Posted May 13th, 2010 by Admin

Lesson 2: Cubic Expressions: Application to Fluid Transport Systems through Murray’s Law

Project Unit Title: Moving and Pumping Through Fluids at Different Sizes

Unit Theme:

Fluid dynamics is a complex discipline that describes the flow of fluid in and around objects. Students will get a glimpse of fluid dynamics through three lessons related to real and irrational numbers, cubic functions, density and viscosity, and scale models. Most importantly, students will experience how mathematics can be used to model natural phenomena.

Lesson Theme:

Mathematics can be used to describe the flow of fluid through closed circulatory systems in living organisms such as plants and mammals.

Title:

Cubic Expressions: Application to Fluid Transport Systems through Murray’s law

Introduction/Theme:

The students will view a schematic image of a leaf and the veins of the plant that circulate fluid throughout the leaf. Using a simplified version of Murray’s law, the students will be able to mathematically model the fluid transport system of some plants and animals.

Learning Outcomes:

The student will be able to use cube and cube roots to model real-life phenomenon. Students will be able to mathematically model an efficient circulatory system like the one in the human body and in some plants.

Curriculum Alignment: Grade 8 Mathematics

Goal 3, Objective 3.02 – apply geometric properties and relationships, including the Pythagorean theorem, to solve problems
- Use estimates of irrational numbers in appropriate situations
Goal 5, Objective 5.04 – solve equations using the inverse relationships of addition and subtraction, multiplication and division, squares and square roots, and cubes and cube roots

Classroom Time Required:

One 50-55 minute class period if the students complete the project outside of class, or two 50-55 minute class periods if the students complete the project in class.

Materials Needed:

Handout of the warm-up (included in the lesson) – 1 per student
Leaf diagram – 1 per student
Activity sheet/project rubric – 1 per student
Calculators – 1 per student
Clay – have this available for students to create their model of Murray’s law if they complete it during class (amount depends on whether students work individually or in groups)
Play-do - have this available for students to create their model of Murray’s law if they complete it during class (amount depends on whether students work individually or in groups)
Pipe cleaners - have this available for students to create their model of
Murray’s law if they complete it during class (amount depends on whether students work individually or in groups)

Technology Resources:

Each group of students will need a calculator. An LCD projector can be used to show a scanned in leaf. IMAGE J can also be downloaded and used to measure the radii of actual vein leaves.

Pre-activities:

The students should have a solid understanding of the cube and cube root of a number and be able to solve equations involving cubes and cube roots. This warm-up includes six questions on simplifying and calculating square roots. This can be done individually at the beginning of class or students can complete this in their groups before beginning the activity.

Activities:

The students should complete a warm-up activity on cubes and cube roots. The warm-up may be used to refresh equations involving cubes and cube roots.
Divide the students into teams of 3-4.
A discussion on how mathematics is used for modeling real life phenomena should take place. The students should come up with a variety of examples (frequency from Lesson 1, scale models to represent large or small objects, viscosity to determine how fast Ketchup runs out of the bottle, the time it will take to fly from New York to Los Angeles with a head wind, the probability of hitting a certain region on a target). Ask the students to think about the human body and how mathematics may be related to the design and structure of the body. Students may discuss ideas such as proportionality of body parts, blood pressure, height/weight/body mass index. Ask the students if they think it might be important to model systems and parts of the human body (proportion of bone lengths, blood pressure, etc). Would it be helpful to model the structure of the circulatory system? Have the students look at their arms/hands and describe the circulatory system. Guide the students to describe the branches and various diameters of the veins and the different types of vessels. These include arteries, arterioles, capillaries, venules, and veins. Tell the students that they are going to model the efficient circulatory system that humans and some plants have developed by looking at the veins in a leaf. Their goal is to create a function/equation that models the circulatory system of humans and fluid transport system of some plants.
The students should have a few minutes to review the leaf schematic. After they have reviewed the image, have students discuss what information has been provided (radii of the vein branches).
In their teams, allow the students to explore various relationships with the radii. Offer them hints such as creating a table, conducting different operations with the numbers, etc. The students should offer suggestions to their group.
Depending on the level of the class, more or less assistance may be needed. Some students may need the hint of adding the numbers and then trying to cube them. The teacher may offer as much assistance as needed to prevent the students from becoming frustrated or giving up.
Coach the students until they discover the formula . Have a group of students present their findings. To guide the students on steps 5-6, consider asking the students to choose one set area from the schematic (one main leaf and two branches). Ask the students if they can find a relationship or equation that will allow them to take the two radii of the branches to equal the radii of the main branch. Ask the students if they remember the Pythagorean Theorem. Does the Pythagorean Theorem work for this data? Ask the students to think about equations involving cubes and cube roots like problem #5 and #6 from the warm-up. These questions may guide the students to discover the relationship between the radii of the branching veins and the radii of the main vein.
The teacher will reveal that the formula is Murray’s law. The teacher should lead a discussion about the efficiency of this system and why it is important to plants. Teacher note: Maximizing the water flow increases photosynthesis which yields faster growth. Energy used by the plant is minimized. For more advanced students, discuss how the water flow is maximized and the area of the transport system is minimized. The teacher should lead the discussion to the human body. What conclusions can be drawn about the blood flow, starting in the large arteries to smaller arteries, to vessels and then to capillaries? Note that the circulatory systems of crabs, sponges, mollusks, and earthworms can also be modeled using Murray’s law. Extend the discussion to water flow and delivery systems of water in urban areas. Would it be beneficial to understand Murray’s law and apply the equation when developing large water transport systems?
The students will create a model using play-do, pipe cleaners, straws, etc of an efficient transport system that follows Murray’s law. This may be done in groups or individually and may be completed during class or outside of class.

Assessment:

The assessment will occur through the use of rubric for the student-created model of Murray’s law. If the students create the model outside of class, the rubric needs to be modified to include the project only.

Modifications:

This plan is intended for students placed in eighth grade math or a math equivalent to pre-algebra. Providing more information during the inquiry stage of the lesson could modify this plan. The teacher could provide more direct hints during this time and could provide Murray’s law for the students. A leaf schematic with different numbers could be provided for students that are not able to create a project. This would reduce the project to calculating the cubes and cube roots.

Supplemental Information

Article on Murray’s law

Plant plumbing is more human than once thought. (2003). University of Utah Public Relations.
http://unews.utah.edu/p/?r=031406-52

This article discusses the discovery of Murray’s law and the research that led to the law being applied to humans. The article is appropriate for an 8th grade student.

Critical Vocabulary:

Murray’s law – an equation that models an efficient fluid transport system. Murray’s law states that the radii of vessels will narrow according from large vessels to smaller vessels. The formula for this law is to find the sum of the cubes of the radii of the branching vessels. This sum should approximate the radius cubed of the larger “parent” vessel. (Teacher note: A diagram of a vein splitting and labeling the larger “parent” vessel and the smaller vessels may help to convey this idea.)

r³ = r₁³ + r₂³ + r₃³ + …

Websites and Resources

Download Image J Software - http://rsbweb.nih.gov/ij/

Image J allows you to scan a picture and measure the picture according to pixel and actual distance.

The full leaf on the leaf schematic was taken from the following website.
Hefner, P. (n.d.) Beauty in science and spirit, forward.
http://paul.carr2.home.comcast.net/~paul.carr2/BBookDescription2.htm