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Breast Cancer: An Exponential Process

Author: Vicki Baker


Philadelphia High School for Girls

Year: 2021

Seminar: Cancer Biology and Technology

Grade Level: 10

Keywords: Algebra 2, Breast cancer, cancer, Elementary Functions, exponential functions, logarithmic functions, Pennsylvania, Philadelphia

School Subject(s): Algebra, Math

This curriculum unit explores the basic science and classification of cancers by cell type, tissue and organ origin, the difference between normal and cancer cells, how the cells change to become cancer cells, progress, and metastasize. The unit content includes cancer treatment, the science of treatment resistance and the use of body’s immune system to assist in treatment and cancer therapy and the effects of preventative measures such as smoking, diet, exercise, and heredity on contracting cancer. The classroom activities connect the growth and propagation of cancer with exponential and logarithmic functions. The curriculum unit is designed for high school Algebra 2 and Elementary Functions students.

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Full Unit Text
Content Objectives


In my seminar, “Cancer Biology and Technology”, we were provided with the current thinking of how cancer begins and what technology exists to diagnose and treat it. During our cancer exploration we learned the basic science and classification of cancers by cell type, tissue and organ origin, the difference between normal and cancer cells, how the cells change to become cancer cells, progress, and metastasize. Cancer is extremely complicated. Our studies included cancer treatment, the science of treatment resistance and the use of body’s immune system to assist in treatment and cancer therapy. Our studies included the human elements of diet, exercise, and heredity on contracting cancer as well as cancer’s progress through discovery, treatment, remission, and reemergence from dormancy sometimes after hiding for years.

Problem Statement

I teach high school math at The Philadelphia High School for Girls, a special admission, magnet, high school with a rich history of academic excellence. Founded in 1848 to “prepare teachers for the common schools of Philadelphia,” Girls’ High, as it is affectionately known, was the first municipally supported secondary school for girls in the United States and was called the Girls’ Normal School. In 1893, the Philadelphia High School for Girls separated from the Girls’ Normal School, and the foundation for today’s college preparatory curriculum was laid. The school continues its legacy as a school for academically talented girls, providing young women with outstanding opportunities for scholarship, leadership, and service. Its motto, “Vincit qui se vincit” (He conquers who conquers himself), is a key centering point for our students maturing into young woman of purpose and honor (Cutler, 2013).

Girls’ High students represent the variety of schools in the City of Philadelphia having applied and gained admission from diverse neighborhoods and middle schools of every type (public, private, parochial, magnet and neighborhood) from all over the city. They are as ethnically diverse as the City of Philadelphia having students from African American, White, Latinx, Asian, Native American, and Mixed-Race households. Our diverse population while representative of Philadelphia neighborhoods across Philadelphia has become increasingly dominated by black and brown students of multiple economic levels, cultures, and ethnicities. Their demographics are also consistent with the poverty recorded for the City of Philadelphia and the School District of Philadelphia.

The Pew Charitable Trusts 2018 Report, “Philadelphia’s Poor: Experiences from Below the Poverty Line” indicates that Philadelphia is home to nearly 400,000 people living below the poverty line. These statistics are corroborated in the Princeton University report which says, “Poverty effects a quarter of Philadelphians, with the burden falling heaviest on the city’s minorities (particularly Hispanics) and children at 39% and 38% impoverished, respectively.

To be living in poverty means supporting a family of four on an income of $24,300. Raising a family in a city, below the level of poverty is difficult, but for 12.2% of Philadelphians these difficulties are compounded below the deep poverty line supporting a family of four on an income of less than $12,1502 . This means that 40% of the city’s poor, are living in deep poverty. To further put these statistics into perspective, the poverty rate for the United States is 15.8%, a rate Philadelphia has never approached in the past decade .

The 2020-2021 Girls’ High Economically Disadvantaged (CEP Rate)of 98% represents an 8% increase compared to a year ago. This measure is calculated by taking the percentage of students identified as economically disadvantages, multiplying by a USDA defined CEP factor of 1.6 and capped at 100%

The student population at Girls’ High as described by demographics, shares many of the challenges of the city overall. Students have excelled academically which makes them different than many Philadelphia students living in poverty.

Girls’ High School Demographics
Girls’ High Economically Disadvantaged

(CEP Rate)

Percent of Students with IEP’s

Source: School Profiles, The School District of Philadelphia,

The prevalence of English language learner has also increased at Girls’ High. The number of ELL students while unchanged between 2020-2021 and 2019-2020, has doubled since 2018- 2019.

