/month
8
(15) Which of the following tanks would hold 40 gallons of water?
(A) A rectangular tank that is 2 ft wide by 2.5 ft long by 1 ft high.
(B) A cylindrical tank that is 1 ft in diameter and is 5 ft. high.
(C) An inverted conical tank that is 5 ft high and 2 ft in diameter.
(D) A cubic tank that has an edge of 1.75 ft.
(16)
Jackie made a rectangular prism to hold her earrings. The net
of the rectangular prism is shown below. Use a ruler to
measure the dimensions of the prism to the nearest ¼ inch.
Which is closest to the volume of this rectangular prism?
(A) 4 in 3
(B) 1.2 in 3
(C) 8.5 in 3
(D) 13.5 in 3
(17) Janice uses a rectangular box to store her art supplies. The
dimensions of the rectangular box are 22.5 inches by
14 inches by 11.5 inches.
What is the volume of this box to the nearest tenth of a
cubic foot?
(A) 3622.5 ft 3
(B) 301.9 ft 3
(C) 25.2 ft 3
9
(D) 2.1 ft 3
(18) Mr. Kelly’s company manufactures a cylindrical soup can
that has a diameter of 6 inches and a volume of 226 cubic
inches. If the diameter stays the same and the height is
doubled, what will happen to the can’s volume?
(A) It will remain the same.
(C) It will triple.
(B) It will double.
(D) It will quadruple.
(19) The figure below shows a conical cup containing water. The
water depth can be represented by x, and the area of the water
surface can be represented by A. As the water depth changes,
the area of the water surface changes, as shown in the table
below. Which equation best represents the relationship
between the area of the water surface and the water depth?
Which equation best represents the relationship between the area
of the water surface and the water depth?
(A) A x – 1)2 in. 2
16
(C) A x2 in. 2
16
(B) A x in. 2
2
(D) A xin. 2
16
10
(20) Look at the solid sphere and the cylinder containing water
shown in Figure 1.
Figure 2 shows the sphere submerged in the water inside the
cylinder.
Which is closest to the height of the water level in Figure 2?
(A) 13 cm (B) 17 cm
(C) 15 cm
11
(D) 11 cm
Lab Set-Up
The teacher will fill the water tank before class starts in the morning.
Lab Procedure (Day 2):
Have students start with the ½ inch pipe. Have them turn the water on
and start the timer at the same time. One student could be the time keeper
and the other person could turn on the water. The water will be collected
and measured at a collection container at the other end of the pipe. The
following is a sample of a data collection table that could be used in this
experiment.
Water pipe size
⅜ inch
⅜ inch
⅜ inch
½ inch
½ inch
½ inch
¾ inch
¾ inch
¾ inch
Δ Water Volume
Δ Time
Water Flow Rate
When students arrive in class, put the students in groups of three students.
One student will be the timer. He/she will start timing when the valve is
opened up to start the water flow. The second person will measure the
volume of the water collected. The timer will stop the clock after the
required time intervals. The time intervals will be approximately
3 sec, 5 sec, 7 sec. The third person in the group will be the recorder.
He/she will record the information into the data table for the group. For
instance, in a class of 15 students, there will be 5 groups doing 9 trials that
average one minute in length. This will take approximately 45 minutes,
which gives the students about 5 minutes to get their data together and
distribute the data to each group member. If the class is larger, some of the
trials can be completed the following day.
12
Lab Procedure (Day 3):
The students will get together in their groups and finish
calculating the water flow rates. The students will then plot the
data on a graph, where the independent variable is time (x-axis),
and the dependent variable is the volume of the water (y-axis).
The flow rate is the slope of the line.
Example of graph:
60
Water Volume in ounces
50
40
30
20
10
2
4
6
8
10
12
14
16
18
20
22
24
26
Time in seconds
Each pipe diameter should be graphed in a different color.
For instance,
⅜ inch is the blue line
½ inch is the green line
¾ inch is the red line
Once the points are plotted, draw the “best-fit” line between the
points. The slope of the line is the water flow rate.
13
Once the water flow rate is calculated and graphed. The next
exercise is to convert ounces/second into gallons/minute.
128 ounces = 1 gal.
To convert oz/sec to gal/min
48 ounces
x
128 ounces/gal
60 seconds/min
3 seconds
= 7.5 gal/min
If there are 231 in3 in one gal. of water , what is the water flow in cubic
inches per minute?
Calculate the cross sectional area of each pipe:
Area of a circle = π r2
⅜ inch pipe:
½ inch pipe:
¾ inch pipe:
Divide the Water Flow rate in in3/sec. by the cross-sectional area
of the pipe. This is the velocity of the water flow.
14
Discussion Questions:
(1) Which pipe had the highest flow rate in gal/min?
(2) In which pipe did the water flow the fastest?
(3) How would you describe the water flow in the ⅜ inch pipe?
(4) How would you describe the water flow in the ½ inch pipe?
(5) How would you describe the water flow in the ¾ inch pipe?
