6.6: What is the influence of peers and youth culture inside and outside the classroom?
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By Sarah Pope
Introduction
Middle school and high school years are filled with activities. Students can engage in sports, clubs, and perhaps, the year book staff or the school news paper. Unfortunately, many students do not. They face greater challenges than the debate team. They face and fall short against peer pressure. Peer pressure can have both a positive and negative effect on a student. Adolescents base much of their behavior (positive or negative) on either attempting to fit in with a peer group or because they are excluded or rejected by a peer group (usually results in negative behavior). Middle and High school is a time when students are forming their self-image and feelings of ridicule and rejection towards them from other students can lead to feeling of isolation and loneliness. In order to compensate for these feelings students will attempt to find an outlet for their frustration (Lashbrook, 2000). Three common outlets are the proverbial "sex, drugs, and rock and roll". They are compelled to consume alcohol and other drugs; have intercourse at an early age; even reenact the lyrics in their music. Ultimately, conforming to their social culture, many teens succumb to “what their friends are doing”. Their school is where they come to learn. They learn about mathematics, science, English and geography, but they should also learn the facts on drugs, sex and their prevention. As teachers, it is our duty to educate the youth of America. Yet, we can also strive to understand, relate to and reassure the students. All of us have been there before. We bring to the classroom wisdom and experience that students have yet to learn. We can support our pupils and help them conquer the stress that they face daily. First, we must educate ourselves and the students. This will be a prime arsenal needed in this battle of temptation.
Drugs in the school
The use of drugs in the public school system is growing at an alarming rate. According to a survey conducted by the National Center on Addiction and Substance Abuse at Columbia University, 51% of high school students and 20% of middle school students, said that drugs are used, sold, and kept in their school (Loesevitz, 2008). This not only increases the likelihood that students will engage in substance abuse, but also the probability that students will have a “disrupted educational process” (Loesevitz,2007 paragraph 4). Consequently, this will lead to lower grades, increased failure, and ultimately dropping out of school, which leads to future problems in the road ahead of them. Requiring the schools and teachers to be proactive on this regards, and encourage their students to “be above the influence" before they are introduced to that lifestyle, will have a profound impact on adolescence drug and alcohol use (abovetheinfluence.com). Project Northland, a study done in Minnesota, concluded that intervention, social behavior curricula, peer leadership, and parental involvement, beginning in the sixth grade, reduced the number of students in high school abusing drugs (Perry, et al. , 1996). Seventy-five percent of our nation’s schools offer Drug Abuse Resistance Education (D.A.R.E.) programs, but that education focuses on younger students (dare.com/home/about_dare.asp). The students who face peer pressure to consume alcohol and other drugs do not have such opportunities (Cook, 2007). Legislators are beginning to get involved, and asking schools to start school-based drug programs in their district (Tobler, et al. , 2000). A great way to get the truth about drugs across to teens could be as simple as a mandatory “afterschool special." There the students will learn ways to say no to drugs, taught threw a skit or reenactment of a collaborative student created script. Here the kids would also find support, which will help them face their biggest obstacle, each other. However, there is help for those that, despite intervention attempts, fall victim to drug and alcohol abuse. Teen Help is a website available to teens that will guide them away from the downward spiral of alcohol and substance abuse. Also, the shoulder of a caring teacher or counselor can also have an enormous impact on students desperately in need of guidance.
Authors Lizabeth Crawford from Bradley University in IL, and Katherine Novak from Butler University IN, concluded in an article featured in the February issue of The Family Journal that students from broken homes are more likely to engage in substance abuse. These young adults do not receive the support they need at home so they turn to their peers. They get that sense of a family with their friends, this relationship is a double edge sword; where one influences the other (Pope self experience). Substance and alcohol abuse are not the only subjects of peer pressure, but sex as well.
Sex in the classroom
Spin the bottle, an innocent game of kissing is just one example of how kids encourage each other's sexual behavior (Jaccard, 2005). Where do kids learn such games: their older siblings or parents? Where do they learn anything on the subject of sex? Schools have implemented a morsel of sex education in to the curriculum, but many questions go unanswered. Teens will turn to each other or pornography for their answers (http;//focusas.com /sexual behavior.html), two sources that will give distorted answers. In primary school, children often learn about puberty (parent willing) in the classroom. When they arrive in middle school, they learn about reproductive health. That is the extent of public school's sex education (Pope self experience). There are not any classes on how to say no to early intercourse, or what to do if you become pregnant. Kids need to know the following facts about sex and its risks. “According to the U.S. Centers for Disease Control and Prevention (CDC), about 25% of sexually active teenagers get a sexually transmitted disease (STD) every year, and 80% of infected teens don’t even know they have an STD, passing the diseases along to unsuspecting partners. When it comes to AIDS, the data is even more chilling -- of the new HIV infections each year, about 50% occur in people under the age of 25.” (http;//focusas.com /sexual behavior.html) “Each year, almost 750,000 teenage women aged 15-19 become pregnant.”(www.guttmacher.org/pubs/2006/09/12/USTPstats.pdf) “The abortion rate among teen's ages 15-17 are 14.5 abortions per 1,000 females in 2000…More than one-third of all teenage pregnancies end in abortion. ( http://www.childtrendsdatabank.org/indicators/27TeenAbortions.cfm ) Teens will develop their own cultural normality on sexual behavior, thus teaching them the right way to say no will benefit their well being. Is it the lack of sexual knowledge coupled with a curiosity for sex that urges them to have intercourse? Or is it the music they are listening to?
It is in the Music
In the digital age of the twenty-first century, there is no problem getting a hold of popular music. In fact, the average teen listens to 1.5 to 2.5 hours of it daily. Music is important to a growing teen’s identity (prdiatrics.aappublications.org). Many social groups are formed around the genre of music its members listen to. However different the music, they all have a common thread: similar themes, like sex run threw them all.  A national survey done by the American Academy of Pediatrics, on 1461 teenagers found that, when teens listen to music with “degrading sexual lyrics” they are more prone to engage in a range of sexual activities; they also found that reducing the amount of exposure to music with this type of content could help delay the onset of sexual behavior” (prdiatrics.aappublications.org). Music has evolved within the past four decades. There are new genre like “gangsta rap”, and “heavy metal," both groups that offer very explicit lyrics to its listener. An alarming conclusion, to a study preformed in Sweden, stated that adolescents that begin to listen to rock music at an early age were more likely to be influenced by their peers (prdiatrics.aappublications.org). Like the old saying goes “if you hang out with ducks, you’re bound to get in the water.” The American Academy of Pediatrics urges adults to stress the importance of music and lyric selection in young teen’s music. Music, to the adolescent, helps them deal with many issues they are facing like, friendship, growing up, having fun and struggling (Dunbar, 2008). There are no laws to censor the air waves; only designation of explicit lyrics on the cover of CDs. Teachers can do their part by not allowing it in the classroom, a standard rule in most schools, yet a commonly broken rule. They can also, influence the students to be more selective in their musical choices.
Conclusion
Not all students will rely on their peers for support. Many find guidance and help at home. For the children that are not as fortunate, we as the extended family can have a great impact on their teenage lives, as well as their future. Heading these kids off at the curb and nipping any problems in the bud, will have an enormous impact on their future. I decided to write this article because I was a troubled teen. After my mother remarried and my little brother was born, I did not receive the supervision I needed. Consequently, I ran to my friends for comfort. I abused drugs and cut class. At the end of my high school years, I did graduate, but I was six months pregnant. I don’t think I would have made it to graduation at all if it had not been for the influence of caring teachers. They encouraged me while in class, as well as to continue with my education. Now, nearly ten years later, I have an associate degree and I am currently working on a bachelor’s degree. Someday I will be that caring teacher that will change someone’s life. A new trend among students is forming because of devoted teachers. They are reassessing their music selection and the risks and dangers of sex and drugs. With continued education and mentoring programs, students will be prepared not only for their graduation day, but also to say no to negative influences, and possibly help a friend.
Exercise \(\PageIndex{1}\)
1. Based on the article, who are teens turning to for unanswered sex questions?
A) Their really cool parents.
B) Their involved schools.
C) Their friends and pornography.
D) The local library.
2. D.A.R.E. stands for what?
A) Drugs Are Really Expensive.
B) Drug Abuse Resistance Education.
C) Drug Assistance Retrieval Education.
D) Does Abuse Really Exist?
3. As a child in the public school system, how would I learn about my changing body?
A) Through a sex education class given in Elementary School.
B) Through a class project assigned by the teacher.
C) By reading the graffiti on the bathroom stalls.
D) By reading a note based around class.
4.
A) Have a slammin' party.
B) Rob a liquor store.
C) Dance real funny.
D) Have premarital sex.
5. When assigned to teach a school based drug prevention program in the classroom, my main focus is what?
A) Educate students on drug abuse.
B) Help the students start a drug ring in school.
C) Teach students the definition of prevention.
D) Tell the students where to score the best "stuff".
- Answer
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1. C) Their friends and pornography.
2. B) Drug Abuse Resistance Education.
3. A) through a sex education class given in Elementary School.
4. D) Have premarital sex.
5. A) Educate students on drug abuse.
References
Above the influence information, retrieved January 30, 2008, http://abovetheinfluence.com .
Crawford, Lizabeth, Novak, Kathrine (2008). Parent—Child Relations and Peer Associations as Mediators of the Family Structure—Substance Use Relatioship. Journal of Family Issues, v.29 no.2, pgs. 155.
Cook, Glenn. (2007). Hard Times for Today’s Teens, Families. American School Board Journal, v.195, no.2, pgs. 6-7.
D.A.R.E. Information, Retrieved January 30, 2008, dare.com/home/about_dare.asp
Dunbar, Sthephen. Teen Sex Apparently Driven By Music, Not Libido. (2008, January 26). New York Times. Retrieved January 26, 2007, via Google.Com
Jaccard, James. (2005). Peer Influences on Risk Behavior: An Anaylsis of the Effects of a Close Friend. Developmental Psychology, v. 41 no. 1, pgs. 135-147.
Lashbrook, Jeffery T. (2000). Fitting In: Exploring the Emotional Dimension of Adolescent Peer Pressure. Adolescence, v. 35 no. 140, pgs. 747-757.
Loesevitz, Michael (2007). Random Drug Testing in Public Schools. Journal of Law & Education, v. 36 no. 3, pgs.453-460.
Martino, S., Collins, R., Elliott, M., Strachman, A., Kanouse, D., Berry, S. (1995). Exposure to Degrading Versus Non-degrading Music Lyrics and Sexual Behavior Among Youth. Retrieved January 26, 2007, prdiatrics.aappublications.org.
