10.5: Writing Process- Creating a Position Argument
sing Learning Outcomes
By the end of this section, you will be able to:
- Demonstrate brainstorming processes and tools as means to discover topics, ideas, positions, and details.
- Apply recursive strategies for organizing drafting, collaborating, peer reviewing, revising, rewriting, and editing.
- Compose a position argument that integrates the writer’s ideas with those from appropriate sources.
- Give and act on productive feedback to works in progress.
- Apply or challenge common conventions of language or grammar in composing and revising.
Now is the time to try your hand at writing a position argument. Your instructor may provide some possible topics or a singular topic. If your instructor allows you to choose your own topic, consider a general subject you feel strongly about and whether you can provide enough support to develop that subject into an essay. For instance, suppose you think about a general subject such as “adulting.” In looking back at what you have learned while becoming an adult, you think of what you wish you had known during your early teenage years. These thoughts might lead you to brainstorm about details of the effects of money in your life or your friends’ lives. In reviewing your brainstorming, you might zero in on one topic you feel strongly about and think it provides enough depth to develop into a position argument. Suppose your brainstorming leads you to think about negative financial concerns you or some of your friends have encountered. Thinking about what could have helped address those concerns, you decide that a mandated high school course in financial literacy would have been useful. This idea might lead you to formulate your working thesis statement —first draft of your thesis statement—like this: To help students learn how to make sensible financial decisions, a mandatory class in financial literacy should be offered in high schools throughout the country.
Once you decide on a topic and begin moving through the writing process, you may need to fine-tune or even change the topic and rework your initial idea. This fine-tuning may come as you brainstorm, later when you begin drafting, or after you have completed a draft and submitted it to your peers for constructive criticism. These possibilities occur because the writing process is recursive —that is, it moves back and forth almost simultaneously and maybe even haphazardly at times, from planning to revising to editing to drafting, back to planning, and so on.
Figure \(10.5\) Because the writing process is recursive, you can go from any step to another at any time to improve your paper. (CC BY 4.0; Rice University & OpenStax)
Summary of Assignment
Write a position argument on a controversial issue that you choose or that your instructor assigns to you. If you are free to choose your own topic, consider one of the following:
- The legal system would be strengthened if ______________________.
- The growing use of technology in college classrooms is weakening _____________.
- For safety reasons, public signage should be _________________.
- For entrance into college, standardized testing _________________________.
- In relation to the cost of living, the current minimum wage _______________________.
- During a pandemic, America __________________________.
- As a requirement to graduate, college students __________________________.
- To guarantee truthfulness of their content, social media platforms have the right to _________________.
- To ensure inclusive and diverse representation of people of all races, learning via virtual classrooms _________________.
- Segments of American cultures have differing rules of acceptable grammar, so in a college classroom ___________________.
In addition, if you have the opportunity to choose your own topic and wish to search further, take the lead from trailblazer Charles Blow and look to media for newsworthy “trends.” Find a controversial issue that affects you or people you know, and take a position on it. As you craft your argument, identify a position opposing yours, and then refute it with reasoning and evidence. Be sure to gather information on the issue so that you can support your position sensibly with well-developed ideas and evidence.
Another Lens. To gain a different perspective on your issue, consider again the people affected by it. Your position probably affects different people in different ways. For example, if you are writing that the minimum wage should be raised, then you might easily view the issue through the lens of minimum-wage workers, especially those who struggle to make ends meet. However, if you look at the issue through the lens of those who employ minimum-wage workers, your viewpoint might change. Depending on your topic and thesis, you may need to use print or online sources to gain insight into different perspectives.
For additional information about minimum-wage workers, you could consult
- printed material available in your college library;
- databases in your college library; and
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online sources. For instance, you could use a search engine to find details about ◦ pros and cons of raising the minimum wage;
- what happens after the minimum wage is raised;
- how to live on a minimum-wage salary;
- how a raise in minimum wage is funded; and
- minimum wage in various U.S. states.
To gain more insight about your topic, adopt a stance that opposes your original position and brainstorm ideas from that viewpoint. Begin by gathering evidence that would help you refute your previous stance and appeal to your audience.
Quick Launch: Working Thesis Frames and Organization of Ideas
After you have decided on your topic, the next step is to arrive at your working thesis. You probably have a good idea of the direction your working thesis will take. That is, you know where you stand on the issue or problem, but you are not quite sure of how to word your stance to share it with readers. At this point, then, use brainstorming to think critically about your position and to discover the best way to phrase your statement.
For example, after reading an article discussing different state-funded community college programs, one student thought that a similar program was needed in Alabama, her state. However, she was not sure how the program worked. To begin, she composed and answered “reporters’ questions” such as these:
- What does a state-funded community college program do? pays for part or all of the tuition of a two-year college student
- Who qualifies for the program? high school graduates and GED holders
- Who benefits from this? students needing financial assistance, employers, and Alabama residents
- Why is this needed? some can’t afford to go to college; tuition goes up every year; colleges would be more diverse if everyone who wanted to go could afford to go
- Where would the program be available? at all public community colleges
- When could someone apply for the program? any time
- How can the state fund this? use lottery income, like other states
The student then reviewed her responses, altered her original idea to include funding through a lottery, and composed this working thesis:
To provide equal educational opportunities for all residents, the state of Alabama should create a lottery to completely fund tuition at community colleges.
Remember that a strong thesis for a position should
- state your stance on a debatable issue;
- reflect your purpose of persuasion; and
- be based on your opinion or observation.
When you first consider your topic for an argumentative work, think about the reasoning for your position and the evidence you will need—that is, think about the “because” part of your argument. For instance, if you want to argue that your college should provide free Wi-Fi for every student, extend your stance to include “because” and then develop your reasoning and evidence. In that case, your argument might read like this: Ervin Community College should provide free Wi-Fi for all students because students may not have Internet access at home.
Note that the “because” part of your argument may come at the beginning or the end and may be implied in your wording.
As you develop your thesis, you may need help funneling all of your ideas. Return to the possibilities you have in mind, and select the ideas that you think are strongest, that recur most often, or that you have the most to say about. Then use those ideas to fill in one of the following sentence frames to develop your working thesis. Feel free to alter the frame as necessary to fit your position. While there is no limit to the frames that are possible, these may help get you started.
________________ is caused/is not caused by ________________, and _____________ should be done.
Example: A declining enrollment rate in college is caused by high tuition rates, and an immediate freeze on the cost of tuition should be applied.
______________ should/should not be allowed (to) ________________ for a number of reasons.
Example: People who do not wear masks during a pandemic should not be allowed to enter public buildings for a number of reasons.
Because (of) ________________, ___________________ will happen/continue to happen.
Example: Because of a lack of emphasis on STEAM (Science, Technology, Engineering, the Arts, and Mathematics) education in public schools, America will continue to lag behind many other countries.
_____________ is similar to/nothing like ________________ because ______________.
Example: College classes are nothing like high school classes because in college, more responsibility is on the student, the classes are less frequent but more intense, and the work outside class takes more time to complete.
______________ can be/cannot be thought of as __________________ because ______________.
Example: The Black Lives Matter movement can be thought of as an extension of the Civil Rights movement from the 1950s and 1960s because it shares the same mission of fighting racism and ending violence against Black people.
Next, consider the details you will need to support your thesis. The Aristotelian argument structure, named for the Greek philosopher Aristotle, is one that may help you frame the draft of your position argument. For this method, use something like the following chart. In Writing Process: Creating a Position Argument, you will find a similar organizer that you can copy and use for your assignment.
Figure \(10.6\) Position argument essay planning chart. Follow the numbers sequentially to plan your draft. (CC BY 4.0; Rice University & OpenStax)
Drafting: Rhetorical Appeals and Types of Supporting Evidence
To persuade your audience to support your position or argument, consider various rhetorical appeals— ethos, logos, pathos, and kairos—and the types of evidence to support your sound reasoning. See Reasoning Strategies: Improving Critical Thinking for more information on reasoning strategies and types of evidence.
Rhetorical Appeals
To establish your credibility, to show readers you are trustworthy, to win over their hearts, and to set your issue in an appropriate time frame to influence readers, consider how you present and discuss your evidence throughout the paper.
- Appeal to ethos . To establish credibility in her paper arguing for expanded mental health services, a student writer used these reliable sources: a student survey on mental health issues, data from the International Association of Counseling Services (a professional organization), and information from an interview with a campus mental health counselor.
- Appeal to logos . To support her sound reasoning, the student writer approached the issue rationally, using data and credible evidence to explain the current situation and its effects.
- Appeal to pathos . To show compassion and arouse audience empathy, the student writer shared the experience of a student on her campus who struggled with anxiety and depression.
- Appeal to kairos . To appeal to kairos, the student emphasized the immediate need for these services, as more students are now aware of their particular mental health issues and trying to deal with them.
The way in which you present and discuss your evidence will reflect the appeals you use. Consider using sentence frames to reflect specific appeals. Remember, too, that sentence frames can be composed in countless ways. Here are a few frames to get you thinking critically about how to phrase your ideas while considering different types of appeals.
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Appeal to ethos:
According to __________________, an expert in ______________, __________________ should/should not happen because ________________________.
Appeal to ethos: Although ___________________is not an ideal situation for _________________, it does have its benefits. -
Appeal to logos:
If ____________________ is/is not done, then understandably, _________________ will happen.
Appeal to logos: This information suggests that ____________________ needs to be investigated further because ____________________________. -
Appeal to pathos:
The story of _____________________ is uplifting/heartbreaking/hopeful/tragic and illustrates the need for ____________________.
Appeal to pathos: ___________________ is/are suffering from ________________, and that is something no one wants. -
Appeal to kairos:
_________________ must be addressed now because ________________.
Appeal to kairos: These are times when ______________ ; therefore, _____________ is appropriate/ necessary.