How did it compare to the ⅜ inch pipe?
(6) What does each “best-fit” line for each pipe represent?
What does the slope of each line represent?
(7) Can you make a prediction for the water flow in a small ¼ pipe?
(8) Can you think of an everyday example of water flow?
15
Day 4: The Power Point Presentation will be given to the
students and the Countercurrent Flow Limitation
Experiment will be explained. I will talk to the students
about all of the different applications of water flow there
are in every day life. For example, most of the students
have probably gone to a water park this summer and may
be able to make the connection between the water flow and
the velocity of the water.
For the Algebra II students, I will give them some word
problems with more than one pipe draining a tank of water.
For the Pre-Calculus students, I will give them some
half-life problems and explain how radioactive elements
decay. For the Calculus students, we will look at the
inverted cone related rate problems and how the water level
will fall faster, as the water nears the bottom of the cone. I
have included some examples of these types of problems at
the end of this lesson.
Volume Demonstrations:
I want to demonstrate how to calculate the volume of all of
the different types of water tanks and have them convert
from cubic units to gallons using 231 in3 = 1 gal.
Rectangular
Cubic
Cylindrical
Conical
Spherical
V=lwh
V = x3
V=πr2h
V=⅓πr2h
V = 4/3 π r 3
16
Day 5:
Post-Test of Basic Knowledge
of Water Flow
Vocabulary Terms:
(1) One gallon of water contains _______ oz.
(A) 32 oz
(B)
64 oz.
(C) 128 oz
(D) 256 oz.
(2) How many cubic inches are in one gallon?
(A) 216 in3
(B) 231 in3
(C) 246 in3
(D) 343 in3
(3) How many cubic inches are there in one cubic foot?
(A) 36 in3
(B) 144 in3
(C) 864 in3
(D) 1728 in3
(4) In an algebraic equation, the variable x is the
Dependent / Independent variable.
(Circle the correct term)
(5) In an algebraic equation, the variable y is the
Dependent / Independent variable.
(Circle the correct term)
17
(6) Given the following set of data points:
Time (seconds)
3
5
7
Volume (oz)
32
64
80
Write an equation for the “best-fit” line.
(7) Using the data in the previous problem, what is the
slope of the line.
(8) Using the data from problem # 6, what is the average
rate of change in oz/sec.?
(9) In a ¾ inch pipe, what is the cross-sectional area of the pipe in
square inches?
(A) 1.767 in2
(B) 1.326 in2
(C) 0.884 in2
(D) 0.442 in2
(10) If water is flowing out of a 100 gal. cylindrical tank at
5 gal/min, how long will it take to empty the tank?
(11) How fast will the water level fall in a tank that has a
diameter of 3 feet and a height of 2 feet, if the tank is
draining at 8 gal/min?
(Write your answer to the nearest tenth of an inch/minute)
18
(12) Calculate the water flow in gal/min, if the pipe size
is ½ inch and the velocity of the water is 6 ft/sec.
(A) 3.672 gal/min (B) 7.344 gal/min (C) 11.016 gal/min (D) 14.688 gal/min
(13) If 8 oz. of water is leaking out of a faucet in 2 hours,
How much water will leak out of the faucet in gal/day?
(A) ½ gal/day (B) ¾ gal/day
(14)
(C) 1 gal/day (D) 1.5 gal/day
If 16 oz of water is leaking out of a faucet in 2 hours,
How much water will leak out in one month.
(Use 1 month = 30 days)
(A) 22.5 gal/month (B) 45 gal/month
(C) 67.5 gal/month (D) 90 gal/month
(15) Which of the following tanks would hold 40 gallons of water?
(A) A rectangular tank that is 2 ft wide by 2.5 ft long by 1 ft high.
(B) A cylindrical tank that is 1 ft in diameter and is 5 ft. high.
(C) An inverted conical tank that is 5 ft high and 2 ft in diameter.
(D) A cubic tank that has an edge of 1.75 ft.
19
(16)
Jackie made a rectangular prism to hold her earrings. The net
of the rectangular prism is shown below. Use a ruler to
measure the dimensions of the prism to the nearest ¼ inch.
Which is closest to the volume of this rectangular prism?
(A) 4 in 3
(B) 1.2 in 3
(C) 8.5 in 3
(D) 13.5 in 3
(17) Janice uses a rectangular box to store her art supplies. The
dimensions of the rectangular box are 22.5 inches by
14 inches by 11.5 inches.
What is the volume of this box to the nearest tenth of a
cubic foot?
(A) 3622.5 ft 3
(B) 301.9 ft 3
(C) 25.2 ft 3
(D) 2.1 ft 3
(18) Mr. Kelly’s company manufactures a cylindrical soup can
that has a diameter of 6 inches and a volume of 226 cubic
inches. If the diameter stays the same and the height is
doubled, what will happen to the can’s volume?
(A) It will remain the same.
(C) It will triple.
20
(B) It will double.
(D) It will quadruple.