Perry, C., Williams, C., Veblen-Mortenson, S., Toomey, T., Komro, K., Anstine, B., McGovern, P., Finnegan, J., Forster, J., Wagenaar, A., Wolfson, M. (1996). Project Northland: Outcomes of a Communitywide Alcohol Use Prevention Program During Early Adolescence. American Journal Of Public Health, v. 86 no. 7.
Teen Abortions. Retrieved January 30, 2008 via, http://www.childtrendsdatabank.org/indicators/27TeenAbortions.cfm
Teen Help, Schools for Troubled Teens. January 26, 2008, teenhelp.us
Tobler, N., Streke, A., Stackpole, K., Roona, M., Orchorn, P.,Marshall, D. (2000). School-Based Adolescent drug Prevention Programs: 1998 Meta-Analysis. Journal of Primary Prevention, v. 20, no. 4
U.S. Teenage Pregnancy Statistics National and State Trends and Trends by Racs and Ethnicity(2006). Guttmacher Institute, New York. Retrieved January 30, 2008 Via www.guttmacher.org/pubs/2006/09/12/USTPstats.pdf
U.S. Teenage Sexual Behavior. January 26, 2008. http://focusas.com/sexualbehavior.html .


3.5: Reading- The Disadvantages of Mixed Economies
The disadvantages of mixed economies can be understood through examining criticisms of social democracy.
KEY points
- One disadvantage of mixed economies is that they tend to lean more toward government control and less toward individual freedoms.
- While most modern forms of government are consistent with some form of mixed economy, the mixed economy is most commonly associated with social democratic parties or nations run by social democratic governments.
- Some critics of contemporary social democracy argue that when social democracy abandoned Marxism it also abandoned socialism and has become, in effect, a liberal capitalist movement.
- Marxian socialists argue that because social democratic programs retain the capitalist mode of production they also retain the fundamental issues of capitalism, including cyclical fluctuations, exploitation and alienation.
- The democratic socialist critique of social democracy states that capitalism could never be sufficiently “humanized” and any attempt to suppress the economic contradictions of capitalism would only cause them to emerge elsewhere.
- Market socialists criticize social democracy for maintaining a property-owning capitalist class, which has an active interest in reversing social democratic policies and a disproportionate amount of power over society to influence governmental policy as a class.
Terms
- Social democracy : a moderate political philosophy or ideology that aims to achieve socialistic goalswithin capitalist society such as by means of a strong welfare state and regulation of private industry.
- Regulation : A law or administrative rule, issued by an organization, used to guide or prescribe the conduct of members of that organization.
- Mixed economy : Mixed economy is an economic system in which both the state and private sector direct the economy, reflecting characteristics of both market economies and planned economies.
Examples
- Many pubs in Britain are suffering due to drinking and smoking regulations imposed by the government for the good of society. As a result, many question whether pubs have a future.
Disadvantages of Social Democratic Policy In a Mixed Economy
While most modern forms of government are consistent with some form of mixed economy, given the broad range of economic systems that can be described by the term, the mixed economy is most commonly associated with social democratic parties or nations run by social democratic governments. In contemporary terms, “social democracy” usually refers to a social corporatist arrangement and a welfare state in developed capitalist economies.
Critics of contemporary social democracy argue that when social democracy abandoned Marxism it also abandoned socialism and has become, in effect, a liberal capitalist movement. They argue that this has made social democrats similar to center-left, but pro-capitalist groups, such as the U.S. Democratic Party .
Marxian socialists argue that because social democratic programs retain the capitalist mode of production they also retain the fundamental issues of capitalism, including cyclical fluctuations, exploitation and alienation. Social democratic programs intended to ameliorate capitalism, such as unemployment benefits or taxation on profits and the wealthy, create contradictions of their own through limiting the efficiency of the capitalist system by reducing incentives for capitalists to invest in production.
Others contrast social democracy with democratic socialism by defining the former as an attempt to strengthen the welfare state and the latter as an alternative socialist economic system to capitalism. The democratic socialist critique of social democracy states that capitalism could never be sufficiently “humanized” and any attempt to suppress the economic contradictions of capitalism would only cause them to emerge elsewhere. For example, attempts to reduce unemployment too much would result in inflation, and too much job security would erode labor discipline. In contrast to social democracy, democratic socialists advocate a post-capitalist economic system based either on market socialism combined with workers self-management, or on some form of participatory-economic planning.
Social democracy can also be contrasted with market socialism. While a common goal of both systems is to achieve greater social and economic equality, market socialism does so by changes in enterprise ownership and management, whereas social democracy attempts to do so by government-imposed taxes and subsidies on privately owned enterprises. Market socialists criticize social democracy for maintaining a property-owning capitalist class, which has an active interest in reversing social democratic policies and a disproportionate amount of power over society to influence governmental policy as a class.
GLOSSARY
Active
Having the power or quality of acting; causing change; communicating action or motion; acting.
Benefit
An advantage, help or aid from something. Employee benefits and (especially in British English) benefits in kind (also called fringe benefits, perquisites, perqs or perks) are various non-wage compensations provided to employees in addition to their normal wages or salaries.
Capitalism
A socio-economic system based on the abstraction of resources into the form of privately-owned money, wealth, and goods, with economic decisions made largely through the operation of a market unregulated by the state.
Economic system
An economic system is the combination of the various agencies, entities (or even sectors as described by some authors) that provide the economic structure that defines the social community.
Economy
Collective focus of the study of money, currency and trade, and the efficient use of resources. The system of production and distribution and consumption. The overall measure of a currency system; as the national economy.
Efficiency
The extent to which a resource, such as electricity, is used for the intended purpose; the ratio of useful work to energy expended. The extent to which time is well used for the intended task. Improved efficiency was a principle goal of progressives, one they thought attainable by the application of scientific and rational thought to social problems.
Enterprise
A company, business, organization, or other purposeful endeavor.
Fluctuation
A motion like that of waves; a moving in this and that direction.
Goal
A desired result that one works to achieve.
Good
An object produced for market.
Incentive
Something that motivates, rouses, or encourages. It is used to motivate individuals (often, employees) for better performance by providing financial or other types of rewards. An anticipated reward or aversive event available in the environment. Something that motivates an individual to perform an action.
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time. An increase in the general level of prices or in the cost of living. An increase in the quantity of money, leading to a devaluation of existing money.
Interest
The price paid for obtaining, or price received for providing, money or goods in a credit transaction, calculated as a fraction of the amount or value of what was borrowed. A great attention and concern from someone or something; intellectual curiosity.
Lead
Potential opportunity for a sale or transaction, a potential customer.
Management
The act of getting people together to accomplish desired goals and objectives using available resources efficiently and effectively. Administration; the process or practice of managing administration; the process or practice of managing. Administration; the process or practice of managing.
Market
A group of potential customers for one’s product. One of the many varieties of systems, institutions, procedures, social relations and infrastructures whereby parties engage in exchange.
Mixed economies
A system in which both the state and private sector direct the way goods and services are bought and sold.
Planning
The act of formulating a course of action, or of drawing up plans.
Price
The price is the amount a customer pays for the product. The quantity of payment or compensation given by one party to another in return for goods or services. The cost required to gain possession of something.
Product
Any tangible or intangible good or service that is a result of a process and that is intended for delivery to a customer or end user. Anything, either tangible or intangible, offered by the firm as a solution to the needs and wants of the consumer; something that is profitable or potentially profitable; goods or a service that meets the requirements of the various governing offices or society.
Profits
Collective form of profit.
Recession
A period of reduced economic activity. A significant decline in economic activity spread across the economy, lasting more than a few months, normally visible in real GDP, real income, employment, and industrial production.
Security
Proof of ownership of stocks, bonds, or other investment instruments. The condition of not being threatened, especially physically, psychologically, emotionally, or financially.
Socialism
Any of various economic and political philosophies that support social equality, collective decision-making, distribution of income based on contribution and public ownership of productive capital and natural resources, as advocated by socialists. A political philosophy based on principles of community decision making, social equality, and the avoidance of economic and social exclusion, with preference to community goals over individual ones. The intermediate phase of social development between capitalism and full communism. This is a strategy whereby the state has control of all key resource-producing industries and manages most aspects of the economy, in contrast to laissez-faire capitalism.
System
A whole composed of relationships among the members. The part of the universe being studied, arbitrarily defined to any size desired.
Welfare state
A social system in which the state takes overall responsibility for the welfare of its citizens, providing health care, education, unemployment compensation, and social security.
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11.7C: Clonal Selection of Antibody-Producing Cells
- Describe the clonal selection hypothesis in regards to the production of B cells
The clonal selection hypothesis has become a widely accepted model for how the immune system responds to infection and how certain types of B and T lymphocytes are selected for destruction of specific antigens invading the body.
Four predictions of the clonal selection hypothesis
- Each lymphocyte bears a single type of receptor with a unique specificity (by V(D)J recombination).
- Receptor occupation is required for cell activation.
- The differentiated effector cells derived from an activated lymphocyte will bear receptors of identical specificity as the parental cell.
- Those lymphocytes bearing receptors for self molecules will be deleted at an early stage.
In 1954, Danish immunologist Niels Jerne put forward a hypothesis which stated that there is already a vast array of lymphocytes in the body prior to any infection. The entrance of an antigen into the body results in the selection of only one type of lymphocyte to match it and produce a corresponding antibody to destroy the antigen. This selection of only one type of lymphocyte results in it being cloned or reproduced by the body extensively to ensure there are enough antibodies produced to inhibit and prevent infection. Australian immunologist Frank Macfarlane Burnet, with input from David W. Talmage, worked on this model and was the first to name it “clonal selection theory. ” Burnet explained immunological memory as the cloning of two types of lymphocyte. One clone acts immediately to combat infection whilst the other is longer lasting, remaining in the immune system for a long time, which results in immunity to that antigen. In 1958, Sir Gustav Nossal and Joshua Lederberg showed that one B cell always produces only one antibody, which was the first evidence for clonal selection theory.
B cells exist as clones. All B cells derive from a particular cell, and as such, the antibodies and their differentiated progenies can recognize and/or bind the same specific surface components composed of biological macromolecules ( epitope ) of a given antigen. Such clonality has important consequences, as immunogenic memory relies on it. The great diversity in immune response comes about due to the up to 109 clones with specificities for recognizing different antigens. Upon encountering its specific antigen, a single B cell, or a clone of cells with shared specificity, divides to produce many B cells. Most of such B cells differentiate into plasma cells that secrete antibodies into blood that bind the same epitope that elicited proliferation in the first place. A small minority survives as memory cells that can recognize only the same epitope. However, with each cycle, the number of surviving memory cells increases. The increase is accompanied by affinity maturation which induces the survival of B cells that bind to the particular antigen with high affinity. This subsequent amplification with improved specificity of immune response is known as secondary immune response. B cells that have not been activated by antigen are known as naive lymphocytes; those that have met their antigen, become activated, and have differentiated further into fully functional lymphocytes are known as effector B lymphocytes.