Academic Rationale

Girls’ High students are a group of highly able learners who bring energy and the thirst for learning when they interact with each other and question both the content and relevance of the work they are given.  Even with their strong educational background, my students struggle to apply and extend the mathematical concepts to real life word problems. According to Christopher Masullo in the December 6, 2017 Education Week article, “What’s the Problem with Word Problems?”, “Word problems tend to be complicated in part because of their descriptive language. Students often do not understand what exactly they are being asked, especially when the problem includes abstract concepts. Other issues arise when students lack the fundamentals of math and cannot formulate a plan for solving or separating an equation’s steps.”.

While strong math students in previous settings, across all grades, my students’ math skills are often rudimentary and reflect rote learning. Their math knowledge consists of algorithms and formulas of which they have little practical knowledge, and they do not know how to apply their math skills to real world applications. My students read the problem and focus narrowly on the question. They focus on remembering “the right way,” the “formula” or “algorithm,” or “the prior lesson” required to solve the problem. While my students have solid math skills and a strong grasp of math fundamentals, they lack the ability to apply these skills to new math content and to solve word problems. They fail to learn deeply with an ability to apply the recent skill later when it is applied to a more complex problem.

Tracing mathematical roots back to the earlier grade levels, teachers often use key words often help younger students create a problem-solving plan. The word “more” is taught as a clue for when to add and similarly, the word “fewer” indicates subtraction. The difficulties with word problem reoccur when there is a lack of key words to follow. In more advanced problems, each problem is unique and a single, overarching algorithm to solve them does not exist. (Masullo, 2017)

To engage my students so that they can apply math concepts to real world problems, I will use cancer as a background since most if not all my students have encountered cancer in their lives. Whether relatives, friends, neighbors, or celebrities their lives much like mine have been touched by cancer. My curriculum unit will focus on breast cancer. Following a description of the general development of cancer, this curriculum unit will hone into breast cancer its prevention, causes, development, treatment, prevalence, and advocacy in the African American community. Girls’ High student body is 78% black and brown and overall has a 96% non-white population.

My curriculum unit will focus on breast cancer and the disease among African American women. I am focusing on breast cancer because it has the highest incidence of cancer in PA and over the last 5 years is the 2nd highest cause of cancer-related death. Since 2007, the number of women aged 50 and over who have died of breast cancer has continued to decrease. The number of women under age 50 who have died of breast cancer has stayed steady. (, 2021) I want to both educate about the disease and its prevalence as well as highlight opportunities for advocacy, education and self-awareness or actualization as tools to help in their personal journey.  Topics to be covered include how the disease begins and grows and why it is so prevalent and deadly. We will also discuss preventative screening approaches that allow women to screen for breast cancer using monthly self-breast exams. This will offer them some control over their health and bodies. I plan to talk about methods for testing for breast cancer. I understand from my own experience that effective diagnosis even with mammography within communities of color is problematic due to characteristics such as dense breasts. African American women die from breast cancer including an extremely aggressive strain, at higher rates. While there are some limits to mammography there is good evidence that across all populations that screening helps reduce mortality. Recent technology advances such a digital breast tomosyntheses and MRI help in screening women with dense breasts and /or higher risk populations.

According to the Breast Cancer Prevention Partners, an organization dedicated to reducing the risk of developing breast cancer, African American women face both disproportionate exposure to breast carcinogens and the highest risk of serious health impacts from the disease. Some products marketed to Black women, some used for generations and many popular with black women of all ages, contain ingredients known to contribute to developing breast cancer. Examples include skin lighteners, hair relaxers, Brazilian blowout treatments and acrylic nails, to name a few.

My curriculum unit will provide opportunities for students to experience the math content with real life data. My plan is to provide historical are well as local data that students can analyze while learning about the disease. I also hope to communicate when do they start? What are the trends and why it is important to young people?

Content Narrative

The content narrative will summarize the basic science and classification of cancers by cell type, tissue, and organ origin to provide a background on the biology of how cancer develops. This will include the difference between normal and cancer cells, how the cells change to become cancer cells, how they progress, and metastasize. The next section will discuss the diagnosis and treatment of cancer. Finally, the human elements of diet, exercise, and heredity on contracting cancer as well as cancer’s progress through discovery, treatment, remission, and reemergence from dormancy sometimes after hiding for years.

Introduction to Cancer Biology

To understand cancer, the first question to answer is, what is cancer? Cancer is a disease of ageing. While young children, even infants can develop cancer, their cancers are rare and often more treatable than cancers of the adult population.  The median age of a cancer diagnosis is 66 years old. Women receive diagnoses at earlier ages than men but the male rate of diagnosis increases rapidly and surpasses that of women in the 55–64-year-old range.

Normal cells, shown below as the “normal epithelium”, exist for decades before becoming abnormal in the dysplasia/adenoma phase. These abnormal cells can remain in this form or can mutate and become carcinoma cells. When cancer progresses, abnormal cells progress to become cancer cells. On average, cancer is diagnosed later in life because it develops over time and is only discovered once the cells have grown to a point where they can be detected.