(19) The figure below shows a conical cup containing water. The
water depth can be represented by x, and the area of the water
surface can be represented by A. As the water depth changes,
the area of the water surface changes, as shown in the table
below. Which equation best represents the relationship
between the area of the water surface and the water depth?
Which equation best represents the relationship between the area
of the water surface and the water depth?
(A) A x – 1)2 in. 2
16
(C) A x2 in. 2
16
(B) A x in. 2
2
(D) A xin. 2
16
21
(20) Look at the solid sphere and the cylinder containing water
shown in Figure 1.
Figure 2 shows the sphere submerged in the water inside the
cylinder.
Which is closest to the height of the water level in Figure 2?
(A) 13 cm (B) 17 cm
(C) 15 cm
22
(D) 11 cm
Follow Up Projects
Pre-Calculus
In this project the students will consider everything that they have
learned about water flow. They will now design their own water
flow system. I would require everyone to use ½ inch pipe and
place a maximum width and length of 2 feet. The amount of
water would need to be less than 1 gallon of water. Their design
could be a model of a water slide with varying angles of pipes or
they could design an irrigation system, which emphasizes how
much water they want to go to each field. This project will be
done the following week that I have taught the method of
calculating water flow and also they will need to calculate the
velocity of the water that is going through their system. They will
need to write an equation for water flow and also for the velocity
of the water going through the pipe.
Algebra II
Water Conservation Project
Find a faucet that is leaking in your house. Place a container under
the drip and measure the amount of water after about one hour.
From this measurement, calculate the amount of water that will be
lost down the drain in one day, in one month, in one year.
Using the amount that is charged for water usage, estimate the
amount of money that would be saved if the faucet were repaired.
Calculus Students
Derivative
For my Calculus students, I will want them to do a related-rate
project. For instance, I would want them to calculate the rate of
change at certain levels in the water tank.
23
Algebra II
Problems involving water flow:
(1)
If one pipe can empty a 40 gal water tank in 5 minutes and
a second pipe can empty a 40 gal water tank in 8 minutes,
how long will it take for both pipes to empty the water
tank?
(2)
If one pipe can empty a 40 gal water tank in 8 minutes and
both pipes can empty the tank in 6 minutes, how long will
it take the second pipe to empty the tank if the other pipe
is not open?
(3)
If three pipes can empty a 40 gal water tank in 4 minutes,
and the first pipe can empty the tank in 12 minutes and the
second pipe can empty the tank in 16 minutes, how long
will it take the third pipe to empty the tank, if the other
two pipes are not working?
24
Pre-Calculus Half-Life Problems
Use the formula:
N = N0 e kt
(1)
(2)
(3)
If there are 20 mg of Iodine-135 and the half-life is
6.7 hours, how much will be left after 3 hours?
(a)
Find the decay constant (k) for iodine.
(b)
Find the amount left.
If there is 30 mg of Radon-225 and the half-life is
14.9 days, how much will be left after one week?
(a)
Find the decay constant (k) for radon.
(b)
Find the amount left.
If there is 60 mg of Cobalt originally and the half-life is
5.26 years, how long will it take until there are only 40 mg
of cobalt remaining?
(a)
Find the decay constant (k) for cobalt.
(b) Find the time it takes to decay.
25
AP Calculus
Related-Rate Problem
Given an Inverted Cone Tank
Radius = 6 ft.
Height = 18 ft.
r/h = ⅓
r=⅓h
Volume = ⅓π r2h
V = 1/3 π (1/9 h2)h
V = 1/27 π h3
dV/dt = 1/9 π h2 dh/dt
Modeling Activity: Take a clear plastic cone with a 6 cm
radius and a height of 18 cm. Have a small pin hole at the
bottom for the water to run out. Using a piece of notebook
paper, tape a small strip of notebook paper along the side of
the cone, so that the height can be measured at each time
interval. Pour water into the cone, holding your finger on
the hole at the bottom. Remove your finger, the same time
as the timer starts. Mark the height of the water at
20 second intervals until the cone is completely empty.
Write down the time when the water stopped running out.
Measure the vertical height at each line that is marked on
the piece of paper. Now compare the change in height at
each time interval. The dh/dt should be increasing as the
water moves toward the bottom of the cone.
26
Summary
At the conclusion of the follow-up activities and lesson, the
students will be able to apply their knowledge of water
flow to every day situations and have a basic understanding
of how the nuclear power plant uses this flow to keep the
nuclear facility safe. They will also learn how important
water flow is to our environment and can practice water
conservation. They will also be able to collect data, make
graphs and interpret their results. These are important
skills in the area of mathematics. At the end of this lesson,
they should be able to make decisions based on the data
that they have collected. At some point, I would like the
students to have the opportunity to go out and take actual
measurements of the water flow in rivers and streams.
They are currently surveying beach erosion through their
Advanced Physics Class and this could be an extension of
this activity. It is my hope that the students will have an
increased awareness of their environment and make
decisions that will be environmentally friendly.
27