Key Points
- In 1954, immunologist Niels Jerne put forth the hypothesis that there is already a vast array of lymphocytes in the body before infection. The entrance of an antigen into the body results in the selection of only one type of lymphocyte to match it and produce a corresponding antibody to destroy it.
- B cells exist as clones derived from a particular cell. Thus the antibodies and their differentiated progenies can recognize and/or bind the same specific surface components composed of biological macromolecules of a given antigen. Clonality has important consequences for immunogenic memory.
- The clonal selection hypothesis states that an individual B cell expresses receptors specific to the distinct antigen, determined before the antibody ever encounters the antigen.
Key Terms
- clonal selection : An hypothesis which states that an individual lymphocyte (specifically, a B cell) expresses receptors specific to the distinct antigen, determined before the antibody ever encounters the antigen. Binding of Ag to a cell activates the cell, causing a proliferation of clone daughter cells.
- clone : A group of identical cells derived from a single cell.


10.6: Perils of Experimental Research
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Experimental research is one of the most difficult of research designs, and should not be taken lightly. This type of research is often best with a multitude of methodological problems. First, though experimental research requires theories for framing hypotheses for testing, much of current experimental research is atheoretical. Without theories, the hypotheses being tested tend to be ad hoc, possibly illogical, and meaningless. Second, many of the measurement instruments used in experimental research are not tested for reliability and validity, and are incomparable across studies. Consequently, results generated using such instruments are also incomparable. Third, often experimental research uses inappropriate research designs, such as irrelevant dependent variables, no interaction effects, no experimental controls, and non-equivalent stimulus across treatment groups. Findings from such studies tend to lack internal validity and are highly suspect. Fourth, the treatments (tasks) used in experimental research may be diverse, incomparable, and inconsistent across studies, and sometimes inappropriate for the subject population. For instance, undergraduate student subjects are often asked to pretend that they are marketing managers and asked to perform a complex budget allocation task in which they have no experience or expertise. The use of such inappropriate tasks, introduces new threats to internal validity (i.e., subject’s performance may be an artefact of the content or difficulty of the task setting), generates findings that are non-interpretable and meaningless, and makes integration of findings across studies impossible.
The design of proper experimental treatments is a very important task in experimental design, because the treatment is the raison d’etre of the experimental method, and must never be rushed or neglected. To design an adequate and appropriate task, researchers should use prevalidated tasks if available, conduct treatment manipulation checks to check for the adequacy of such tasks (by debriefing subjects after performing the assigned task), conduct pilot tests (repeatedly, if necessary), and if in doubt, use tasks that are simple and familiar for the respondent sample rather than tasks that are complex or unfamiliar.
In summary, this chapter introduced key concepts in the experimental design research method and introduced a variety of true experimental and quasi-experimental designs. Although these designs vary widely in internal validity, designs with less internal validity should not be overlooked and may sometimes be useful under specific circumstances and empirical contingencies.


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attention-deficit/hyperactivity disorder (ADHD)
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neurodevelopmental disorder that includes symptoms of inattention, hyperactivity, impulsivity and deficits in executive functions
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balanced literacy approach
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approach to teaching reading that combines elements of the phonics-based and whole language approaches
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concrete operational thinking
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third stage of Piaget’s theory on cognitive development during which children begin to understand basic cognitive principles and concepts such as cause and effect, relationships, size, and distance
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constructivism
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idea that students play an active role in acquiring knowledge by constructing it through experience, rather than just passively receiving information
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crystallized intelligence
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existing knowledge that individuals have developed during their life through education and experience
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dyscalculia
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learning disability that involves difficulties in math
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dysgraphia
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learning disability that involves difficulties in writing
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dyslexia
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learning disability that involves difficulties in reading
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exceptional learner
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child who has one or more of a variety of special learning needs
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fluid intelligence
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ability to use logic and to solve problems in new ways
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Flynn effect
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effect that describes the significant increases of scores on intelligence tests over time
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general intelligence factor (g)
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general cognitive factor that underlies multiple cognitive skills
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giftedness
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variation in intelligence marked by cognitive flexibility, cognitive performance, specific ability areas, and an IQ above 130, may be referred to by other terms
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individualized education program (IEP)
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written plan that spells out the specific educational goals and services that have been individualized for a student with a disability
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intellectual disability
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disability that consists of limits in intellectual functioning, often indicated by an IQ score less than 70 and challenges in adaptive functioning
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intelligence
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description of an individual’s ability to adapt to the world around them
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intelligence quotient (IQ)
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score used to quantify human intelligence
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learning disability
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condition that impacts learning in a specific academic area
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least-restrictive environment
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principle that states that all children with a disability should receive general education in an environment that is as similar as possible to the one for children without disabilities
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mental age
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age at which a person is performing based on an intelligence test
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metacognition
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knowledge about how we think and learn and how we use that awareness to become better thinkers and learners
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metalinguistic awareness
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awareness of the qualities of language, allowing individuals to think about and evaluate language
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metamemory
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understanding of how memory works
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phonics approach
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approach to teaching language that teaches children to translate letters into sounds and to combine individual sounds to form words
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self-efficacy
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individual’s confidence in their ability to successfully solve a problem or complete a task
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seriation
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ability to put objects in order, such as by size or color
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triarchic theory of intelligence
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theory of intelligence that proposes that there are three types of intelligence: analytical, creative and practical
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whole-language approach
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approach to teaching reading that uses natural context such as books rather than focusing on the sounds that make up words


7.2C: Vitamin E
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Vitamin E refers to a group of compounds that include both tocopherols and tocotrienols. Of the many different forms of vitamin E, γ-tocopherol is the most common form found in the North American diet. γ-Tocopherol can be found in corn oil, soybean oil, margarine, and dressings. α-tocopherol, the most biologically active form of vitamin E, is the second-most common form of vitamin E in the diet. This variant can be found most abundantly in wheat germ oil, sunflower, and safflower oils. As a fat-soluble antioxidant, it interrupts the propagation of reactive oxygen species that spread through biological membranes or through a fat when its lipid content undergoes oxidation by reacting with more-reactive lipid radicals to form more stable products. Regular consumption of more than 1,000 mg (1,500 IU) of tocopherols per day may be expected to cause hypervitaminosis E, with an associated risk of vitamin K deficiency and consequently of bleeding problems.
Sample of α-tocopherol, one of the various forms of vitamin E
Functions
Vitamin E has many biological functions, including its role as a fat-soluble antioxidant.
- As an antioxidant, vitamin E acts as a peroxyl radical scavenger, disabling the production of damaging free radicals in tissues, by reacting with them to form a tocopheryl radical, which will then be reduced by a hydrogen donor (such as vitamin C) and thus return to its reduced state.[16] As it is fat-soluble, it is incorporated into cell membranes, which protects them from oxidative damage. Vitamin E has also found use as a commercial antioxidant in ultra high molecular weight polyethylene (UHMWPE) used in hip and knee implants by resisting oxidation.
- As an enzymatic activity regulator, for instance, protein kinase C (PKC), which plays a role in smooth muscle growth, can be inhibited by α-tocopherol. α-Tocopherol has a stimulatory effect on the dephosphorylation enzyme, protein phosphatase 2A, which in turn, cleaves phosphate groups from PKC, leading to its deactivation, bringing the smooth muscle growth to a halt.
- Vitamin E also has an effect on gene expression. Macrophages rich in cholesterol are found in atherosclerotic tissue. Scavenger receptor CD36 is a class B scavenger receptor found to be up-regulated by oxidized low density lipoprotein (LDL) and binds it. Treatment with α-tocopherol was found to downregulate the expression of the CD36 scavenger receptor gene and the scavenger receptor class A (SR-A) and modulates expression of the connective tissue growth factor (CTGF). The CTGF gene, when expressed, is responsible for the repair of wounds and regeneration of the extracellular tissue lost or damaged during atherosclerosis.
- Vitamin E also plays a role in eye and neurological functions, and inhibition of platelet coagulation.
- Vitamin E also protects lipids and prevents the oxidation of polyunsaturated fatty acids.
Although most vitamin E supplementation studies used α-tocopherol individually, this design of studying only one isoform of vitamin E may introduce errors in interpreting overall vitamin E effects; for example, using only α-tocopherol in studies of inflammation can reduce serum γ- and δ-tocopherol concentrations. Moreover, a 2013 review involving single long-term supplementation with α-tocopherol showed that many clinical studies revealed an inverse relationship between supplementation and cardiovascular disease risk or mortality, but other studies showed no effect.
Deficiency
Vitamin E deficiency can cause:
- spinocerebellar ataxia
- myopathies
- peripheral neuropathy
- ataxia
- skeletal myopathy
- retinopathy
- impairment of the immune response
- red blood cell destruction
Supplementation
Vitamin E supplementation has not been shown to have significant benefit for people who are healthy, and appears to be harmful. It does not improve blood sugar control in an unselected group of people with diabetes mellitus[35] or decrease the risk of stroke.[36] Daily supplementation of vitamin E does not decrease the risk of prostate cancer, and may increase it. Studies on its role in age-related macular degeneration are ongoing, though if it is of a combination of dietary antioxidants used to treat the condition it may increase the risk. Routine supplementation with vitamin E during pregnancy has been shown to offer no benefit to the mother or the child. Vitamin E has been reported to cause more side effects, such as abdominal pain in pregnant women, and also the increased risk of having early rupture of membranes at term.
Vitamin E, along with β-carotene and vitamin C, has no protective effect on reducing the risk of cataract, cataract extraction, progression of cataract, and slowing the loss of visual acuity.
Toxicity
The LD 50 , or the toxic dose required to kill 50% of group of rats and mice, respectively is 4000 mg of VitaminE E/kg of rat and 4000 mg of Vitamin E/kg of mouse.[43] Comparatively speaking, and at lethal doses, Vitamin E is less toxic than table salt and acetaminophen and it is more toxic than ethanol and Vitamin C. Vitamin E can act as an anticoagulant, increasing the risk of bleeding problems. As a result, many agencies have set a tolerable upper intake levels (UL) at 1,000 mg (1,500 IU) per day.[1] In combination with certain other drugs such as aspirin, hypervitaminosis E can be life-threatening. Hypervitaminosis E may also counteract vitamin K, leading to a vitamin K deficiency.