Cancer develops in stages. First a cell mutates generally without any knowledge of the person. The first cancer stage, hyperplasia, occurs when a cell begins to grow too much. At this stage, the cells still appear to be normal.  The next phase, dysplasia, is the first stage where cells grow rapidly and appear to be abnormal. The cancer cells are present in an organ and as they grow to the in-situ cancer stage the tumor continues to grow within the organ. Finally, as more cancer cells grow, the tumor cells invade blood vessels and the malignant cells continue to grow.

Mutations in two classes of genes are responsible for beginning the cells’ change to cancer: oncogenes and tumor suppressor genes. This change happens between to normal epithelium and dysplasia/adenoma stages.

Oncogenes are mutated genes that contribute to cancer development. In their normal, unmutated state, oncogenes are called proto-oncogenes, and they regulate normal cell division and cell death When a proto-oncogene becomes mutated, the cell makes extra proto-oncogene copies leading to uncontrolled cell division and tumors. (National genome research)

Tumor suppressor normal genes that control cell growth and division. They repair DNA mistakes or tell cells when to die (a process known as apoptosis or programmed cell death). When tumor suppressor genes do not work properly, cells can grow out of control, which can lead to cancer. (American Cancer society).

The next question is what causes cells to mutate? There are two caused of gene mutation environment and genetic inheritance (BRCA12). Environmental factors are ultraviolot light from sunlight, radiation and carcinogens such are found in tobacco asbestos, nickel, cadmium, radon, vinyl chloride, benzidene, and benzene. Gene mutations caused by genetic inheritance (BRCA12) are specific to inherited DNA and may or may not develop.

As the tumor cells invade blood vessels and the malignant cells continue to grow the cells leave the organ and travel to other parts of the body via the blood or lymphatic system. People who die from cancer generally die from metastatic disease.

What Causes Cancer?

Many milestones in have been reached in the study of cancer.

The diagrams above show the current understanding regarding the progression of cancer. In the early 200’s after the cells began uncontrolled growth the prevailing thought was that tumor cells proliferated evading growth  suppressor cells, evaded adjacent cells and metastasised. The cells continues to grow in new locations forming new blood vessels and continuing to grow and proliferate evading cell death signals.(left hand side) The diagram on the right-hand side is similar showing advances in the details of each of the previous steps from the left. The circa 2011 diagram identifies the cells the cancer evades, that would normally cause death, as T and B cells, the mechanisms for cell propagation and metastasis and more clearly defined and the molecular cell characteristics identify effects on the  genome and cancer cell’s ability to reprogram normal cellular metabolism for growth.

Theories on how cancer develops are being studies around the world. The Alfred Knudson tow hit theory of cancer theorizes that a norml cell is hit twice before mutation. This represents another advance which provides more information to scientists studying caner and developing treatments.

How is Cancer Diagnosed and Treated?

Cancer Diagnosis

If a patients’ symptoms suggest cancer is present, tests are available many which are specific to different types of cancer. Three types of tests are discussed here: lab tests, imaging tests and biopsies.

Lab tests of bodily substances can be used to diagnose cancer. Testing of blood or  tissue samples are looking for tumor markers, substances produced by cancer cells in response to the presence of cancer.

Imaging tests create pictures of areas inside the body and can include CT scans, MRI, nuclear scan, bone scan, PET scan, mammogram, ultrasound, and x-rays depending on the indicated  type of cancer . CT scans use x-ryas to take pictures of organs and create a 3-D image. A dye or contrast material is frequently used for highlight areas in the study.  MRI tests use powerful magnets and radio waves to take pictures of the body in slices. The slices are used to create detailed images of the body. Nuclear scans use  radioactive material injected into the body to create pictures of bones or organs. Bone scans are specialized nuclear scans specific to bone cancer.  (Berg, 2012) PET scan is another type of nuclear scan that uses radioactive glucose to look for glucose build up in the body. Cancer cells absorb more glucose than normal cells. Mammogram is used to find cancer in the breast tissue. , Ultrasound uses high-energy sound waves which echo off bodily tissue. A computer used the echoes to create sonograms, pictures of the area studied. A transducer covered with a warm gel moves on the skin over the part of the body that is being examined. X-rays create pictures of the body using low levels of radiation.

A surgical procedure called a biopsy can also be used to diagnose cancer. A doctor removes a sample of tissue using a needle, endoscopy, or surgery. (National cancer institute)

There are multiple types of biopsies. A standard biopsy samples tissue in an invasive surgical procedure where tissue is collected under anesthesia . A liquid biopsy is a minimally invasive surgical procedure where tissues are drawn into a liquid for a quick comprehensive tissue profile.

