6.5: Writing Process- Creating a Proposal
By the end of this section, you will be able to:
- Describe the elements of the rhetorical situation for your proposal.
- Apply prewriting strategies to discover a problem to write about.
- Gather and synthesize information from appropriate sources.
- Draft a thesis statement and create an organizational plan.
- Compose a proposal that develops your ideas and integrates evidence from sources.
- Implement strategies for drafting, peer reviewing, and revising.
Sometimes writing a paper comes easily, but more often writers work hard to generate ideas and evidence, organize their thoughts, draft, and revise. Experienced writers do their work in multiple steps, and most engage in a recursive process that involves thinking and rethinking, writing and rewriting, and repeating steps multiple times as their ideas develop and sharpen. In broad strokes, most writers go through the following steps to achieve a polished piece of writing:
- Planning and Organization. Your proposal will come together more easily if you spend time at the start considering the rhetorical situation, understanding your assignment, gathering ideas and evidence, drafting a thesis statement, and creating an organizational plan.
- Drafting. When you have a good grasp of the problem and solution you are going to write about and how you will organize your proposal, you are ready to draft.
- Review. With a first draft in hand, make time to get feedback from others. Depending on the structure of your class, you may receive feedback from your instructor or your classmates. You can also work with a tutor in the writing center on your campus, or you can ask someone else you trust, such as a friend, roommate, or family member, to read your writing critically and give honest feedback.
- Revising. After reviewing feedback from your readers, plan to revise. Focus on their comments: Is your thesis clear? Do you need to make organizational changes to the proposal? Do you need to explain or connect your ideas more clearly?
Considering the Rhetorical Situation
Like other kinds of writing projects, a proposal starts with assessing the rhetorical situation —the circumstance in which a writer communicates with an audience of readers about a subject. As a proposal writer, you make choices based on the purpose for your writing, the audience who will read it, the genre , and the expectations of the community and culture in which you are working. The brainstorming questions in Table \(6.1\) can help you begin:
| Rhetorical Situation Element | Brainstorming Questions |
Your Responses |
|
Topic Are you free to choose your own problem and solution to write about, or is your topic specified? |
What do you want to know more about? What requirements do you have? Do you need to do research? | |
|
Purpose What is the purpose of the proposal? |
Is the purpose to examine a problem and explain possible solutions? Or is it to recommend a specific solution? | |
|
Audience Who will read your writing? |
Who is your primary audience—your instructor? Your classmates? Other students or administrators on your campus? People in your community? How will you shape your writing to connect most effectively with this audience? Do you need to consider secondary audiences, such as people outside of class? If so, who are those readers? | |
|
Presentation In what format should you prepare your proposal? |
Should you prepare a written proposal or use another medium? Should you include visuals and other media along with text, such as figures, charts, graphs, photographs, audio, or video? What other presentation requirements do you need to be aware of? | |
|
Context How do the time period and location affect decisions you make about your proposal? |
What problems in your city, county, state, area, nation, or the world need a solution? What current events or new information might relate to the problem? Is your college or university relevant to the problem? | |
|
Culture and Community What social or cultural assumptions do you or oyour audience have? |
How will you show awareness of your community’s social and cultural expectations in your writing? |
Summary of Assingment
Write a proposal that discusses a problem you want to learn more about and that recommends a solution. The problem you choose must be a current problem, even though it may have been a problem for many years. The problem must also affect many people, and it must have an actual solution or solutions that you can learn about through research. In other words, the problem cannot be unique to you, and the solution you recommend cannot be one you only imagine; both the problem and the solution must be grounded in reality.
One way to get ideas about a problem to write about is to read a high-quality newspaper, website, or social media account for a week.  You’ll begin to get a handle on problems in your community or state that people care about. If you read a paper or website with a national or international audience, you’ll learn about problems that affect people in other places.
You will need to consult and cite at least five reliable sources. They can be scholarly, but they do not have to be. They must be credible, trustworthy, and unbiased. Possible sources include articles from reputable newspapers, magazines, and academic and professional journals; reputable websites; government sources; and visual sources. Depending on your topic, you may want to conduct a survey, an interview, or an experiment. See Research Process: Accessing and Recording Information and Annotated Bibliography: Gathering, Evaluating, and Documenting Sources for information about creating and finding sources. Your proposal can include a visual or media source if it provides appropriate, relevant evidence.
Another Lens. Another way to approach a proposal assignment is to consider problems that affect you directly and affect others. Perhaps you are concerned about running up student loan debt. Or perhaps you worry about how to pay your rent while earning minimum wage. These concerns are valid and affect many college students around the United States. Another way is to think about problems that affect others. Perhaps students in your class or on your campus have backgrounds and experiences that differ from yours— what problems or challenges might they have encountered during their time in college that you don’t know about?
As you think about the purpose and audience for your proposal, think again about the rhetorical situation, specifically about the audience you want to reach and the mode of presentation best suited to them and your purpose. For example, say you’re dissatisfied with the process for electing student leaders on your campus. If your purpose is to identify the problems in the process and propose a change, then your audience would include other students, the group or committee that oversees student elections, and perhaps others. To reach other students who might also be dissatisfied, you might write an article, editorial, or letter for the campus newspaper, social media page, or website, depending on how students on your campus get news. In addition, you might organize a meeting of other students to get their input on the problem. To reach the decision makers, which may include elected students, faculty, and administrators, you might need to prepare an oral presentation and a slide deck.
Below in Figure \(6.7\) are three slides from Shawn Krukowski’s proposal that he adapted for a presentation: the title slide, a slide on one aspect of the problem, and a slide introducing one of the proposed solutions.
Figure \(6.7\) Presentation Slides (CC BY 4.0; Rice University & OpenStax)
Quick Launch: Finding a Problem to Write About
A proposal must address a real-life problem and present one or more workable solutions. Usually, problems worth writing about are not easily solved; if they were, they would no longer be considered problems. Indeed, problems in proposals are often complex, and solutions are often complicated and involve trade-offs. Sometimes people disagree about whether the problem is a problem at all and whether any proposed solutions are viable solutions.
Exploring a Problem
One way to generate ideas about a problem is to brainstorm. To explore a topic for your proposal, use a graphic organizer like Table \(6.2\) to write responses to the following statements and questions:
| Exploring Questions |
Your responses |
| What problem am I interested in learning about? | |
| How do I know this is a problem? | |
| What are a few examples of the problem? | |
| What causes the problem? | |
| Who is affected by the problem? | |
| What are some negative effects of the problem? | |
| Why should the problem be solved? | |
| What are the potential consequences of the problem if nothing is done? | |
| What are some realistic solutions to the problem? |
For example, perhaps you’re considering a career in information technology, and you’re taking an IT class. You might be interested in exploring the problem of data breaches. A data breach is a real-world problem with possible solutions, so it passes the first test of being an actual problem with possible solutions. Your responses to the questions above might look something like those in Table \(6.3\):
| Exploring Questions | Sample Responses |
| A problem I’m interested in learning more about is… | Data breaches |
| How I know data breaches are a problem… | In my class, we’re spending a lot of time on data security and breaches. Also, data breaches are in the news almost constantly, and a Google search turns up many that don’t make the news. |
| What are a few examples of data breaches? | I’ve heard about people getting their information stolen. I’ve heard about foreign governments, like Russia, stealing national security information and trying to interfere in recent elections. In my class, I’m learning about businesses that have customer and employee information stolen. |
| What causes data breaches? | Hackers have a variety of methods to break into websites, to get people to click on links, and to lure people to give out information. |
| Who is affected by data breaches? | Individual people, businesses, utility companies, schools and universities, governments (local, state, and national)—pretty much anyone can be affected. |
| What are some negative effects of data breaches? | Identity theft, financial theft, national security, power shutdowns, and election interference. Data breaches cause chaos and cost a lot of money. |
| Why should the problem of data breaches be solved? | People need the personal, financial, and medical information they share with businesses and other organizations to remain private. Businesses and organizations need to keep their operations secure. Governments need to keep national security secrets from getting into the hands of people who want to harm them. |
| What are the potential consequences of data breaches if nothing is done? | People will continue to be victims of identity theft and all that goes with it, including losing money, and they will lose trust in institutions they used to trust, like banks, hospitals, and the government. |
| What are some realistic solutions to data breaches? | Increased security for individuals, like two-factor authentication, stronger passwords, and education to avoid getting scammed. For companies and governments, stronger protections on websites. I need to learn more. |
Narrowing and Focusing
Many problems for a proposal can be too broad to tackle in a single paper. For example, the sample above reveals that data breaches are indeed a problem but that several aspects can be explored. If you tried to cover all the aspects, you would be left writing general paragraphs with little specific information. The topic needs to be narrowed and focused.
The data breaches example above could be narrowed to the following problems—and possibly even more. Note that the questions start to zero in on possible solutions, too. In your own writing, as you brainstorm, try placing subtopics you discover into their own categories and asking more questions, as shown in Table \(6.4\).
| Narrowing a Problem | Focusing Questions |
| Problem: Hackers get into computer systems and steal information. | What tactics do hackers use? What do they steal? What do they do with what they steal? How successful are they? How can hacking be stopped or reduced? |
| Problem: Data breaches harm individuals. | What happens to people when their information is stolen? In what ways is their information used? What should people do when they discover their information has been stolen? Are there steps they can take to protect their information? |
| Problem: Data breaches harm businesses, organizations, and medical systems. | What happens to businesses, organizations, and medical systems when hackers break in? What kind of information is stolen? How is the information used? What do businesses, organizations, and medical systems do, or what should they do, when they know about a breach? How can data breaches be prevented? |
| Problem: Data breaches harm governments. | What happens to governments when hackers break into their systems? What kind of information is stolen? How is the information used? What should governments do when they know about a breach? How can data breaches be prevented? |
Sample Proposal Topics
The following broad topics are potentially suitable as a start for a proposal. Choose one of these or one of your own, and ask the exploring questions. Then look at your responses, and ask focusing questions. Continue to focus until you have a specific problem that you can discuss in sufficient depth and offer a concrete solution or solutions.