Dietary Reference Intake
The Food and Nutrition Board (FNB) of the U.S. Institute of Medicine updated Estimated Average Requirements (EARs) and Recommended Dietary Allowances (RDAs) for vitamin E in 2000. The current EAR for vitamin E for women and men ages 14 and up is 12 mg/day. The RDA is 15 mg/day. RDAs are higher than EARs so as to identify amounts that will cover people with higher than average requirements. RDA for pregnancy equals 15 mg/day. RDA for lactation equals 19 mg/day. For infants up to 12 months the Adequate Intake (AI) is 4–5 mg/day and for children ages 1–13 years the RDA increases with age from 6 to 11 mg/day. The FNB also sets Tolerable Upper Intake Levels (ULs) for vitamins and minerals when evidence is sufficient. In the case of vitamin E the UL is 1,000 mg/day.
For U.S. food and dietary supplement labeling purposes the amount in a serving is expressed as a percent of Daily Value (%DV). For vitamin E labeling purposes 100% of the Daily Value was 30 mg, but as of May 2016 it has been revised to 15 mg. A table of the pre-change adult Daily Values is provided at Reference Daily Intake. Food and supplement companies have until July 28, 2018 to comply with the change.
Vitamin E and atherosclerosis
Atherosclerosis is a disease condition that refers to the buildup of plaque, which is a substance containing lipid and cholesterol (mainly the low-density lipoprotein or LDL cholesterol) on the inner layer of the arterial lumen. With the existing plaque, instead of being smooth and elastic, the layers become thickened and irregular and the lumen of the artery become narrower. This vessel-narrowing effect lead to a reduction of blood circulation and can lead to or worsen the condition of hypertension.
There are currently multiple theories explaining factors causing and affecting the cholesterol plaque build up within arteries with the most popular theory indicating that the rate of build up is affected by the oxidation of the LDL cholesterol. LDL cholesterol is one of the five major groups of lipoproteins with one of the physiological roles being lipid transportation. A typical LDL particle contain 2,700 fatty acid molecules and half of them are poly-unsaturated fatty acids, which are very oxidation sensitive. Once the oxidation of LDL occur, it will start a series of undesirable effects starting from the increase production of inflammatory cytokines by stimulating the endothelial cells and monocytes, followed by increased production of tissue factors, production of macrophages and monocytes, which eventually lead to the formation of foam cells and accelerated development of atherosclerosis. With the presence of adequate concentration of vitamin E, which is a very potent fat-soluble antioxidant, it can inhibit the oxidation of LDL, and this inhibition contributes protection against the development of atherosclerosis and can stabilize the existing plaque.
According to one meta-analysis, nine cohort studies showed that high intake of tocopherol was associated with a lower risk of cardiovascular diseases compared with lower intake. In this study, higher dietary, supplementation and combined vitamin E intake was also associated with lower disease incidents. In 1993, a study of 39,919 male health professionals aged 40 to 75 showed that consumption of more than 60 IU of vitamin E (any form) per day was associated with a lower incidence of coronary heart disease compared with less than 7.5 IU/day intake. This study also showed an inverse association between vitamin E supplementation and the incidence of heart disease.
A 2015 systematic review of clinical trials concluded that vitamin E supplementation alone improved endothelial function as determined by measurements of forearm blood flow, but when combined with vitamin C supplementation, it did not.[63] A meta-analysis of clinical trials showed no significant association between vitamin E supplementation and cardiovascular mortality.
Contributors and Attributions
- Wikipedia (CC SA BY 4.0)


Prepared childbirth refers to being not only in good physical condition to help provide a healthy environment for the baby to develop, but also helping individuals to prepare to accept their new roles as parents. Additionally, parents can receive information and training that will assist them for delivery and life with the baby. The more future parents can learn about childbirth and the newborn, the better prepared they will be for the adjustment they must make to a new life.
Approaches to Childbirth
There are many different approaches to childbirth that influence how expectant parents prepare. The following table describes a few of these:
The emphasis of this method is on teaching the woman to be in control in the process of delivery. It includes learning muscle relaxation, breathing through contractions, having a focal point (usually a picture to look at) during contractions and having a support person who goes through the training process with the mother and serves as a coach during delivery.
The Leboyer Method
This method involves giving birth in a quiet, dimly lit room and allowing the newborn to lie on the mother’s stomach with the umbilical cord intact for several minutes while being given a warm bath.
Dick-Read Method / Mongan Method / Hypnobirthing
This method comes from the suggestion that the fear of childbirth increases tension and makes the process of childbearing more painful. It emphasizes the use of relaxation and proper breathing with contractions as well as family support and education.
Bradley Method
“The Bradley Method focuses on preparing the mother for a natural childbirth coached by her partner. They learn techniques to reduce the perception of pain and stay relaxed. The emphasis is on being prepared for an unassisted vaginal birth without medication.” 2
Alexander Technique
This is a technique that can be used during childbirth that involves training to stop habitual reactions to pain, such as tensing muscles and increase conscious awareness and control over posture and movement. This involves being able to move freely and stay upright during labor and using body positioning that is beneficial to the labor process.3
Waterbirth
Involves immersion in warm water. Proponents believe this method is safe and provides many benefits for both mother and infant, including pain relief and a less traumatic birth experience for the baby. However, critics argue that the procedure introduces unnecessary risks to the infant such as infection and water inhalation.4
Lotus Birth
Or umbilical cord nonseverance – UCNS, is the practice of leaving the umbilical cord uncut after childbirth so that the baby is left attached to the placenta until the cord naturally separates at the umbilicus. This usually occurs within 3–10 days after birth. The practice is performed mainly for spiritual purposes of the parents, including for the perceived spiritual connection between placenta and newborn.5
Silent Birth
Sometimes known as quiet birth, is a birthing procedure advised by L. Ron Hubbard and advocated by Scientologists in which "everyone attending the birth should refrain from spoken words as much as possible." 6
Medicated Childbirth
Health care providers can provide pain relief during labor with different types of medication, including epidurals, spinal blocks, combined spinal-epidurals, and systemic and local analgesia. There are benefits and side effects of each.7
Choosing Location of Childbirth & Who Will Deliver
The vast majority of births occur in a hospital setting. However, one percent of women choose to deliver at home (Martin, Hamilton, Osterman, Curtin, & Mathews, 2015). Women who are at low risk for birth complications can successfully deliver at home. More than half (67%) of home deliveries are by certified nurse midwives. Midwives are trained and licensed to assist in delivery and are far less expensive than the cost of a hospital delivery. However, because of the potential for a complication during the birth process, most medical professionals recommend that delivery take place in a hospital. In addition to home births, one-third of out-of-hospital births occur in freestanding clinics, birthing centers, in physician’s offices, or other locations.


7.3E: Exercises
Practice Makes Perfect
Add and Subtract Rational Expressions with a Common Denominator
In the following exercises, add.
1. \(\dfrac{2}{15}+\dfrac{7}{15}\)
- Answer
-
\(\dfrac{3}{5}\)
2. \(\dfrac{7}{24}+\dfrac{11}{24}\)
3. \(\dfrac{3c}{4c−5}+\dfrac{5}{4c−5}\)
- Answer
-
\(\dfrac{3c+5}{4c−5}\)
4. \(\dfrac{7m}{2m+n}+\dfrac{4}{2m+n}\)
5. \(\dfrac{2r^2}{2r−1}+\dfrac{15r−8}{2r−1}\)
- Answer
-
\(r+8\)
6. \(\dfrac{3s^2}{3s−2}+\dfrac{13s−10}{3s−2}\)
7. \(\dfrac{2w^2}{w^2−16}+\dfrac{8w}{w^2−16}\)
- Answer
-
\(\dfrac{2w}{w−4}\)
8. \(\dfrac{7x^2}{x^2−9}+\dfrac{21x}{x^2−9}\)
In the following exercises, subtract.
9. \(\dfrac{9a^2}{3a−7}−\dfrac{49}{3a−7}\)
- Answer
-
\(3a+7\)
10. \(\dfrac{25b^2}{5b−6}−\dfrac{36}{5b−6}\)
11. \(\dfrac{3m^2}{6m−30}−\dfrac{21m−30}{6m−30}\)
- Answer
-
\(\dfrac{m−2}{2}\)
12. \(\dfrac{2n^2}{4n−32}−\dfrac{18n−16}{4n−32}\)
13. \(\dfrac{6p^2+3p+4}{p^2+4p−5}−\dfrac{5p^2+p+7}{p^2+4p−5}\)
- Answer
-
\(\dfrac{p+3}{p+5}\)
14. \(\dfrac{5q^2+3q−9}{q^2+6q+8}−\dfrac{4q^2+9q+7}{q^2+6q+8}\)
15. \(\dfrac{5r^2+7r−33}{r^2−49}−\dfrac{4r^2+5r+30}{r^2−49}\)
- Answer
-
\(\dfrac{r+9}{r+7}\)
16. \(\dfrac{7t^2−t−4}{t^2−25}−\dfrac{6t^2+12t−44}{t^2−25}\)
Add and Subtract Rational Expressions whose Denominators are Opposites
In the following exercises, add or subtract.
17. \(\dfrac{10v}{2v−1}+\dfrac{2v+4}{1−2v}\)
- Answer
-
\(4\)
18. \(\dfrac{20w}{5w−2}+\dfrac{5w+6}{2−5w}\)
19. \(\dfrac{10x^2+16x−7}{8x−3}+\dfrac{2x^2+3x−1}{3−8x}\)
- Answer
-
\(x+2\)
20. \(\dfrac{6y^2+2y−11}{3y−7}+\dfrac{3y^2−3y+17}{7−3y}\)
21. \(\dfrac{z^2+6z}{z^2−25}−\dfrac{3z+20}{25−z^2}\)
- Answer
-
\(\dfrac{z+4}{z−5}\)
22. \(\dfrac{a^2+3a}{a^2−9}−\dfrac{3a−27}{9−a^2}\)
23. \(\dfrac{2b^2+30b−13}{b^2−49}−\dfrac{2b^2−5b−8}{49−b^2}\)
- Answer
-
\(\dfrac{4b−3}{b−7}\)
24. \(\dfrac{c^2+5c−10}{c^2−16}−\dfrac{c^2−8c−10}{16−c^2}\)
Find the Least Common Denominator of Rational Expressions
In the following exercises, a. find the LCD for the given rational expressions b. rewrite them as equivalent rational expressions with the lowest common denominator.