Cancer Treatments

There are four cancer treatment methods: surgery, chemotherapy, targeted therapy, and radiation. Surgery removes the tumors by resecting primary tumors. Chemotherapy kills rapidly dividing cells. Targeted therapy blocks a specific protein of pathway that identified as key to cancer growth. Radiation causes damage to the tumor’s DNA and triggers the cancer cells to die.

Health Factors Related to Cancer

The American Cancer Society (ACS) in their Guideline for Diet and Physical Activity for Cancer Prevention outlines their cancer prevention recommendations. The first and most important health risk factor is tobacco use. Tobacco is followed by body weight, diet, and physical activity.

Whether smoking cigarettes or using smokeless products, tobacco can cause cancer and then block your body from fighting it:(CDC). Poisons in tobacco weakens the body’s immune system which makes it harder to kill cancer cells. The body is then unable to stop the growth of cancer cells. Tobacco can also damage DNA and the natural cell coding that manages cell growth and duplication.

For people who do not use tobacco, changes are body weight, diet, and physical activity can help mitigate cancer development. At least 18% of all cancers diagnosed in the US could be prevented by reducing excess body weight, increasing physical activity, limiting alcohol consumption, and improving nutrition. Avoiding these risk factors throughout life may greatly reduce a persons’ lifetime risk of developing or dying from cancer. These actions are affected by the social, physical, economic, and regulatory environments where a person lives.

Curriculum Unit Background

Breast Anatomy

Breast anatomy involves an arrangement of ducts, lobes, ligaments, and muscle. Each breast contains 15 to 20 lobes of glandular tissue arranged in a circle pattern like the petals of a daisy. The lobes are further divided into smaller lobules that produce milk for breastfeeding. Small ducts that look like tubes move the milk to a reservoir beneath the nipple. The nipple is in the middle of the areola, which is the darker area surrounding the nipple.

The breast exchanges fluid in the lymph nodes. Lymph nodes are small, bean-shaped organs that help fight infection and are found throughout the body. They produce and filter a colorless fluid called lymph, which contains white blood cells known as lymphocytes (immune cells involved in defending against infections and such diseases as cancer). Lymph vessels filter and carry lymph fluid from the breast to the lymph nodes. Clusters of lymph nodes near the breast are in the armpit (known as axillary lymph nodes), above the collarbone, in the neck, and in the chest. Breast cancers can form in the ducts and the lobes.

(Memorial Sloan Kettering Cancer Center

Breast Cancer

Breast cancer starts when cells in the breast (such as cells lining the ducts and lobules) begin to grow abnormally. The abnormal cells could be in the lining of the milk ducts and/or lobules. These abnormal cells if mutated to cancer cells can grow out of control and invade surrounding cells. If it continues to grow it can spread beyond the breast to other areas of the body and become life threatening.

The figure shows a breast cancer tumor. During the development of cancer, the normal balance between cell division and cell loss is disrupted. The malignant cells divide far faster than new cells are needed. Since each division of a malignant cell results in 2 newly formed cells that retain their capacity to divide, there is an overall increase in the total number of dividing cells. (Winchester Breast cancer can invade through nearby tissue or spread through the body via the lymphatic system and blood.

Breast Cancer in Pennsylvania

Female breast cancer along with lung and bronchus and prostate cancer, are two of the three most commonly diagnosed invasive cancers in Pennsylvania. In 2011, nearly 10,600 women were diagnosed with invasive breast cancer. Detection and prevention have played a critical roll in stabilizing breast cancer in Pennsylvania. Here are some statewide data from the Bureau of Health Promotion & Risk Reduction report, The Burden of Cancer in Pennsylvania: Calculating Costs Understanding Impacts Exploring Interventions, which describes breast cancer in Pennsylvania.

The percentage of women in Pennsylvania who had a mammogram in the last two years was slightly lower than the national average in 2010 after m meeting or exceeding national trends prior to the year 2000 when it reached its peak at 77.8 percent. . Overall, the rate of mammography for 2010 is 73.9 percent for Pennsylvania and 75.2 percent for the United States.

The chart on the left shows that compared to the national average, Black women 40 years and over had the highest rate of mammography in 2012. Their rate of testing was 80.5 percent for Pennsylvania and 78.3 percent nationally. Multiracial women showed the lowest screening rates at 49.6 percent for Pennsylvania and 60.1 percent nationally.

The report on mammograms by eduction and income show that in Pennsylvania, the lowest rate of mammogram screening was found in women with lower incomes and less than a high school education at 63.8 percent, and those making less than $15,000, at 65.0 percent. These rates are slightly higher than the national rates where the rate for women with the lowest level of education was 62.7 percent and 60.4 percent for those with the least amount of annual income.