- Health fields: cost of medical and dental care for uninsured people, management of chronic conditions and diseases, infection control, vaccinations, access to mental health care, drug use and addiction, sports injuries, workplace safety
- Education: gaps in academic achievement, curriculum, recruitment and retention of staff and/or students, buildings and grounds, graduation rates, cocurricular activities
- Environment: forest management and fires, hurricanes and other extreme storms, water and air pollution, sustainable development, invasive species, waste management, recycling and composting, community gardening
- Engineering and computer science: robotics, vehicles and transportation, digital divide, online privacy, misinformation and misbehavior on social media, video games
- Business and manufacturing: quality improvement, process improvement, cost control, communication, social media, pay equity, fundraising, sourcing of materials, net-zero energy processes, workplace safety
- Policy and politics: public institutions, such as public schools, libraries, transportation systems, and parks; taxes, fees, and services; donations to political campaigns; healthcare, such as Medicare and Medicaid; social security; unemployment insurance; services for active military and veterans; immigration policy
- Society and culture: social media and free speech; inequality in housing, employment, education, and more; cancel culture; bullying; wealth and poverty; support for the arts; athletes and sports; disparities related to race, sex, gender identity and expression, age, and/or ability
Gathering Information
Proposals are rooted in information and evidence; therefore, most proposal assignments require you to conduct research. Depending on your assignment, you may need to do formal research, an activity that involves finding sources and evaluating them for reliability, reading them carefully and taking notes, and citing all words you quote and ideas you borrow. See Research Process: Accessing and Recording Information and Annotated Bibliography: Gathering, Evaluating, and Documenting Sources for detailed instruction on conducting research. If you are proposing a solution to a problem in your local community or on your campus, you may need to conduct primary research as well, such as a survey or interviews with people who live or work there.
Whether you conduct in-depth research or do background reading, keep track of the ideas that come to you and the information you learn. You can write or dictate notes using an app on your phone or computer, or you can jot notes in a journal if you prefer pen and paper. Then, when you are ready to begin to organize what you have learned, you will have a record of your thoughts and information. Always track the source of the information you gather, whether from your reading or a person you interviewed, so that you can return to that source if you need more information and can credit the source in your paper.
Kinds of Evidence
You will use evidence to demonstrate that the problem is real and worthy of being solved and that your recommended solution is workable. Choose evidence for your proposal that is rooted in facts. In addition, choose evidence that best supports the angle you take on your topic and meets your instructor’s requirements. Cite all evidence you use from a source. Consider the following kinds of evidence and examples of each:
-
Definition:
an explanation of a key word, idea, or concept.
The Personal Data Notification & Protection Act of 2017 defines a security breach as "a compromise of the security, confidentiality, or integrity of, or the loss of, computerized data that results in... (i) the unauthorized acquisition of sensitive personally identifiable information; or (ii) access to sensitive personally identifiable information that is for an unauthroized purpose, or in excess of authorization." -
Example:
an illustration of an idea or concept.
Every month, university staff members receive a fake phishing email from the IT department. The goal is to train employees of the university to be critical readers of every email they receive. -
Expert opinion:
a statement by a professional whose opinion is respected in the field.
In The Sixth Extinction, science writer Elizabeth Kolbert observes that humans are making the choice about “which evolutionary pathways will remain and open and which will be forever closed” (268). -
Fact:
information that is true and can be proven correct or accurate.
Statements of fact are built on evidence and data. In March and April of 2020, 43 states in the United States issued orders directing residents to stay home except for essential activities. -
Interview:
a person-to-person, phone, or remote conversation that involves an interviewer posing questions to another person or group of people.
During an interview, I asked about parents’ decisions to vaccinate their children. One pediatrician said, “The majority of parents see the benefits of immunizations for their children and for public health. For those who don’t, I talk to them and try to understand why they feel the way they do.” -
Quotation:
the exact words of an author or speaker.
According to the Federal Aviation Administration, SpaceX was required to conduct a “comprehensive review of the company’s safety culture, operational decision-making, and process discipline,” in addition to investigating the crash of its prototype spacecraft (Chang). -
Statistics:
numerical fact or item of data.
According to the Environmental Protection Agency, more than 40 million tons of food waste were generated in 2017, comprising 15.2% of all trash sent to landfills (DeSilver). -
Survey:
a structured interview in which respondents are all asked the same questions and their answers are tabulated and interpreted. Surveys reveal attitudes, beliefs, or habits of the general public or segments of the population.
In a survey of adults conducted in July 2020, 64 percent of respondents said that social media have a mostly negative effect on American society (Auxier). - Visuals and other media: graphs, figures, tables, photographs, diagrams, charts, maps, videos, audio recordings, etc.
Thesis and Organization
Drafting a Thesis
When you have a solid grasp of the problem and solution, try drafting a thesis . A thesis is the main idea that you will convey in your proposal and to which all the paragraphs in the paper should relate. In a proposal, you will likely express this main idea in a thesis statement of one or two sentences toward the end of the introduction.
For example, in the thesis statement Shawn Krukowski wrote for his proposal on climate change, he identifies the problem and previews the solutions he presents:
What is needed to slow climate change is unified action in two key areas—mitigation and adaptation—spurred by government leadership in the United States and a global commitment to addressing the problem immediately.
Here is another example that identifies a problem and multiple solutions:
The number of women employed in the IT field is decreasing every year, a trend that can be changed with a multifaceted approach that includes initiatives in middle schools, high schools, and colleges; active recruitment; mentoring programs; and flexible work arrangements.
After you draft a thesis statement, ask these questions and revise it as needed:
- Is it engaging? A thesis for a proposal should pique readers’ interest in the problem and possible solutions.
- Is it precise and specific? If you are interested in curbing the spread of invasive plant species, for example, your thesis should indicate which environment the plant or plants are invading and that you are proposing ways to stop the spread.
Organizing Your Ideas
A proposal has a recognizable shape, starting with an introduction, followed by discussions of the problem, possible solutions, potential objections to the solutions, and a conclusion with a recommendation. A graphic organizer like Table \(6.5\) can help you organize your ideas and evidence.
| Proposal Section | Content | Your Notes |
| Introduction (usually one paragraph, but can be two) |
Draw readers in with an overview. Provide necessary background here or in the description of the problem, defining terms as needed. State the thesis. |
|
| Description of the Problem (one or more paragraphs) |
Explain the problem, establishing it as a problem in need of a solution. Develop the paragraph(s) with evidence. |
|
| Possible Solution or Solutions (one or more paragraphs) |
Describe possible solutions, one at a time. Develop the paragraph(s) with evidence. |
|
| Conclusion and Recommendations (one or more paragraphs) | Conclude with a summary of the points and recommend a solution or a course of action. |
Drafting a Proposal
With a tentative thesis, an organization plan, and evidence, you are ready to begin drafting your proposal. For this assignment, you will discuss a problem, present possible solutions, address objections to the solutions, and conclude with a recommendation.
Introduction
You may choose to write the introduction first, last, or midway through the drafting process. Whenever you choose to write it, use it to draw readers in. Make the proposal topic clear, and be concise. End the introduction with your thesis statement.
Opening a proposal with an overview of your topic is a reliable strategy, as shown in the following student-written example on women working in IT. The thesis statement, which appeared earlier in this section, is underlined:
People who work in the information technology (IT) field often start their careers fixing computers and other electronic devices for others. Through experience and education, an IT worker’s career path can branch out to specialize in everything from programming new software to setting up and maintaining networks. The IT field is growing because of the constant development of technology, and the demand for employees also is growing. Yet the number of women employed in the IT field is decreasing every year, a trend that can be changed with a multifaceted approach that includes initiatives in middle schools, high schools, and colleges; active recruitment; mentoring programs; and flexible work arrangements.
Body Paragraphs: Problem, Solutions, Objections
The body paragraphs of your proposal should present the problem, the solution or solutions, and potential objections to the proposed solution(s). As you write these paragraphs, consider using the point , evidence , and analysis pattern:
- The point is the central idea of the paragraph, usually given in a topic sentence stated in your own words at or toward beginning of the paragraph.
- With the evidence you provide, you develop the paragraph and support the point given in the topic sentence. Include details, examples, quotations, paraphrases, and summaries from sources. In your sentences and paragraphs, synthesize the evidence you give by showing the connections between sources. See Position Argument: Practicing the Art of Rhetoric and Argumentative Research: Enhancing the Art of Rhetoric with Evidence for more information on quoting, summarizing, paraphrasing, and synthesizing.
- The analysis comes at the end of the paragraph. In your own words, draw a conclusion about the evidence you have provided and relate it to the topic sentence.
The paragraphs that follow show the point-evidence-analysis pattern in practice.
Body Paragraphs: Problem
Follow the introduction with a discussion of the problem. Using paragraph structure, define the problem and discuss it, drawing on evidence from your sources. This paragraph (or paragraphs) should answer these questions: What is the problem? Why is this a problem? The following example, from the proposal on women working in IT, answers the first question:
The information technology (IT) field is continuously expanding, with many more positions available than workers to fill them. In fact, the pool of IT professionals was so small that in 2001, Congress raised the visa limit in an effort to fill the gap with employees from overseas (Varma, 2002). And yet the number of women represented in the occupation is decreasing. From 1990 to 2020, the percentage of women in IT declined from 31 percent to 25 percent, even though women make up 47 percent of all employed adults in the United States. According to White (2021), only 19 percent of women pursue a computer science major in college, compared to 27 percent in 1997. Of those women who graduated with a computer science degree, 38 percent are working in the field compared to 56 percent of men, a statistic that indicates women are not staying in the field. Although gender diversity supposedly is valued in the workplace, the underrepresentation of women in IT is clearly a problem.
The writer then goes on to answer the second question: Why is this a problem? The writer discusses stereotypes, lack of encouragement and role models, workplace culture, pay, and prospects for advancement (not shown here).
Body Paragraphs: Solutions
After presenting and explaining the problem, use specific information from the sources you consulted to present the solution or solutions you have discovered through your research. If you are proposing more than one solution, present them one at a time, using headings as appropriate.
The solutions section will likely be the longest part of your proposal. Below are two paragraphs from the proposal about women working in IT. Note how the first paragraph introduces the solutions and how the second paragraph uses evidence to develop the first proposed solution. Also note the informative boldface headings.
The following suggestions are ways to encourage women to enter IT and build their careers, with the eventual goal of achieving gender balance in the field. The solutions discussed include encouraging interest in computer technology among girls in middle school and high school, actively recruiting college-age women to study IT, and within the field, mentoring women and expanding workplace flexibility to improve retention.
The National Center for Women & Information Technology (NCWIT) is an organization that encourages girls in middle school and high school to explore their interest in IT. One program, the NCWIT’s Aspirations in Computing, supports women in high school by showing them that they can succeed in technology and introducing them to other students with similar interests. The same program matches middle-school girls with female high-school and college students and awards scholarships for computing and programming competitions. In addition, internships and IT courses in middle school and high school provide opportunities to learn what a career in IT entails, with or without a degree in IT. Opportunities like these give girls and women support and a sense of belonging. The paragraphs that follow (not shown here) continue the discussion of the possible solutions.