25. \(\dfrac{5}{x^2−2x−8},\dfrac{2x}{x^2−x−12}\)
- Answer
-
a. \((x+2)(x−4)(x+3)\)
b. \(\dfrac{5x+15}{(x+2)(x−4)(x+3)}\),
\(\dfrac{2x^2+4x}{(x+2)(x−4)(x+3)}\)
26. \(\dfrac{8}{y^2+12y+35},\dfrac{3y}{y^2+y−42}\)
27. \(\dfrac{9}{z^2+2z−8},\dfrac{4z}{z^2−4}\)
- Answer
-
a. \((z−2)(z+4)(z−4)\)
b. \(\dfrac{9z−36}{(z−2)(z+4)(z−4)}\),
\(\dfrac{4z^2−8z}{(z−2)(z+4)(z−4)}\)
28. \(\dfrac{6}{a^2+14a+45},\dfrac{5a}{a^2−81}\)
29. \(\dfrac{4}{b^2+6b+9},\dfrac{2b}{b^2−2b−15}\)
- Answer
-
a. \((b+3)(b+3)(b−5)\)
b. \(\dfrac{4b−20}{(b+3)(b+3)(b−5)}\),
\(\dfrac{2b^2+6b}{(b+3)(b+3)(b−5)}\)
30. \(\dfrac{5}{c^2−4c+4},\dfrac{3c}{c^2−7c+10}\)
31. \(\dfrac{2}{3d^2+14d−5},\dfrac{5d}{3d^2−19d+6}\)
- Answer
-
a. \((d+5)(3d−1)(d−6)\)
b. \(\dfrac{2d−12}{(d+5)(3d−1)(d−6)}\),
\(\dfrac{5d^2+25d}{(d+5)(3d−1)(d−6)}\)
32. \(\dfrac{3}{5m^2−3m−2},\dfrac{6m}{5m^2+17m+6}\)
Add and Subtract Rational Expressions with Unlike Denominators
In the following exercises, perform the indicated operations.
33. \(\dfrac{7}{10x^2y}+\dfrac{4}{15xy^2}\)
- Answer
-
\(\dfrac{21y+8x}{30x^2y^2}\)
34. \(\dfrac{1}{12a^3b^2}+\dfrac{5}{9a^2b^3}\)
35. \(\dfrac{3}{r+4}+\dfrac{2}{r−5}\)
- Answer
-
\(\dfrac{5r−7}{(r+4)(r−5)}\)
36. \(\dfrac{4}{s−7}+\dfrac{5}{s+3}\)
37. \(\dfrac{5}{3w−2}+\dfrac{2}{w+1}\)
- Answer
-
\(\dfrac{11w+1}{(3w−2)(w+1)}\)
38. \(\dfrac{4}{2x+5}+\dfrac{2}{x−1}\)
39. \(\dfrac{2y}{y+3}+\dfrac{3}{y−1}\)
- Answer
-
\(\dfrac{2y^2+y+9}{(y+3)(y−1)}\)
40. \(\dfrac{3z}{z−2}+\dfrac{1}{z+5}\)
41. \(\dfrac{5b}{a^2b−2a^2}+\dfrac{2b}{b^2−4}\)
- Answer
-
\(\dfrac{b(5b+10+2a^2)}{a^2(b−2)(b+2)}\)
42. \(\dfrac{4}{cd+3c}+\dfrac{1}{d^2−9}\)
43. \(\dfrac{−3m}{3m−3}+\dfrac{5m}{m^2+3m−4}\)
- Answer
-
\(-\dfrac{m}{m+4}\)
44. \(\dfrac{8}{4n+4}+\dfrac{6}{n^2−n−2}\)
45. \(\dfrac{3r}{r^2+7r+6}+\dfrac{9}{r^2+4r+3}\)
- Answer
-
\(\dfrac{3(r^2+6r+18)}{(r+1)(r+6)(r+3)}\)
46. \(\dfrac{2s}{s^2+2s−8}+\dfrac{4}{s^2+3s−10}\)
47. \(\dfrac{t}{t−6}−\dfrac{t−2}{t+6}\)
- Answer
-
\(\dfrac{2(7t−6)}{(t−6)(t+6)}\)
48. \(\dfrac{x−3}{x+6}−\dfrac{x}{x+3}\)
49. \(\dfrac{5a}{a+3}−\dfrac{a+2}{a+6}\)
- Answer
-
\(\dfrac{4a^2+25a−6}{(a+3)(a+6)}\)
50. \(\dfrac{3b}{b−2}−\dfrac{b−6}{b−8}\)
51. \(\dfrac{6}{m+6}−\dfrac{12m}{m^2−36}\)
- Answer
-
\(\dfrac{−6}{m−6}\)
52. \(\dfrac{4}{n+4}−\dfrac{8n}{n^2−16}\)
53. \(\dfrac{−9p−17}{p^2−4p−21}−\dfrac{p+1}{7−p}\)
- Answer
-
\(\dfrac{p+2}{p+3}\)
54. \(\dfrac{−13q−8}{q^2+2q−24}−\dfrac{q+2}{4−q}\)
55. \(\dfrac{−2r−16}{r^2+6r−16}−\dfrac{5}{2−r}\)
- Answer
-
\(\dfrac{3}{r−2}\)
56. \(\dfrac{2t−30}{t^2+6t−27}−\dfrac{2}{3−t}\)
57. \(\dfrac{2x+7}{10x−1}+3\)
- Answer
-
\(\dfrac{4(8x+1)}{10x−1}\)
58. \(\dfrac{8y−4}{5y+2}−6\)
59. \(\dfrac{3}{x^2−3x−4}−\dfrac{2}{x^2−5x+4}\)
- Answer
-
\(\dfrac{x−5}{(x−4)(x+1)(x−1)}\)
60. \(\dfrac{4}{x^2−6x+5}−\dfrac{3}{x^2−7x+10}\)
61. \(\dfrac{5}{x^2+8x−9}−\dfrac{4}{x^2+10x+9}\)
- Answer
-
\(\dfrac{1}{(x−1)(x+1)}\)
62. \(\dfrac{3}{2x^2+5x+2}−\dfrac{1}{2x^2+3x+1}\)
63. \(\dfrac{5a}{a−2}+\dfrac{9}{a}−\dfrac{2a+18}{a^2−2a}\)
- Answer
-
\(\dfrac{5a^2+7a−36}{a(a−2)}\)
64. \(\dfrac{2b}{b−5}+\dfrac{3}{2b}−\dfrac{2b−15}{2b^2−10b}\)
65. \(\dfrac{c}{c+2}+\dfrac{5}{c−2}−\dfrac{10c}{c^2−4}\)
- Answer
-
\(\dfrac{c−5}{c+2}\)
66. \(\dfrac{6d}{d−5}+\dfrac{1}{d+4}+\dfrac{7d−5}{d^2−d−20}\)
67. \(\dfrac{3d}{d+2}+\dfrac{4}{d}−\dfrac{d+8}{d^2+2d}\)
- Answer
-
\(\dfrac{3(d+1)}{d+2}\)
68. \(\dfrac{2q}{q+5}+\dfrac{3}{q−3}−\dfrac{13q+15}{q^2+2q−15}\)
Add and Subtract Rational Functions
In the following exercises, find a. \(R(x)=f(x)+g(x)\) b. \(R(x)=f(x)−g(x)\).
69. \(f(x)=\dfrac{−5x−5}{x^2+x−6}\) and \( g(x)=\dfrac{x+1}{2−x}\)
- Answer
-
a. \(R(x)=−\dfrac{(x+8)(x+1)}{(x−2)(x+3)}\)
b. \(R(x)=\dfrac{x+1}{x+3}\)
70. \(f(x)=\dfrac{−4x−24}{x^2+x−30}\) and \( g(x)=\dfrac{x+7}{5−x}\)
71. \(f(x)=\dfrac{6x}{x^2−64}\) and \(g(x)=\dfrac{3}{x−8}\)
- Answer
-
a. \(R(x)=\dfrac{3(3x+8)}{(x−8)(x+8)}\)
b. \(R(x)=\dfrac{3}{x+8}\)
72. \(f(x)=\dfrac{5}{x+7}\) and \( g(x)=\dfrac{10x}{x^2−49}\)
Writing Exercises
73. Donald thinks that \(\dfrac{3}{x}+\dfrac{4}{x}\) is \(\dfrac{7}{2x}\). Is Donald correct? Explain.
- Answer
-
Answers will vary.
74. Explain how you find the Least Common Denominator of \(x^2+5x+4\) and \(x^2−16\).
75. Felipe thinks \(\dfrac{1}{x}+\dfrac{1}{y}\) is \(\dfrac{2}{x+y}\).
a. Choose numerical values for
x
and
y
and evaluate \(\dfrac{1}{x}+\dfrac{1}{y}\).
b. Evaluate \(\dfrac{2}{x+y}\) for the same values of
x
and
y
you used in part a..
c. Explain why Felipe is wrong.
d. Find the correct expression for \(1x+1y\).
- Answer
-
a. Answers will vary.
b. Answers will vary.
c. Answers will vary.
d. \(\dfrac{x+y}{x}\)
76. Simplify the expression \(\dfrac{4}{n^2+6n+9}−\dfrac{1}{n^2−9}\) and explain all your steps.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. After reviewing this checklist, what will you do to become confident for all objectives?


4.3: The Wronskian
Suppose that having determined that two solutions of (4.2.1) are \(x = X_1(t)\) and \(x = X_2(t)\), we attempt to write the general solution to (4.2.1) as (4.2.2) . We must then ask whether this general solution will be able to satisfy the two initial conditions given by \[\label{eq:1}x(t_0)=x_0,\quad\overset{.}{x}(t_0)=u_0.\]
Applying these initial conditions to (4.2.2) , we obtain \[\begin{align}c_1X_1(t_0)+c_2X_2(t_0)&=x_0,\nonumber \\ c_1\overset{.}{X}_1(t_0)+c_2\overset{.}{X}_2(t_0)&=u_0,\label{eq:2}\end{align}\] ) which is observed to be a system of two linear equations for the two unknowns \(c_1\) and \(c_2\). Solution of \(\eqref{eq:2}\) by standard methods results in \[c_1=\frac{x_0\overset{.}{X}_2(t_0)-u_0X_2(t_0)}{W},\quad c_2=\frac{u_0X_1(t_0)-x_0\overset{.}{X}_1(t_0)}{W},\nonumber\] where \(W\) is called the Wronskian and is given by \[\label{eq:3}W=X_1(t_0)\overset{.}{X}_2(t_0)-\overset{.}{X}_1(t_0)X_2(t_0).\]
Evidently, the Wronskian must not be equal to zero \((W\neq 0)\) for a solution to exist.