The map on the right shows the percentage of Pennsylvania women, 40 years and older, who had mammograms in the past year. Philadelphia women received mammograms during 2012 at a higher rate, 64%, compared to the statewide average 60 percent. Philadelphia County had the highest percentage of mammograms in the state.

Over the last 10 years, female breast cancer incidence rates for all women have remained steady without dramatic changes. However, until 2011, the black females breast cancer incidence rates remain higher than any other ethnicity.

Breast Cancer in Philadelphia

This section will provide an overview of breast cancer in Philadelphia: how the disease develops and grows and the technology available for its detection. The data source is, “Cancer and Cancer Health Disparities in Philadelphia – Urban Health Collaborative.” Drexel Urban Health Collaborative, 2020,  The next, section provides personal preventative screening approaches that allow women to use monthly self-breast exams to begin a practice offering some control over their health and bodies.

The most common type of cancer deaths among women in Philadelphia in 2016 were from lung and bronchus disease. The second highest cause of death was breast cancer.

While the most common cancer death was from lung and bronchus disease, breast cancer had the highest number of new cases. By ethnicity, new cases among black women per 100,000 was almost equivalent to white women.

A summary of 2016 deaths by ethnicity show significantly more black women died from both all cancers and breast cancer than white or Hispanic women per 100,000 people.

Lowering the Risk of Developing Breast Cancer

What dos the term risk factor mean? A risk factor is “an activity or action that increases your chances of getting a disease, such as breast cancer” (American Cancer society). Having a risk factor, or even many risk factors, does not indicate you are certain to develop the disease. The risk factor categories related to developing breast cancer are similar to those outlined in the general cancer section: drinking alcohol, being overweight or obese, tobacco use and lack of physical activity. Emphasis is placed on some key concerns in each category.

Drinking Alcohol

Alcohol use is one of the most important preventable risk factors for cancer in general breast cancer specifically. Alcohol use accounts for about 6% of all cancers and 4% of all cancer deaths in the United States. Alcohol use has been linked with cancers of the mouth, throat, voice box, esophagus, liver, colon and rectum and breast. For each of these cancers, the more alcohol you drink, the higher your cancer risk.

Drinking alcohol is clearly linked to an increased risk of breast cancer. Even small amounts of alcohol consumption can increase breast cancer risk and the risk increases with the amount of alcohol consumed. Women who consume one alcoholic drink a day have a small (about 7% to 10%) increase in risk compared with non-drinkers, while women who have 2 to 3 drinks a day have about a 20% higher risk than non-drinkers. TO mitigate against developing breast cancer, women should not drink alcohol. For women who do drink, they should have no more than 1 drink a day.

Healthy Weight and Regular Exercise

The American Cancer Society recommends women stay at a healthy weight throughout their life and avoid excess weight gain by balancing your food intake with physical activity. Exactly how physical activity might reduce breast cancer risk has not been conclusive, but it is recommended that adults get 150 to 300 minutes of moderate intensity or 75 to 150 minutes of vigorous intensity activity each week (or a combination of these). Getting to or going over the upper limit of 300 minutes is ideal. It is important to adapt a healthy weight and regular exercise routine early in life so that it is a natural part of one’s lifestyle. It is much harder to change life-style practices later in life. (CA, 2020

Exponential Growth and Disease

Cell division in a human cell is well documented in mitosis. One cell divides to create two new cells. This is the normal cell propagation process that repeats itself in abnormal cells as well. Each division of an abnormal cell whether bacterial or  malignant also results in two newly formed cells and each cell retains its capacity to divide on its own. In each case, there is an overall increase from one cell to two after the first cell divides.

The mathematical category for numbers increasing in this manner is an exponential model. Students have been introduced to exponential equations in algebra 1. Exponential functions will be modeled using cancer, bacteria, and communicable diseases such as Covid-19.

Teaching Strategies

Students will learn to:

  • Determine expressions and values using mathematical procedures and rules.
  • Connect representations of linear, exponential, and logarithmic functions.
  • Justify their reasoning and explain their solutions mathematically and by extension, creatively.
  • Use correct notation, language, and mathematical conventions to communicate results or solutions.

My curriculum unit will focus on exponential functions – the math that models both growth of cancer cells. The cancer information in the cancer background details general information as well as  breast cancer because breast cancer  has the highest incidence of cancer in PA and over the last 5 years is the 2nd highest cause of cancer-related death. The math lessons link the reproduction of cancer cells and the spread of viruses.  The lessons are designed to provide hands-on learning as students create their own exponential function graphs and create graphs with macaroni. These approaches allow for collaborative work as well as an approach conducive to auditory, cooperative, and kinesthetic learning styles.

The natural extension for exponential functions is logarithmic functions, the inverse of exponential functions. The data from the exponential lesson is used in the second lesson to help reinforce the relationship and introduce the rate of change students can estimate using a line of best fit.

The last lesson breaks from the ordinary classroom lesson by asking students to create a comic, commercial or flyer of their own choosing to encourage personal behaviors known to help prevent breast cancer.

Classroom Activities

Lesson 1: Exponential Growth

Teaching Objectives: To model exponential functions using real life models.

Essential Question: How are exponential functions utilized in real life situations.

Lesson Objective(s):

  1. Students will graph exponential growth functions.
  2. Students will use exponential models to solve real-life problems.

Previous Learning: Students studied exponential functions in Algebra 1, so they should be able to evaluate and graph y = abx.

New Vocabulary: exponential function, exponential growth function, growth factor, asymptote, exponential decay function, decay factor

Previous Vocabulary: properties of exponents


  1. 1 sheet of graph paper, grid spacing 1 line per centimeter. Can create graph paper at a website like (
  2. X-Y Table for graph


Activity 1:

  1. Give each student or student group a sheet of graph paper the table below.
  2. Students should create a graph using the table.

Question: What type of function does the graph represent? How do you know? What is the equation of the function graphed in this activity?

Activity 2:


  1. 1 sheet of graph paper, grid spacing 1 line per centimeter. Can create graph paper at a website like (
  2. Orzo enriched macaroni (1 cm long), about 50 kernels in zip-lock snack bags
  3. 2-sided tape or glue
  4. 1-sided tape
  5. Tweezers (eyebrow type, 1 per student or groups of students)


  1. Give each student or student group a sheet of graph paper and a bag of orzo macaroni, tweezers.
  2. Given the equation y = 2x
  3. Create a table for x = 0, 1, 2, 3, 4
  4. Using the macaroni, students place orzo to represent the height of the y values for each x value. (See below). Tip: Place a strip of two-sided tape on each x from 1 to 4. Use the tweezers to place the number of orzo macaroni pieces in place; cover the line with 1-sided tape.

Question: What is the significance of the orzo macaroni shape? How are the resulting graph and the orzo macaroni related? Why does the graph have the shape?

Sketch the graph of the curve, using the top of the macaroni for each coordinate.

Activity 3: Exponential growth and epidemics and disease

Cancer and viral infections are similar in the manner that they spread. Students will model cancer growth and viral growth using their own data.


Watch the video Simulating an epidemic using this link:

Students will use the internet to find data on the spread of the coronavirus in Philadelphia or the US represented in a graph.

Example: Covid 19 spread in Philadelphia

Philadelphia Inquirer 2020

How is this graph similar to the graphs in activities one and two? How is the exponential model for a virus similar to the spread of cancer? How is it different?

Lesson 2: Logarithms – Making Exponential Functions look Linear.

Teaching Objectives:

  1. Students evaluate logarithms using the three properties of logarithms.
  2. Students will graph the logarithmic equations and compare them to exponential functions.

Essential Question: How can you use the inverse relationships of exponential functions to create of logarithmic functions?

Lesson Objectives:

  1. Students will use the properties of logarithms to evaluate logarithms.
  2. Students will use the properties of logarithms to expand or condense logarithmic expressions.
  3. Students will graph logarithms and compare the graphs to exponential graphs.

Lesson Information:


What is an exponent? The exponent of a number indicates how man times the number will be multiplied by itself.

Example: 23  = 2 x 2 x 2 x2 = 8 two is multiplied 3 times and gives the answer 8

What is a logarithm? A logarithm indicates how many times one number is multiplied to get the other number. In this way exponential and logarithmic are inverses.


we are looking for the exponent

? = the exponent or 3

Three properties of logarithms:

Using the properties:                      First rewrite the problem.

= ½                    Property 3

= ½          Property 1

Change base: If the logarithm is not in base 10, the answer can be derived using the change base formula.


  1. Use the table from lesson 1, activity 1 for the equation y = 2x, and the table for x = 0, 1, 2, 3, 4
  2. Write each exponential equation for each x and y value.
  3. Convert the exponential equation to the logarithmic equation.
  4. Graph the resulting logarithmic equation on graph paper.

Describe the difference between the exponential equation and the logarithmic equation. What other type of equation could the logarithmic represent (linear)? What is the slope of the line? What does the slope represent if the graph were a model for cancer? Viral infection?

Lesson 3: Becoming an advocate for cancer prevention.

Essential Question: How can students learn about the causes of breast cancer and use that knowledge to help students take preventative steps against breast cancer?

Learning Objectives:

  1. Student will demonstrate the knowledge of breast cancer prevention.
  2. Students will use creativity and technology to summarize risks of contracting breast cancer.

Previous Learning: Student will have prior knowledge about how cancer develops and specifically the preventative measures women can engage in help protect themselves against breast cancer. (See information in the curriculum unit background and appendix to create slides or information sheets).


If you read the comics, you may be familiar with Doonesbury a comic created by Garry Trudeau in the 1960’s and 1970’s. Doonesbury began as a continuation of Bull Tales, which appeared in the Yale University student newspaper, the Yale Daily News, from 1968 to 1970. It focused on local campus events at Yale. The comic strip began in 1968 and is a nationally syndicated comic still published daily and on Sundays in local newspapers. It has focused on presidents, politicians, social issues, lifestyles cultures and many other topics over the years. Each comic strip is developed using characters common to the strip or with new character created for a specific topic. The topics continue for days, weeks and sometimes months.

In 1988, after years of litigation, there was a $280 billion settlement with the tobacco industry called the Master Settlement Agreement (MSA) intended to offset states’ Medicaid and other health care costs related to illnesses and deaths from tobacco use. Many people assumed the money would be used for youth focused anti-smoking campaigns. The Doonesbury comic strip created a character, Mr. Butts, to tell the story of the MSA, highlight nicotine addiction and advocate against smoking among young people. I am providing it as an example of the type of message your publication should deliver.

In this assignment you will create a document to help publicize methods available to prevent breast cancer. Your document should use the information we have studied about lowering the risk of developing breast cancer. Below are some excerpts from the Mr. Butts Doonesbury comic strip. Additional examples are found in the appendix of Mr. Butts, as well as additional characters depicting the danger of other substances.

Introducing Doonesbury’s Mr. Butts

The Assignment

In this assignment you will create a cartoon (at least 4 frames), a TV commercial or TV commercial storyboard (at least 4 frames), or poster/flier 8.5 x 11” size) to communicate the ways a woman can prevent developing breast cancer.

As a reminder, here are the ways a women can lower the risk of developing breast cancer:

·       Do not use tobacco.

·       Maintain a healthy weight.

·       Get a mammogram (35t and over).

·       Limit alcohol intake (max 1 glass per day).

·       Exercise 300 minutes each week.

·       Do a monthly breast exam.


  • (10 pts) Your publication should include a minimum of 4 of the preventative measures.
  • (15 pts) Explanation: At the end of your work, explain your publication choices

Why did your select your characters?

What does each represent?

Define any of the vocabulary words you learned about cancer prevention.

  • (5 pts) Neat appearance/easy to read.



“A Healthy Diet and Physical Activity Can Help Reduce Your Cancer Risk.” CA: A Cancer Journal for Clinicians, vol. 70, no. 4, 2020, pp. 272–273.

Berg, Wendie A., et al. “Detection of Breast Cancer with Addition of Annual Screening Ultrasound or a Single Screening MRI to Mammography in Women with Elevated Breast Cancer Risk.” JAMA: The Journal of the American Medical Association, vol. 307, no. 13, 2012, pp. 1394–1404.

“Breast Cancer – Statistics.” Cancer.Net, 25 June 2012,

“Cancer and Cancer Health Disparities in Philadelphia – Urban Health Collaborative.” Drexel.Edu, 9 Sept. 2020,

CancerConnect. “Racial Disparity in Breast Cancer Mortality Is Partially Explained.” CancerConnect, 8 Mar. 2009,

“Cancerous Cell Growth and Development.” Winchesterhospital.Org, Accessed 22 June 2021.

CDCTobaccoFree. “Smoking and Cancer.” Cdc.Gov, 2 Apr. 2021,

Centers for Disease Control and Prevention (US), et al. Report of the Surgeon General: How Tobacco Causes Disease: The Biology and Behavioral Basis for Tobacco Attributable Disease. A Report of the Surgeon General. Edited by Health and Human Services Department, U.S. Government Printing Office, 2011.

Currie, Janet. “Healthy, Wealthy, and Wise: Socioeconomic Status, Poor Health in Childhood, and Human Capital Development.” Journal of Economic Literature, vol. 47, no. 1, 2009, pp. 87–122.

Cutler, W. (2012). Public Education: The School District of Philadelphia Encyclopedia of Greater Philadelphia. Retrieved from

“Factors with Unclear Effects on Breast Cancer Risk.” Cancer.Org, Accessed 24 June 2021.

“How Cancer Is Diagnosed.” Cancer.Gov, 3 Sept. 2015,

“How Does Breast Cancer Start & Spread?” Org.Au, 14 Aug. 2019,

Gwu.Edu, The Burden of Cancer in Pennsylvania: Calculating Costs Understanding Impacts Exploring Interventions; Bureau of Health Promotion & Risk Reduction | Room 1008 H&W Building | 625 Forster Street | Harrisburg, PA 17120-0701 2000-2010 Accessed 21 June 2021.

Kim, Pilyoung, et al. “Effects of Childhood Poverty and Chronic Stress on Emotion Regulatory Brain Function in Adulthood.” Proceedings of the National Academy of Sciences of the United States of America, vol. 110, no. 46, 2013, pp. 18442–18447.

Masullo, Christopher. “What’s the Problem with Word Problems? (Opinion).” Education Week, Education Week, 6 Dec. 2017,

McCoy, Dana Charles, et al. “Neighborhood Crime and School Climate as Predictors of Elementary School Academic Quality: A Cross-Lagged Panel Analysis.” American Journal of Community Psychology, vol. 52, no. 1–2, 2013, pp. 128–140.

“Oncogene.” Genome.Gov, Accessed 23 June 2021.

“Philadelphia’s Poor: Experiences from below the Poverty Line.” Pewtrusts.Org, Accessed 22 June 2021.

“Public Schools Percent of Low-Income Reports.” Education.Pa.Gov, Accessed 23 June 2021.

Richardson, Lisa C., et al. “Timeliness of Breast Cancer Diagnosis and Initiation of Treatment in the National Breast and Cervical Cancer Early Detection Program, 1996-2005.” American Journal of Public Health, vol. 100, no. 9, 2010, pp. 1769–1776.

Rock, Cheryl L., et al. “American Cancer Society Guideline for Diet and Physical Activity for Cancer Prevention.” CA: A Cancer Journal for Clinicians, vol. 70, no. 4, 2020, pp. 245–271.

Stead, Lesley A., et al. “Triple-Negative Breast Cancers Are Increased in Black Women Regardless of Age or Body Mass Index.” Breast Cancer Research: BCR, vol. 11, no. 2, 2009, p. R18.

Strober, Jordan W., and Matthew J. Brady. “Dietary Fructose Consumption and Triple-Negative Breast Cancer Incidence.” Frontiers in Endocrinology, vol. 10, 2019, p. 367.

Sun, Yi-Sheng, et al. “Risk Factors and Preventions of Breast Cancer.” International Journal of Biological Sciences, vol. 13, no. 11, 2017, pp. 1387–1397.

Yi, Myungsun, and Young Mi Ryu. “The Effects of Breast Health Education in Women with Breast Cancer.” Research Journal of Pharmacy and Technology, vol. 10, no. 7, 2017, p. 2295.

 Teacher Internet Resources: Cancer related websites

Sister’s Network – National African American breast cancer survivorship organization cancer resources

African American Breast Cancer Alliance

National Breast and Cervical Cancer Early Detection Program (NBCCEDP)

Cancer Connect,cancer%20death%20than%20White%20women.

Doctor Approved Info from Amsco

American Cancer Society Journals – Guideline for Diet and Physical, Activity for Cancer Prevention, Article American Cancer Society Diet and Exercise

The Burden of Cancer in PA – Comprehensive cancer report for Pennsylvania

An Analysis of Cancer Incidence in PA Counties 2008-2012 – 4-year cancer report

Komen Adolescent Breast Cancer Curriculum (Latina)

Breast Cancer Prevention Partners

African American Women and Breast Cancer

Breast Cancer Education for Young Women – Howard University

Higher Population-Based Incidence Rates of Triple-Negative Breast Cancer Among Young African American Women: Implications for Breast Cancer Screening Recommendations, Amirikia, K, Mills, P, Bush, Published in Cancer. Volume 117, Issue 12, pp. 2747-2753.

Students Video Resources

Cancer Quest Cell Division – explanation of cell division

National Cancer Institute – 3-part series on cancer

How cancer grows

How cells divide

Oncogenes & tumor suppression genes


Academic Standards:

The Common Core Standards for High School functions involving linear, quadratic, and exponential models are designed to provide real-life modeling of algebraic functions. Each of the lessons and activities focus on modeling functions and reasoning with the outcome. The use of graphs and tables as well as comparing the functions and their graphs is utilized throughout the curriculum unit.

CCSS.MATH.CONTENT.HSF.LE.A.1         Distinguish between situations that can be modeled with linear functions and with exponential functions.

CCSS.MATH.CONTENT.HSF.LE.A.1.A Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

CCSS.MATH.CONTENT.HSF.LE.A.1.B Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

CCSS.MATH.CONTENT.HSF.LE.A.1.C Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

CCSS.MATH.CONTENT.HSF.LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

CCSS.MATH.CONTENT.HSF.LE.A.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

CCSS.MATH.CONTENT.HSF.LE.A.4 For exponential models, express as a logarithm the solution to ab  = d where a, c, and d are numbers and the base b is 2, 10, or  e; evaluate the logarithm using graphing technology.

CCSS.MATH.CONTENT.HSF.LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.





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