The paragraphs that follow (not shown here) continue the discussion of the possible solutions.
Body Paragraphs: Objections
Depending on the problem and solution, consider the objections readers may raise, and explain why your proposal is necessary and worthwhile. For example, the proposal on women in IT does not discuss objections because few people would object to the writer’s proposal. Shawn Krukowski, however, in his proposal on climate change, includes a section on objections to taking action. He focuses the discussion on people who deny that climate change is a problem. Would you do the same? Consider whether this section of Shawn’s proposal might have been stronger had he addressed objections to the solutions he proposed—mitigation and adaptation—instead of objections to the problem.
Despite scientific evidence, some people and groups deny that climate change is real or, if they admit it exists, insist it is not a valid concern. Those who think climate change is not a problem point to Earth’s millennia-long history of changing climate as evidence that life has always persisted. Most of the change, however, predates human civilization, which has benefited from thousands of years of stable climate. The rapid change since the Industrial Revolution is unprecedented in human history.
Those who deny climate change or its dangers seek primarily to relax or remove pollution standards and regulations in order to protect, or maximize profit from, their industries. To date, their lobbying has been successful. For example, the world’s fossil-fuel industry received $5.3 trillion in 2015 alone, while the U.S. wind-energy industry received $12.3 billion in subsidies between 2000 and 2020 (Green America, 2020).
Conclusion and Recommendation
The conclusion and recommendation section of your proposal is the part in which you interpret your findings and make a recommendation or give a call to action. At this point, focus on the solution that will best solve the problem, suggesting or summarizing specific actions.
Below is the recommendation section from the proposal about women in IT. In the full conclusion (not shown here), the writer summarizes the main points of the proposal. In the recommendation paragraph that follows, the writer calls for specific actions:
Many researchers have studied why few women choose IT as a career and why some decide to leave the field. Although the numbers cannot be improved immediately, the following changes in school and the workplace could recruit and retain more women in IT:
- Include technology education courses and formal IT programs in middle- and high-school curricula to give girls and young women opportunities to develop an interest at an early age.
- Develop internship and mentor programs in high schools and colleges to combat stereotyping and encourage women to enter the field.
- Develop and encourage workplace mentor programs, flexible work options, and open communication for professional growth and retention.
With time and effort, these actions may result in more women seeing themselves in long-term IT careers.
References or Works Cited Page
Including any data you gathered through primary research, such as a survey you created and administered, interviews you conducted, or observational notes you took, you must cite the sources you consulted. These sources appear in the text of your proposal and in a bibliography at the end. The paragraphs in the previous section, including Shawn Krukowski’s proposal, use APA documentation style. For more on documenting sources, see Index and Guide to Documentation, MLA Documentation and Format, and APA Documentation and Format.
Abstract or Executive Summary
An abstract (or executive summary) summarizes your proposal. The purpose is to present information briefly and economically so that readers can decide whether they want to read further. Include your main points, but not the evidence.
Although an abstract or executive summary comes first in a proposal, it is advisable to write it after you have completed your proposal and are certain of your main points. The example below is the abstract from the proposal about women in IT.
The purpose of this proposal is to raise awareness of the small number of women working in the information technology (IT) field, to examine the factors that contribute to discouraging women from entering IT, and to propose ways to draw women into the field and retain them. Although the IT field is growing, the number of women employed within it remains low. Women may be reluctant to pursue a career in IT because of stereotypes, few role models, and lack of encouragement. Women who have already established a career in IT report leaving the field for these reasons, as well as family responsibilities and lack of advancement. There are several potential ways to raise the number of women in IT. Encouraging interest in computer technology among girls in middle school and high school, recruiting college-age women to study IT, mentoring young professional women, and improving workplace flexibility will, over time, break down stereotypes and increase the number of women in the IT field.
Peer Review: Getting Feedback from Readers
With a complete draft in hand, you may engage in peer review with your classmates, giving feedback to each other about the strengths and weaknesses of your drafts. For peer review within a class, your instructor may provide a list of questions or a form for you to complete as you work together.
Conferencing in Writing Groups
Other people can provide feedback on your writing beside your classmates. If you have an on-campus writing center, it is well worth your time to make an online or in-person appointment with a tutor at any point in your writing process. You will get valuable comments and improve your ability to review your own writing. Another way to get fresh eyes on your writing is to ask a friend or family member to read your draft. To get useful feedback, provide a list of questions or a form such as the one shown in Table 6.6 for them to complete as they read.
| Question for Reviewer | Comment or Suggestion |
| Does the introduction interest you in the problem? | |
| Can you find the thesis statement? Underline it for the writer. | |
| Does the thesis state the problem and suggest that a solution will be proposed? | |
| Can you identify a paragraph or paragraphs that define and explain a problem? | |
| Can you identify a paragraph or paragraphs that propose one or more solutions to the problem? | |
| Does the writer address possible objections to the solution or solutions? If not, are there objections the writer should consider? | |
| Does each paragraph start with a point stated in the writer’s own words, and does that point relate to the thesis? Mark paragraphs that don’t have a clear point. | |
| Does each paragraph support the main point of the paragraph with details and evidence, such as facts, statistics, or examples? Mark paragraphs that need more support and/or explanation. | |
| Does each paragraph end with analysis, in the writer’s own words, that draws a conclusion? Mark paragraphs that need analysis. | |
| Is there a clear conclusion that makes a recommendation and connects to the thesis statement? | |
| Where do you get lost or confused? Mark anything that is unclear. | |
| Does the proposal flow from one point to the next? | |
| Does the essay have a clear, descriptive title? |
Revising Your Proposal
A strong college paper is rarely written in a single draft, so build in time to revise your work. Take time with the comments you receive from your readers, and read your own work with a critical eye.
Figure \(6.8\) Whether you choose to do all your work on a computer or do some by hand, writing is nonetheless recursive, and revising is a major part of the process. (credit: Geschäftsfrau sitzt am Laptop und schreibt auf Notizblock mit Stift by Ivan Radic/flickr, CC BY 2.0)
Responding to Reviewers' Feedback
When you receive feedback from readers—whether from your instructor, your classmates, a writing tutor, or someone else—read each comment carefully to understand what the reader is communicating. Do your best not to become defensive, and be open to suggestions for improvement. Remind yourself that your readers are trying to help. As someone who hasn’t thought about your proposal as much as you have, a new reader can often see strengths and weaknesses that you cannot. Analyze each response, and decide whether acting on a suggestion will make your writing better. Remember that you remain the author, and you make the final call on your writing.
As you read, keep track of the comments your readers make. Pay special attention to strengths and weaknesses that more than one reader identifies. Use that information to improve later assignments as well as your proposal.
Revising on Your Own
The following revising strategies can help you read your draft critically and carefully:
- Read your draft aloud. Read the entire text from the beginning slowly and carefully, marking spots that need revision. Reading in this way allows you to see areas that need clarification, explanation, or development that you may have missed when you wrote the first draft. You can also have someone read your draft aloud to you.
- Make a paragraph outline. The most common unit of thought in writing is the paragraph, a group of sentences set off from other groups because they focus on a single idea. Writing a paragraph outline creates a map of your whole paper that can help you determine whether the organization is effective or needs changing. Number each paragraph and write a phrase describing its topic or focus. Check that each paragraph has a topic sentence that states the main idea of the paragraph.
- Test your evidence. Check whether each piece of evidence is factual and supports the main idea of the paragraph. Check that each piece of evidence is introduced, woven into your sentences, and cited.
- Listen for your voice. In most college papers, your language should sound like a real person. If your instructor requires a formal style for the assignment, the language should be objective and in third-person point of view.
- Let go if you need to. View change as good. Learn to let go of words, sentences, paragraphs, and maybe even your entire first draft. Sometimes the best way to revise is to start fresh. The knowledge you have built in writing a first draft will serve you well if you need to start over.
- Create a new file for each revision. Each time you revise a draft, save the new version with a new file name so that you don’t lose your previous work. That way, you can return to an earlier version of your draft if you are not happy with the revision.
- Edit and proofread. When you are satisfied with the overall shape of your paper, reread it once again to check for sentence-level errors in grammar, punctuation, spelling, and source citations.
Taking it Public: Publishing or Presenting Your Proposal
Publishing is a final step in the writing process. You may want to consider publishing your full proposal in your campus newspaper (or rewriting it as a letter to the editor) if your topic is related to your school. Or you may want to present it to an organization or committee on campus that can help you make your solution a reality. If your topic is related to the community in which you live, consider submitting your proposal to the local newspaper or presenting it at a city council meeting. (Note that if you decide to present your proposal orally, you’ll need to figure out in advance the procedure for speaking or getting on a meeting agenda.) If your topic is more general and involves substantial research, consider submitting your proposal to one of these journals that publish undergraduate research work in all fields:
- American Journal of Undergraduate Research ( https://openstax.org/r/americanjournalofugreasearch )
- Midwest Journal of Undergraduate Research ( https://openstax.org/r/midwestjournalugreasearch )
- PURSUE Undergraduate Research Journal ( https://openstax.org/r/pursuejournalugreasearch )


13.2: Amino acids
41
Harper College Chemistry Department
Amino acids are the monomers that comprise proteins. Each amino acid has the same fundamental structure, which consists of a central carbon atom, or the alpha ( α ) carbon, bonded to an amino group (NH 2 ), a carboxyl group (COOH), and to a hydrogen atom. Every amino acid also has another atom or group of atoms bonded to the central atom known as the R group ( Figure 1 ).
Scientists use the name “amino acid” because these acids contain both amino group and carboxylic acid group in their basic structure. As we mentioned, there are 20 common amino acids present in proteins. Nine of these are essential amino acids in humans because the human body cannot produce them and we obtain them from our diet. For each amino acid, the R group (or side chain) is different ( Figure 2 ).
Which categories of amino acid would you expect to find on a soluble protein’s surface and which would you expect to find in the interior? What distribution of amino acids would you expect to find in a protein embedded in a lipid bilayer?
The chemical nature of the side chain determines the amino acid’s nature (that is, whether it is acidic, basic, polar, or nonpolar). For example, the amino acid glycine has a hydrogen atom as the R group. Amino acids such as valine, methionine, and alanine are nonpolar or hydrophobic in nature, while amino acids such as serine, threonine, and cysteine are polar and have hydrophilic side chains. The side chains of lysine and arginine are positively charged, and therefore these amino acids are also basic amino acids. Proline has an R group that is linked to the amino group, forming a ring-like structure. Proline is an exception to the amino acid’s standard structure since its amino group is not separate from the side chain ( Figure 3 ).