For examples, the two solutions \[X_1(t)=A\sin\omega t,\quad X_2(t)=B\sin\omega t,\nonumber\] have a zero Wronskian at \(t=t_0\), as can be shown by computing \[\begin{aligned} W&=(A\sin\omega t_0)(B\omega\cos\omega t_0)-(A\omega\cos\omega t_0)(B\sin\omega t_0) \\ &=0;\end{aligned}\] while the two solutions \[X_1(t)=\sin\omega t,\quad X_2(t)=\cos\omega t,\nonumber\] with \(\omega\neq 0\), have a nonzero Wronskian at \(t=t_0\), \[\begin{aligned}W&=(\sin\omega t_0)(-\omega\sin\omega t_0)-(\omega\cos\omega t_0)(\cos\omega t_0) \\ &=-\omega.\end{aligned}\]
When the Wronskian is not equal to zero, we say that the two solutions \(X_1(t)\) and \(X_2(t)\) are linearly independent. The concept of linear independence is borrowed from linear algebra, and indeed, the set of all functions that satisfy (4.2.1) can be shown to form a two-dimensional vector space.


15.7: Putting It Together- Capital Budgeting Decisions
When a company is faced with a large, capital budgeting decision, they have multiple ways to evaluate the potential options.
By first vetting possible projects based on your company guidelines and standards, then using one or more of several methods to analyze the decisions, it is hoped that a company can make the most profitable decision in the long term.
The payback method, net present value method, internal rate of return method and the simple rate of return method are all potential ways to work through these decisions. Learning the steps involved in each of these methods can help you, as a manager, to make good decisions, or to offer other managers insight into capital project decision making.


11.10: Presenting an Appraisal
- Discuss how to present an appraisal
One of the best ways to ensure an effective—and legally defensible—appraisal is to prepare in advance. For a traditional annual appraisal, well in advance. In her Muse article “A First-Time Manager’s Guide to Performance Reviews,” Pepperdine University Career Center Director Amy Adams lays out a one year preparation plan. As Adams notes “Planning will help you deliver more comprehensive [read: effective] feedback.” [1] An employee should never be blindsided by a review. If you’ve been providing feedback on a regular basis, the appraisal will be constructive or, on the downside, end in closure. Ongoing dialog also builds trust and a willingness to hear feedback as developmental coaching rather than a personal attack.
Here’s a step-by-step guide to presenting an appraisal:
- Schedule an appraisal meeting in advance. An appraisal should be presented—ideally, in person or at least via videoconference—and discussed, not handed off for signature.
- Optional, but recommended: Ask the employee to complete and submit a self-evaluation prior to the appraisal meeting. This will allow your employee time to reflect and will give you a heads up on any significant differences in opinion.
- Decide on the desired outcome and plan accordingly. Assuming you’ve been provided feedback on an ongoing basis, the substance of the meeting won’t be a surprise. Have the information or resources you need on hand or on call.
-
Develop your talking points.
- Focus on the most significant success factors and, based on the data you’ve collected, identify specifics that warrant recognition as well as development opportunities.
- Relate your findings to organizational or departmental goals.
- Based on the above, identify next steps, including future SMART goals and associated action items. The objective is for the employee to come out of the meeting with a clear sense of where they stand, a renewed sense of purpose and a clear sense of direction.
-
Conduct the appraisal meeting
- Provide your employee with a written copy of his or her appraisal. Discuss your observations and expectations. Invite comments. Keep feedback focused on performance, rather than the person and emphasize opportunities for improvement.
- At a minimum, the meeting should include the employee’s acknowledgement of the appraisal. Ideally, the meeting will be a springboard to the next review or appraisal period.
-
Adams, Amy. "
A First-Time Manager's Guide to Performance Reviews
." The Muse. Accessed August 20, 2019. ↵
Contributors and Attributions
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24.5D: Urethra
The urethra is a muscular tube that connects the bladder with the outside of the body and removes urine from the body.
LEARNING OBJECTIVES
Describe the location and function of the urethra as part of the urinary system
KEY TAKEAWAYS
Key Points
- The external urethral sphincter is a striated muscle that allows voluntary control over urination.
- Since the urethra is much shorter in females, they are much more susceptible to bacterial infections or a urinary tract infection.
- The urethra has smooth muscle tissue, erectile tissue, and mucous membrane tissue.
- Semen is carried into the urethra through the vas deferens.

The Urethra
The urethra is a tube that connects the urinary bladder to the genitals for the removal of fluids from the body. The external urethral sphincter is a striated muscle that allows voluntary control over urination by controlling the flow of urine from the bladder into the urethra.
The Female Urethra
Female urethra : A depiction of a female urethra.
In females, the urethra is shorter relative to males, and emerges above the vaginal opening. Its lining is composed primarily of non-keratinized stratified squamous epithelium that becomes transitional near the bladder. The urethra consists of three layers of tissues:
- The muscular layer is a continuation of the smooth muscle of the bladder.
- The erectile urethral tissue is specialized tissue that may become engorged with blood during sexual arousal.
- The mucuous urethral tissue are mucous membranes that line the interior of the urethra. The mucuous membranes are especially important for protecting the urethral tissues from the corrosive effects of urine.
The urethral sphincter separates the bladder from the urethra. Somatic (conscious) innervation of the external urethral sphincter is supplied by the pudendal nerve, which allows the sphincter to open and close. As the urethra is shorter in females relative to males, they are more vulnerable to bacterial urinary tract infections.
 Semen does not flow through the bladder or the rest of the urinary tract, instead it is a fluid made of sperm cells and other fluids that passes through a few different glands from the testes to the urethra through the vas deferens. Aside from semen, and the male urethra’s greater length, it is structurally similar to the female urethra.
Male urethra : A detailed view of the male urethra.


1.0: Introduction
Before we set out on our “cool, brisk walk," let’s get oriented. What is discrete mathematics, anyway? Why is it called that? What does it encompass? And what is it good for?
Let’s take the two words of the subject, in reverse order. First, math . When most people hear “math," they think “numbers." After all, isn’t math the study of quantity? And isn’t that the class where we first learned to count, add, and multiply?
Mathematics certainly has its root in the study of numbers — specifically, the “natural numbers" (the integers from 1 on up) that fascinated the ancient Greeks. Yet math is broader than this, almost to the point where numbers can be considered a special case of something deeper. In this book, when we talk about trees, sets, or formal logic, there might not be a number in sight.
Math is about abstract, conceptual objects that have properties, and the implications of those properties. An “object" can be any kind of “thought material" that we can define and reason about precisely. Much of math deals with questions like, “suppose we defined a certain kind of thing that had certain attributes. What would be the implications of this, if we reasoned it all the way out?" The “thing" may or may not be numerical, whatever it turns out to be. Like a number, however, it will be crisply defined, have certain known aspects to it, and be capable of combining with other things in some way.
Fundamental to math is that it deals with the abstract . Abstract, which is the opposite of concrete, essentially means something that can’t be perceived with the senses. A computer chip is concrete: you can touch it, you can see it. A number is not; nor is a function, a binary tree, or a logical implication. The only way to perceive these things is with the power of the mind. We will write expressions and draw pictures of many of our mathematical structures in order to help visualize them, and nearly everything we study will have practical applications whereby the abstractness gets grounded in concreteness for some useful purpose. But the underlying mathematical entity remains abstract and ethereal — only accessible to the mind’s eye. We may use a pencil to form the figure “5" on a piece of paper, but that is only a concrete manifestation of the underlying concept of “five-ness." Don’t mistake the picture or the symbol for the thing itself, which always transcends any mere physical representation.
The other word in the name of our subject is “discrete" (not to be confused with “discreet," which means something else entirely). The best way to appreciate what discrete means is to contrast it with its opposite, continuous. Consider the following list:
| Discrete | Continuous |
|---|---|
| whole numbers ( \(\mathbb{Z}\) ) | real numbers ( \(\mathbb{R}\) ) |
int
|
double
|
| digital | analog |
| quantum | continuum |
| counting | measuring |
| number theory | analysis |
| \(\Sigma\) | \(\int\) |
| – | \(\frac{d}{dx}\) |
What do the left-hand entries have in common? They describe things that are measured in crisp, distinct intervals, rather than varying smoothly over a range. Discrete things jump suddenly from position to position, with rigid precision. If you’re 5 feet tall, you might some day grow to 5.3 feet; but though there might be 5 people in your family, there will never be 5.3 (although there could be 6 someday).
The last couple of entries on this list are worth a brief comment. They are math symbols, some of which you may be familiar with. On the right side — in the continuous realm — are \(\int\) and \(\frac{d}{dx}\) , which you’ll remember if you’ve taken calculus. They stand for the two fundamental operations of integration and differentiation. Integration, which can be thought of as finding “the area under a curve," is basically a way of adding up a whole infinite bunch of numbers over some range. When you “integrate the function \(x^2\) from 3 to 5," you’re really adding up all the tiny, tiny little vertical slivers that comprise the area from \(x=3\) on the left to \(x=5\) on the right. Its corresponding entry in the left-column of the table is \(\Sigma\) , which is just a short-hand for “sum up a bunch of things." Integration and summation are equivalent operations, it’s just that when you integrate, you’re adding up all the (infinitely many) slivers across the real-line continuum. When you sum, you’re adding up a fixed sequence of entries, one at a time, like in a loop. \(\Sigma\) is just the discrete “version" of \(\int\) .
The same sort of relationship holds between ordinary subtraction (“–") and differentiation ( \(\frac{d}{dx}\) ). If you’ve plotted a bunch of discrete points on \(x\) - \(y\) axes, and you want to find the slope between two of them, you just subtract their \(y\) values and divide by the ( \(x\) ) distance between them. If you have a smooth continuous function, on the other hand, you use differentiation to find the slope at a point: this is essentially subtracting the tiny tiny difference between two supremely close points and then dividing by the distance between them. Thus subtraction is just the discrete “version" of \(\frac{d}{dx}\) .
