Level II: Draft Interdisciplinary Mathematics/Science Investigation, "An Earful Investigation"

Developers: Anna Graeber and Randy McGinnis, UMCP

Science-technolgy-society design:

Tier one: Focus on the structure of a body's organ (ear) and infections (viral and bacterial)

Tier two : Focus on the impact of humans on the living environment (antibiotics and resistance within the body and in society)

Note: The inspiration for this lesson came from the Saturday, March 4, 1994 Washington Post article entitled "When Children Get an Earful" (p. A7 ff). Popular press issues can often be springboards for interdisciplinary topics.

An "EarFul" Investigation



Topics: comparing ratios (written in any form), finding a percent of a number

Linkage With Mathematics Curriculum Standards

Mathematics goals which are targeted in this activity that are identified in the NCTM Standards (Grades 5 - 8):

Understand and apply ratios, proportions, and percents in a wide variety of situations

Develop number sense for .. decimals, fractions, percents

Investigate relationships among fractions, decimal, and percents

Discuss mathematical ideas and make conjectures and convincing arguments.

Prerequisite knowledge: Fraction, meaning of percent, writing percents as fractions, construction of circle graph, some notion of ratio.

Background Information. Ratios can express the size of a part to the size of a part, or as in fractions the size of a part to the size of the whole. It is only the context and the verbiage of a statement that indicate which type of ratio is being expressed.

Two notations for ratio are common a/b or a:b. Like ratios can be expressed as and derived like equivalent fractions.

Ratios can be written expressed as decimal or percentage equivalents by dividing the denominator by the numerator.

The relative size of a ratio can be judged using benchmarks associated with fractions or percents. Two ratios can be compared by writing both in the same form (fractional, percent ,decimal) and then comparing or by finding the equivalent of one in the form of the other and then comparing.

To find a given percent of a number or what percent one number is of another, or the number given the value of a certain percent, solve the corresponding proportional statement of equality. (Cross multiplication should not be the first technique taught. Use unit ratios or nice multiples.)


Topics: Structure of the ear, bacteria, bacteria/viral infections, antibiotics, resistance

Linkage With Science Curriculum Standards

Science topic areas (goals or outcomes) which are targeted in this activity that are identified in major science reform publications:


Viruses, bacteria, fungi, and parasites may infect the human body and interfere with normal body functions. (grades 6-8, p. 145)

Human beings have many similarities and differences (Human organism, Grades, 6-8, p. 129)

Specialized cells and the molecules they produce identify and destroy microbes that get inside the body (Grades 6-8, p. 137)

Each culture has distinctive patterns of behavior, usually practiced by most of the people who grow up in it (grades 6-8, p. 155)

Know that hypothesis are valuable, even if they turn out not to be true, if they lead to fruitful investigations (Grades 6-8, p. 287)

National Science Educational Standards

As a result of their activities in grades 5-8, all students should develop an understanding of: structure and function in living systems (p.8)

Maryland School Performance Assessment Program, 1992

Life Science

*The life science program (6-8) includes an emphasis on investigation of the structure and function of the human body and the impact of human behavior on other living organisms and the environment (Concepts of Science, Outcome 1, Content Areas, p. 9)

Prerequisite knowledge : nature of bacteria, nature of virus

Background Information.

Middle ear infections (otitis mediia) most commonly result when bacteria or viruses invade the nose or throat and make the short journey through the auditory tube (formerly called the eustation tube) to the middle ear. There the body's white blood cells attack the invading microbes which produces pus that drains down the auditory tube. As the infection proceeds, however, it causes the auditory tube to become blocked which results in fluid building up in the middle ear. This produces an earache and sometimes deafness. Middle ear infections of this type are common among young children since the auditory tube is relatively short , and young children spent much time lying down. Middle ear infections can also result when enlarged adenoids block the entrance to the auditory tube. Mucus will not then be able to drain from the middle ear and may become very sticky and glue like.

The structure of the ear consists of three parts--the outer, middle, and inner ear --which enables the ear to function as a hearing and balancing organ. The eardrum, a thin membrane, separates the outer ear from the middle ear. The middle ear, where otitis media occurs, is closed to external invaders except through the auditory tube. The auditory tube connects the middle ear and the throat. The middle ear contains three small bones, the hammer, the anvil, and the stirrup, which transfer sound vibrations from the eardrum to the inner ear where the cochlea is located. The cochlea contains fluid and cila of specialized hearing cells. As the cilia bend due to the movement of the cochlea fluid produced by sound vibrations they send out signals along the auditory nerve to the brain. These impulses are perceived as sound. The inner ear also contains three small curved tubes called the semicircular canals. Cells inside those tubes send messages to the brain which result in the body being able to keep its sense of balance. The semicircular canals can also be affected when the middle ear is infected. This results in a loss of balance or dizziness.

Thus the structure of the ear, the age of the person, and the body's reaction to invading microbes all contribute to the formation of middle ear infections and the ear's impairment to perform its functions.


Farb, Stanley N. (1980). The ear, nose and throat book: A doctor's guide to better health. New York: Appleton-Century-Crofts.

Stoppard, Miriam (1986). Body & Child A to Z Handbook. Tucson:

The Body Press.


Different folk explanations for ear infections and their treatment among different cultures; inclusion of students with disabilities in lesson


Tier One


On the Island of Earf there are about 98,700 people. During 1995, many people had middle ear infections. In fact, the island's Director of Health is concerned because middle ear infections are now three times more prevalent than they were in 1975. They now account for 15% of all prescriptions written for antibiotics. Your team is part of the staff at one of the island's six health clinics.

In your team, discuss and in your clinic's log outline major ideas:

1. a What do you know about middle ear infections, what causes them, and how they are treated?

b. Also, what do you know about the structure of the ear?,

2. a. Do you know if "15% of all prescriptions written for antibiotics" is a large number of prescriptions? Explain. Restate "15% of the nation's antibiotic prescriptions" in two ways --using a fraction or a ratio. Explain how you arrived at your answers.

b. What do you believe ear infections are "now three times more prevalent than in 1975" means?


Math component:

One-fifteenth of all antibiotic prescriptions are for ear infections.

Fifteen of every doctor's antibiotic prescriptions are for ear infections.

Since there are now more people now than there were in 1975 there are more ear infections.

15% can't be very many prescriptions.

15 to 85 is the ratio of antibiotic prescriptions for ear aches to all antibiotic prescriptions.

What does the % mean?

Don't I need to know how many antibiotic prescriptions were written?

Science component:

Ear infections are caused by the eardrum being clogged as a result of not cleaning the ear.

Ear infections occur because water gets in the ear.

Ear infections are caused by the ear drum being inflamed.

Ear infections occur when you have colds.

Ear infections occur when the ear canal gets a rash.

Ear infections make you dizzy.

Ear infections are cured by reducing or decreasing the things you are doing that is affecting the ear.

Physicians cure ear infections by putting medicine in the ear canal.

The ear is cured by cleaning it out.

The ear infection is always cured by taking antibiotics.

What is the structure of the ear and how does that relate to ear infections?


Potential Activities:

Activity I Teams or the class investigate the following topics as indicated by the prior knowledge displayed.

A. The structure of the ear.

Modeling structure of ear.

B. The bodies reaction to invasion by bacteria.

Looking at bacteria under the microscope (shapes of bacteria)

Observing decomposition using bacteria (different environments)

Culturing bacteria in petri dishes.

C. Ratios expressed in the forms a/b, a:b and as percents.

D. The comparison of two ratios by expressing both in the same format.

Modeling fractions on hundreds squares

E. Exploration of qualitative proportions in the infection setting -- In 1994 there were about the fewer ear infections treated in Clinic Area A than in Clinic area B There were more ear infections caused by viruses in Area A than area B.. Which ratio is larger ear infections caused by virus : ear infections in Area A or ear infections caused by virus : ear infections in Area B ?

F. Problem Solving with Percents

Modeling the product of two percents

Modeling the percent of a number

When the activities are over and a whole class discussion has considered questions such as those below, the class can move on to the "An Earful of Microbes Activity" :

What role does the structure of the ear play in ear infections?

Why might you expect that antibiotics might not help all ear infections?

How do you tell a part to part ratio from a part to whole ratio?

What benchmarks can you use to help you estimate the relative size of percents?

Possible Resources for Activities:

Videocassette of structure of the ear (e.g. Ear Infections in Children, Home Health Videos, Medical Video Encyclopedia, Inc.)

Model of the ear (e.g. , The Human Ear, Anamod Series No. 13005)

Written materials on the ear (life sciences textbook, tradebooks including, The Story of Your Ear (Dr. Alvin Silverstein and Virginia Silverstein), Ears are for hearing (Paul Showers), What happens when you listen (Joy Richardson), Chris gets ear tubes (Betty Pace)

Computer software on the ear, bacteria, viruses (e.g. Human Life Processes II: Systems Level, Biology Series, IBM

Models of fractions and percents, Explorer calculators

Activity II: At this point, students complete the activities called "An Earful of Microbes" and "An Island of Microbes."

"An Earful of Microbes"

Materials Needed : Each team of students needs a Clinic Log, and each student needs a copy of the student pages.

Also the teacher will need to prepare 42 Paper cups, (42) Sticky Labels (15 labeled Virus; 14 labeled Bacteria--Sp; 4 labeled Bacteria--Mc; and 9 labeled Bacteria Hi

Each team of students needs a Clinic Log, a calculator, a protractor, and each student needs a copy of the student pages.


We know that ear infections are caused by microbes that invade the ear. Some treatments are effective for one type of microbe but not for other microbes. The proportion of ear infecting microbes of a given type in Earf Island's population varies from time to time as it does in other locations. The Director of Health for the island has requested each clinic to collect information so that he can compile data on the causes of ear infections on the entire island The Director of Health will also want the data from each clinic.

Your team is to request information from the lab. What would you ask your clinic's laboratory to collect for you if you want to know what microbes are causing ear infections in your clinic's assigned region? Explain your answer . Record in the clinic's log.

In order to investigate the causes of ear infections, we will ask your team to pick up its' sample from the central laboratory.

Elect someone in your group to go to the designated area and pick up your clinic's microbe cultures and bring them to your group. Look inside each cup and in your clinic's log tally the data in a table. The table should show both the microtype, the name of the microbe, and the number in your clinic's sample

For your team's sample:

Write the ratio of number viral ear infections to number of ear infections.

If viral infections and bacterial infections are the only microbes you found, what must the ratio of bacterial ear infections to ear infections be and why?

Now, as part of your report to the Director of Health, complete the circle graph below to fairly accurately reflect the fraction of the samples that was bacterial

Also with you team discuss how you would compile and graphically display the information the Director of Health wants. Record the main ideas in the log.

An Island of Microbes Activity

Equipment needed: same as for "An Earful of Microbes" activity


Eventually the Director of Health decides that he wants the pooled data (data from all of the clinics combined) displayed as a bar graph. The graph is to show the number of cases caused by each type of bacterial as well as by viruses. Given that each clinic's team has identical labels for their cultures, what might be an easy way to construct the bar graph for the Director? Discuss with the whole class.

Once your class has agreed on figures for the whole island sample,

Write the ratio of number viral infections to number of ear infections.

Write the ratio of number bacterial infections to number of ear infections.

Write the ratio of number Hi bacterial infections to number bacterial of ear infections.

Do the same for the number of Mc and Sp bacterial infections.

Why would the Director of Health want this type of information from the pooled data?

The Director of Health is interested in comparing the whole island data to the following data from the neighboring continent.

35% of ear infections are caused by viruses.

Of the ear infections caused by bacteria:

about 50% are caused by Sp

about 30% are caused by Hi

about 20% are caused by Mc

Are virus caused infections as prevalent in the whole island sample as on the continent? How do you know?

Of the bacteria caused ear infections, are the _______ bacteria caused ear infections as prevalent in the whole island sample as on the continent? How do you know?.

Is it correct to say that for the data from the continent a ratio 35 to 50 describes the ratio of ear infections caused by viruses to ear infections caused by bacteria? Tell why or why not?

If over the course of a year a doctor on the island had treated 48 cases of ear infection, about how many would you expect (based on the national sample) to be caused by the Mc bacteria? Explain how you reached your answer.

Which sample had ratios closer to the continental sample, your clinic's or the whole island? Is this the answer you would expect? Why or why not?


Prior to continuing on to tier two of the lesson, there should be an opportunity to for students to look back at their earlier beliefs about earaches and ratios and compare them to their current beliefs. This may be prompted by displaying the original newsprint listings. Students can evaluate their original statements as correct, containing some truth, or erroneous. It will be helpful to discuss the sources of erroneous beliefs -- was there an apparent logic--- an overgeneralization from other situations, a belief caused by everyday language?


Develop an alternative assessment based on an analogous situation of a body infection caused by either bacteria or a virus, such as conjunctivitis; Have students make presentations on the structure and function of the ear and explain how middle ear infections are caused.

Tier Two [to be developed]



Physical science--sound, including the speed of sound in different media

Mathematics--Sampling, statistics,

Math/Science integration-- exponential (bacterial) growth