A single upper case letter or a three-letter abbreviation represents amino acids. For example, the letter V or the three-letter symbol val represent valine. Just as some fatty acids are essential to a diet, some amino acids also are necessary. These essential amino acids in humans include isoleucine, leucine, and cysteine. Essential amino acids refer to those necessary to build proteins in the body, but not those that the body produces. Which amino acids are essential varies from organism to organism.
The sequence and the number of amino acids ultimately determine the protein’s shape, size, and function. A covalent bond, or peptide bond , attaches to each amino acid, which a dehydration reaction forms. One amino acid’s carboxyl group and the incoming amino acid’s amino group combine, releasing a water molecule. The resulting bond is the peptide bond ( Figure 3.24 ).
The products that such linkages form are peptides. As more amino acids join to this growing chain, the resulting chain is a polypeptide. Each polypeptide has a free amino group at one end. This end the N terminal, or the amino terminal, and the other end has a free carboxyl group, also the C or carboxyl terminal. While the terms polypeptide and protein are sometimes used interchangeably, a polypeptide is technically a polymer of amino acids, whereas the term protein is used for a polypeptide or polypeptides that have combined together, often have bound non-peptide prosthetic groups, have a distinct shape, and have a unique function. After protein synthesis (translation), most proteins are modified. These are known as post-translational modifications. They may undergo cleavage, phosphorylation, or may require adding other chemical groups. Only after these modifications is the protein completely functional.
LINK TO LEARNING
Click through the steps of protein synthesis in this interactive tutorial .
EVOLUTION CONNECTION
The Evolutionary Significance of Cytochrome c
Cytochrome c is an important component of the electron transport chain, a part of cellular respiration, and it is normally located in the cellular organelle, the mitochondrion. This protein has a heme prosthetic group, and the heme’s central ion alternately reduces and oxidizes during electron transfer. Because this essential protein’s role in producing cellular energy is crucial, it has changed very little over millions of years. Protein sequencing has shown that there is a considerable amount of cytochrome c amino acid sequence homology among different species. In other words, we can assess evolutionary kinship by measuring the similarities or differences among various species’ DNA or protein sequences.
Scientists have determined that human cytochrome c contains 104 amino acids. For each cytochrome c molecule from different organisms that scientists have sequenced to date, 37 of these amino acids appear in the same position in all cytochrome c samples. This indicates that there may have been a common ancestor. On comparing the human and chimpanzee protein sequences, scientists did not find a sequence difference. When researchers compared human and rhesus monkey sequences, the single difference was in one amino acid. In another comparison, human to yeast sequencing shows a difference in the 44th position.


6.1: Grading Has Always Made Writing Better
-
- Last updated
- Save as PDF
- Cheryl E. Ball & Drew M. Loewe ed.
- West Virginia University via Digital Publishing Institute and West Virginia University Libraries
Author: Mitchell R. James, Lakeland Community College.
 It may not rest on your hip, potent and rigid like a cop’s gun, but in the long run it’s more powerful.”—Larry Tjarks
There are a number of problems surrounding the ubiquitous practice of grading student writing. In Schools Without Failure , William Glasser notes that grading tends to be perceived by students as various levels of failure. In addition, Marie Wilson argues that a focus on failure leads teachers to approach student writing in search of deficiencies instead of strengths, which puts students in a state of preventative or corrective mindsets when trying to learn. These mindsets are especially troubling for students in writing classes, where errors must be made in order for students to grow and develop.
Another problem with grading, Brian Huot notes in (Re) Articulating Writing Assessment for Teaching and Learning , is that it rarely communicates anything of value to students. When I take a narrative that a student has written in one of my courses— something that has evolved through several drafts and has greatly improved—and I tell that student the paper is an 85%, what am I saying? 85% of what? Am I saying the narrative is in the top 85% of the class, the top 85% of narratives written by all college freshmen in the U.S., or in the top 85% of all the narratives I’ve ever read? Or maybe I’m comparing what was executed in the narrative to a rubric, and I’m suggesting the student met 85% of the objectives on the rubric, such as effective dialogue, strong verbs, and detailed description. But might a narrative that uses all three objectives still be a poorly written narrative?
The breakdown of communication inherent in this kind of summative - only , end-of-the-paper/project grading is a grave issue. As a case in point, Liesel K. O’Hagan and colleagues demonstrate the lack of useful information gleaned by students when grading is implemented in a classroom. As a part of the study, one student wrote, “I don’t even understand what the grade means on my paper. The top says something like a B and then all the comments say positive things and then there are all these errors marked. Then the person next to me wrote only half as much as I did and has even more errors marked and he got an A. It just doesn’t make any sense to me.”
So why are we still so dependent on grading? The simplest answer is growth in student numbers. Education used to be only for the wealthy and privileged. That changed at the start of the 20th century, and continues through our present time with such acts as mandatory attendance laws, the GI Bill, and the growth of open-enrollment colleges. As student numbers and diversity rose in the classroom, the models of grading we use today came to fruition, and those that had been used before were relegated to near obscurity. However, it might be in the past where we can find the answers to the present question: If grading writing is counterproductive, what else can we do?
The grading process in place before the late 19th century hinged more on direct contact between student work, course content, the student, and the teacher. For example, in English classes, teachers would respond to student writing in both written and spoken form. There were many levels of communication between the student and teacher, which provided more opportunity for the student to gain an understanding and command of course content. In addition, a student’s success depended on demonstrating the skills taught. If students could demonstrate the necessary skills (reading, writing, or speaking) then, and only then, did they pass the course. This more attentive and interactive approach is akin to what occurs in assessment .
Assessment and grading are not synonymous. Grading is a silent, one-way evaluation, where a teacher assigns a letter, rife with a set of socio-cultural significances, to a piece of student writing. Assessment, on the other hand, provides the opportunity for two kinds of evaluation—formative and summative.
Formative evaluation—done typically by responding to in-process student writing several times during the semester—replaces the punishment or praise of student learning, typically demonstrated
through grading a final product or test, with a process that encourages communication as a part of learning. When using formative evaluation, teachers and students speak with one another often. In addition, formative evaluation creates safe spaces for student learning because students are not focused on trying to avoid failure but, instead, are searching for insight and growth. As grades lose their power, the desire to evade punishment or failure can dissolve into the desire to seek knowledge and learn something new. Finally, because of the communicative nature of formative evaluation, students develop the capacity to talk about and, in some instances, even teach the material themselves as they work with their peers to explain what they know.
Summative evaluation follows extensive formative evaluation. Summative evaluation is superior to grading because it assesses a student’s ability to meet a priori criteria without the use of a letter grade. Summative evaluation methods such as student self-reflection on the learning process, ungraded portfolio assessment, and contract grading all provide the opportunity for teachers to assess and respond to student learning free of the socio-political, socio-economic letter grade.
Unfortunately, like most teachers, I have to provide grades in the summative sense. If I don’t submit a letter grade at the end of a semester, I will not have a job. But providing end-of-semester grades doesn’t preclude providing formative assessment that can help students revise a text or project so they will better understand why they might receive an 85% as a final grade. If I had a choice by my institution whether to provide summative grades, however, I wouldn’t do it again. In short, the enterprise of grading student writing should be replaced by a combination of formative and summative evaluation.
Further Reading
To learn more about grading, assessment, and higher education, read Stephen Tchudi’s Alternatives to Grading Student Writing , Brian Huot’s (Re)Articulating Writing Assessment , and William Glasser’s Schools Without Failure . If looking for the most contemporary material on the subject of grading in education, consult the work of Mark Barnes, who has published a number of books such as Assessment 3.0 and has an intriguing TED Talk on the need to eliminate grading altogether. Carnegie Mellon University’s Assess Teaching and Learning website also contains informative definitions for and practices of assessment techniques.
Keywords
assessment, formative assessment, grading, grading alternatives, summative assessment
Author Bio
Mitchell R. James is a fiction writer, poet, and assistant professor of composition and literature at Lakeland Community College. He received a B.A. in literature from Eastern Illinois University, and both an M.A. in literature and a Ph.D. in composition from Indiana University of Pennsylvania. Mitch’s research interests are classical rhetoric, interdisciplinary approaches to creative writing, formative and summative assessment, alternatives to grading, and responding to student writing. Mitch has been writing, researching, and teaching in these topics for many passionate years. Track Mitch down on Twitter @mrjames5527.


1.5: The Scattering Amplitude and Scattering Cross Section
We can use the results of the previous section to systematically characterize the outcomes of a scattering experiment. Let the incident wavefunction be a plane wave,
\[\psi_i(\mathbf{r}) = \Psi_i \, e^{i\mathbf{k}_i\cdot\mathbf{r}},\]
in \(d\) -dimensional space. Here, \(\Psi_i \in \mathbb{C}\) is the incident wave amplitude , and \(\mathbf{k}_i\) is the incident momentum. Let \(k = |\mathbf{k}_i|\) denote its magnitude, so that the particle energy is \(E = \hbar^2k^2/2m\) . We adopt coordinates \((r,\Omega)\) , where \(r\) is the distance from the origin. For 1D, \(\Omega \in \pm\) which specifies the choice of “forward” or “backward” scattering; for 2D polar coordinates, \(\Omega = \phi\) ; and for 3D spherical coordinates, \(\Omega = (\theta,\phi)\) .
Far from the origin, the scattered wavefunction reduces to the form
\[\psi_s(\mathbf{r})\; \overset{r\rightarrow\infty}{\longrightarrow}\; \Psi_i \, r^{\frac{1-d}{2}} \, e^{ikr} \, f(\Omega).\]
The complex function \(f(\Omega)\) , called the scattering amplitude , is the fundamental quantity of interest in scattering experiments. It describes how the particle is scattered in various directions, depending on the inputs to the problem (i.e., \(\mathbf{k}_i\) and the scattering potential).
Sometimes, we write the scattering amplitude using the alternative notation
\[f(\mathbf{k}_i \rightarrow \mathbf{k}_f), \;\;\;\mathrm{where}\;\; \mathbf{k}_f = k \hat{\mathbf{r}}.\]
This emphasizes firstly that the incident wave-vector is \(\mathbf{k}_i\) ; and secondly that the particle is scattered in some direction which can be specified by either the unit position vector \(\hat{\mathbf{r}}\) , or equivalently by the momentum vector \(\mathbf{k}_f = k \hat{\mathbf{r}}\) , or by the angular coordinates \(\Omega\) .