Don’t worry, you don’t need to have fully understood any of the integration or differentiation stuff I just talked about, or even to have taken calculus yet. I’m just trying to give you some feel for what “discrete" means, and how the dichotomy between discrete and continuous really runs through all of math and computer science. In this book, we will mostly be focusing on discrete values and structures, which turn out to be of more use in computer science. That’s partially because as you probably know, computers themselves are discrete, and can only store and compute discrete values. There can be many of them — megabytes, gigabytes, terabytes — but each value stored is fundamentally comprised of bits, each of which has a value of either 0 or 1. This is unlike the human brain, by the way, whose neuronal synapses communicate based on the continuous quantities of chemicals present in their axons. So I guess “computer" and “brain" are another pair of entries we could add to our discrete vs. continuous list.
There’s another reason, though, why discrete math is of more use to computer scientists than continuous math is, beyond just the bits-and-bytes thing. Simply put, computers operate algorithmically. They carry out programs in step-by-step, iterative fashion. First do this, then do that, then move on to something else. This mechanical execution, like the ticking of a clock, permeates everything the computer can do, and everything we can tell it to do. At a given moment in time, the computer has completed step 7, but not step 8; it has accumulated 38 values, but not yet 39; its database has exactly 15 entries in it, no more and no less; it knows that after accepting this friend request, there will be exactly 553 people in your set of friends. The whole paradigm behind reasoning about computers and their programs is discrete, and that’s why we computer scientists find different problems worth thinking about than most of the world did a hundred years ago.
But it’s still math. It’s just discrete math. There’s a lot to come, so limber up and let me know when you’re ready to hit the road.


9.1: Four Color Theorem
The Four-Color Theorem states that in any plane surface with regions in it (people think of them as maps ), the regions can be colored with no more than four colors in such a way that two regions that have a common border do not get the same color. They are called adjacent (next to each other) if they share a segment of the border, not just a point. [1]
This was one of the first theorems to be proved by a computer , using a proof by exhaustion . In proof by exhaustion, the conclusion is established by dividing it into cases and proving each one separately. The number of cases sometimes may be very large. For example, the first proof of the Four-Color Theorem was a proof by exhaustion with 1,936 cases (in 1976). This proof was controversial because most of the cases were checked by a computer program, not by traditional mathematical arguments. The shortest known proof of the Four-Color Theorem today still has over 600 cases.
Even though the problem was first presented in 1852 as a problem to color political maps of countries , mapmakers are not particularly interested in it. According to an article by the math historian Kenneth May , “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of map-making do not mention the four-color property.”
Many simpler maps can be colored using three colors. The fourth color is required for some maps, such as one in which one region is surrounded by an odd number of others, which touch each other in a cycle. One such example is given in the image. The Five-Color Theorem states that five colors are enough to color a map. It has a short, elementary proof and was proved in the late 19th century ( Heawood 1890 ). Proving that four colors suffice turned out to be significantly more difficult. A number of false proofs and false counterexamples have appeared since the first statement of the Four-Color Theorem at University College London in England.
To be able to correctly state and solve the problem, it is necessary to clarify some aspects. First, all points that belong to three or more countries must be ignored.
For the purpose of the theorem every "country" has to be a connected region , or contiguous. In the real world, this is not true: Alaska as part of the United States , Nakhchivan as part of Azerbaijan , and Kaliningrad as part of Russia are not contiguous. Because the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map, such as the one shown above: in this map, the two regions labeled A belong to the same country and must be the same color. This map then requires five colors since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others. If A consisted of three regions, six or more colors might be required. In this manner, it is possible to construct maps that require an arbitrarily high number of colors. A similar construction also applies if a single color is used for all bodies of water, as is usual on real maps.
An easier-to-state version of the theorem uses graph theory . The set of regions of a map can be represented more abstractly as an undirected graph that has a vertex for each region and an edge for every pair of regions that share a boundary segment. [3] This graph is planar , meaning that these edges and vertices can be drawn on paper without any edges crossing each other. A planar graph can be formed from every map in this way. In graph-theoretic terminology, the Four-Color Theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short, "every planar graph is four-colorable" ( Thomas 1998 , p. 849; Wilson 2002 ).
Reference
- References (22)
Contributors and Attributions
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Saburo Matsumoto
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2.2: Multiplication of Matrices
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The next important matrix operation we will explore is multiplication of matrices. The operation of matrix multiplication is one of the most important and useful of the matrix operations. Throughout this section, we will also demonstrate how matrix multiplication relates to linear systems of equations.
First, we provide a formal definition of row and column vectors.
Definition \(\PageIndex{1}\): Row and Column Vectors
Matrices of size \(n\times 1\) or \(1\times n\) are called vectors . If \(X\) is such a matrix, then we write \(x_{i}\) to denote the entry of \(X\) in the \(i^{th}\) row of a column matrix, or the \(i^{th}\) column of a row matrix.
The \(n\times 1\) matrix \[X=\left[ \begin{array}{c} x_{1} \\ \vdots \\ x_{n} \end{array} \right]\nonumber \] is called a column vector. The \(1\times n\) matrix \[X = \left[ \begin{array}{ccc} x_{1} & \cdots & x_{n} \end{array} \right]\nonumber \] is called a row vector .
We may simply use the term vector throughout this text to refer to either a column or row vector. If we do so, the context will make it clear which we are referring to.
In this chapter, we will again use the notion of linear combination of vectors as in Definition 9.2.2 . In this context, a linear combination is a sum consisting of vectors multiplied by scalars. For example, \[\left[ \begin{array}{r} 50 \\ 122 \end{array} \right] = 7\left[ \begin{array}{r} 1 \\ 4 \end{array} \right] +8\left[ \begin{array}{r} 2 \\ 5 \end{array} \right] +9\left[ \begin{array}{r} 3 \\ 6 \end{array} \right]\nonumber \] is a linear combination of three vectors.
It turns out that we can express any system of linear equations as a linear combination of vectors. In fact, the vectors that we will use are just the columns of the corresponding augmented matrix!
Definition \(\PageIndex{2}\): The Vector Form of a System of Linear Equations
Suppose we have a system of equations given by \[\begin{array}{c} a_{11}x_{1}+\cdots +a_{1n}x_{n}=b_{1} \\ \vdots \\ a_{m1}x_{1}+\cdots +a_{mn}x_{n}=b_{m} \end{array}\nonumber \] We can express this system in vector form which is as follows: \[x_1 \left[ \begin{array}{c} a_{11}\\ a_{21}\\ \vdots \\ a_{m1} \end{array} \right] + x_2 \left[ \begin{array}{c} a_{12}\\ a_{22}\\ \vdots \\ a_{m2} \end{array} \right] + \cdots + x_n \left[ \begin{array}{c} a_{1n}\\ a_{2n}\\ \vdots \\ a_{mn} \end{array} \right] = \left[ \begin{array}{c} b_1\\ b_2\\ \vdots \\ b_m \end{array} \right]\nonumber \]
Notice that each vector used here is one column from the corresponding augmented matrix. There is one vector for each variable in the system, along with the constant vector.
The first important form of matrix multiplication is multiplying a matrix by a vector. Consider the product given by \[\left[ \begin{array}{rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array} \right] \left[ \begin{array}{r} 7 \\ 8 \\ 9 \end{array} \right]\nonumber \] We will soon see that this equals \[7\left[ \begin{array}{c} 1 \\ 4 \end{array} \right] +8\left[ \begin{array}{c} 2 \\ 5 \end{array} \right] +9\left[ \begin{array}{c} 3 \\ 6 \end{array} \right] =\left[ \begin{array}{c} 50 \\ 122 \end{array} \right]\nonumber \]
In general terms, \[\begin{aligned} \left[ \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array} \right] &= \ x_{1}\left[ \begin{array}{c} a_{11} \\ a_{21} \end{array} \right] +x_{2}\left[ \begin{array}{c} a_{12} \\ a_{22} \end{array} \right] +x_{3}\left[ \begin{array}{c} a_{13} \\ a_{23} \end{array} \right] \\ &=\left[ \begin{array}{c} a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3} \\ a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3} \end{array} \right] \end{aligned}\] Thus you take \(x_{1}\) times the first column, add to \(x_{2}\) times the second column, and finally \(x_{3}\) times the third column. The above sum is a linear combination of the columns of the matrix. When you multiply a matrix on the left by a vector on the right, the numbers making up the vector are just the scalars to be used in the linear combination of the columns as illustrated above.
Here is the formal definition of how to multiply an \(m\times n\) matrix by an \(n\times 1\) column vector.
Definition \(\PageIndex{3}\): Multiplication of Vector by Matrix
Let \(A=\left[ a_{ij} \right]\) be an \(m\times n\) matrix and let \(X\) be an \(n\times 1\) matrix given by \[A=\left[ A_{1} \cdots A_{n}\right], X = \left[ \begin{array}{r} x_{1} \\ \vdots \\ x_{n} \end{array} \right]\nonumber \]
Then the product \(AX\) is the \(m\times 1\) column vector which equals the following linear combination of the columns of \(A\): \[x_{1}A_{1}+x_{2}A_{2}+\cdots +x_{n}A_{n} = \sum_{j=1}^{n}x_{j}A_{j}\nonumber \]
If we write the columns of \(A\) in terms of their entries, they are of the form \[A_{j} = \left[ \begin{array}{c} a_{1j} \\ a_{2j} \\ \vdots \\ a_{mj} \end{array} \right]\nonumber \] Then, we can write the product \(AX\) as \[AX = x_{1}\left[ \begin{array}{c} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{array} \right] + x_{2}\left[ \begin{array}{c} a_{12} \\ a_{22} \\ \vdots \\ a_{m2} \end{array} \right] +\cdots + x_{n}\left[ \begin{array}{c} a_{1n} \\ a_{2n} \\ \vdots \\ a_{mn} \end{array} \right]\nonumber \]
Note that multiplication of an \(m \times n\) matrix and an \(n \times 1\) vector produces an \(m \times 1\) vector.
Here is an example.
Example \(\PageIndex{1}\): A Vector Multiplied by a Matrix
Compute the product \(AX\) for \[A = \left[ \begin{array}{rrrr} 1 & 2 & 1 & 3 \\ 0 & 2 & 1 & -2 \\ 2 & 1 & 4 & 1 \end{array} \right], X = \left[ \begin{array}{r} 1 \\ 2 \\ 0 \\ 1 \end{array} \right]\nonumber \]
Solution
We will use Definition \(\PageIndex{3}\) to compute the product. Therefore, we compute the product \(AX\) as follows. \[\begin{aligned} & 1\left[ \begin{array}{r} 1 \\ 0 \\ 2 \end{array} \right] + 2\left[ \begin{array}{r} 2 \\ 2 \\ 1 \end{array} \right] + 0\left[ \begin{array}{r} 1 \\ 1 \\ 4 \end{array} \right] + 1 \left[ \begin{array}{r} 3 \\ -2\\ 1 \end{array} \right] \\ &= \left[ \begin{array}{r} 1 \\ 0 \\ 2 \end{array} \right] + \left[ \begin{array}{r} 4 \\ 4 \\ 2 \end{array} \right] + \left[ \begin{array}{r} 0 \\ 0 \\ 0 \end{array} \right] + \left[ \begin{array}{r} 3 \\ -2\\ 1 \end{array} \right] \\ &= \left[ \begin{array}{r} 8 \\ 2 \\ 5 \end{array} \right]\end{aligned}\]
Using the above operation, we can also write a system of linear equations in matrix form . In this form, we express the system as a matrix multiplied by a vector. Consider the following definition.