Types of Supporting Evidence
Depending on the point you are making to support your position or argument, certain types of evidence may be more effective than others. Also, your instructor may require you to include a certain type of evidence. Choose the evidence that will be most effective to support the reasoning behind each point you make to support your thesis statement. Common types of evidence are these:
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Anecdotes:
short narrative.
Renada G., a junior at Powell College South, worked as a waitress for 15 hours a week during her first three semesters of college. But in her sophomore year, when her parents were laid off during the pandemic, Renada had to increase her hours to 35 per week and sell her car to stay in school. Her grades started slipping, and she began experiencing symptoms of depression and anxiety. When she called the campus health center to make an appointment for counseling, Renada was told she would have to wait two weeks before she could be seen. -
Definition:
explanation that emphasizes the meaning of an idea, term, or concept.
Here is part of how Lyndon B. Johnson defined the Great Society: “But most of all, the Great Society is not a safe harbor, a resting place, a final objective, a finished work. It is a challenge constantly renewed, beckoning us toward a destiny where the meaning of our lives matches the marvelous products of our labor.” -
Description:
evidence that portrays a person, place, thing, or idea with sensory or other vivid details.
Bowen Lake is nestled in verdant foothills, lush with tall grasses speckled with wildflowers. Around the lake, the sweet scent of the purple and yellow flowers fills the air, and the fragrance of the hearty pines sweeps down the hillsides in a westerly breeze. Wood frogs’ and crickets’ songs suddenly stop, as the blowing of moose calling their calves echoes across the lake’s soundless surface. Or this was the scene before the deadly destruction of fires caused by climate change. -
Example:
evidence that illustrates an idea.
When elaborating on America’s beauty being in danger, Johnson says, “The water we drink, the food we eat, the very air that we breathe, are threatened with pollution. Our parks are overcrowded, our seashores overburdened. Green fields and dense forests are disappearing.” -
Expert opinion:
evidence or viewpoints provided by a professional in the field or someone whose ideas are respected on the subject.
Speaking about President Lyndon B. Johnson and the Vietnam War, noted historian and Johnson biographer Doris Kearns Goodwin said, “It seemed the hole in his heart from the loss of work was too big to fill.” -
Fact:
information that is true and can be proven correct or accurate.
Charles Blow has worked at the Shreveport Times , The Detroit News , National Geographic , and The New York Times . -
Interview:
evidence gathered firsthand from a source person, usually in a person-to-person conversation, by phone, or through a remote meeting.
When interviewed by George Rorick and asked about the identities of his readers, Charles Blow said that readers’ emails do not elaborate on descriptions of who the people are. However, “the kinds of comments that they offer are very much on the thesis of the essay.” -
Quotation:
exact words repeated by someone other than the original author or speaker.
In his speech, Lyndon B. Johnson says, “The Great Society is a place where every child can find knowledge to enrich his mind and to enlarge his talents.” -
Statistics:
numerical fact or item of data, usually from a study.
To support the need for change in classrooms, Johnson uses these statistics: “Each year more than 100,000 high school graduates, with proved ability, do not enter college because they cannot afford it. And if we cannot educate today’s youth, what will we do in 1970 when elementary school enrollment will be five million greater than 1960?” - Visuals: graphs, photographs, charts, or maps used in addition to written or spoken information.
Figure \(10.7\) Bar graph of gene counts. Graphics like this one present information visually and concisely. (credit: “Between a Chicken and a Grape: Estimating the Number of Human Genes,” by Pertea, Mihaela, and Steven L. Salzberg/flickr, CC BY 2.0)
Brainstorm for Supporting Points
Use one or more brainstorming techniques, such as a web diagram as shown in Figure 10.8 or the details generated from “because” statements, to develop ideas or particular points in support of your thesis. Your goal is to get as many ideas as possible. At this time, do not be concerned about how ideas flow, whether you will ultimately use an idea (you want more ideas than will end up in your finished paper), spelling, or punctuation.
Figure \(10.8\) Idea web (CC BY 4.0; Rice University & OpenStax)
When you have finished, look over your brainstorming. Then circle three to five points to incorporate into your draft. Also, plan to answer “ reporters’ questions ” to provide readers with any needed background information. For example, the student writing about the need for more mental health counselors on her campus created and answered these questions:
- What is needed? More mental health counseling is needed for Powell College South.
- Who would benefit from this? The students and faculty would benefit.
- Why is this needed? The college does not have enough counselors to meet all students’ needs.
- Where are more counselors needed? More counselors are needed at the south campus.
- When are the counselors needed? Counselors need to be hired now and be available both day and night to accommodate students’ schedules.
- How can the college afford this? Instead of hiring daycare workers, the college could use students and faculty from the Early Childhood Education program to run the program and use the extra money to pay the counselors.
Using Logic
In a position argument, the appropriate use of logic is especially important for readers to trust what you write. It is also important to look for logic in material you read and possibly cite in your paper so that you can determine whether writers’ claims are reasonable. Two main categories of logical thought are inductive reasoning and deductive reasoning .
- Inductive reasoning moves from specific to broad ideas. You begin by collecting details, observations, incidents, or facts; examining them; and drawing a conclusion from them. Suppose, for example, you are writing about attendance in college classes. For three weeks, you note the attendance numbers in all your Monday, Wednesday, and Friday classes (specific details), and you note that attendance is lower on Friday than on the other days (a specific detail). From these observations, you determine that many students prefer not to attend classes on Fridays (your conclusion).
- Deductive reasoning moves from general to specific ideas. You begin with a hypothesis or premise , a general concept, and then examine possibilities that would lead to a specific and logical conclusion. For instance, suppose you think that opportunities for foreign students at your college are inadequate (general concept). You examine the specific parts of that concept (e.g., whether your college provides multicultural clubs, help with language skills, or work-study opportunities) and determine that those opportunities are not available. You then determine that opportunities for foreign students are lacking at your college.
Logical Fallacies and Propaganda
Fallacies are mistakes in logic. Readers and writers should be aware of these when they creep into writing, indicating that the points the writers make may not be valid. Two common fallacies are hasty generalizations and circular arguments. See Glance at Genre: Rhetorical Strategies for more on logical fallacies.
- A hasty generalizationis a conclusion based on either inadequate or biased evidence. Consider this statement: “Two students in Math 103 were nervous before their recent test; therefore, all students in that class must have text anxiety.” This is a hasty generalization because the second part of the statement (the generalization about all students in the class) is inadequate to support what the writer noted about only two students.
- A circular argument is one that merely restates what has already been said. Consider this statement: “The Hate U Give is a well-written book because Angie Thomas, its author, is a good novelist.” The statement that Thomas is a good novelist does not explain why her book is well written.
In addition to checking work for fallacies, consider propaganda, information worded so that it endorses a particular viewpoint, often of a political nature. Two common types of propaganda are bandwagon and fear.
- In getting on the bandwagon , the writer encourages readers to conform to a popular trend and endorse an opinion, a movement, or a person because everyone else is doing so. Consider this statement: “Everyone is behind the idea that 7 a.m. classes are too early and should be changed to at least 8 a.m. Shouldn’t you endorse this sensible idea, too?”
- In using fear, the writer presents a dire situation, usually followed by what could be done to prevent it. Consider this statement: “Our country is at a turning point. Enemies threaten us with their power, and our democracy is at risk of being crushed. The government needs a change, and Paul Windhaus is just the man to see we get that change.” This quotation appeals to fear about the future of the country and implies that electing a certain individual will solve the predicted problems.
Figure \(10.9\) Propaganda is often used during times of war or crisis and is often presented visually. This World War I poster from the U.S. Navy appeals for men to enlist. It points out men in the navy have an opportunity to “see the world, save money, learn a trade, and serve their country” but avoids mentioning the likelihood of being in lifethreatening situations during combat. (credit: “World War I Posters” by Ruttan, Charles E./Library of Congress Prints and Photographs Division, Public Domain)
Organize the Paper
To begin, write your thesis at the top of a blank page. Then select points from your brainstorming and reporters’ questions to organize and develop support for your thesis. Keep in mind that you can revise your thesis whenever needed.
To begin organizing her paper on increased mental health services on her college campus, the student wrote this thesis at the top of a page:
Because mental health is a major concern at Powell College South, students could benefit from expanding the services offered.
Next she decided the sequence in which to present the points. In a position or an argument essay, she could choose one of two methods: thesis-first organization or delayed-thesis organization.
Thesis-First Organization
Leading with a thesis tells readers from the beginning where you stand on the issue. In this organization, the thesis occupies both the first and last position in the essay, making it easy for readers to remember.
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Introduce the issue and assert your thesis.
Make sure the issue has at least two debatable sides. Your thesis establishes the position from which you will argue. Writers often state their thesis as the last sentence in the first paragraph, as the student writer has done:
The problem of mental health has become front-page news in the last two months. Hill’s Herald, Powell College South’s newspaper, reported 14 separate incidents of students who sought counseling but could not get appointments with college staff. Since mental health problems are widespread among the student population, the college should hire more health care workers to address this problem. -
Summarize the counterclaims.
Before elaborating on your claims, explain the opposition’s claims. Including this information at the beginning gives your argument something to focus on—and refute—throughout the paper. If you ignore counterclaims, your argument may appear incomplete, and readers may think you have not researched your topic sufficiently. When addressing a counterclaim, state it clearly, show empathy for those who have that view, and then immediately refute it with support developed through reasoning and evidence. Squeezing the counterclaims between the thesis and the evidence reserves the strongest places—the opening and closing—for your position.
Counterclaim 1: Powell College South already employs two counselors, and that number is sufficient to meet the needs of the student population.
Counterclaim 2: Students at Powell College South live in a metropolitan area large enough to handle their mental health needs. -
Refute the counterclaims.
Look for weak spots in the opposition’s argument, and point them out. Use your opponent’s language to show you have read closely but still find problems with the claim. This is the way the writer refuted the first counterclaim:
While Powell College South does employ two counselors, those counselors are overworked and often have no time slots available for students who wish to make appointments. -
State and explain your points, and then support them with evidence.
Present your points clearly and precisely, using Reasoning Strategies: Improving Critical Thinking to explain and cite your evidence. The writer plans to use a problem-solution reasoning strategy to elaborate on these three points using these pieces of evidence:
Point 1: Wait times are too long.
Kay Payne, one of the campus counselors, states that the wait time for an appointment with her is approximately 10 days.
Point 2: Mental health issues are widespread within the student community.
In a recent on-campus student survey, 75 percent of 250 students say they have had some kind of mental health issues at some point in their life.
Point 3: The staff-to-student ratio is too high.
The International Accreditation of Counseling Services states that the recommended ratio is one full-time equivalent staff member for every 1,000 to 1,500 students. -
Restate your position as a conclusion.
Near the end of your paper, synthesize your accumulated evidence into a broad general position, and restate your thesis in slightly different language.
The number of students who need mental health counseling is alarming. The recent news articles that attest to their not being able to schedule appointments add to the alarm. While Powell College South offers some mental health counseling, the current number of counselors and others who provide health care is insufficient to handle the well-being of all its students. Action must be taken to address this problem.
Delayed-Thesis Organization
In this organizational pattern, introduce the issue and discuss the arguments for and against it, but wait to take a side until late in the essay. By delaying the stance, you show readers you are weighing the evidence, and you arouse their curiosity about your position. Near the end of the paper, you explain that after carefully considering both pros and cons, you have arrived at the most reasonable position.
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Introduce the issue.
Here, the writer begins with action that sets the scene of the problem.
Tapping her foot nervously, Serena looked at her watch again. She had been waiting three hours to see a mental health counselor at Powell College South, and she did not think she could wait much longer. She had to get to work. -
Summarize the claims for one position.
Before stating which side you support, explain how the opposition views the issue. This body paragraph presents evidence about the topic of more counselors:
Powell College South has two mental health counselors on staff. If the college hires more counselors, more office space will have to be created. Currently Pennington Hall could accommodate those counselors. Additional counselors would allow more students to receive counseling. -
Refute the claims you just stated.
Still not stating your position, point out the other side of the issue.
While office space is available in Pennington Hall, that location is far from ideal. It is in a wooded area of campus, six blocks from the nearest dorm. Students who would go there might be afraid to walk through the woods or might be afraid to walk that distance. The location might deter them from making appointments. -
Now give the best reasoning and evidence to support your position.
Because this is a delayed-thesis organization, readers are still unsure of your stance. This section should be the longest and most carefully documented part of the paper. After summarizing and refuting claims, the writer then elaborates on these three points using problem-solution reasoning supported by this evidence as discussed in Reasoning Strategies: Improving Critical Thinking, implying her position before moving to the conclusion, where she states her thesis.
Point 1: Wait times are too long.
Kay Payne, one of the campus counselors, states that the wait time for an appointment with her is approximately 10 days.
Point 2: Mental health issues are widespread within the student community.
In a recent on-campus student survey, 75 percent of 250 students say they have had some kind of mental health issues at some point in their life.
Point 3: The staff-to-student ratio is too high.
The International Association of Counseling Services states that one full-time equivalent staff member for every 1,000 to 1,500 students is the recommended ratio. -
State your thesis in your conclusion.
Your rhetorical strategy is this: after giving each side a fair hearing, you have arrived at the most reasonable conclusion.
According to the American Psychological Association, more than 40 percent of all college students suffer from some form of anxiety. Powell College South students are no different from college students elsewhere: they deserve to have adequate mental health counseling.
Compose
Drafting begins when you organize your evidence or research notes and then put them into some kind of written form. As you write, focus on building body paragraphs through the techniques presented in Reasoning Strategies: Improving Critical Thinking that show you how to support your position and then add evidence. Using a variety of evidence types builds credibility with readers. Remember that the recursiveness of the writing process allows you to move from composing to gathering evidence and back to brainstorming ideas or to organizing your draft at any time. Move around the writing process as needed.
Keep in mind that a first draft is just a beginning—you will revise it into a better work in later drafts. Your first draft is sometimes called a discovery draft because you are discovering how to shape your paper: which ideas to include and how to support those ideas. These suggestions and graphic organizer may be helpful for your first draft:
- Write your thesis at the top of the paper.
- Compose your body paragraphs: those that support your argument through reasoning strategies and those that address counterclaims.
- Leave your introduction, conclusion, and title for later drafts.
Use a graphic organizer like Table \(10.1\) to focus points, reasoning, and evidence for body paragraphs. You are free to reword your thesis, reasoning, counterclaim(s), refutation of counterclaim(s), concrete evidence, and explanation/elaboration/clarification at any time. You are also free to adjust the order in which you present your reasoning, counterclaim(s), and refuting of counterclaim(s).
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Composing Your First Draft Position Argument |
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Name ________________________________________________________________________ |
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Thesis: In your thesis, remember to include 1) an explanation of the issue and 2) your position about what should happen regarding the issue. Issue: My position: Thesis as a single declarative sentence: ________ Background information: Reporters’ questions: |
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Point 1 in support of thesis: Reasoning: explanation/elaboration/clarification: Concrete evidence: |
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Point 2 in support of thesis: Reasoning: explanation/elaboration/clarification: Concrete evidence: |
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Point 3 in support of thesis: Reasoning: explanation/elaboration/clarification: Concrete evidence: |
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Points 4 & 5 in support of thesis: Reasoning: explanation/elaboration/clarification: Concrete evidence: |
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Counterclaim: Reasoning: explanation/elaboration/clarification: |
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Refuting of counterclaim: Reasoning: explanation/elaboration/clarification: |
Develop a Writing Project through Multiple Drafts
Your first draft is a kind of experiment in which you are concerned with ideas and with getting the direction and concept of the paper clear. Do not think that your first draft must be perfect; remind yourself that you are just honing your work. In most serious writing, every phase of the process can be considered recursive, helping you shape the best paper possible.
Peer Review: Critical Thinking and Counterclaims
After you have completed the first draft, begin peer review. Peer reviewers can use these sentence starters when thinking critically about overall strengths and developmental needs.
- One point about your position that I think is strong is ______ because ________.
- One point about your position that I think needs more development is _____ because _______.
- One area that I find confusing is _____________; I was confused about _______.
- One major point that I think needs more explanation or detail is _______.
- In my opinion, the purpose of your paper is to persuade readers _______.
- In my opinion, the audience for your paper is _______.
- One area of supporting evidence that I think could use more development is _______.
- One counterclaim you include is ________________.
- Your development of the counterclaim is __________________ because ________________.
Refuting Counterclaims
Peer reviewers are especially helpful with position and argument writing when it comes to refuting counterclaims. Have your peer reviewer read your paper again and look for supporting points and ideas to argue against , trying to break down your argument. Then ask your reviewer to discuss the counterclaims and corresponding points or ideas in your paper. This review will give you the opportunity to think critically about ways to refute the counterclaims your peer reviewer suggests.
Revising: Reviewing a Draft and Responding to Counterclaims
Revising means reseeing, rereading, and rethinking your thoughts on paper until they fully match your intention. Mentally, it is conceptual work focused on units of meaning larger than the sentence. Physically, it is cutting, pasting, deleting, and rewriting until the ideas are satisfying. Be ready to spend a great deal of time revising your drafts, adding new information and incorporating sources smoothly into your prose.
The Revising Process
To begin revising, return to the basic questions of topic ( What am I writing about? ), purpose ( Why am I writing about this topic? ), audience ( For whom am I writing? ), and culture ( What is the background of the people for whom I am writing? ).
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Questions about the topic.
Be sure your topic and thesis focus on your position, and omit extra material. Answer these questions:
- What is the general scope of my topic? __________________________________
- What is my thesis? __________________________________________________
- Does my thesis focus on my topic? ______________________________________
- Does my thesis clearly state my position? ______________________
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Questions about purpose.
It is often easier to locate your purpose—or lack of purpose—after you have written a draft or two. Answer these questions:
- What do I hope to accomplish in writing about this topic? ____________________
- Do all parts of the paper advance this purpose? ____________________________
- Does my paper focus on my argument or position? _________________________
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Questions about audience.
Make sure your paper is aimed accurately at your readers. Answer these questions:
- What does my audience know about this subject? _______________________
- What does my audience need to know to understand the point of my paper? _______________________________________________________________
- What questions or objections do I anticipate from my audience? ___________ ________________________________________________________________
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Questions about culture.
Your paper should reflect consideration of cultural differences, if any, between you and your audience. Answer these questions:
- What is the culture of the people for whom I am writing? Do all readers share the same culture? ________________________________________________________
- How do my beliefs, values, and customs differ from those of my audience? ____________________________________________________________________
- How do the cultures of the authors of sources I cite differ from my culture or the culture(s) of my audience? ______________________________________________
Because of the recursivity of the writing process, returning to these questions will help you fine-tune the language and structure of your writing and target the support you develop for your audience.
Responding to Counterclaims
The more complex the issue, the more opposing sides it may have. For example, a writer whose position is that Powell College South Campus should offer daycare to its students with children might find opposition for different reasons. Someone may oppose the idea out of concern for cost; someone else may support the idea if the daycare is run on a volunteer basis; someone else may support the idea if the services are offered off campus.
As you revise, continue studying your peer reviewer’s comments about counterclaims. If you agree with any counterclaim, then say so in the paragraph in which you address counterclaims. This agreement will further establish your credibility by showing your fairness and concern for the issue. Look over your paper and peer review comments, and then consider these questions:
- In what ways do you address realistic counterclaims? What other counterclaims should you address? Should you add to or replace current counterclaims?
- In what ways do you successfully refute counterclaims? What other refutations might you include?
- Are there any counterclaims with which you agree? If so, how do you concede to them in your paper? In what ways does your discussion show fairness?
After completing your peer review and personal assessment, make necessary revisions based on these notes. See Annotated Student Sample an example of a student’s argumentative research essay. Note how the student
- presents the argument;
- supports the viewpoint with reasoning and evidence;
- includes support in the form of facts, opinions, paraphrases, and summaries;
- provides citations (correctly formatted) about material from other sources in the paper;
- uses ethos, pathos, and logos throughout the paper; and
- addresses counterclaims (dissenting opinions).


8.8: Glossary
Ballads – a song form used often in folk music, which is used to tell a story that usually contains a moral or lesson.
Bebop - a style of small group jazz developed in the late 1940s, which featured fast moving harmonies, angular melodies, and highly complex rhythms
Big Band – large jazz ensembles (15-20 members) popular in the 1930’s and 1940’s. The term “Big Band” also refers to the era in which these bands were popular.
Bluegrass – a variation of country music featuring fiddle, guitar, mandolin, bass guitar, and the five-string banjo that developed largely in the Appalachian region
Blues – a style of music that, at the turn of the twentieth century, began to form out of African American work songs, field hollers, and spirituals. Today, the word “blues” is used loosely and can refer to feeling sad or down, to any song played in a bluesy style, or more specifically, to a song that follows a blues form, which is a twelve-bar strophic song form.
Broadway Musical – a style of Musical Theatre, which integrated a cohesive plot with songs and dances that advanced that plot. Broadway specifically refers to the street of the same name in New York City that became known for this style.
Children’s Song – a type of folk song designed to teach a simple lesson. They are often simple to sing and easy to remember.
Contemporary Country – a mixture of rock rhythm sections and a singer singing with a country accent about many of the same topics that traditional country singers have used over the decades.
Contemporary R&B – generally refers to music with jazz, gospel, and funk roots that uses electronic instruments, drums, horns, and vocals.
Country Music – a term describing a broad variety of musical styles including Bluegrass, Hillbilly Music, and Contemporary Country. Generally speaking, most types of music that fall under this category originated in the American South (although it also encompasses Western Swing and cowboy songs) and features a singing style with a distinctly rural southern accent, as well as an instrumentation that favors string instruments such as the banjo, guitar, or fiddle.
Dance Music – music written for dancing. The instrumentation of various types of folk dance music varies with the style.
Dixieland – an early form of jazz developed in New Orleans during the turn of the twentieth century featuring syncopated rhythms, improvised solos and harmonies, as well as a common instrumentation that included trumpet, clarinet, saxophone, trombone, tuba, banjo, piano, guitar, and drums.
Folk Music – a term used to describe a wide variety of musical forms that developed within different cultures, often for different reasons. Folk music is often passed down not through written music, but orally from one generation to another.
Hillbilly Music – an early form of country music, Hillbilly Music was an alternative to the jazz and dance music of the 1920s and was portrayed as wholesome music of the “good old days.”
Honky Tonk Music – a country combo style that became quite popular in the 1940s and 1950s. Originally performed in saloons known as “honky tonks,’ many of the songs dealt with subjects associated with honky tonks such as infidelity and drinking.
Improvisation – the act of creating melodies and harmonies on the spot without reading the music off a page.
Minstrel Show – an American form of theatre developed in the nineteenth century and featuring white performers in blackface performing in a variety show, which depicted black characters as happy participants in romanticized versions of the American slave south.
Musical Theatre – a type of dramatic performance that tells a story through dialogue, with singing and dancing added to support and move the plot along.
New Orleans Jazz
– (see Dixieland)
Operetta
– a “light opera” developed in the nineteenth century that required classically
trained singers, but featured less complex music than a typical opera.
Protest Song – a type of folk song written to directly, or by suggestion, voice complaints about some injustice.
Ragtime – a musical genre developed near the turn of the twentieth century that featured syncopated rhythms. The style became nationally popular after being widely published as sheet music.
Rap – a form of spoken word delivered over a beat. It can be improvised or written out in advance.
Rhythm and Blues (R&B) – a term originally referring to music recorded by black musicians and intended for use by the African American community. The term has evolved throughout the years and encompasses several different musical styles, including soul, funk and now contemporary R&B.
Rock and Roll – a style of music that grew out of Rhythm and Blues and came into prominence during the 1950s. The style features a strong backbeat and often features electric guitar, bass and drums. The style is now known as “rock” has spawned many subgenres.
Sampling – a technique in which a clip of a preexisting song is isolated and looped, often as a background for a rapper
Scratching – the technique of improvising a rhythmic solo on one turntable over a beat Swing – a term used to describe the rhythmic bounce that characterizes the jazz style. The
term can also refer to the big band music of the 1930s and 1940s.
Syncopation – the act of disrupting the normal pattern of accents in a piece of music by emphasizing what would normally be weak beats.
Western Swing – a style of country music that developed in western cities and borrowed instruments from the dance band such as saxophones, trombones, trumpets, piano, bass, and drums.
Work Songs – a type of folk song devised to help groups of people perform physical work. The music usually uses the tempo of the work itself and was sung by lumberjacks, railroad workers, and prison chain gangs, among others.


1.22.1: Volume: Partial Molar: General Analysis
At temperature \(\mathrm{T}\), pressure \(\mathrm{p}\) and equilibrium, the volume of a closed system containing i-chemical substances where the amounts can be independently varied, is defined by the following equation.
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \ldots \ldots \mathrm{n}_{\mathrm{i}}\right] \nonumber \]
Or, in general terms according to Euler’s theorem,
\[\mathrm{V}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}} \nonumber \]
where
\[\mathrm{V}_{\mathrm{j}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i} \neq \mathrm{j}}} \nonumber \]
The general differential of equation (b) has the following form.
\[\mathrm{dV}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mathrm{dV} \mathrm{V}_{\mathrm{j}}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{V}_{\mathrm{j}} \, \mathrm{dn} \mathrm{n}_{\mathrm{j}} \nonumber \]
The general differential of equation (a) has the following form
\[\mathrm{dV}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dT}+\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i} \neq \mathrm{j}}} \, \mathrm{dn}_{\mathrm{j}} \nonumber \]
Comparison of equations (d) and (e) shows that
\[0=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dT}-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mathrm{dV} \mathrm{j}_{\mathrm{j}} \nonumber \]
Equation (f) is the Gibbs-Duhem Equation with respect to the volumetric properties of a closed system at equilibrium.
Application
A given closed system contains \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The system is at equilibrium where \(\mathrm{G}\) is a minimum, the affinity for spontaneous change \(\mathrm{A}\) is zero and the composition-organisation \(\xi^{\mathrm{eq}}\). The dependent variable volume \(\mathrm{V}\) is defined using a set of independent variables; equation (g).
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right] \nonumber \]
Equation (k) has an interesting property. If we multiply the extensive variables \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\) by a factor \(\mathrm{k}\), the volume of the system equals (\(\mathrm{V}. \mathrm{~k}\)). In terms of Euler’s Theorem [1], the variable \(\mathrm{V}\) linked to the variables \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\) is a homogeneous function of the first degree. The important consequence is the following key relation.
\[\mathrm{V}=\mathrm{n}_{1} \, \mathrm{V}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}} \nonumber \]
where
\[\mathrm{V}_{1}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})} \nonumber \]
and
\[\mathrm{V}_{\mathrm{j}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(1)} \nonumber \]
We do not have to specify the conditions ‘at constant \(\mathrm{T}\) and \(\mathrm{p}\)’ in conjunction with equation (h) which is a mathematical identity.
Footnote
[1] Degree of Homogeneity
At temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the volume of a closed system containing \(\mathrm{n}_{j}\) moles of each chemical substance \(j\) is given by
\[\mathrm{V}=\mathrm{V}\left[\mathrm{n}_{1}, \mathrm{n}_{2} \ldots \ldots \ldots \ldots . \ldots \mathrm{n}_{\mathrm{k}}\right] \nonumber \]
The property volume has unit degree of homogeneity. That is to say – if the amount of each substance is increased by a factor \(\lambda\) then the volume increases by the same factor. Thus
\[\mathrm{V}\left[\lambda \mathrm{n}_{1}, \lambda \mathrm{n}_{2} \ldots \ldots \ldots \ldots . . \lambda \mathrm{n}_{\mathrm{k}}\right]=\lambda \, \mathrm{V}\left[\mathrm{n}_{1}, \mathrm{n}_{2} \ldots \ldots \ldots \ldots . \ldots \mathrm{n}_{\mathrm{k}}\right] \nonumber \]


7.1: The Ellipse
In this section, you will:
- Write equations of ellipses in standard form.
- Graph ellipses centered at the origin.
- Graph ellipses not centered at the origin.
- Solve applied problems involving ellipses.
Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in Figure \(\PageIndex{1}\), is such a room. 1 It is an oval-shaped room called a whispering chamber because the shape makes it possible for sound to travel along the walls. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.
Writing Equations of Ellipses in Standard Form
A conic section, or conic , is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in Figure \(\PageIndex{2}\)
Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points \((x,y)\) in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci ).
We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See Figure \(\PageIndex{3}\).
Every ellipse has two axes of symmetry. The longer axis is called the major axis , and the shorter axis is called the minor axis . Each endpoint of the major axis is the vertex of the ellipse (plural: vertices ), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See Figure \(\PageIndex{4}\).
In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the x - and y -axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.
To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.
Deriving the Equation of an Ellipse Centered at the Origin
To derive the equation of an ellipse centered at the origin, we begin with the foci (−c,0) and (c,0). The ellipse is the set of all points (x,y) such that the sum of the distances from (x,y) to the foci is constant, as shown in Figure \(\PageIndex{5}\).
If \((a, 0)\) is a vertex of the ellipse, the distance from \((-c, 0)\) to \((a, 0)\) is \(a-(-c)=a+c\). The distance from \((c, 0)\) to \((a, 0)\) is \(a-c\). The sum of the distances from the foci to the vertex is
\[
(a+c)+(a-c)=2 a
\]
If \((x, y)\) is a point on the ellipse, then we can define the following variables:
\[
\begin{array}
d_1=\text { the distance from }(-c, 0) \text { to }(x, y) \\[6pt]
d_2=\text { the distance from }(c, 0) \text { to }(x, y)
\end{array} \notag
\]
By the definition of an ellipse, \(d_1+d_2\) is constant for any point \((x, y)\) on the ellipse. We know that the sum of these distances is \(2 a\) for the vertex \((a, 0)\). It follows that \(d_1+d_2=2 a\) for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.
|
\(d_1+d_2=\sqrt{(x-(-c))^2+(y-0)^2}+\sqrt{(x-c)^2+(y-0)^2}=2 a\) |
Distance formula |
| \(\sqrt{(x+c)^2+y^2}+\sqrt{(x-c)^2+y^2}=2 a\) | Simplify expressions. |
| \(\sqrt{(x+c)^2+y^2}=2 a-\sqrt{(x-c)^2+y^2}\) | Move radical to opposite side. |
| \((x+c)^2+y^2=\left[2 a-\sqrt{(x-c)^2+y^2}\right]^2\) | Square both sides. |
| \(x^2+2 c x+c^2+y^2=4 a^2-4 a \sqrt{(x-c)^2+y^2}+(x-c)^2+y^2\) | Expand the squares. |
| \(x^2+2 c x+c^2+y^2=4 a^2-4 a \sqrt{(x-c)^2+y^2}+x^2-2 c x+c^2+y^2\) | Expand remaining squares. |
| \(2 c x=4 a^2-4 a \sqrt{(x-c)^2+y^2}-2 c x\) | Combine like terms. |
|
\(4 c x-4 a^2=-4 a \sqrt{(x-c)^2+y^2}\) |
Isolate the radical. |
| \(c x-a^2=-a \sqrt{(x-c)^2+y^2}\) | Divide by 4 . |
| \({\left[c x-a^2\right]^2=a^2\left[\sqrt{(x-c)^2+y^2}\right]^2}\) | Square both sides. |
| \(c^2 x^2-2 a^2 c x+a^4=a^2\left(x^2-2 c x+c^2+y^2\right)\) | Expand the squares. |
| \(c^2 x^2-2 a^2 c x+a^4=a^2 x^2-2 a^2 c x+a^2 c^2+a^2 y^2\) | Distribute \(a^2\). |
| \(a^2 x^2-c^2 x^2+a^2 y^2=a^4-a^2 c^2\) | Rewrite. |
| \(x^2\left(a^2-c^2\right)+a^2 y^2=a^2\left(a^2-c^2\right)\) | Factor common terms. |
| \(x^2 b^2+a^2 y^2=a^2 b^2\) | Set \(b^2=a^2-c^2\). |
| \(\dfrac{x^2 b^2}{a^2 b^2}+\dfrac{a^2 y^2}{a^2 b^2}=\dfrac{a^2 b^2}{a^2 b^2}\) | Divide both sides by \(a^2 b^2\). |
| \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) | Simplify. |
Thus, the standard equation of an ellipse is \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\). This equation defines an ellipse centered at the origin. If a>b, the ellipse is stretched further in the horizontal direction, and if \(b>a\), the ellipse is stretched further in the vertical direction.
Writing Equations of Ellipses Centered at the Origin in Standard Form
Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we’ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.
The key features of the ellipse are its center, vertices, co-vertices, foci, and lengths and positions of the major and minor axes. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.
STANDARD FORMS OF THE EQUATION OF AN ELLIPSE WITH CENTER \((0,0)\)
The standard form of the equation of an ellipse with center \((0,0)\) and major axis on the \(x\)-axis is
\[
\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1
\]
where
- \(a>b\)
- the length of the major axis is \(2 a\)
- the coordinates of the vertices are \(( \pm a, 0)\)
- the length of the minor axis is \(2 b\)
- the coordinates of the co-vertices are \((0, \pm b)\)
- the coordinates of the foci are \(( \pm c, 0)\), where \(c^2=a^2-b^2\). See Figure 6 a
The standard form of the equation of an ellipse with center \((0,0)\) and major axis on the \(y\)-axis is
\[
\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1
\]
where
- \(a>b\)
- the length of the major axis is \(2 a\)
- the coordinates of the vertices are \((0, \pm a)\)
- the length of the minor axis is \(2 b\)
- the coordinates of the co-vertices are \(( \pm b, 0)\)
- the coordinates of the foci are \((0, \pm c)\), where \(c^2=a^2-b^2\). See Figure \(6 \mathbf{b}\)
Note that the vertices, co-vertices, and foci are related by the equation \(c^2=a^2-b^2\). When we are given the coordinates of the foci and vertices of an ellipse, we can use this relationship to find the equation of the ellipse in standard form.
Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.
1. Determine whether the major axis lies on the \(x\) - or \(y\)-axis.
a. If the given coordinates of the vertices and foci have the form \(( \pm a, 0)\) and \(( \pm c, 0)\) respectively, then the major axis is the \(x\)-axis. Use the standard form \(\dfrac{x^2}{a^2}+d\dfrac{y^2}{b^2}=1\).
b. If the given coordinates of the vertices and foci have the form \((0, \pm a)\) and \((0, \pm c)\), respectively, then the major axis is the \(y\)-axis. Use the standard form \(\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1\).
2. Use the equation \(c^2=a^2-b^2\), along with the given coordinates of the vertices and foci, to solve for \(b^2\).
3. Substitute the values for \(a^2\) and \(b^2\) into the standard form of the equation determined in Step 1 .
Writing the Equation of an Ellipse Centered at the Origin in Standard Form
What is the standard form equation of the ellipse that has vertices \(( \pm 8,0)\) and foci \(( \pm 5,0)\) ?
- Answer
-
The foci are on the \(x\)-axis, so the major axis is the \(x\)-axis. Thus, the equation will have the form
\[
\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1
\]The vertices are \(( \pm 8,0)\), so \(a=8\) and \(a^2=64\).
The foci are \(( \pm 5,0)\), so \(c=5\) and \(c^2=25\).
We know that the vertices and foci are related by the equation \(c^2=a^2-b^2\). Solving for \(b^2\), we have:
\[
\begin{array}{l}
c^2=a^2-b^2 \\
25=64-b^2 \quad \text { Substitute for } c^2 \text { and } a^2 . \\
b^2=39 \quad \text { Solve for } b^2 \text {. } \\
\end{array}
\]Now we need only substitute \(a^2=64\) and \(b^2=39\) into the standard form of the equation. The equation of the ellipse is \(\dfrac{x^2}{64}+\dfrac{y^2}{39}=1\).
What is the standard form equation of the ellipse that has vertices \((0, \pm 4)\) and foci \((0, \pm \sqrt{15})\) ?
Can we write the equation of an ellipse centered at the origin given coordinates of just one focus and vertex?
Yes. Ellipses are symmetrical, so the coordinates of the vertices of an ellipse centered around the origin will always have the form \(( \pm a, 0)\) or \((0, \pm a)\). Similarly, the coordinates of the foci will always have the form \(( \pm c, 0)\) or \((0, \pm c)\). Knowing this, we can use \(a\) and \(c\) from the given points, along with the equation \(c^2=a^2-b^2\), to find \(b^2\).
Writing Equations of Ellipses Not Centered at the Origin
Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translated hh units horizontally and \(k\) units vertically, the center of the ellipse will be \((h,k)\). This translation results in the standard form of the equation we saw previously, with \(x\) replaced by \((x−h)\) and \(y\) replaced by \((y−k)\).
STANDARD FORMS OF THE EQUATION OF AN ELLIPSE WITH CENTER \((H, K)\)
The standard form of the equation of an ellipse with center \((h, k)\) and major axis parallel to the \(x\)-axis is
\[
\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1
\]
where
- \(a>b\)
- the length of the major axis is \(2 a\)
- the coordinates of the vertices are \((h \pm a, k)\)
- the length of the minor axis is \(2 b\)
- the coordinates of the co-vertices are \((h, k \pm b)\)
- the coordinates of the foci are \((h \pm c, k)\), where \(c^2=a^2-b^2\). See Figure \(7 \mathbf{a}\)
The standard form of the equation of an ellipse with center \((h, k)\) and major axis parallel to the \(y\)-axis is
\[
\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1
\]
where
- \(a>b\)
- the length of the major axis is \(2 a\)
- the coordinates of the vertices are \((h, k \pm a)\)
- the length of the minor axis is \(2 b\)
- the coordinates of the co-vertices are \((h \pm b, k)\)
- the coordinates of the foci are \((h, k \pm c)\), where \(c^2=a^2-b^2\). See Figure \(7 \mathbf{b}\)
Just as with ellipses centered at the origin, ellipses that are centered at a point \((h, k)\) have vertices, covertices, and foci that are related by the equation \(c^2=a^2-b^2\). We can use this relationship along with the midpoint and distance formulas to find the equation of the ellipse in standard form when the vertices and foci are given.
Given the vertices and foci of an ellipse not centered at the origin, write its equation in standard form.
1. Determine whether the major axis is parallel to the \(x\) - or \(y\)-axis.
a. If the \(y\)-coordinates of the given vertices and foci are the same, then the major axis is parallel to the \(x\)-axis. Use the standard form \(\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\).
b. If the \(x\)-coordinates of the given vertices and foci are the same, then the major axis is parallel to the \(y\)-axis. Use the standard form \(\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1\).
2. Identify the center of the ellipse \((h, k)\) using the midpoint formula and the given coordinates for the vertices.
3. Find \(a^2\) by solving for the length of the major axis, \(2 a\), which is the distance between the given vertices.
4. Find \(c^2\) using \(h\) and \(k\), found in Step 2, along with the given coordinates for the foci.
5. Solve for \(b^2\) using the equation \(c^2=a^2-b^2\).
6. Substitute the values for \(h, k, a^2\), and \(b^2\) into the standard form of the equation determined in Step 1.
Writing the Equation of an Ellipse Centered at a Point Other Than the Origin
What is the standard form equation of the ellipse that has vertices \((-2,-8)\) and \((-2,2)\) and foci \((-2,-7)\) and \((-2,1)\) ?
- Answer
-
The \(x\)-coordinates of the vertices and foci are the same, so the major axis is parallel to the \(y\)-axis. Thus, the equation of the ellipse will have the form
\[
\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1
\]First, we identify the center, \((h, k)\). The center is halfway between the vertices, \((-2,-8)\) and \((-2,2)\). Applying the midpoint formula, we have:
\[
\begin{aligned}
(h, k) & =\left(\dfrac{-2+(-2)}{2}, \dfrac{-8+2}{2}\right) \\
& =(-2,-3)
\end{aligned}
\]Next, we find \(a^2\). The length of the major axis, \(2 a\), is bounded by the vertices. We solve for \(a\) by finding the distance between the \(y\)-coordinates of the vertices.
\[
\begin{array}{c}
2 a=2-(-8) \\
2 a=10 \\
a=5
\end{array}
\]So \(a^2=25\).
Now we find \(c^2\). The foci are given by \((h, k \pm c)\). So, \((h, k-c)=(-2,-7)\) and \((h, k+c)=(-2,1)\). We substitute \(k=-3\) using either of these points to solve for \(c\).
\[
\begin{array}{c}
k+c=1 \\
-3+c=1 \\
c=4
\end{array}
\]So \(c^2=16\).
Next, we solve for \(b^2\) using the equation \(c^2=a^2-b^2\).
\[
\begin{array}{c}
c^2=a^2-b^2 \\
16=25-b^2 \\
b^2=9
\end{array}
\]Finally, we substitute the values found for \(h, k, a^2\), and \(b^2\) into the standard form equation for an ellipse:
\[
\dfrac{(x+2)^2}{9}+\dfrac{(y+3)^2}{25}=1
\]
What is the standard form equation of the ellipse that has vertices \((-3,3)\) and \((5,3)\) and foci \((1-2 \sqrt{3}, 3)\) and \((1+2 \sqrt{3}, 3) ?\)
Graphing Ellipses Centered at the Origin
Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1, a>b\) for horizontal ellipses and \(\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1, a>b\) for vertical ellipses.
Given the standard form of an equation for an ellipse centered at \((0,0)\), sketch the graph.
1. Use the standard forms of the equations of an ellipse to determine the major axis, vertices, covertices, and foci.
a. If the equation is in the form \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\), where \(a>b\), then
- the major axis is the \(x\)-axis
- the coordinates of the vertices are \(( \pm a, 0)\)
- the coordinates of the co-vertices are \((0, \pm b)\)
- the coordinates of the foci are \(( \pm c, 0)\)
b. If the equation is in the form \(\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1\), where \(a>b\), then
- the major axis is the \(y\)-axis
- the coordinates of the vertices are \((0, \pm a)\)
- the coordinates of the co-vertices are \(( \pm b, 0)\)
- the coordinates of the foci are \((0, \pm c)\)
2. Solve for \(c\) using the equation \(c^2=a^2-b^2\).
3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.
Graph the ellipse given by the equation, \(\dfrac{x^2}{9}+\dfrac{y^2}{25}=1\). Identify and label the center, vertices, covertices, and foci.
- Answer
-
First, we determine the position of the major axis. Because \(25>9\), the major axis is on the \(y\)-axis. Therefore, the equation is in the form \(\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1\), where \(b^2=9\) and \(a^2=25\). It follows that:
- the center of the ellipse is \((0,0)\)
- the coordinates of the vertices are \((0, \pm a)=(0, \pm \sqrt{25})=(0, \pm 5)\)
- the coordinates of the co-vertices are \(( \pm b, 0)=( \pm \sqrt{9}, 0)=( \pm 3,0)\)
- the coordinates of the foci are \((0, \pm c)\), where \(c^2=a^2-b^2\) Solving for \(c\), we have:
\[
\begin{array}{l}
c= \pm \sqrt{a^2-b^2} \\
= \pm \sqrt{25-9} \\
= \pm \sqrt{16} \\
= \pm 4
\end{array}
\]Therefore, the coordinates of the foci are \((0, \pm 4)\).
Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. SeeFigure \(\PageIndex{8}\)Figure \(\PageIndex{8}\):
Graph the ellipse given by the equation \(\dfrac{x^2}{36}+\dfrac{y^2}{4}=1\). Identify and label the center, vertices, covertices, and foci.
Graphing an Ellipse Centered at the Origin from an Equation Not in Standard Form
Graph the ellipse given by the equation \(4 x^2+25 y^2=100\). Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.
- Answer
-
First, use algebra to rewrite the equation in standard form.
\[
\begin{array}{c}
4 x^2+25 y^2=100 \\
\dfrac{4 x^2}{100}+\dfrac{25 y^2}{100}=\dfrac{100}{100} \\
\dfrac{x^2}{25}+\dfrac{y^2}{4}=1
\end{array}
\]Next, we determine the position of the major axis. Because \(25>4\), the major axis is on the \(x\)-axis. Therefore, the equation is in the form \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\), where \(a^2=25\) and \(b^2=4\). It follows that:
- the center of the ellipse is \((0,0)\)
- the coordinates of the vertices are \(( \pm a, 0)=( \pm \sqrt{25}, 0)=( \pm 5,0)\)
- the coordinates of the co-vertices are \((0, \pm b)=(0, \pm \sqrt{4})=(0, \pm 2)\)
- the coordinates of the foci are \(( \pm c, 0)\), where \(c^2=a^2-b^2\). Solving for \(c\), we have:
\[
\begin{array}{l}
c= \pm \sqrt{a^2-b^2} \\
= \pm \sqrt{25-4} \\
= \pm \sqrt{21}
\end{array}
\]Therefore the coordinates of the foci are \(( \pm \sqrt{21}, 0)\).
Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. -
Graph the ellipse given by the equation \(49x^2+16y^2=784\). Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci
Graphing Ellipses Not Centered at the Origin
When an ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point, \((h, k)\), we use the standard forms \(\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1, a>b\) for horizontal ellipses and \(\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1, a>b\) for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.
Given the standard form of an equation for an ellipse centered at \((h, k)\), sketch the graph.
1. Use the standard forms of the equations of an ellipse to determine the center, position of the major axis, vertices, co-vertices, and foci.
a. If the equation is in the form \(\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\), where \(a>b\), then the center is \((h, k)\)
- the major axis is parallel to the \(x\)-axis
- the coordinates of the vertices are \((h \pm a, k)\)
- the coordinates of the co-vertices are \((h, k \pm b)\)
- the coordinates of the foci are \((h \pm c, k)\)
b. If the equation is in the form \(\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1\), where \(a>b\), then the center is \((h, k)\)
- the major axis is parallel to the \(y\)-axis
- the coordinates of the vertices are \((h, k \pm a)\)
- the coordinates of the co-vertices are \((h \pm b, k)\)
- the coordinates of the foci are \((h, k \pm c)\)
2. Solve for \(c\) using the equation \(c^2=a^2-b^2\).
3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.
Graphing an Ellipse Centered at \((h, k)\)
Graph the ellipse given by the equation, \(\dfrac{(x+2)^2}{4}+\dfrac{(y-5)^2}{9}=1\). Identify and label the center, vertices, co-vertices, and foci.
- Answer
-
First, we determine the position of the major axis. Because \(9>4\), the major axis is parallel to the \(y\) axis. Therefore, the equation is in the form \(\dfrac{(x-h)^2}{b^2}+\dfrac{(y-k)^2}{a^2}=1\), where \(b^2=4\) and \(a^2=9\). It follows that:
- the center of the ellipse is \((h, k)=(-2,5)\)
- the coordinates of the vertices are \((h, k \pm a)=(-2,5 \pm \sqrt{9})=(-2,5 \pm 3)\), or \((-2,2)\) and \((-2,8)\)
- the coordinates of the co-vertices are \((h \pm b, k)=(-2 \pm \sqrt{4}, 5)=(-2 \pm 2,5)\), or \((-4,5)\) and \((0,5)\)
- the coordinates of the foci are \((h, k \pm c)\), where \(c^2=a^2-b^2\). Solving for \(c\), we have:
\[
\begin{array}{l}
c= \pm \sqrt{a^2-b^2} \\
= \pm \sqrt{9-4} \\
= \pm \sqrt{5}
\end{array}
\]Therefore, the coordinates of the foci are \((-2,5-\sqrt{5})\) and \((-2,5+\sqrt{5})\).
Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.
-
Graph the ellipse given by the equation \(\dfrac{(x-4)^2}{36}+\dfrac{(y-2)^2}{20}=1\). Identify and label the center, vertices, co-vertices, and foci.
Given the general form of an equation for an ellipse centered at \((h, k)\), express the equation in standard form.
1. Recognize that an ellipse described by an equation in the form \(a x^2+b y^2+c x+d y+e=0\) is in general form.
2. Rearrange the equation by grouping terms that contain the same variable. Move the constant term to the opposite side of the equation.
3. Factor out the coefficients of the \(x^2\) and \(y^2\) terms in preparation for completing the square.
4. Complete the square for each variable to rewrite the equation in the form of the sum of multiples of two binomials squared set equal to a constant, \(m_1(x-h)^2+m_2(y-k)^2=m_3\), where \(m_1, m_2\), and \(m_3\) are constants.
5. Divide both sides of the equation by the constant term to express the equation in standard form.
Graphing an Ellipse Centered at \((h, k)\) by First Writing It in Standard FormGraph the ellipse given by the equation \(4 x^2+9 y^2-40 x+36 y+100=0\). Identify and label the center, vertices, co-vertices, and foci.
- Answer
-
We must begin by rewriting the equation in standard form.
\[
4 x^2+9 y^2-40 x+36 y+100=0
\]Group terms that contain the same variable, and move the constant to the opposite side of the equation.
\[
\left(4 x^2-40 x\right)+\left(9 y^2+36 y\right)=-100
\]Factor out the coefficients of the squared terms.
\[
4\left(x^2-10 x\right)+9\left(y^2+4 y\right)=-100
\]Complete the square twice. Remember to balance the equation by adding the same constants to each side.
\[
4\left(x^2-10 x+25\right)+9\left(y^2+4 y+4\right)=-100+100+36
\]Rewrite as perfect squares.
\[
4(x-5)^2+9(y+2)^2=36
\]Divide both sides by the constant term to place the equation in standard form.
\[
\dfrac{(x-5)^2}{9}+\dfrac{(y+2)^2}{4}=1
\]Now that the equation is in standard form, we can determine the position of the major axis. Because \(9>4\), the major axis is parallel to the \(x\)-axis. Therefore, the equation is in the form \(\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1\), where \(a^2=9\) and \(b^2=4\). It follows that:
the center of the ellipse is \((h, k)=(5,-2)\)
the coordinates of the vertices are \((h \pm a, k)=(5 \pm \sqrt{9},-2)=(5 \pm 3,-2)\), or \((2,-2)\) and \((8,-2)\)
the coordinates of the co-vertices are \((h, k \pm b)=(5,-2 \pm \sqrt{4})=(5,-2 \pm 2)\), or \((5,-4)\) and \((5,0)\)
the coordinates of the foci are \((h \pm c, k)\), where \(c^2=a^2-b^2\). Solving for \(c\), we have:\[
\begin{array}{l}
c= \pm \sqrt{a^2-b^2} \\
= \pm \sqrt{9-4} \\
= \pm \sqrt{5}
\end{array} \notag
\]Therefore, the coordinates of the foci are \((5-\sqrt{5},-2)\) and \((5+\sqrt{5},-2)\).
Next we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse as shown in Figure \(\PageIndex{11}\). -
Graph the ellipse given by the equation \(\dfrac{(x-4)^2}{36}+\dfrac{(y-2)^2}{20}=1\). Identify and label the center, vertices, co-vertices, and foci.
Solving Applied Problems Involving Ellipses
Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See Figure \(\PageIndex{12}\). In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper.
Locating the Foci of a Whispering Chamber
The Statuary Hall in the Capitol Building in Washington, D.C. is a whispering chamber. Its dimensions are 46 feet wide by 96 feet long as shown in Figure \(\PageIndex{13}\): .
- What is the standard form of the equation of the ellipse representing the outline of the room? Hint: assume a horizontal ellipse, and let the center of the room be the point (0,0).
- If two senators standing at the foci of this room can hear each other whisper, how far apart are the senators? Round to the nearest foot.
- Answer
-
a. We are assuming a horizontal ellipse with center \((0,0)\), so we need to find an equation of the form \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\), where \(a>b\). We know that the length of the major axis, \(2 a\), is longer than the length of the minor axis, \(2 b\). So the length of the room, 96 , is represented by the major axis, and the width of the room, 46 , is represented by the minor axis.
- Solving for \(a\), we have \(2 a=96\), so \(a=48\), and \(a^2=2304\).
- Solving for \(b\), we have \(2 b=46\), so \(b=23\), and \(b^2=529\).
Therefore, the equation of the ellipse is \(\dfrac{x^2}{2304}+\dfrac{y^2}{529}=1\).
b. To find the distance between the senators, we must find the distance between the foci, \(( \pm c, 0)\), where \(c^2=a^2-b^2\). Solving for \(c\), we have:
\(c^2=a^2-b^2\)
\(c^2=2304-529\) Substitute using the values found in part (a). \(c= \pm \sqrt{2304-529}\) Take the square root of both sides. \(c= \pm \sqrt{1775}\) Subtract \(c \approx \pm 42\) Round to the nearest foot.
Suppose a whispering chamber is 480 feet long and 320 feet wide.
a. What is the standard form of the equation of the ellipse representing the room? Hint: assume a horizontal ellipse, and let the center of the room be the point (0,0).(0,0).
b. If two people are standing at the foci of this room and can hear each other whisper, how far apart are the people? Round to the nearest foot
Media
Access these online resources for additional instruction and practice with ellipses.
Conic Sections: The Ellipse
Graph an Ellipse with Center at the Origin
Graph an Ellipse with Center Not at the Origin


13.6: Practice
Problems 34-36 ask you to solve problems about a strange veterinarian who created three mystifying machines.
Cat Machine : Place a cat in the input bin of this machine, press the button, and out jump two dogs and a mouse.
Dog Machine : This machine converts a dog into a cat and a mouse.
Mouse Machine : This machine can convert a mouse into a cat and three dogs.
Each machine can also operate in reverse. For example, if you have two dogs and a mouse, you can use the first machine to convert them into a cat.
The veterinarian hands you two cats, and asks you to convert them into exactly three dogs (no extra dogs and no other animals). Can you do it? If yes, say what process you would use. If no, say why not.
The veterinarian hands you one dog. He says he only wants cats, but he doesn’t care how many. Can you help him? How?
The veterinarian hands you one cat. He says he only wants dogs, but he doesn’t care how many. Can you help him? How?
Problems 37-40 present several growing patterns made with toothpicks. For each problem you work on, do the following:
- Describe in words and pictures how you see the pattern growing.
- Calculate the number of toothpicks you would need to build the 10th figure in the pattern. Justify your answer based on how the pattern grows.
- Calculate the number of toothpicks you would need to build the 100th figure in the pattern.
- Describe how you can figure out the number of toothpicks in any figure in the pattern. Be sure to justify your answer based on how the pattern grows.
- Could you make one of the figures in the pattern using exactly 25 toothpicks? If yes, which figure? If no, why not? Justify your answer.
- Could you make one of the figures in the pattern using exactly 100 toothpicks? If yes, which figure? If no, why not? Justify your answer.
In a mobile , the arms must be perfectly balanced for it to hang properly. The artist Alexander Calder was famous for his artistic mobiles.You can view some of his amazing work here . Click “Explore Works.”
Problems 41-42 present you with mobile puzzles. In these puzzles:
- Objects that are the same shape have the same weight. (So all circles weigh the same, all squares weigh the same, etc.)
- Assume the strings and rods that hold the objects together don’t factor into the total weight.
- Each arm of the mobile must have exactly the same weight.
In this puzzle:
- The total weight is 36 grams.
- All shapes weigh less than 10 grams.
- All of the weights are whole numbers.
- One circle weighs more than one square.
Find the weight of each piece. Is there more than one answer? How do you know you are right?
In this puzzle, the total weight is 54 grams.
Find the weight of each piece. Is there more than one answer? How do you know you are right?


5.4: Conductance
When students first see the parallel resistance equation, the natural question to ask is, “Where did that thing come from?” It is truly an odd piece of arithmetic, and its origin deserves a good explanation.
What is the Difference Between Resistance and Conductance?
Resistance, by definition, is the measure of friction a component presents to the flow of electrons through it. Resistance is symbolized by the capital letter “R” and is measured in the unit of “ohm.” However, we can also think of this electrical property in terms of its inverse: how easy it is for electrons to flow through a component, rather than how difficult . If resistance is the word we use to symbolize the measure of how difficult it is for electrons to flow, then a good word to express how easy it is for electrons to flow would be conductance .
Mathematically, conductance is the reciprocal, or inverse, of resistance:
The greater the resistance, the less the conductance, and vice versa. This should make intuitive sense, resistance and conductance being opposite ways to denote the same essential electrical property. If two components’ resistances are compared and it is found that component “A” has one-half the resistance of component “B,” then we could alternatively express this relationship by saying that component “A” is twice as conductive as component “B.” If component “A” has but one-third the resistance of component “B,” then we could say it is three times more conductive than component “B,” and so on.
Carrying this idea further, a symbol and unit were created to represent conductance. The symbol is the capital letter “G” and the unit is the mho , which is “ohm” spelled backwards (and you didn’t think electronics engineers had any sense of humor!). Despite its appropriateness, the unit of the mho was replaced in later years by the unit of siemens (abbreviated by the capital letter “S”). This decision to change unit names is reminiscent of the change from the temperature unit of degrees Centigrade to degrees Celsius , or the change from the unit of frequency c.p.s. (cycles per second) to Hertz . If you’re looking for a pattern here, Siemens, Celsius, and Hertz are all surnames of famous scientists, the names of which, sadly, tell us less about the nature of the units than the units’ original designations.
As a footnote, the unit of siemens is never expressed without the last letter “s.” In other words, there is no such thing as a unit of “siemen” as there is in the case of the “ohm” or the “mho.” The reason for this is the proper spelling of the respective scientists’ surnames. The unit for electrical resistance was named after someone named “Ohm,” whereas the unit for electrical conductance was named after someone named “Siemens,” therefore it would be improper to “singularize” the latter unit as its final “s” does not denote plurality.
Back to our parallel circuit example, we should be able to see that multiple paths (branches) for current reduces total resistance for the whole circuit, as electrons are able to flow easier through the whole network of multiple branches than through any one of those branch resistances alone. In terms of resistance , additional branches result in a lesser total (current meets with less opposition). In terms of conductance , however, additional branches results in a greater total (electrons flow with greater conductance):
Total Parallel Resistance
Total parallel resistance is less than any one of the individual branch resistances because parallel resistors resist less together than they would separately:
Total Parallel Conductance
Total parallel conductance is greater than any of the individual branch conductances because parallel resistors conduct better together than they would separately:
To be more precise, the total conductance in a parallel circuit is equal to the sum of the individual conductances:
If we know that conductance is nothing more than the mathematical reciprocal (1/x) of resistance, we can translate each term of the above formula into resistance by substituting the reciprocal of each respective conductance:
Solving the above equation for total resistance (instead of the reciprocal of total resistance), we can invert (reciprocate) both sides of the equation:
So, we arrive at our cryptic resistance formula at last! Conductance (G) is seldom used as a practical measurement, and so the above formula is a common one to see in the analysis of parallel circuits.
Review
- Conductance is the opposite of resistance: the measure of how easy it is for electrons to flow through something.
- Conductance is symbolized with the letter “G” and is measured in units of mhos or Siemens .
- Mathematically, conductance equals the reciprocal of resistance: G = 1/R


4.6: Body Systems
Organs do not work in isolation but function in cooperation with other organs and body structures to bring about the MRS GREN functions necessary to keep an animal alive. For example the stomach can only work in conjunction with the mouth and oesophagus (gullet). These provide it with the food it breaks down and digests. It then needs to pass the food on to the intestines etc. for further digestion and absorption. The organs involved with the taking of food into the body, the digestion and absorption of the food and elimination of waste products are collectively known as the digestive system.
The 11 body systems
-
Skin
- The skin covering the body consists of two layers, the epidermis and dermis . Associated with these layers are hairs, feathers, claws, hoofs, glands and sense organs of the skin.
-
Skeletal System
- This can be divided into the bones of the skeleton and the joints where the bones move over each other.
-
Muscular System
- The muscles, in conjunction with the skeleton and joints, give the body the ability to move.
-
Cardiovascular System
- This is also known as the circulatory system. It consists of the heart, the blood vessels and the blood. It transports substances around the body.
-
Lymphatic System
- This system is responsible for collecting and “cleaning” the fluid that leaks out of the blood vessels. This fluid is then returned to the blood system. The lymphatic system also makes antibodies that protect the body from invasion by bacteria etc. It consists of lymphatic vessels, lymph nodes, the spleen and thymus glands.
-
Respiratory System
- This is the system involved with bringing oxygen in the air into the body and getting rid of carbon dioxide, which is a waste product of processes that occur in the cell. It is made up of the trachea, bronchi, bronchioles, lungs, diaphragm, ribs and muscles that move the ribs in breathing.
-
Digestive System
- This is also known as the gastrointestinal system , alimentary system or gut . It consists of the digestive tube and glands like the liver and pancreas that produce digestive secretions. It is concerned with breaking down the large molecules in foods into smaller ones that can be absorbed into the blood and lymph. Waste material is also eliminated by the digestive system.
-
Urinary System
- This is also known as the renal system . It removes waste products from the blood and is made up of the kidneys, ureters and bladder.
-
Reproductive System
- This is the system that keeps the species going by making new individuals. It is made up of the ovaries, uterus, vagina and fallopian tubes in the female and the testes with associated glands and ducts in the male.
-
Nervous System
- This system coordinates the activities of the body and responses to the environment. It consists of the sense organs (eye, ear, semicircular canals, and organs of taste and smell), the nerves, brain and spinal cord.
-
Endocrine System
- This is the system that produces chemical messengers or hormones. It consists of various endocrine glands (ductless glands) that include the pituitary, adrenal, thyroid and pineal glands as well as the testes and ovary.
Contributors and Attributions


3.1: Interfacing to Canvas via LTI
Linking a campus Canvas instance to ADAPT via Learning Tools Interoperability (LTI) is an automatic process, but requires some activity from the Canvas Administrator (and hence approval from the respective campus).
Procedure
The first is to contact the ADAPT team (via "Contact us" on ADAPT front page ) to get a specific URL for their school.
After a campus specific tag is assigned, to link Canvas to ADAPT, the campus Canvas Administrator goes this website:
where my-key will have to be manually changed for the specific campus assigned by the ADAPT team. For example, for University of California, Davis this would be:
or for Western Washington University is would be
The procedure is outlined on this page and various elements are pre-populated on the screen and they copy/paste into their Developer Key config form. A video this procedure is below.
Question and concerns about this procedure can be address via<EMAIL_ADDRESS>.


7.2.2: Glycolysis
Even exergonic, energy-releasing reactions require a small amount of activation energy to proceed. However, consider endergonic reactions, which require much more energy input because their products have more free energy than their reactants. Within the cell, where does energy to power such reactions come from? The answer lies with an energy-supplying molecule called adenosine triphosphate, or ATP. ATP is a small, relatively simple molecule, but within its bonds contains the potential for a quick burst of energy that can be harnessed to perform cellular work. This molecule can be thought of as the primary energy currency of cells in the same way that money is the currency that people exchange for things they need. ATP is used to power the majority of energy-requiring cellular reactions.
ATP in Living Systems
A living cell cannot store significant amounts of free energy. Excess free energy would result in an increase of heat in the cell, which would denature enzymes and other proteins, and thus destroy the cell. Rather, a cell must be able to store energy safely and release it for use only as needed. Living cells accomplish this using ATP, which can be used to fill any energy need of the cell. How? It functions as a rechargeable battery.
When ATP is broken down, usually by the removal of its terminal phosphate group, energy is released. This energy is used to do work by the cell, usually by the binding of the released phosphate to another molecule, thus activating it. For example, in the mechanical work of muscle contraction, ATP supplies energy to move the contractile muscle proteins.
ATP Structure and Function
At the heart of ATP is a molecule of adenosine monophosphate (AMP), which is composed of an adenine molecule bonded to both a ribose molecule and a single phosphate group (Figure \(\PageIndex{1}\)). Ribose is a five-carbon sugar found in RNA and AMP is one of the nucleotides in RNA. The addition of a second phosphate group to this core molecule results in adenosine di phosphate (ADP); the addition of a third phosphate group forms adenosine tri phosphate (ATP).
Figure \(\PageIndex{1}\): The structure of ATP shows the basic components of a two-ring adenine, five-carbon ribose, and three phosphate groups.The addition of a phosphate group to a molecule requires a high amount of energy and results in a high-energy bond. Phosphate groups are negatively charged and thus repel one another when they are arranged in series, as they are in ADP and ATP. This repulsion makes the ADP and ATP molecules inherently unstable. The release of one or two phosphate groups from ATP, a process called hydrolysis, releases energy.
Glycolysis
You have read that nearly all of the energy used by living things comes to them in the bonds of the sugar, glucose. Glycolysis is the first step in the breakdown of glucose to extract energy for cell metabolism. Many living organisms carry out glycolysis as part of their metabolism. Glycolysis takes place in the cytoplasm of most prokaryotic and all eukaryotic cells.
Glycolysis begins with the six-carbon, ring-shaped structure of a single glucose molecule and ends with two molecules of a three-carbon sugar called pyruvate. Glycolysis consists of two distinct phases. In the first part of the glycolysis pathway, energy is used to make adjustments so that the six-carbon sugar molecule can be split evenly into two three-carbon pyruvate molecules. In the second part of glycolysis, ATP and nicotinamide-adenine dinucleotide (NADH) are produced (Figure \(\PageIndex{2}\)).
If the cell cannot catabolize the pyruvate molecules further, it will harvest only two ATP molecules from one molecule of glucose. For example, mature mammalian red blood cells are only capable of glycolysis, which is their sole source of ATP. If glycolysis is interrupted, these cells would eventually die.
Figure \(\PageIndex{2}\): In glycolysis, a glucose molecule is converted into two pyruvate molecules.Summary
ATP functions as the energy currency for cells. It allows cells to store energy briefly and transport it within itself to support endergonic chemical reactions. The structure of ATP is that of an RNA nucleotide with three phosphate groups attached. As ATP is used for energy, a phosphate group is detached, and ADP is produced. Energy derived from glucose catabolism is used to recharge ADP into ATP.
Glycolysis is the first pathway used in the breakdown of glucose to extract energy. Because it is used by nearly all organisms on earth, it must have evolved early in the history of life. Glycolysis consists of two parts: The first part prepares the six-carbon ring of glucose for separation into two three-carbon sugars. Energy from ATP is invested into the molecule during this step to energize the separation. The second half of glycolysis extracts ATP and high-energy electrons from hydrogen atoms and attaches them to NAD + . Two ATP molecules are invested in the first half and four ATP molecules are formed during the second half. This produces a net gain of two ATP molecules per molecule of glucose for the cell.
Glossary
- ATP
- (also, adenosine triphosphate) the cell’s energy currency
- glycolysis
- the process of breaking glucose into two three-carbon molecules with the production of ATP and NADH
Contributors and Attributions
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Samantha Fowler (Clayton State University), Rebecca Roush (Sandhills Community College), James Wise (Hampton University). Original content by OpenStax (CC BY 4.0; Access for free at https://cnx.org/contents/b3c1e1d2-83...4-e119a8aafbdd ).