From the scattering amplitude, we define two other important quantities of interest:
Definition: Scattering Cross Sections
\[\begin{align} \frac{d\sigma}{d\Omega} &= \big|f(\Omega)\big|^2 \qquad\quad\quad \text{(the } \textbf{differential scattering cross section}) \\ \sigma &= \int d\Omega\; \big|f(\Omega)\big|^2 \quad\; \text{(the }\textbf{total scattering cross section}). \end{align}\]
In the second equation, \(\int d\Omega\) denotes the integral(s) over all the angle coordinates; for 1D, this is instead a discrete sum over the two possible directions, forward and backward.
The term “cross section” comes from an analogy with the scattering of classical particles. Consider the probablity current density associated with the scattered wavefunction:
\[\mathbf{J}_s = \frac{\hbar}{m} \mathrm{Im}\big[\psi_s^*\nabla\psi_s\big].\]
Let us focus only on the \(r\) -component of the current density, in the \(r\rightarrow\infty\) limit:
\[\begin{align}\begin{aligned}J_{s,r}\; &\;\;=\;\;\;\, \frac{\hbar}{m}\, \mathrm{Im}\left[\psi_s^* \frac{\partial}{\partial r}\psi_s\right] \\ &\overset{r\rightarrow\infty}{\longrightarrow}\;\, \frac{\hbar}{m}\; |\Psi_i|^2 \; |f(\Omega)|^2 \mathrm{Im}\left[\left(r^{\frac{1-d}{2}} \, e^{ikr}\right)^* \frac{\partial}{\partial r} \left(r^{\frac{1-d}{2}} \, e^{ikr}\right)\right] \\ & \;\;=\;\;\;\, \frac{\hbar \,k}{m}\; |\Psi_i|^2 \; |f(\Omega)|^2 \,r^{1-d}.\end{aligned}\end{align}\]
The total flux of outgoing probability is obtained by integrating \(J_{s,r}\) over a constant- \(r\) surface:
\[I_s \;=\; \int d\Omega\; r^{d-1} \, J_{s,r} \;=\; \frac{\hbar k}{m} \;|\Psi_i|^2 \; \int d\Omega\; \big|f(\Omega)\big|^2. \label{scatterI}\]
We can assign a physical interpretation to each term in this result. The first factor, \(\hbar k/m\) , is the particle’s speed (i.e., the group velocity of the de Broglie wave). The second factor, \(|\Psi_i|^2\) , is the probability density of the incident wave, which has units of \([x^{-d}]\) (i.e., inverse \(d\) -dimensional “volume”). The product of these two factors represents the incident flux ,
\[J_{i} = \frac{\hbar k}{m} \;|\Psi_i|^2.\]
This has units of \([x^{1-d}t^{-1}]\) (i.e., rate per unit “area”).
Let us re-imagine this incident flux \(J_i\) as a stream of classical particles, and the scatterer as a “hard-body” scatterer that only interacts with those particles striking it directly:
In this classical picture, the rate at which the incident particles strike the scatterer is
\[I_s = J_i \, \sigma,\]
where \(\sigma\) is the exposed cross-sectional area of the scatterer. Comparing this expression to Equation \(\eqref{scatterI}\), we see that \(\int d\Omega\,|f|^2\) plays a role analogous to the classical hard-body cross-sectional area. We hence call
\[\sigma \equiv \int d\Omega\,|f|^2\]
the total scattering cross section . Moreover, the integrand \(|f|^2\) is called the differential scattering cross section , for it represents the rate, per unit of solid angle, at which particles are scattered in a given direction. The total and differential scattering cross sections are the principal observable quantities that can be obtained from scattering experiments.


1.13: End of Chapter Problems II
The following problems on dosage calculations I are to be completed with unit 2.
Dosage Calculations I
Please set up your calculations using the dimensional analysis.
1. Epinephrine is used to treat peanut allergies. A 5 year old patient needs an immediate treatment of 0.12 mg of epinephrine. The solution is available in concentration of 0.50 mg/ml. Calculate the volume in ml required for the patient.
2. A physician ordered 100 mg of Demerol. Demerol is available as 50 mg per tablet. How many tablets should the nurse administer?
3. The physician ordered 20 mg of Valium. Valium is available as 10 mg per tablet. How many tablets should the nurse administer?
4. The recommended daily allowance (RDA) for magnesium for 19-30 yrs old+ men is 400 mg. Magnesium supplements come in 200 mg capsules. How many capsules need to be taken to meet RDA?
5. The safe dosage of an IV antibiotic for children weighing more than 2.0kg is 60.mg per kilogram of body weight. How many mg should be administered to a child weighing 16 kg?
6. The tranquilizer valium is sold in 2.0 mL syringes that contain 50.0 mg of the drug per 1.0 mL of liquid. If a physician prescribes 25 mg of this drug, how many mL should be administered?
7. An antibiotic is sold in 3.0 mL ampoules that contain 60.0 mg of the drug. How many mL of the antibiotic should be withdrawn from the ampoule if 45 mg are to be administered to a patient?


1.5: Subtraction of Whole Numbers
- understand the subtraction process
- be able to subtract whole numbers
- be able to use a calculator to subtract one whole number from another whole number
Subtraction
Subtraction is the process of determining the remainder when part of the total is removed.
Suppose the sum of two whole numbers is 11, and from 11 we remove 4. Using the number line to help our visualization, we see that if we are located at 11 and move 4 units to the left, and thus remove 4 units, we will be located at 7. Thus, 7 units remain when we remove 4 units from 11 units.
The minus symbol (-) is used to indicate subtraction. For example, 11 − 4 indicates that 4 is to be subtracted from 11.
The number immediately in front of or the minus symbol is called the minuend , and it represents the original number of units.
The number immediately following or below the minus symbol is called the subtrahend , and it represents the number of units to be removed .
The result of the subtraction is called the difference of the two numbers. For example, in 11 − 4 = 7
Subtraction as the Opposite of Addition
Subtraction can be thought of as the opposite of addition. We show this in the problems in Sample Set A.
8 - 5 = 3 since 3 + 5 = 8.
9 - 3 = 6 since 6 + 3 = 9.
Practice Set A
Complete the following statements.
7 - 5 = since + 5 = 7
- Answer
-
7 - 5 = 2 since 2 + 5 = 7
Practice Set A
9 - 1 = since + 1 = 9
- Answer
-
9 - 1 = 8 since 8 + 1 = 9
Practice Set A
17 - 8 = since + 8 = 17
- Answer
-
17 - 8 = 9 since 9 + 8 = 17
The Subtraction Process
We'll study the process of the subtraction of two whole numbers by considering the difference between 48 and 35.
which we write as 13.
To subtract two whole numbers,
The process
-
Write the numbers vertically, placing corresponding positions in the same column.
\(\begin{array} {r} {48} \\ {\underline{-35}} \end{array}\) -
Subtract the digits in each column. Start at the right, in the ones position, and move to the left, placing the difference at the bottom.
\(\begin{array} {r} {48} \\ {\underline{-35}} \\ {13} \end{array}\)
Perform the following subtractions.
\(\begin{array} {r} {275} \\ {\underline{-142}} \\ {133} \end{array}\)
\(\begin{array} {l} {5 - 2 = 3.} \\ {7 - 4 = 3.} \\ {2 - 1 = 1.} \end{array}\)
\(\begin{array} {r} {46,042} \\ {\underline{-\ \ 1,031}} \\ {45,011} \end{array}\)
\(\begin{array} {l} {2 - 1 = 1.} \\ {4 - 3 = 1.} \\ {0 - 0 = 0.} \\ {6 - 1 = 5.} \\ {4 - 0 = 4.} \end{array}\)
Find the difference between 977 and 235.
Write the numbers vertically, placing the larger number on top. Line up the columns properly.
\(\begin{array} {r} {977} \\ {\underline{-235}} \\ {742} \end{array}\)
The difference between 977 and 235 is 742.
In Keys County in 1987, there were 809 cable television installations. In Flags County in 1987, there were 1,159 cable television installations. How many more cable television installations were there in Flags County than in Keys County in 1987?
We need to determine the difference between 1,159 and 809.
There were 350 more cable television installations in Flags County than in Keys County in 1987.
Practice Set B
Perform the following subtractions.
\(\begin{array} {r} {534} \\ {\underline{-203}} \end{array}\)
- Answer
-
331
Practice Set B
\(\begin{array} {r} {857} \\ {\underline{-\ \ 43}} \end{array}\)
- Answer
-
814
Practice Set B
\(\begin{array} {r} {95,628} \\ {\underline{-34,510}} \end{array}\)
- Answer
-
61,118
Practice Set B
\(\begin{array} {r} {11,005} \\ {\underline{-\ \ 1,005}} \end{array}\)
- Answer
-
10,000
Practice Set B
Find the difference between 88,526 and 26,412.
- Answer
-
62,114
In each of these problems, each bottom digit is less than the corresponding top digit. This may not always be the case. We will examine the case where the bottom digit is greater than the corresponding top digit in the next section.
Subtraction Involving Borrowing
It often happens in the subtraction of two whole numbers that a digit in the minuend (top number) will be less than the digit in the same position in the subtrahend (bottom number). This happens when we subtract 27 from 84.
\(\begin{array} {r} {84} \\ {\underline{-27}} \end{array}\)
We do not have a name for 4 − 7. We need to rename 84 in order to continue. We'll do so as follows:
Our new name for 84 is 7 tens + 14 ones.
= 57
Notice that we converted 8 tens to 7 tens + 1 ten, and then we converted the 1 ten to 10 ones. We then had 14 ones and were able to perform the subtraction.
The process of borrowing (converting) is illustrated in the problems of Sample Set C.
- Borrow 1 ten from the 8 tens. This leaves 7 tens.
- Convert the 1 ten to 10 ones.
- Add 10 ones to 4 ones to get 14 ones.
- Borrow 1 hundred from the 6 hundreds. This leaves 5 hundreds.
- Convert the 1 hundred to 10 tens.
- Add 10 tens to 7 tens to get 17 tens.
Practice Set C
Perform the following subtractions. Show the expanded form for the first three problems.
\(\begin{array} {r} {53} \\ {\underline{-35}} \end{array}\)
- Answer
-
Practice Set C
\(\begin{array} {r} {76} \\ {\underline{-28}} \end{array}\)
- Answer
-
Practice Set C
\(\begin{array} {r} {872} \\ {\underline{-565}} \end{array}\)
- Answer
-
Practice Set C
\(\begin{array} {r} {441} \\ {\underline{-356}} \end{array}\)
- Answer
-
85
Practice Set C
\(\begin{array} {r} {775} \\ {\underline{-\ \ 66}} \end{array}\)
- Answer
-
709
Practice Set C
\(\begin{array} {r} {5,663} \\ {\underline{-2,559}} \end{array}\)
- Answer
-
3,104
Borrowing More Than Once
Sometimes it is necessary to borrow more than once . This is shown in the problems in Sample Set D.
Perform the Subtractions. Borrowing more than once if necessary
- Borrow 1 ten from the 4 tens. This leaves 3 tens.
- Convert the 1 ten to 10 ones.
- Add 10 ones to 1 one to get 11 ones. We can now perform 11 − 8.
- Borrow 1 hundred from the 6 hundreds. This leaves 5 hundreds.
- Convert the 1 hundred to 10 tens.
- Add 10 tens to 3 tens to get 13 tens.
- Now 13 − 5 = 8.
- 5 − 3 = 2.
- Borrow 1 ten from the 3 tens. This leaves 2 tens.
- Convert the 1 ten to 10 ones.
- Add 10 ones to 4 ones to get 14 ones. We can now perform 14 − 5.
- Borrow 1 hundred from the 5 hundreds. This leaves 4 hundreds.
- Convert the 1 hundred to 10 tens.
- Add 10 tens to 2 tens to get 12 tens. We can now perform 12 − 8 = 4.
- Finally, 4 − 0 = 4.
\(\begin{array} {r} {71529} \\ {\underline{-\ \ 6952}} \end{array}\)
After borrowing, we have
Practice Set D
Perform the following subtractions.
\(\begin{array} {r} {526} \\ {\underline{-358}} \end{array}\)
- Answer
-
168
Practice Set D
\(\begin{array} {r} {63,419} \\ {\underline{-\ \ 7,779}} \end{array}\)
- Answer
-
55,640
Practice Set D
\(\begin{array} {r} {4,312} \\ {\underline{-3,123}} \end{array}\)
- Answer
-
1,189
Borrowing from Zero
It often happens in a subtraction problem that we have to borrow from one or more zeros.
Consider the problem \(\begin{array} {r} {503} \\ {\underline{-\ \ 37}} \end{array}\)
Since we do not have a name for 3 − 7, we must borrow from 0.
Since there are no tens to borrow, we must borrow 1 hundred. One hundred = 10 tens.
We can now borrow 1 ten from 10 tens (leaving 9 tens). One ten = 10 ones and 10 ones + 3 ones = 13 ones.
Now we can suggest the following method for borrowing from a single zero.
Borrowing from a Single Zero To borrow from a single zero,
- Decrease the digit to the immediate left of zero by one.
- Draw a line through the zero and make it a 10.
- Proceed to subtract as usual.
Perform this subtraction.
\(\begin{array} {r} {503} \\ {\underline{-\ \ 37}} \end{array}\)
The number 503 contains a single zero
-
The number to the immediate left of 0 is 5. Decrease 5 by 1.
5 - 1 = 4
- Draw a line through the zero and make it a 10.
-
Borrow from the 10 and proceed.
\(\begin{array} {l} {\text{1 ten + 10 ones}} \\ {\text{10 ones + 3 ones = 13 ones}} \end{array}\)
Practice Set E
Perform each subtraction.
\(\begin{array} {r} {906} \\ {\underline{-\ \ 18}} \end{array}\)
- Answer
-
888
Practice Set E
\(\begin{array} {r} {5102} \\ {\underline{-\ \ 559}} \end{array}\)
- Answer
-
4,543
Practice Set E
\(\begin{array} {r} {9055} \\ {\underline{-\ \ 386}} \end{array}\)
- Answer
-
8,669
Consider the problem \(\begin{array} {r} {5000} \\ {\underline{-\ \ \ \ 37}} \end{array}\)
In this case, we have a group of zeros.
Since we cannot borrow any tens or hundreds, we must borrow 1 thousand. One thousand = 10 hundreds.
We can now borrow 1 hundred from 10 hundreds. One hundred = 10 tens.
We can now borrow 1 ten from 10 tens. One ten = 10 ones.
From observations made in this procedure we can suggest the following method for borrowing from a group of zeros.
Borrowing from a Group of zeros
To borrow from a group of zeros,
- Decrease the digit to the immediate left of the group of zeros by one.
- Draw a line through each zero in the group and make it a 9, except the rightmost zero, make it 10.
- Proceed to subtract as usual.
Perform each subtraction.
\(\begin{array} {r} {40,000} \\ {\underline{-\ \ \ \ \ 125}} \end{array}\)
Solution
The number 40,000 contains a group of zeros.
-
The number to the immediate left of the group is 4. Decrease 4 by 1.
4 - 1 = 3 - Make each 0, except the rightmost one, 9. Make the rightmost 0 a 10.
- Subtract as usual.
\(\begin{array} {r} {8,000,006} \\ {\underline{-\ \ \ \ \ 41,107}} \end{array}\)
Solution
The number 8,000,006 contains a group of zeros.
-
The number to the immediate left of the group is 8. Decrease 8 by 1.
8 - 1 = 7 - Make each zero, except the rightmost one, 9. Make the rightmost 0 a 10.
-
To perform the subtraction, we’ll need to borrow from the ten.
\(\begin{array} {l} {\text{1 ten = 10 ones}} \\ {\text{10 ones + 6 ones = 16 ones}} \end{array}\)
Practice Set F
Perform each subtraction.
\(\begin{array} {r} {21,007} \\ {\underline{-\ \ 4,873}} \end{array}\)
- Answer
-
16,134
Practice Set F
\(\begin{array} {r} {10,004} \\ {\underline{-\ \ 5,165}} \end{array}\)
- Answer
-
4,839
Practice Set F
\(\begin{array} {r} {16,000,000} \\ {\underline{-\ \ \ \ \ 201,060}} \end{array}\)
- Answer
-
15,789,940
Calculators
In practice, calculators are used to find the difference between two whole numbers.
Find the difference between 1006 and 284.
| Display Reads | ||
| Type | 1006 | 1006 |
| Press | −− | 1006 |
| Type | 284 | 284 |
| Press | = | 722 |
The difference between 1006 and 284 is 722.
(What happens if you type 284 first and then 1006? We'll study such numbers in Chapter 10.)
Practice Set G
Use a calculator to find the difference between 7338 and 2809.
- Answer
-
4,529
Practice Set G
Use a calculator to find the difference between 31,060,001 and 8,591,774.
- Answer
-
22,468,227
Exercises
For the following problems, perform the subtractions. You may check each difference with a calculator.
Exercise \(\PageIndex{1}\)
\(\begin{array} {r} {15} \\ {\underline{-\ \ 8}} \end{array}\)
- Answer
-
7
Exercise \(\PageIndex{2}\)
\(\begin{array} {r} {19} \\ {\underline{-\ \ 8}} \end{array}\)
Exercise \(\PageIndex{3}\)
\(\begin{array} {r} {11} \\ {\underline{-\ \ 5}} \end{array}\)
- Answer
-
6
Exercise \(\PageIndex{4}\)
\(\begin{array} {r} {14} \\ {\underline{-\ \ 6}} \end{array}\)
Exercise \(\PageIndex{5}\)
\(\begin{array} {r} {12} \\ {\underline{-\ \ 9}} \end{array}\)
- Answer
-
3
Exercise \(\PageIndex{6}\)
\(\begin{array} {r} {56} \\ {\underline{-12}} \end{array}\)
Exercise \(\PageIndex{7}\)
\(\begin{array} {r} {74} \\ {\underline{-33}} \end{array}\)
- Answer
-
41
Exercise \(\PageIndex{8}\)
\(\begin{array} {r} {80} \\ {\underline{-61}} \end{array}\)
Exercise \(\PageIndex{9}\)
\(\begin{array} {r} {350} \\ {\underline{-141}} \end{array}\)
- Answer
-
209
Exercise \(\PageIndex{10}\)
\(\begin{array} {r} {800} \\ {\underline{-650}} \end{array}\)
Exercise \(\PageIndex{11}\)
\(\begin{array} {r} {35,002} \\ {\underline{-14,001}} \end{array}\)
- Answer
-
21,001
Exercise \(\PageIndex{12}\)
\(\begin{array} {r} {5,000,566} \\ {\underline{-2,441,326}} \end{array}\)
Exercise \(\PageIndex{13}\)
\(\begin{array} {r} {400,605} \\ {\underline{-121,352}} \end{array}\)
- Answer
-
279,253
Exercise \(\PageIndex{14}\)
\(\begin{array} {r} {46,400} \\ {\underline{-\ \ 2,012}} \end{array}\)
Exercise \(\PageIndex{15}\)
\(\begin{array} {r} {77,893} \\ {\underline{-\ \ \ \ \ 421}} \end{array}\)
- Answer
-
77,472
Exercise \(\PageIndex{16}\)
\(\begin{array} {r} {42} \\ {\underline{-18}} \end{array}\)
Exercise \(\PageIndex{17}\)
\(\begin{array} {r} {51} \\ {\underline{-27}} \end{array}\)
- Answer
-
24
Exercise \(\PageIndex{18}\)
\(\begin{array} {r} {622} \\ {\underline{-\ \ 88}} \end{array}\)
Exercise \(\PageIndex{19}\)
\(\begin{array} {r} {261} \\ {\underline{-\ \ 73}} \end{array}\)
- Answer
-
188
Exercise \(\PageIndex{20}\)
\(\begin{array} {r} {242} \\ {\underline{-158}} \end{array}\)
Exercise \(\PageIndex{21}\)
\(\begin{array} {r} {3,422} \\ {\underline{-1,045}} \end{array}\)
- Answer
-
2,377
Exercise \(\PageIndex{22}\)
\(\begin{array} {r} {5,565} \\ {\underline{-3,985}} \end{array}\)
Exercise \(\PageIndex{23}\)
\(\begin{array} {r} {42,041} \\ {\underline{-15,355}} \end{array}\)
- Answer
-
26,686
Exercise \(\PageIndex{24}\)
\(\begin{array} {r} {304,056} \\ {\underline{-\ \ 20,008}} \end{array}\)
Exercise \(\PageIndex{25}\)
\(\begin{array} {r} {64,000,002} \\ {\underline{-\ \ \ \ \ 856,743}} \end{array}\)
- Answer
-
63,143,259
Exercise \(\PageIndex{26}\)
\(\begin{array} {r} {4,109} \\ {\underline{-\ \ \ 856}} \end{array}\)
Exercise \(\PageIndex{27}\)
\(\begin{array} {r} {10,113} \\ {\underline{-\ \ 2,079}} \end{array}\)
- Answer
-
8,034
Exercise \(\PageIndex{28}\)
\(\begin{array} {r} {605} \\ {\underline{-\ \ 77}} \end{array}\)
Exercise \(\PageIndex{27}\)
\(\begin{array} {r} {59} \\ {\underline{-26}} \end{array}\)
- Answer
-
33
Exercise \(\PageIndex{28}\)
\(\begin{array} {r} {36,107} \\ {\underline{-\ \ 8,314}} \end{array}\)
Exercise \(\PageIndex{29}\)
\(\begin{array} {r} {92,526,441,820} \\ {\underline{-59,914,805,253}} \end{array}\)
- Answer
-
32,611,636,567
Exercise \(\PageIndex{30}\)
\(\begin{array} {r} {1,605} \\ {\underline{-\ \ 881}} \end{array}\)
Exercise \(\PageIndex{31}\)
\(\begin{array} {r} {30,000} \\ {\underline{-26,062}} \end{array}\)
- Answer
-
3,938
Exercise \(\PageIndex{32}\)
\(\begin{array} {r} {600} \\ {\underline{-216}} \end{array}\)
Exercise \(\PageIndex{33}\)
\(\begin{array} {r} {90,000,003} \\ {\underline{-\ \ \ 726,048}} \end{array}\)
- Answer
-
8,273,955
For the following problems, perform each subtraction.
Exercise \(\PageIndex{34}\)
Subtract 63 from 92.
The word "from" means "beginning at." Thus, 63 from 92 means beginning at 92, or 92 - 63.
Exercise \(\PageIndex{35}\)
Subtract 35 from 86.
- Answer
-
51
Exercise \(\PageIndex{34}\)
Subtract 382 from 541.
Exercise \(\PageIndex{35}\)
Subtract 1,841 from 5,246.
- Answer
-
3,405
Exercise \(\PageIndex{36}\)
Subtract 26,082 from 35,040.
Exercise \(\PageIndex{37}\)
Find the difference between 47 and 21.
- Answer
-
26
Exercise \(\PageIndex{38}\)
Find the difference between 1,005 and 314.
Exercise \(\PageIndex{39}\)
Find the difference between 72,085 and 16.
- Answer
-
72,069
Exercise \(\PageIndex{40}\)
Find the difference between 7,214 and 2,049.
Exercise \(\PageIndex{41}\)
Find the difference between 56,108 and 52,911.
- Answer
-
3,197
Exercise \(\PageIndex{42}\)
How much bigger is 92 than 47?
Exercise \(\PageIndex{43}\)
How much bigger is 114 than 85?
- Answer
-
29
Exercise \(\PageIndex{44}\)
How much bigger is 3,006 than 1,918?
Exercise \(\PageIndex{45}\)
How much bigger is 11,201 than 816?
- Answer
-
10,385
Exercise \(\PageIndex{46}\)
How much bigger is 3,080,020 than 1,814,161?
Exercise \(\PageIndex{47}\)
In Wichita, Kansas, the sun shines about 74% of the time in July and about 59% of the time in November. How much more of the time (in percent) does the sun shine in July than in November?
- Answer
-
15%
Exercise \(\PageIndex{48}\)
The lowest temperature on record in Concord, New Hampshire in May is 21°F, and in July it is 35°F. What is the difference in these lowest temperatures?
Exercise \(\PageIndex{49}\)
In 1980, there were 83,000 people arrested for prostitution and commercialized vice and 11,330,000 people arrested for driving while intoxicated. How many more people were arrested for drunk driving than for prostitution?
- Answer
-
11,247,000
Exercise \(\PageIndex{50}\)
In 1980, a person with a bachelor's degree in accounting received a monthly salary offer of $1,293, and a person with a marketing degree a monthly salary offer of $1,145. How much more was offered to the person with an accounting degree than the person with a marketing degree?
Exercise \(\PageIndex{51}\)
In 1970, there were about 793 people per square mile living in Puerto Rico, and 357 people per square mile living in Guam. How many more people per square mile were there in Puerto Rico than Guam?
- Answer
-
436
Exercise \(\PageIndex{52}\)
The 1980 population of Singapore was 2,414,000 and the 1980 population of Sri Lanka was 14,850,000. How many more people lived in Sri Lanka than in Singapore in 1980?
Exercise \(\PageIndex{53}\)
In 1977, there were 7,234,000 hospitals in the United States and 64,421,000 in Mainland China. How many more hospitals were there in Mainland China than in the United States in 1977?
- Answer
-
57,187,000
Exercise \(\PageIndex{54}\)
In 1978, there were 3,095,000 telephones in use in Poland and 4,292,000 in Switzerland. How many more telephones were in use in Switzerland than in Poland in 1978?
For the following problems, use the corresponding graphs to solve the problems.
Exercise \(\PageIndex{55}\)
How many more life scientists were there in 1974 than mathematicians?
- Answer
-
165,000
Exercise \(\PageIndex{56}\)
How many more social, psychological, mathematical, and environmental scientists were there than life, physical, and computer scientists?
Exercise \(\PageIndex{57}\)
How many more prosecutions were there in 1978 than in 1974?
- Answer
-
74
Exercise \(\PageIndex{58}\)
How many more prosecutions were there in 1976-1980 than in 1970-1975?
Exercise \(\PageIndex{59}\)
How many more dry holes were drilled in 1960 than in 1975?
- Answer
-
4,547
Exercise \(\PageIndex{60}\)
How many more dry holes were drilled in 1960, 1965, and 1970 than in 1975, 1978 and 1979?
For the following problems, replace the ☐ with the whole number that will make the subtraction true.
Exercise \(\PageIndex{61}\)
\(\begin{array} {r} {14} \\ {\underline{-☐}} \\ {3} \end{array}\)
- Answer
-
11
Exercise \(\PageIndex{62}\)
\(\begin{array} {r} {21} \\ {\underline{-☐}} \\ {14} \end{array}\)
Exercise \(\PageIndex{63}\)
\(\begin{array} {r} {35} \\ {\underline{-☐}} \\ {25} \end{array}\)
- Answer
-
10
Exercise \(\PageIndex{64}\)
\(\begin{array} {r} {16} \\ {\underline{-☐}} \\ {9} \end{array}\)
Exercise \(\PageIndex{65}\)
\(\begin{array} {r} {28} \\ {\underline{-☐}} \\ {16} \end{array}\)
- Answer
-
12
For the following problems, find the solutions.
Exercise \(\PageIndex{66}\)
Subtract 42 from the sum of 16 and 56.
Exercise \(\PageIndex{67}\)
Subtract 105 from the sum of 92 and 89.
- Answer
-
76
Exercise \(\PageIndex{68}\)
Subtract 1,127 from the sum of 2,161 and 387.
Exercise \(\PageIndex{69}\)
Subtract 37 from the difference between 263 and 175.
- Answer
-
51
Exercise \(\PageIndex{70}\)
Subtract 1,109 from the difference between 3,046 and 920.
Exercise \(\PageIndex{71}\)
Add the difference between 63 and 47 to the difference between 55 and 11.
- Answer
-
60
Exercise \(\PageIndex{72}\)
Add the difference between 815 and 298 to the difference between 2,204 and 1,016.
Exercise \(\PageIndex{73}\)
Subtract the difference between 78 and 43 from the sum of 111 and 89.
- Answer
-
165
Exercise \(\PageIndex{74}\)
Subtract the difference between 18 and 7 from the sum of the differences between 42 and 13, and 81 and 16.
Exercise \(\PageIndex{75}\)
Find the difference between the differences of 343 and 96, and 521 and 488.
- Answer
-
214
Exercises for Review
Exercise \(\PageIndex{76}\)
In the number 21,206, how many hundreds are there?
Exercise \(\PageIndex{77}\)
Write a three-digit number that has a zero in the ones position.
- Answer
-
330 (answers may vary)
Exercise \(\PageIndex{78}\)
How many three-digit whole numbers are there?
Exercise \(\PageIndex{79}\)
Round 26,524,016 to the nearest million.
- Answer
-
27,000,000
Exercise \(\PageIndex{80}\)
Find the sum of 846 + 221 + 116.


1.2.3: Sources and Types of Law
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The American legal system is made up of many types of codified forms of law, with the United States Constitution being the pre-eminent source of American law. The Constitution establishes the boundaries of federal law, and it must be followed by all citizens, organizations, and entities. It includes Congressional acts, Senate-ratified treaties, executive regulations, and federal case law. The United States Code (“USC”) compiles these laws.
American law mainly originates from constitutional law, statutory law, treaties, administrative regulations, and common law (which includes case law).
The Constitution
The United States Constitution is the foremost law of the land. The Constitution’s first ten amendments are referred to as the Bill of Rights, which offers specific protections of individual liberty and justice. Additionally, the Bill of Rights restricts certain powers of government. The Constitution empowers federal law making by giving Congress the power to enact statutes for certain limited purposes, like regulating interstate commerce. The United States Code officially compiles and codifies the federal statutes.
American Common Law
As discussed in the previous section, the United States follows the common law legal tradition of English law. Judges in the Common Law system help shape the law through their rulings and interpretations. This body of past decisions is known as case law , which is used by judges to inform their own rulings. In fact, judges rely on precedent , i.e., previous court rulings on similar cases, when determining the ruling in their own cases.
An example of how case law works is the case of the State v. Wayfair Inc. (2017 SD 56, 901 N.W.2d 754 (S.D. 2017), cert. granted, 138 S. Ct. 735 (2018)), in which the South Dakota Supreme Court held that a state law requiring internet retailers without an in-state physical presence to remit sales tax was unconstitutional. Unless this ruling is overruled by the United States Supreme Court, then it becomes part of the case law and precedent set in that state, and it will be followed by subsequent rulings when similar cases are filed.
Federal Law
The Constitution empowers federal law making by giving Congress the power to enact statutes for certain limited purposes, like regulating interstate commerce. Federal law preempts conflicting state and local laws. However, federal preemption is not without limits, insofar as states each have their own constitution and are considered sovereign. Therefore, federal law may only preempt state law if it is enacted within the limited powers that are enumerated and granted to Congress in the Constitution.
Broad interpretations of the Constitution’s Commerce and Spending Clauses have expanded the reach of federal law into many areas. Indeed, its reach in some areas, such as aviation and railroads, is now so broad that it preempts virtually all state law. In others areas, such as family law, lawmaking continues to be left to the states. Finally, a number of powerful federal and state laws coexist in areas such as antitrust, trademark, employment law, and others.
Statutes
When a bill becomes a federal law, it is assigned a law number and prepared for publication by the Office of the Federal Register (OFR) of the National Archives and Records Administration (NARA). Public laws are also given legal statutory citation by the OFR and are incorporated into the United States Code (USC).
Regulations
Laws differ from regulations in that laws are passed by either the U.S. Congress or state congresses. Regulations, by contrast, are standards and rules adopted by administrative agencies that govern how laws will be enforced.
Federal agencies often enjoy broad rulemaking authority when Congress acts to grant them this power. Called “regulations,” these agency rules normally carry the force of law, as long as they demonstrate a reasonable interpretation of the relevant statutes. For example, the Environmental Protection Agency (EPA) has established regulations for businesses and their emission and disposal of pollutants to protect the environment. The EPA has the authority to enforce these regulations when a business violates them, and such enforcement is usually done by fining the company or by using other means.
The Administrative Procedure Act (APA) enables the adoption of regulations, which are codified and incorporated into the Code of Federal Regulations (CFR). Federal agencies frequently draft and distribute forms, manuals, policy statements, letters, and rulings. Though these may be considered as persuasive authority by the courts, they do not carry the same force as law. In other words, if a person or business questions a regulation of a government agency, saying it is unconstitutional, and that party is successful in proving it, then the regulation is not enforced and the agency will need to revise it or remove it.
State Law
America, as diverse as its fifty states, is also governed by fifty different state constitutions, state governments, and state courts. Each has its own legislative, executive, and judicial branches. States are empowered to create legislation that is related to matters not preempted by the federal Constitution and federal laws. Most cases involve state law issues and are litigated in state courts.
Local Law
In addition to federal and state law, municipalities, towns or cities, and counties may enact their own laws that do not conflict with state or federal laws.
As demonstrated, American law does not draw from one source alone; instead, it is derived from many sources.
Contributors and Attributions
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The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>. Download for free at https://openstax.org/details/books/b...w-i-essentials .


3.2: Missing Assumptions
We need to be able to identify (and then to incorporate into our argument maps) assumptions that are part of the argument, but that weren’t explicitly stated. These are called Hidden Premises , Missing Assumptions , Suppressed Premises/Assumptions , etc. Regardless of the name, these are cases where an argument in fact relies on a claim that it doesn’t state as a premise. There is a claim that must be true if the inference is to make sense, but isn’t explicitly claimed to be true by the argument as it is written or spoke.
Identifying Hidden Assumptions
It’s a bit tricky, but it might be one of the most important and practical skills you’ll learn in this class. How do we figure out when an argument has one or more hidden premises and how do we identify what those premises are?
Well, one answer is simply that formal logic does the job for us when we’re dealing with deductive arguments. This is the math-like system of argument analysis we’ll learn later in the semester. But for now, we need some informal tools to allow us to identify hidden premises or assumptions. This is not only a skill you can build without knowing logic, but is also a skill that extends to inductive reasoning as well and so is far more broadly applicable.
Check out this argument:
Flowers smell nice
???
\(\therefore\) Let’s plant some flowers
[that little triangle of dots is a “therefore” sign]
We can’t quite get from “smelling nice” to “things we’ll plant” without an assumption which links these two ideas. Notice how flowers are in both the premise and the conclusion, so we don’t need to link the topic of flowers together with the topic of flowers.
Abstractly, is A is related to B, and C is related to B, then what we need is something linking A and C so that we can bridge the gap between A and B being associated with one another to C and B being associated with one another.
Less abstractly, if we have three topics: flowers, smelling nice, and things to plant; then we need something linking smelling nice and things to plant so that we know the fact that flowers smell nice is a compelling reason to think that flowers are the kind of thing we should be planting. We know that flowers smell nice and we’re trying to get to flowers should be planted. Any idea what the hidden link might be?
The hidden assumption is something like “we should plant things that smell nice.” Can you see how that completes the inference? Check out some more examples and see if you can figure out what is going on in each example: why do we need the extra premise for the inference to work?
These wildfires are out of control! So global warming is real.
The hidden assumption is something like “global warming is the best explanation for an increase in wildfires.”
We should believe only what is reasonable, so we should reject theism.
The hidden assumption is something like “belief in theism is unreasonable.”
No one believes the Earth is flat anymore, so it’s a silly belief.
The hidden assumption here is something like “any belief that no one currently holds is a silly belief.”
Here’s a step-by-step process for identifying hidden assumptions:
A step-by-step process:
1. First, identify the inference or sub-inference with the hidden assumption
- Which one is “incomplete”?
2. Then, look at the premises of the inference and identify the “terms” or topics discussed in each premise
- Each premise is usually a claim which links two topics together.
3. Then, ask how we can link the terms that aren’t yet linked.
- This requires a bit of imagination and instinct, but you can do it!
4. Finally, write the assumption that links the unlinked terms.
Now that you’ve identified a hidden assumption or more, perform the following two steps:
5. Check to be sure your argument now works
- Does the argument now have a link between each topic? Is there a path from the topics in the premises to the topics in the conclusion?
6. Perform the “negative test” on your assumption
- If you negate your hidden assumption, you should end up with an argument that makes no sense. If the argument with the negated premise makes sense, then you haven’t identified a hidden assumption (i.e. the argument was fine without your assumption).
Let’s take a look at how this works with some real arguments.
I think we should invade North Korea. Look, the Kim Jong dynasty is simply never going to give up on their goal of being a nuclear power.
Okay, this inference is really “fast” meaning that it skates over a few hidden assumptions and so doesn’t seem to work all by itself. It’s like it rushes straight to the finish line without actually running the course. Let’s break it down step by step.
Step 1
There’s only one apparent premise and one apparent conclusion, so identifying the inference in question is easy.
Step 2
The “terms” or topics of this inference are:
A. We should invade
B. North Korea
C. Kim Jong dynasty
D. Nuclear Arms
Step 3
How do we connect these topics? First, we need to connect “North Korea” with the “Kim Jong dynasty”.
Then we’ll need to connect “being a nuclear power” with “we should invade.”
Step 4
Let’s try these assumptions and see how the argument works out:
1. The Kim Jong dynasty is never going to give up on their goal of being a nuclear power .
2. The Kim Jong dynasty is going to lead North Korea for the foreseeable future.
3. Any country that aspires to be a nuclear power is one we should invade .
4. Therefore, we should invade North Korea .
This inference is more complete and connects the topics together more completely, but it rests on one very shaky premise. Can you identify which one?
Yes, that’s right. You are very smart. Premise 3 is pretty wacky, right? What if Argentina decided it wanted to be a Nuclear Power? Should we invade them? I would hope not. They’re a pretty harmless nation at the moment.
So, we have a choice. Either decide that the argument is pretty weak and reject it out of hand, or we can exercise the Principal of Charity to try to interpret the argument to be as plausible as possible. We should always interpret arguments—especially the ones we’re skeptical of or disagree with—to be as rational and plausible as possible.
With that in mind, let’s change this argument up a bit so that it makes a bit more sense. We might not agree with the argument in the end, but at least we will have understood the argument in the best possible light. We will have seen what the most plausible argument for that conclusion on the basis of similar premises is.
If we ignore premise 3, the weak premise, and try to replace it with a few premises which make more specific and believable claims, we’ll be in a better spot. We’ll need to tie together some new topics. Premise 3 was supposed to make a link between “aspires to be a nuclear power” and “we should invade.” There’s a bit of conceptual “distance” between these ideas, though, so we shouldn’t just posit a principle like premise 3 above which directly links them. That would be too easy a principle to reject. Instead, we’ll travel the distance between these ideas in a few steps rather than a giant leap.
How about we connect “aspires to be a nuclear power” to “they’re dangerous”?
Then we can get from “they’re dangerous” to “we should invade.” That sounds more plausible, right?
1. The Kim Jong dynasty is never going to give up on their goal of being a nuclear power .
2. The Kim Jong dynasty is going to lead North Korea for the foreseeable future.
3. North Korea under Kim Jong rule would be an immediate existential danger to its neighbors and the rest of the world if they ever became a nuclear power .
4. If we invade North Korea , then we prevent that danger .
5. Therefore, we should invade North Korea .
Now the argument seems to hang together a bit more clearly. We have a clear path from the Kim Jong dynasty through to a nuclear North Korea, to the danger that poses and therefore a motivation for invading, all the way to the claim that we should invade. It’s still shaky reasoning, but it’s approaching the strongest version of the original argument.
There’s still technically something missing. Between 4 and 5 we’ve missed a premise.
In order to get from an “is” claim to an “ought” claim, often you’ll need a general normative principle. That is, we need a general rule which allows us to move from a simple statement of supposed fact (premise 4) to a prescription for what we should do (the conclusion, #5). This one will do:
4a. If we can prevent immediate existential danger to whole countries then we must/should act so as to prevent that danger.
That actually seems pretty plausible, right? So in this case the missing premise wasn’t so shaky (you might disagree, though).
We can then perform the negative test on our hidden assumptions and figure out if the argument falls apart without them. If we deny 4a, then we can’t get from 4 to 5. Does that make sense? If so,
Mapping Hidden Assumptions
Mapping Hidden Assumptions is relatively simple.
A hidden assumption will always offer conjoint support for its conclusion/sub-conclusion.
Think about it: if hidden assumptions are things that must be true for an inference to work, and conjoint premises are premises that must all be true for the inference to work, then it makes sense that any hidden premise will offer conjoint support.
The only difference will be that we’ll use dotted circles instead of regular:
, or , etc.
A few complete examples:
1. We have a right to bodily autonomy
2. Hidden Assumption: (Abortion restrictions infringe on a right to bodily autonomy)
3. Therefore, we have a right to freedom from abortion restrictions
The inference from 1 to 3 makes some sense because we’re all familiar with the abortion debate by now. What we do to make it make sense of it for ourselves, though, is implicitly add in premise two in understanding the inference from 1 to 3.
(1) We’ll never stop climate change, (2) Hidden Assumption: Climate change will intensify fires and storms, so (3) we’re going to have much more fires and storms. (4) Our current system can’t handle even basic disasters. Furthermore, (5) state disaster relief funds are insufficient without help from FEMA’s overburdened funds. So, (6) we need to reform and enhance funding for FEMA immediately.
The inference from 1 to 3 makes little sense if it’s not true that climate change is connected to fires and storms, so we need premise 2 to make that connection.
3, 4, and 5 are independent because each by itself makes sense as a premise for 6.