Definition \(\PageIndex{4}\): The Matrix Form of a System of Linear Equations
Suppose we have a system of equations given by \[\begin{array}{c} a_{11}x_{1}+\cdots +a_{1n}x_{n}=b_{1} \\ a_{21}x_{1}+ \cdots + a_{2n}x_{n} = b_{2} \\ \vdots \\ a_{m1}x_{1}+\cdots +a_{mn}x_{n}=b_{m} \end{array}\nonumber \] Then we can express this system in matrix form as follows. \[\left[ \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array} \right] \left[ \begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{array} \right] = \left[ \begin{array}{c} b_{1}\\ b_{2}\\ \vdots \\ b_{m} \end{array} \right]\nonumber \]
The expression \(AX=B\) is also known as the Matrix Form of the corresponding system of linear equations. The matrix \(A\) is simply the coefficient matrix of the system, the vector \(X\) is the column vector constructed from the variables of the system, and finally the vector \(B\) is the column vector constructed from the constants of the system. It is important to note that any system of linear equations can be written in this form.
Notice that if we write a homogeneous system of equations in matrix form, it would have the form \(AX=0\), for the zero vector \(0\).
You can see from this definition that a vector \[X = \left[ \begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{array} \right]\nonumber \] will satisfy the equation \(AX=B\) only when the entries \(x_{1}, x_{2}, \cdots, x_{n}\) of the vector \(X\) are solutions to the original system.
Now that we have examined how to multiply a matrix by a vector, we wish to consider the case where we multiply two matrices of more general sizes, although these sizes still need to be appropriate as we will see. For example, in Example \(\PageIndex{1}\) , we multiplied a \(3 \times 4\) matrix by a \(4 \times 1\) vector. We want to investigate how to multiply other sizes of matrices.
We have not yet given any conditions on when matrix multiplication is possible! For matrices \(A\) and \(B\), in order to form the product \(AB\), the number of columns of \(A\) must equal the number of rows of \(B.\) Consider a product \(AB\) where \(A\) has size \(m\times n\) and \(B\) has size \(n \times p\). Then, the product in terms of size of matrices is given by \[(m\times\overset{\text{these must match!}}{\widehat{n)\;(n}\times p})=m\times p\nonumber \]
Note the two outside numbers give the size of the product. One of the most important rules regarding matrix multiplication is the following. If the two middle numbers don’t match, you can’t multiply the matrices!
When the number of columns of \(A\) equals the number of rows of \(B\) the two matrices are said to be conformable and the product \(AB\) is obtained as follows.
Definition \(\PageIndex{5}\): Multiplication of Two Matrices
Let \(A\) be an \(m\times n\) matrix and let \(B\) be an \(n\times p\) matrix of the form \[B=\left[ B_{1} \cdots B_{p}\right]\nonumber \] where \(B_{1},...,B_{p}\) are the \(n\times 1\) columns of \(B\). Then the \(m\times p\) matrix \(AB\) is defined as follows: \[AB = A \left[ B_{1} \cdots B_{p}\right] = \left[ (A B)_{1} \cdots (AB)_{p}\right]\nonumber \] where \((AB)_{k}\) is an \(m\times 1\) matrix or column vector which gives the \(k^{th}\) column of \(AB\).
Consider the following example.
Example \(\PageIndex{2}\): Multiplying Two Matrices
Find \(AB\) if possible. \[A = \left[ \begin{array}{rrr} 1 & 2 & 1 \\ 0 & 2 & 1 \end{array} \right], B = \left[ \begin{array}{rrr} 1 & 2 & 0 \\ 0 & 3 & 1 \\ -2 & 1 & 1 \end{array} \right]\nonumber \]
Solution
The first thing you need to verify when calculating a product is whether the multiplication is possible. The first matrix has size \(2\times 3\) and the second matrix has size \(3\times 3\). The inside numbers are equal, so \(A\) and \(B\) are conformable matrices. According to the above discussion \(AB\) will be a \(2\times 3\) matrix. Definition \(\PageIndex{5}\) gives us a way to calculate each column of \(AB\), as follows.
\[\left[ \overset{ \text{First column}}{\overbrace{\left[ \begin{array}{rrr} 1 & 2 & 1 \\ 0 & 2 & 1 \end{array} \right] \left[ \begin{array}{r} 1 \\ 0 \\ -2 \end{array} \right] }},\overset{\text{Second column}}{\overbrace{\left[ \begin{array}{rrr} 1 & 2 & 1 \\ 0 & 2 & 1 \end{array} \right] \left[ \begin{array}{r} 2 \\ 3 \\ 1 \end{array} \right] }},\overset{\text{Third column}}{\overbrace{\left[ \begin{array}{rrr} 1 & 2 & 1 \\ 0 & 2 & 1 \end{array} \right] \left[ \begin{array}{r} 0 \\ 1 \\ 1 \end{array} \right] }}\right]\nonumber \] You know how to multiply a matrix times a vector, using Definition \(\PageIndex{3}\) for each of the three columns. Thus \[\left[ \begin{array}{rrr} 1 & 2 & 1 \\ 0 & 2 & 1 \end{array} \right] \left[ \begin{array}{rrr} 1 & 2 & 0 \\ 0 & 3 & 1 \\ -2 & 1 & 1 \end{array} \right] = \ \left[ \begin{array}{rrr} -1 & 9 & 3 \\ -2 & 7 & 3 \end{array} \right]\nonumber \]
Since vectors are simply \(n \times 1\) or \(1 \times m\) matrices, we can also multiply a vector by another vector.
Example \(\PageIndex{3}\): Vector Times Vector Multiplication
Multiply if possible \(\left[ \begin{array}{r} 1 \\ 2 \\ 1 \end{array} \right] \left[ \begin{array}{rrrr} 1 & 2 & 1 & 0 \end{array} \right] .\)
Solution
In this case we are multiplying a matrix of size \(3 \times 1\) by a matrix of size \(1 \times 4.\) The inside numbers match so the product is defined. Note that the product will be a matrix of size \(3 \times 4\). Using Definition \(\PageIndex{5}\) , we can compute this product as follows \(\: \) \[\left[ \begin{array}{r} 1 \\ 2 \\ 1 \end{array} \right] \left[ \begin{array}{rrrr} 1 & 2 & 1 & 0 \end{array} \right] = \left[ \overset{ \text{First column}}{\overbrace{\left[ \begin{array}{r} 1 \\ 2 \\ 1 \end{array} \right] \left[ \begin{array}{r} 1 \end{array} \right] }},\overset{\text{Second column}}{\overbrace{\left[ \begin{array}{r} 1 \\ 2\\ 1 \end{array} \right] \left[ \begin{array}{r} 2 \end{array} \right] }},\overset{\text{Third column}}{\overbrace{\left[ \begin{array}{r} 1 \\ 2 \\ 1 \end{array} \right] \left[ \begin{array}{r} 1 \end{array} \right] }}, \overset {\text{Fourth column}}{\overbrace{\left[ \begin{array}{r} 1\\ 2\\ 1 \end{array} \right] \left[ \begin{array}{r} 0 \end{array} \right]}} \right]\nonumber \]
You can use Definition \(\PageIndex{3}\) to verify that this product is \[\left[ \begin{array}{cccc} 1 & 2 & 1 & 0 \\ 2 & 4 & 2 & 0 \\ 1 & 2 & 1 & 0 \end{array} \right]\nonumber \]
Example \(\PageIndex{4}\): A Multiplication Which is Not Defined
Find \(BA\) if possible. \[B = \left[ \begin{array}{ccc} 1 & 2 & 0 \\ 0 & 3 & 1 \\ -2 & 1 & 1 \end{array} \right], A = \left[ \begin{array}{ccc} 1 & 2 & 1 \\ 0 & 2 & 1 \end{array} \right]\nonumber \]
Solution
First check if it is possible. This product is of the form \(\left( 3\times 3\right) \left( 2\times 3\right) .\) The inside numbers do not match and so you can’t do this multiplication.
In this case, we say that the multiplication is not defined. Notice that these are the same matrices which we used in Example \(\PageIndex{2}\) . In this example, we tried to calculate \(BA\) instead of \(AB\). This demonstrates another property of matrix multiplication. While the product \(AB\) maybe be defined, we cannot assume that the product \(BA\) will be possible. Therefore, it is important to always check that the product is defined before carrying out any calculations.
Earlier, we defined the zero matrix \(0\) to be the matrix (of appropriate size) containing zeros in all entries. Consider the following example for multiplication by the zero matrix.
Example \(\PageIndex{5}\): Multiplication by the Zero Matrix
Compute the product \(A0\) for the matrix \[A= \left[ \begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array} \right]\nonumber \] and the \(2 \times 2\) zero matrix given by \[0= \left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right]\nonumber \]
Solution
In this product, we compute \[\left[ \begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array} \right] \left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right] = \left[ \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right]\nonumber \]
Hence, \(A0=0\).
Notice that we could also multiply \(A\) by the \(2 \times 1\) zero vector given by \(\left[ \begin{array}{r} 0 \\ 0 \end{array} \right]\). The result would be the \(2 \times 1\) zero vector. Therefore, it is always the case that \(A0=0\), for an appropriately sized zero matrix or vector.


14.7: Other Media Elements
- Evaluate use cases for other miscellaneous media elements
In Types of Sales Promotions in Retail , Direct Marketing Tactics in Retail and Social Media and the IMC Mix you may have read about a number of promotional tools and how marketers might use them as part of Integrated Marketing Communication (IMC). However, you should not infer that these are the best or the only tools that should be considered in any campaign you develop. Instead, the examples we’ve shared are illustrative of how traditional and new media can be leveraged outside and inside the store to help communicate firm, brand and product messages with consumers.
Again, you will want to select the best channels for your campaign, given your budget, your strategy and the competitive environment.
Contributors and Attributions
- Other Media Elements. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution