## Efficiency Means Getting More for Less

### Richard Angelini Sr.

#### Description

Here is a simple experiment. It costs little, the materials will last for 100 years, and it is safe. It is a study of efficiency in nature. Water is measured and predictions are made.

#### Objectives

The student finds measures of length, weight or mass, and capacity or volume using proportional relationships and properties of similar geometric figures.

The student calculates simple mathematical probabilities.

The student explains observed difference between mathematical and experimental results.

The student compares and explains the results of an experiment with the mathematically expected outcomes.

The student knows that biological adaptations include changes in structures, behaviors, or physiology that enhance reproductive success in a particular environment.

The student extends and refines use of accurate records, openness, and replication of experiments to ensure credibility.

#### Materials

Materials for each lab group:
-Water
-Two number 10 tin cans (to hold the water)
-One 250mL beaker or water glass (to pour water)
-Three 30cm lengths of PVC pipe capped on one end: one pipe with a 3/4” diameter; one pipe with a 1 1/2” diameter; one pipe with a 3” diameter
-Pre and Post Test, one per student (See Associated File)

#### Preparations

Note: This makes a good "teacher only presentation" if your kids like to splash water.
1. Go to a home improvement store to get the pipe, caps, and glue. They should cut the pipes to length for you. The cost for each lab group should not be more than \$1.50.
2. Glue a test cap on one end of each pipe so that the pipe becomes a cylindrical vessel capable of holding water. (Note: You will need one 3/4” test cap, one 1 1/2” test cap, and one 3” test cap.) Don't be worried, the associates at the store might put them together for you if you ask. If not, it is easy to do; just ask for instructions.
3. Download the Pre and Post Test in the associated file. Print one teacher copy (with answers), then delete the answers before printing the student copies.

#### Procedures

DIAGNOSTIC
1. Pass out the Pre Test and allow time for students to complete. Collect papers before beginning the Introduction and Demonstration sections.

INTRODUCTION
1. Ask your students to explain why “If a tree trunk is narrower in diameter than all of the tree’s branches added together, how can the tree supply food, support, and water to all its cells?” (If you add up all the diameter measurements of the branches, they total a greater diameter than the diameter of the tree trunk.)

2. Or ask your students, “How does a whale maintain its body heat in the cold waters of the Arctic or Antarctic Oceans?” (Whales enjoy the benefit of a greater mass to surface area ratio, keeping their insides much warmer than small Arctic or Antarctic swimming creatures.)

3. Ask this question, “What does a tree have in common with a Blue Whale?” (It is an increased surface area to volume ratio, for example, greater efficiency.)

4.Are they stumped? Then explain that there is an efficiency of scale involved.

DEMONSTRATION
1. Fill the 3/4” pipe with water.

2. Pour the water from the 3/4” pipe into the 1 1/2” pipe.

3. Repeat steps 1 and 2 until the 1 1/2” pipe is full. Record number of pours.

4. Pour water from the 1 1/2” pipe into the 3” pipe.

5. Repeat steps 1-4 until the 3” pipe is full. Record number of pours.

6. Record the results in the following manner:
3/4” pipe
1 1/2” pipe
3” pipe
(number of pours)
(number of pours)
(number of pours)
(one, of course)
(of 3/4” pipe)
(of 3/4” pipe)

7. Calculate the area of each cylinder surface and record observation of the volume of each cylinder in the following manner:

Write on the chalkboard:
Area = Circumference x 30cm ...or
Area = 3.14 x diameter x 30cm (answer is in squared cm)

(Note: Convert inches to centimeters. Also, we do not calculate the surface area of both ends of the pipe because we wish the students to observe the pipe measurements as one unit of measurement of an object with many units to be measured.)
3/4” pipe
1 1/2” pipe
3” pipe
27.07ccm x 4 = 108.28ccm x 4 = 433.12ccm
57.15sq.cm x 2 = 114.3sq.cm x 2 = 228.6sq.cm

(Right now the student can see that from the 3/4” to the 1 1/2” pipe, the numerator increases by (volume), a multiplier of 4 and the denominator by (area of surface), a multiplier of 2. The same is true for the next step, to the 3” pipe.) At first look, the progression seems to be a doubling process:

4x then 4x then 4x

8. Express as units of measurement of the 3/4” pipe, base units. Now the key to understanding the efficiency of the shape of cylinders is in how the results are displayed! Follow the format below for success. Represent the data collected in a graphic form. Start with 3/4” pipe results as the base line data so it is easy to compare pipes!

Write on the chalkboard:
Pipe:-----------------3/4” pipe-----------1 1/2” pipe-------------3” pipe
Volume of water: 1 unit of volume---? units of volume---? units of volume
Area of pipe:-------1 unit of area------? units of area--------? units of area

Remember: 1 unit of volume and area are defined in terms of the capacity and size of the 3/4” pipe, and all further measurements of volume and area are done in units of the 3/4” pipe’s size and capacity.

Results:
Pipe:-----------------3/4” pipe-----------1 1/2” pipe----------------3” pipe
Volume of water: 1 unit of volume---4 units of volume------16 units of volume
Area of pipe:-------1 unit of area-------2 units of area---------4 units of area

9. Create efficiency ratios. Now to get a visible relationship:

Write on the chalkboard:
3/4” pipe-----1 1/2” pipe-----3” pipe
1 or 1:1------2 or 2:1--------4 or 4:1
1----------------1-----------------1
(Because: 4 divided by 2 is 2 and 16 divided by 4 is 4)

Again note that all the pipe’s volumes and areas are expressed in terms of multiples of the volume and area of the first pipe, the 3/4” pipe, that being one unit of volume and one unit of area. As the diameters of the cylinders increase, the capacity ratio of the cylinders increases. Therefore, the efficiency ratio of the cylinders increases....... LOOK !!

6” pipe-----12” pipe
8 or 8:1----16 or 16:1
1--------------1

The 12” pipe is 16 times more efficient in carrying or storing material than the 3/4” pipe. This is why: The ratio of surface area to total volume for a 3/4” pipe is stated at, 1:1 and for a 12” pipe is, 16:1 . The 12” pipe uses many times less exterior surface to total volume than the 3/4” pipe!!! Just imagine the efficiency of a tree with a 60” trunk! Now you can imagine how any tree can hold up so many branches.

Note: The students love this experiment. My kids got down on the floor and delighted in guessing how many 3/4” pipes it would take to fill the larger pipes. Also, many greatly appreciated being able to see the physical results of experimentation expressed mathematically. It turned into a “hands-on” math class, as well as science class.

10. Teacher assesses activity. (See Assessments and Associated File)

#### Assessments

FORMATIVE:
Divide the class into teams, give points for responses to the steps, for example:
1. Give 5 points for the first team to observe that the water capacity is not increasing arithmetically.
2. Give 10 points to the team that first sees that the volume is increasing exponentially.
3. Observe each team.
a. Give 3 points for a controlled experiment.
b. Give 3 points for each team that decides to record the results in their lab manuals without a teacher prompt.
4. Give 10 points for the first team to suggest a mathematical solution to the question: “What would be the volume of a 36” pipe?”
5. Give 5 points to the team with the first correct answer to number 4 above.

SUMMATIVE:
Assign the Post Test. (See Associated File)

#### Extensions

My special thanks to my friend and colleague Ronald J. Zink, Ed. D. for his review and advice.

#### Attached Files

This file contains the Pre and Post Test with answers.     File Extension: pdf