7.3: Aging and Mental Health
It is estimated that 20% of people age 55 years or older experience some type of mental health concern. The most common conditions include anxiety, severe cognitive impairment, and mood disorders (such as depression or bipolar disorder). Mental health issues are often implicated as a factor in cases of suicide. Older men have the highest suicide rate of any age group. Men aged 85 years or older have a suicide rate of 45.23 per 100,000, compared to an overall rate of 11.01 per 100,000 for all ages.
Social Support
Social support serves major support functions, including emotional support (e.g., sharing problems or venting emotions), informational support (e.g., advice and guidance), and instrumental support (e.g., providing rides or assisting with housekeeping).
- Adequate social and emotional support is associated with reduced risk of mental illness, physical illness, and mortality
- The majority (nearly 90%) of adults age 50 or older indicated that they are receiving adequate amounts of support.
- Adults age 65 or older were more likely than adults age 50–64 to report that they “rarely” or “never” received the social and emotional support they needed (12.2% compared to 8.1%, respectively).
- Approximately one-fifth of Hispanic and other, non-Hispanic adults age 65 years or older reported that they were not receiving the support they need, compared to about one-tenth of older white adults.
- Among adults age 50 or older, men were more likely than women to report they “rarely” or “never” received the support they needed (11.39% compared to 8.49%).
Depression is Not a Normal Part of Growing Older
Depression is a true and treatable medical condition, not a normal part of aging. However older adults are at an increased risk for experiencing depression. If you are concerned about a loved one, offer to go with him or her to see a health care provider to be diagnosed and treated.
Depression is not just having “the blues” or the emotions we feel when grieving the loss of a loved one. It is a true medical condition that is treatable, like diabetes or hypertension.
How Do I Know If It’s Depression?
Someone who is depressed has feelings of sadness or anxiety that last for weeks at a time. He or she may also experience
- Feelings of hopelessness and/or pessimism
- Feelings of guilt, worthlessness and/or helplessness
- Irritability, restlessness
- Loss of interest in activities or hobbies once pleasurable
- Fatigue and decreased energy
- Difficulty concentrating, remembering details and making decisions
- Insomnia, early-morning wakefulness, or excessive sleeping
- Overeating or appetite loss
- Thoughts of suicide, suicide attempts
- Persistent aches or pains, headaches, cramps, or digestive problems that do not get better, even with treatment
How is Depression Different for Older Adults?
- Older adults are at increased risk. We know that about 80% of older adults have at least one chronic health condition, and 50% have two or more. Depression is more common in people who also have other illnesses (such as heart disease or cancer) or whose function becomes limited.
- Older adults are often misdiagnosed and undertreated. Healthcare providers may mistake an older adult’s symptoms of depression as just a natural reaction to illness or the life changes that may occur as we age, and therefore not see the depression as something to be treated. Older adults themselves often share this belief and do not seek help because they don’t understand that they could feel better with appropriate treatment.
How Many Older Adults Are Depressed?
The good news is that the majority of older adults are not depressed. Some estimates of major depression in older people living in the community range from less than 1% to about 5% but rise to 13.5% in those who require home healthcare and to 11.5% in older hospital patients.
Use the interactive maps at this website to determine The State of Mental Health and Aging in America . You can explore different aspects of health by adjusting the Location and Category filters. You can also read this a summary of the information.
Sources
Mental Health Problems in Older Adults: CDC, http://www.cdc.gov/aging/pdf/mental_health.pdf
Depression is Not a Normal Part of Growing Older: http://www.cdc.gov/aging/mentalhealth/depression.htm


6.5: Putting It Together- Recruitment and Selection
Given labor market trends and the accelerating pace of technology, recruiting is becoming a mission-critical. As highlighted in PwC’s Talent Trends 2019 report, 80% of CEOs identified a skills gap as a threat to growth, with those who were “extremely concerned” about the availability of key skills, citing the negative impact on innovation (55%) and people costs (52%), among other factors. [1] PwC notes that respondents level of concern has increased over the last five years with the emergence of new technologies.
To put the upside in perspective, Oracle reports that the financial impact of hiring a top performing employee is estimated to be 10–100 times that of a person’s compensation. Let’s do the math: Hiring a single high-impact employee at an annual salary of $100,000 has the potential of adding $1–$10 million to a company’s revenue every year that person remains with the company. [2] The cost and organizational impact of a poor choice is also significant. The U.S. Department of Labor estimates that the average cost of a bad hiring decision is 30% of the employee’s first year projected earnings. [3] Note, however, that number represents only a fraction of the organizational impact. CFOs surveyed by global staffing firm Robert Half ranked morale (39%) and productivity (34%) effects of a bad hire greater than the monetary (25%) cost. Additionally, a perception that leaders lack competence undermines trust in the organization—a point we’ll discuss further in the Module 15: Corporate Social Responsibility. Perhaps the most damaging effect is on an organization’s top performers; worst case scenario: the pressure and dysfunction could cause them to leave, ending the revenue multiplier effect.
“Getting the right people on the bus” is also critical because personnel costs are often a business’ single largest expense. According to the Society for Human Resource Management, salaries alone can account for 18 to 52 percent of your operating budget. [4] Add in taxes and benefits and the full cost could be in the 40–80% of gross revenue range. [5] Choose well, and your employees can be a source of revenue and competitive advantage. A poor choice can represent a critical liability.
Making the best talent choices also requires organizations to address HR information processing challenges and confront the reality of perceptual errors and unconscious bias. As the UNC Executive Development Team phrases it: “Unconscious biases in the workplace can stymie diversity, recruiting and retention efforts, and unknowingly shape an organization’s culture.” [6] Increasingly, choosing well—and optimizing the recruiting process from cost, effectiveness and candidate experience standpoints—will involve integrating automation, robotics and AI.
For perspective, Swedish recruiters are currently testing the world’s first robot designed to conduct unbiased job interviews. [7] Developed by AI and social robotics firm Furhat Robotics, “Tengai” is a 16” tall computer interface similar to a human head that would be positioned on a table across from a candidate.
Tengai is being trained in collaboration Swedish recruiting firm TNG. As Technology of Business reporter Maddy Savage notes, “the goal is to offer candidates job interviews that are free from any of the unconscious biases that managers and recruiters can often bring to the hiring process, while still making the experience ‘seem human’.” To avoid the transfer of human bias, Tengai is learning from several different recruiters. Additional protocols designed to increase objectivity: dispensing with pre-interview conversation and posing questions in an identical manner, using the same tone and order. Transcripts of each interview are provided to recruiters or managers, eliminating potential bias based on a candidate’s name, face or other information that’s not relevant to job performance. Tengai was scheduled to start interviewing candidates in May 2019. An English-language version of the robot is expected to be available in 2020.
While this is an exciting possibility, it is important to think about both the limitations of technology and the effects of depersonalizing the interview process. Additionally, it’s worth thinking about the fact that while the robot has been created with the goal of eliminating bias, it was created with conventionally white male features.
Of course, we can use what we know about perceptual errors and bias to modify our recruiting processes today. Given the research regarding adverse impact and the implications for organizational success, hacking unconscious bias should be a recruiting—and organizational—priority.
-
"
Talent Trends 2019
." PwC. 2019. Accessed July 26, 2019. ↵
-
"Modern HR in the Cloud: Best Practices for Recruiting the Best Talent." Oracle. Accessed July 26, 2019. ↵
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Cardenas, Rebekah. "What's the Real Cost of a Bad Hire?" HR Exchange. April 2, 2014. Accessed July 26, 2019. ↵
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Deeb, Carol. "
Percent of a Business Budget for Salary
." Chron. Accesed July 26, 2019. ↵
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Ferguson, Grace. "
What Percentage of the Budget Should Be Spent on Payroll?
" Chron. Accessed July, 26, 2019. ↵
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"The Real Effect of Unconscious Bias in the Workplace." UNC Executive Development Blog. October 8, 2015. Accessed July 26, 2019. ↵
-
Savage, Maddy. "
Meet Tengai, a Job Interview Robot Who Won't Judge You
." BBC. March 12, 2019. Accessed July 26, 2019. ↵
Contributors and Attributions
- People. Authored by : Katie White. Provided by : Pixabay. Located at : pixabay.com/illustrations/people-special-different-employment-4009327/. License : CC0: No Rights Reserved . License Terms : Pixabay License
- Robots Could Be Conducting Job Interviews Next Year. Authored by : Mashable. Located at : https://youtu.be/c8rfWRdvv00 . License : All Rights Reserved . License Terms : Standard YouTube License


3.13: Multirule Derivatives
Okay, let’s talk about \(\frac{d}{dx} \ e^{x^2 + x} \sin(x)\). If you’re thinking this looks like a product rule, but it also looks like a chain rule, you’re right. To compute this derivative, we need to do the chain rule and the product rule. This is because it is a multirule problem. Let’s do this example
The way I like to break this down is to consider a little rule and a big rule. In this case, the little rule is the chain rule problem \(\frac{d}{dx} \ e^{x^2 + x}\). If we do this problem, we see that \(f = e^x\), \(f' = e^x\), \(g = x^2 + x\) and \(g' = 2x + 1\). So we have
\[\begin{equation*} \frac{d}{dx} \ e^{x^2 + x} = {\color{red} e^{x^2 + x} (2x + 1)}. \end{equation*}\]
Now we are ready to do the big rule, which is the product rule. At this point we go back to the original problem \(\frac{d}{dx} \ e^{x^2 + x} \sin(x)\). For this product rule, we see \(f = e^{x^2 + x}\), \(g = \sin(x)\), \(g' = \cos(x)\). What is \({\color{red} f'}\)? Why, that’s what we just computed in the equation above! So \({\color{red} f' = e^{x^2 + x}(2x+1)}\). Putting this all together with the product rule \(f g' + g f'\), we have
\[\begin{align*} \frac{d}{dx} \ e^{x^2 + x} \sin(x) & = f g' + g {\color{red} f'} \\ & = \boxed{e^{x^2 + x} \cos(x) + \sin(x) {\color{red} e^{x^2 + x} (2x + 1)}} . \end{align*}\]
little chain rule: \(\frac{d}{dx} \sin(x^2 + x)\)
\(\begin{array}{ll} f = \sin(x) & g = x^2 + x \\ f' = \cos(x) & g' = 2x + 1 \end{array}\)
Result: \({\color{red} \cos(x^2 + x) \cdot (2x+1)}\)
Big quotient rule (aka the whole problem):\(\frac{d}{dx} \frac{x}{\sin(x^2 + x)}\)
\(\begin{array}{ll} f = x & g = \sin(x^2 + x) \\ f' = 1 & g' = {\color{red} \cos(x^2 + x) \cdot (2x+1)} \end{array}\)
Result: \(\boxed{\frac{\sin(x^2 + x) \cdot 1 - x \cos(x^2 + x) \cdot (2x+1)}{(\sin(x^2+x))^2}}\)


5.4: Tort Law
What you’ll learn to do: define tort law, and explain the role of product liability in tort law
In this section you’ll learn about an area of the law that’s especially important for business: tort law and product liability.
Learning Objectives
- Explain the concept of negligence as it relates to tort law
- Explain the three major bases for product liability claims
- Differentiate between compensatory and punitive damages
Tort Law
In common law jurisdictions, a tort is a civil wrong that unfairly causes someone else to suffer loss or harm, resulting in legal liability for the person who commits the tortious act. Although crimes may be torts, the cause of legal action is not necessarily a crime, as the harm may be due to negligence. The following video explains what negligence is.
The victim of the harm can recover his or her loss as damages in a lawsuit. In order to prevail, the plaintiff in the lawsuit, commonly referred to as the injured party, must prove that a breach of duty (i.e., either an action or lack of action) was the legally recognizable cause of the harm.
Legal injuries are not limited to physical injuries and may include emotional, economic, or reputational injuries, as well as violations of privacy, property, or constitutional rights. Torts include such varied topics as auto accidents, false imprisonment, defamation, product liability, copyright infringement, and environmental pollution (toxic torts). While many torts are the result of negligence, tort law also recognizes intentional torts, in which a person has intentionally acted in a way that harms another. In addition, when it comes to product liability, the courts have established a doctrine of “strict liability” for torts arising from injury caused by the use of a company’s product and/or service. Under “strict liability,” the injured party does not have to prove that the company was negligent in order to win a claim for damages.
Tort law is different from criminal law in two ways: (1) torts may result from negligent as well as intentional or criminal actions, and (2) tort lawsuits have a lower burden of proof, such as “preponderance of evidence” rather than “beyond a reasonable doubt.” Sometimes a plaintiff may prevail in a tort case even if the person who allegedly caused harm was acquitted in an earlier criminal trial. For example, O. J. Simpson was acquitted in criminal court of murder but later found liable for the tort of wrongful death.
For businesses, torts that arise from product liability can have devastating consequences. Let’s examine product liability in greater detail.
Product Liability
Product liability is the area of law in which product manufacturers, distributors, and sellers are held responsible for the injuries caused by their products. Traditionally, product liability cases were decided according to the theory of negligence . Negligence is behavior that results in an unintentional injury or causes unintentional harm. Product liability law has evolved, however, and most states have extended product liability into the category of strict liability tort, too. Strict liability torts involve actions that are inherently dangerous and for which a party may be liable no matter how carefully he or she (or it) performs those actions . Regardless of whether the product liability claim is based upon negligence or strict liability, a product liability claim results from either a design defect, a manufacturing defect, or a failure to warn. To better appreciate the issues involved in cases of strict liability, let’s return to where we left you and your summer house-painting business at the beginning of this module.
Having escaped the house-painting business relatively unscathed, you head back home to rethink your options for gainful employment during your summer vacation. You’ve stored your only remaining capital assets—the two ladders and the platform that you’d used for scaffolding—in your father’s garage. One afternoon, your uncle notices them.
Examining one of the ladders, he asks you how much weight it’s designed to hold, and you tell him what the department manager at Ladders ’n’ Things told you: three hundred pounds per rung. Sensing that he might want to buy them, you hasten to add that, although you got them at a cut-rate price because of a little rust, they’re virtually brand-new. As it turns out, your uncle doesn’t want to buy the ladders, but he does offer to pay you $35 an hour to take them to his house and help him put up new roofing. He’s easygoing, he’s family, and he probably won’t sue you for anything, so you jump at the opportunity.
Everything goes smoothly until day two, when you’re working on the scaffolding two stories off the ground. In the process of unwrapping a bundle of shingles, one of the ladders buckles, and the entire platform comes down—landing you on your uncle’s stone patio with a cervical fracture.
Fortunately, there’s no damage to your spinal cord, but you’re in pain and you need surgery. Now it’s your turn to sue somebody. But whom? And for what?
Pursuing a Claim of Product Liability
Not surprising, your lawyer advises you to file an action for product liability—a claim of injury suffered because of a defective product (in your case, the ladder, of course). He goes on to say that in cases of product liability, there are three major bases for product liability claims. Section 2 of the Restatement (Third) of Torts: Products Liability identifies the following: manufacturing defect, design defect, failure to warn (also known as marketing defects). These are explained below:
- Manufacturing defects are those that occur in the manufacturing process and usually involve poor-quality materials or shoddy workmanship.
- Design defects occur where the product design is inherently dangerous or useless (and hence defective) no matter how carefully manufactured; this may be demonstrated either by showing that the product fails to satisfy ordinary consumer expectations as to what constitutes a safe product or that the risks of the product outweigh its benefits.
- Failure-to-warn defects arise in products that carry inherent non-obvious dangers that could be mitigated through adequate warnings to the user, and these dangers are present regardless of how well the product is manufactured and designed for its intended purpose.
In most states, these defects are not legal claims in and of themselves, but they are the grounds for the action (negligence or strict liability). For example, a plaintiff might plead negligent failure to warn or strict liability for defective design.
Grounds of Strict Liability
For the sake of argument, let’s say that your lawyer isn’t very confident about pursuing a claim of negligence against the manufacturer of your ladder. The company doesn’t appear to have been careless in any of the three forms prescribed by law, and it will in any case be difficult to demonstrate the elements required in negligence cases. He suggests instead that you proceed on grounds of strict liability.
In tort law, strict liability is the imposition of liability on a party without a finding of fault (such as negligence or tortious intent). The claimant need only prove that the tort occurred and that the defendant was responsible. The law imputes strict liability to situations it considers to be inherently dangerous. It discourages reckless behavior and needless loss by forcing potential defendants to take every possible precaution. It also has the effect of simplifying and thereby expediting court decisions in these cases. In short, strict liability is based on the following legal theories:
- Certain products put people at risk of injury no matter how much care is taken to prevent injury.
- Consumers should have some means of seeking compensation if they’re injured while using these products.
A classic example of strict liability is the owner of a tiger rehabilitation center. No matter how strong the tiger cages are, if an animal escapes and causes damage and injury, the owner is held liable. Another example is a contractor hiring a demolition subcontractor that lacks proper insurance. If the subcontractor makes a mistake, the contractor is strictly liable for any damage that occurs.
In strict liability situations, although the plaintiff does not have to prove fault, the defendant can raise a defense of “absence of fault”: that is, the defense can argue that the defect was the result of the plaintiff’s actions —not of the product—and that no inference of defect should be drawn solely because an accident occurs. On the other hand, if the plaintiff can prove that the defendant knew about the defect before the damages occurred, in some jurisdictions additional punitive damages can be awarded to the victim.
Compensatory and Punitive Damages
When someone pursues a claim under a tort, the goal (or legal remedy) is usually the award of damages. Damages in tort are generally awarded to restore the plaintiff to the position he or she was in had the tort not occurred.
In law, damages are an award, typically of money, to be paid to a person as compensation for loss or injury. Damages are classified as compensatory (or actual ) damages and punitive damages . Compensatory damages are further categorized into special damages , which are economic losses such as loss of earnings, property damage, and medical expenses, and general damages , which are noneconomic damages such as pain and suffering and emotional distress.
Generally, punitive damages are not awarded in order to compensate the plaintiff but to reform or deter the defendant and similar persons from pursuing a course of action such as that which damaged the plaintiff. Punitive damages are awarded only in special cases where a defendant acted in a blatantly negligent, malicious, or grossly reckless manner.
In the case of Liebeck v. McDonald’s Restaurants (1994), for example, 79-year-old Stella Liebeck spilled McDonald’s coffee in her lap, which resulted in second- and third-degree burns on her thighs, buttocks, groin, and genitals. The burns were severe enough to require skin grafts. Liebeck attempted to have McDonald’s pay her $20,000 medical bills as indemnity for the incident. McDonald’s refused, and Liebeck sued. During the case’s discovery process, internal documents from McDonald’s revealed that the company had received hundreds of similar complaints from customers claiming that McDonald’s coffee caused severe burns. At trial, this led the jury to find that McDonald’s knew their product was dangerous and injuring their customers and that the company had done nothing to correct the problem. The jury decided on $200,000 in compensatory damages , but attributed 20 percent of the fault to Liebeck, reducing her compensation to $160,000. The jury also awarded Liebeck $2.7 million in punitive damages , which, at the time, represented two days’ of McDonald’s coffee sales revenue. The judge later reduced the punitive damages to $480,000. The case is often criticized for the very high amount of damages the jury awarded.
The following videos also explain compensatory and punitive damages.


5.6: Gas Mixtures and Partial Pressures
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Learning Objectives
- To determine the contribution of each component gas to the total pressure of a mixture of gases
In our use of the ideal gas law thus far, we have focused entirely on the properties of pure gases with only a single chemical species. But what happens when two or more gases are mixed? In this section, we describe how to determine the contribution of each gas present to the total pressure of the mixture.
Partial Pressures
The ideal gas law assumes that all gases behave identically and that their behavior is independent of attractive and repulsive forces. If volume and temperature are held constant, the ideal gas equation can be rearranged to show that the pressure of a sample of gas is directly proportional to the number of moles of gas present:
\[P=n \left(\dfrac{RT}{V}\right) = n \times \rm const. \label{10.6.1} \]
Nothing in the equation depends on the nature of the gas—only the amount.
With this assumption, let’s suppose we have a mixture of two ideal gases that are present in equal amounts. What is the total pressure of the mixture? Because the pressure depends on only the total number of particles of gas present, the total pressure of the mixture will simply be twice the pressure of either component. More generally, the total pressure exerted by a mixture of gases at a given temperature and volume is the sum of the pressures exerted by each gas alone. Furthermore, if we know the volume, the temperature, and the number of moles of each gas in a mixture, then we can calculate the pressure exerted by each gas individually, which is its partial pressure, the pressure the gas would exert if it were the only one present (at the same temperature and volume).
To summarize, the total pressure exerted by a mixture of gases is the sum of the partial pressures of component gases . This law was first discovered by John Dalton, the father of the atomic theory of matter. It is now known as Dalton’s law of partial pressures . We can write it mathematically as
\[\begin{align} P_{tot} &= P_1+P_2+P_3+P_4 \ldots \\[4pt] &= \sum_{i=1}^n{P_i} \label{10.6.2} \end{align} \]
where \(P_{tot}\) is the total pressure and the other terms are the partial pressures of the individual gases (up to \(n\) component gases).
For a mixture of two ideal gases, \(A\) and \(B\), we can write an expression for the total pressure:
\[\begin{align} P_{tot} &=P_A+P_B \\[4pt] &=n_A\left(\dfrac{RT}{V}\right) + n_B\left(\dfrac{RT}{V}\right) \\[4pt] &=(n_A+n_B)\left(\dfrac{RT}{V}\right) \label{10.6.3} \end{align} \]
More generally, for a mixture of \(n\) component gases, the total pressure is given by
\[\begin{align} P_{tot} &=(n_1+n_2+n_3+ \; \cdots +n_n)\left(\dfrac{RT}{V}\right)\label{10.6.2a} \\[4pt] &=\sum_{i=1}^n{n_i}\left(\dfrac{RT}{V}\right)\label{10.6.2b} \end{align} \]
Equation \(\ref{10.6.2b}\) restates Equation \(\ref{10.6.3}\) in a more general form and makes it explicitly clear that, at constant temperature and volume, the pressure exerted by a gas depends on only the total number of moles of gas present, whether the gas is a single chemical species or a mixture of dozens or even hundreds of gaseous species. For Equation \(\ref{10.6.2b}\) to be valid, the identity of the particles present cannot have an effect. Thus an ideal gas must be one whose properties are not affected by either the size of the particles or their intermolecular interactions because both will vary from one gas to another. The calculation of total and partial pressures for mixtures of gases is illustrated in Example \(\PageIndex{1}\).
Example \(\PageIndex{1}\): The Bends
Deep-sea divers must use special gas mixtures in their tanks, rather than compressed air, to avoid serious problems, most notably a condition called “the bends.” At depths of about 350 ft, divers are subject to a pressure of approximately 10 atm. A typical gas cylinder used for such depths contains 51.2 g of \(O_2\) and 326.4 g of He and has a volume of 10.0 L. What is the partial pressure of each gas at 20.00°C, and what is the total pressure in the cylinder at this temperature?
Given: masses of components, total volume, and temperature
Asked for: partial pressures and total pressure
Strategy:
- Calculate the number of moles of \(\ce{He}\) and \(\ce{O_2}\) present.
- Use the ideal gas law to calculate the partial pressure of each gas. Then add together the partial pressures to obtain the total pressure of the gaseous mixture.
Solution:
A The number of moles of \(\ce{He}\) is
\[n_{\rm He}=\rm\dfrac{326.4\;g}{4.003\;g/mol}=81.54\;mol \nonumber \]
The number of moles of \(\ce{O_2}\) is
\[n_{\rm O_2}=\rm \dfrac{51.2\;g}{32.00\;g/mol}=1.60\;mol \nonumber \]
B We can now use the ideal gas law to calculate the partial pressure of each:
\[P_{\rm He}=\dfrac{n_{\rm He}RT}{V}=\rm\dfrac{81.54\;mol\times0.08206\;\dfrac{atm\cdot L}{mol\cdot K}\times293.15\;K}{10.0\;L}=196.2\;atm \nonumber \]
\[P_{\rm O_2}=\dfrac{n_{\rm O_2} RT}{V}=\rm\dfrac{1.60\;mol\times0.08206\;\dfrac{atm\cdot L}{mol\cdot K}\times293.15\;K}{10.0\;L}=3.85\;atm \nonumber \]
The total pressure is the sum of the two partial pressures:
\[P_{\rm tot}=P_{\rm He}+P_{\rm O_2}=\rm(196.2+3.85)\;atm=200.1\;atm \nonumber \]
Exercise \(\PageIndex{1}\)
A cylinder of compressed natural gas has a volume of 20.0 L and contains 1813 g of methane and 336 g of ethane. Calculate the partial pressure of each gas at 22.0°C and the total pressure in the cylinder.
- Answer
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\(P_{CH_4}=137 \; atm\); \(P_{C_2H_6}=13.4\; atm\); \(P_{tot}=151\; atm\)
Mole Fractions of Gas Mixtures
The composition of a gas mixture can be described by the mole fractions of the gases present. The mole fraction (\(\chi\)) of any component of a mixture is the ratio of the number of moles of that component to the total number of moles of all the species present in the mixture (\(n_{tot}\)):
\[\chi_A=\dfrac{\text{moles of A}}{\text{total moles}}= \dfrac{n_A}{n_{tot}} =\dfrac{n_A}{n_A+n_B+\cdots}\label{10.6.5} \]
The mole fraction is a dimensionless quantity between 0 and 1. If \(\chi_A = 1.0\), then the sample is pure \(A\), not a mixture. If \(\chi_A = 0\), then no \(A\) is present in the mixture. The sum of the mole fractions of all the components present must equal 1.
To see how mole fractions can help us understand the properties of gas mixtures, let’s evaluate the ratio of the pressure of a gas \(A\) to the total pressure of a gas mixture that contains \(A\). We can use the ideal gas law to describe the pressures of both gas \(A\) and the mixture: \(P_A = n_ART/V\) and \(P_{tot} = n_tRT/V\). The ratio of the two is thus
\[\dfrac{P_A}{P_{tot}}=\dfrac{n_ART/V}{n_{tot}RT/V} = \dfrac{n_A}{n_{tot}}=\chi_A \label{10.6.6} \]
Rearranging this equation gives
\[P_A = \chi_AP_{tot} \label{10.6.7} \]
That is, the partial pressure of any gas in a mixture is the total pressure multiplied by the mole fraction of that gas. This conclusion is a direct result of the ideal gas law, which assumes that all gas particles behave ideally. Consequently, the pressure of a gas in a mixture depends on only the percentage of particles in the mixture that are of that type, not their specific physical or chemical properties. By volume, Earth’s atmosphere is about 78% \(N_2\), 21% \(O_2\), and 0.9% \(Ar\), with trace amounts of gases such as \(CO_2\), \(H_2O\), and others. This means that 78% of the particles present in the atmosphere are \(N_2\); hence the mole fraction of \(N_2\) is 78%/100% = 0.78. Similarly, the mole fractions of \(O_2\) and \(Ar\) are 0.21 and 0.009, respectively. Using Equation \ref{10.6.7}, we therefore know that the partial pressure of N 2 is 0.78 atm (assuming an atmospheric pressure of exactly 760 mmHg) and, similarly, the partial pressures of \(O_2\) and \(Ar\) are 0.21 and 0.009 atm, respectively.
Example \(\PageIndex{2}\): Exhaling Composition
We have just calculated the partial pressures of the major gases in the air we inhale. Experiments that measure the composition of the air we exhale yield different results, however. The following table gives the measured pressures of the major gases in both inhaled and exhaled air. Calculate the mole fractions of the gases in exhaled air.
| Inhaled Air / mmHg | Exhaled Air / mmHg | |
|---|---|---|
| \(P_{\rm N_2}\) | 597 | 568 |
| \(P_{\rm O_2}\) | 158 | 116 |
| \(P_{\rm H_2O}\) | 0.3 | 28 |
| \(P_{\rm CO_2}\) | 5 | 48 |
| \(P_{\rm Ar}\) | 8 | 8 |
| \(P_{tot}\) | 767 | 767 |
Given: pressures of gases in inhaled and exhaled air
Asked for: mole fractions of gases in exhaled air
Strategy:
Calculate the mole fraction of each gas using Equation \(\ref{10.6.7}\).
Solution:
The mole fraction of any gas \(A\) is given by
\[\chi_A=\dfrac{P_A}{P_{tot}} \nonumber \]
where \(P_A\) is the partial pressure of \(A\) and \(P_{tot}\) is the total pressure. For example, the mole fraction of \(\ce{CO_2}\) is given as:
\[\chi_{\rm CO_2}=\rm\dfrac{48\;mmHg}{767\;mmHg}=0.063 \nonumber \]
The following table gives the values of \(\chi_A\) for the gases in the exhaled air.
| Gas | Mole Fraction |
|---|---|
| \({\rm N_2}\) | 0.741 |
| \({\rm O_2}\) | 0.151 |
| \({\rm H_2O}\) | 0.037 |
| \({\rm CO_2}\) | 0.063 |
| \({\rm Ar}\) | 0.010 |
Exercise \(\PageIndex{2}\)
Venus is an inhospitable place, with a surface temperature of 560°C and a surface pressure of 90 atm. The atmosphere consists of about 96% CO 2 and 3% N 2 , with trace amounts of other gases, including water, sulfur dioxide, and sulfuric acid. Calculate the partial pressures of CO 2 and N 2 .
- Answer
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\[P_{\rm CO_2}=\rm86\; atm \nonumber \]
\[P_{\rm N_2}=\rm2.7\;atm \nonumber \]
Summary
The partial pressure of each gas in a mixture is proportional to its mole fraction. The pressure exerted by each gas in a gas mixture (its partial pressure ) is independent of the pressure exerted by all other gases present. Consequently, the total pressure exerted by a mixture of gases is the sum of the partial pressures of the components ( Dalton’s law of partial pressures ). The amount of gas present in a mixture may be described by its partial pressure or its mole fraction. The mole fraction of any component of a mixture is the ratio of the number of moles of that substance to the total number of moles of all substances present. In a mixture of gases, the partial pressure of each gas is the product of the total pressure and the mole fraction of that gas.


19.4: Change Reflected in Thought and Writing
In the late nineteenth century, Americans were living in a world characterized by rapid change. Western expansion, dramatic new technologies, and the rise of big business drastically influenced society in a matter of a few decades. For those living in the fast-growing urban areas, the pace of change was even faster and harder to ignore. One result of this time of transformation was the emergence of a series of notable authors, who, whether writing fiction or nonfiction, offered a lens through which to better understand the shifts in American society.
UNDERSTANDING SOCIAL PROGRESS
One key idea of the nineteenth century that moved from the realm of science to the murkier ground of social and economic success was Charles Darwin’s theory of evolution. Darwin was a British naturalist who, in his 1859 work On the Origin of Species , made the case that species develop and evolve through natural selection, not through divine intervention. The idea quickly drew fire from the Anglican Church (although a liberal branch of Anglicans embraced the notion of natural selection being part of God’s plan) and later from many others, both in England and abroad, who felt that the theory directly contradicted the role of God in the earth’s creation. Although biologists, botanists, and most of the scientific establishment widely accepted the theory of evolution at the time of Darwin’s publication, which they felt synthesized much of the previous work in the field, the theory remained controversial in the public realm for decades.
Political philosopher Herbert Spencer took Darwin’s theory of evolution further, coining the actual phrase “survival of the fittest,” and later helping to popularize the phrase social Darwinism to posit that society evolved much like a natural organism, wherein some individuals will succeed due to racially and ethnically inherent traits, and their ability to adapt. This model allowed that a collection of traits and skills, which could include intelligence, inherited wealth, and so on, mixed with the ability to adapt, would let all Americans rise or fall of their own accord, so long as the road to success was accessible to all. William Graham Sumner, a sociologist at Yale, became the most vocal proponent of social Darwinism. Not surprisingly, this ideology, which Darwin himself would have rejected as a gross misreading of his scientific discoveries, drew great praise from those who made their wealth at this time. They saw their success as proof of biological fitness, although critics of this theory were quick to point out that those who did not succeed often did not have the same opportunities or equal playing field that the ideology of social Darwinism purported. Eventually, the concept fell into disrepute in the 1930s and 1940s, as eugenicists began to utilize it in conjunction with their racial theories of genetic superiority.
Other thinkers of the day took Charles Darwin’s theories in a more nuanced direction, focusing on different theories of realism that sought to understand the truth underlying the changes in the United States. These thinkers believed that ideas and social constructs must be proven to work before they could be accepted. Philosopher William James was one of the key proponents of the closely related concept of pragmatism, which held that Americans needed to experiment with different ideas and perspectives to find the truth about American society, rather than assuming that there was truth in old, previously accepted models. Only by tying ideas, thoughts, and statements to actual objects and occurrences could one begin to identify a coherent truth, according to James. His work strongly influenced the subsequent avant-garde and modernist movements in literature and art, especially in understanding the role of the observer, artist, or writer in shaping the society they attempted to observe. John Dewey built on the idea of pragmatism to create a theory of instrumentalism, which advocated the use of education in the search for truth. Dewey believed that education, specifically observation and change through the scientific method, was the best tool by which to reform and improve American society as it continued to grow ever more complex. To that end, Dewey strongly encouraged educational reforms designed to create an informed American citizenry that could then form the basis for other, much-needed progressive reforms in society.
In addition to the new medium of photography, popularized by Riis, novelists and other artists also embraced realism in their work. They sought to portray vignettes from real life in their stories, partly in response to the more sentimental works of their predecessors. Visual artists such as George Bellows, Edward Hopper, and Robert Henri, among others, formed the Ashcan School of Art, which was interested primarily in depicting the urban lifestyle that was quickly gripping the United States at the turn of the century. Their works typically focused on working-class city life, including the slums and tenement houses, as well as working-class forms of leisure and entertainment (Figure 19.4.1).
Novelists and journalists also popularized realism in literary works. Authors such as Stephen Crane, who wrote stark stories about life in the slums or during the Civil War, and Rebecca Harding Davis, who in 1861 published Life in the Iron Mills , embodied this popular style. Mark Twain also sought realism in his books, whether it was the reality of the pioneer spirit, seen in The Adventures of Huckleberry Finn , published in 1884, or the issue of corruption in The Gilded Age , co-authored with Charles Dudley Warner in 1873. The narratives and visual arts of these realists could nonetheless be highly stylized, crafted, and even fabricated, since their goal was the effective portrayal of social realities they thought required reform. Some authors, such as Jack London, who wrote Call of the Wild , embraced a school of thought called naturalism, which concluded that the laws of nature and the natural world were the only truly relevant laws governing humanity (Figure 19.4.2).
Kate Chopin, widely regarded as the foremost woman short story writer and novelist of her day, sought to portray a realistic view of women’s lives in late nineteenth-century America, thus paving the way for more explicit feminist literature in generations to come. Although Chopin never described herself as a feminist per se, her reflective works on her experiences as a southern woman introduced a form of creative nonfiction that captured the struggles of women in the United States through their own individual experiences. She also was among the first authors to openly address the race issue of miscegenation. In her work Desiree’s Baby , Chopin specifically explores the Creole community of her native Louisiana in depths that exposed the reality of racism in a manner seldom seen in literature of the time.
African American poet, playwright, and novelist of the realist period, Paul Laurence Dunbar dealt with issues of race at a time when most reform-minded Americans preferred to focus on other issues. Through his combination of writing in both standard English and black dialect, Dunbar delighted readers with his rich portrayals of the successes and struggles associated with African American life. Although he initially struggled to find the patronage and financial support required to develop a full-time literary career, Dunbar’s subsequent professional relationship with literary critic and Atlantic Monthly editor William Dean Howells helped to firmly cement his literary credentials as the foremost African American writer of his generation. As with Chopin and Harding, Dunbar’s writing highlighted parts of the American experience that were not well understood by the dominant demographic of the country. In their work, these authors provided readers with insights into a world that was not necessarily familiar to them and also gave hidden communities—be it iron mill workers, southern women, or African American men—a sense of voice.
Mark Twain’s lampoon of author Horatio Algerdemonstrates Twain’s commitment to realism by mocking the myth set out by Alger, whose stories followed a common theme in which a poor but honest boy goes from rags to riches through a combination of “luck and pluck.” See how Twain twists Alger’s hugely popular storyline in this piece of satire.
Author Kate Chopin grew up in the American South and later moved to St. Louis, where she began writing stories to make a living after the death of her husband. She published her works throughout the late 1890s, with stories appearing in literary magazines and local papers. It was her second novel, The Awakening , which gained her notoriety and criticism in her lifetime, and ongoing literary fame after her death (Figure 19.4.3).
The Awakening , set in the New Orleans society that Chopin knew well, tells the story of a woman struggling with the constraints of marriage who ultimately seeks her own fulfillment over the needs of her family. The book deals far more openly than most novels of the day with questions of women’s sexual desires. It also flouted nineteenth-century conventions by looking at the protagonist’s struggles with the traditional role expected of women.
While a few contemporary reviewers saw merit in the book, most criticized it as immoral and unseemly. It was censored, called “pure poison,” and critics railed against Chopin herself. While Chopin wrote squarely in the tradition of realism that was popular at this time, her work covered ground that was considered “too real” for comfort. After the negative reception of the novel, Chopin retreated from public life and discontinued writing. She died five years after its publication. After her death, Chopin’s work was largely ignored, until scholars rediscovered it in the late twentieth century, and her books and stories came back into print. The Awakening in particular has been recognized as vital to the earliest edges of the modern feminist movement.
Excerpts from interviews with David Chopin, Kate Chopin’s grandson, and a scholar who studies her work provide interesting perspectives on the author and her views.
CRITICS OF MODERN AMERICA
While many Americans at this time, both everyday working people and theorists, felt the changes of the era would lead to improvements and opportunities, there were critics of the emerging social shifts as well. Although less popular than Twain and London, authors such as Edward Bellamy, Henry George, and Thorstein Veblen were also influential in spreading critiques of the industrial age. While their critiques were quite distinct from each other, all three believed that the industrial age was a step in the wrong direction for the country.
In the 1888 novel Looking Backward, 2000-1887 , Edward Bellamy portrays a utopian America in the year 2000, with the country living in peace and harmony after abandoning the capitalist model and moving to a socialist state. In the book, Bellamy predicts the future advent of credit cards, cable entertainment, and “super-store” cooperatives that resemble a modern day Wal-Mart. Looking Backward proved to be a popular bestseller (third only to Uncle Tom’s Cabin and Ben Hur among late nineteenth-century publications) and appealed to those who felt the industrial age of big business was sending the country in the wrong direction. Eugene Debs, who led the national Pullman Railroad Strike in 1894, later commented on how Bellamy’s work influenced him to adopt socialism as the answer to the exploitative industrial capitalist model. In addition, Bellamy’s work spurred the publication of no fewer than thirty-six additional books or articles by other writers, either supporting Bellamy’s outlook or directly criticizing it. In 1897, Bellamy felt compelled to publish a sequel, entitled Equality , in which he further explained ideas he had previously introduced concerning educational reform and women’s equality, as well as a world of vegetarians who speak a universal language.
Another author whose work illustrated the criticisms of the day was nonfiction writer Henry George, an economist best known for his 1879 work Progress and Poverty , which criticized the inequality found in an industrial economy. He suggested that, while people should own that which they create, all land and natural resources should belong to all equally, and should be taxed through a “single land tax” in order to disincentivize private land ownership. His thoughts influenced many economic progressive reformers, as well as led directly to the creation of the now-popular board game, Monopoly.
Another critique of late nineteenth-century American capitalism was Thorstein Veblen, who lamented in The Theory of the Leisure Class (1899) that capitalism created a middle class more preoccupied with its own comfort and consumption than with maximizing production. In coining the phrase “conspicuous consumption,” Veblen identified the means by which one class of nonproducers exploited the working class that produced the goods for their consumption. Such practices, including the creation of business trusts, served only to create a greater divide between the haves and have-nots in American society, and resulted in economic inefficiencies that required correction or reform.
Section Summary
Americans were overwhelmed by the rapid pace and scale of change at the close of the nineteenth century. Authors and thinkers tried to assess the meaning of the country’s seismic shifts in culture and society through their work. Fiction writers often used realism in an attempt to paint an accurate portrait of how people were living at the time. Proponents of economic developments and cultural changes cited social Darwinism as an acceptable model to explain why some people succeeded and others failed, whereas other philosophers looked more closely at Darwin’s work and sought to apply a model of proof and pragmatism to all ideas and institutions. Other sociologists and philosophers criticized the changes of the era, citing the inequities found in the new industrial economy and its negative effects on workers.
Review Questions
Which of the following statements accurately represents Thorstein Veblen’s argument in The Theory of the Leisure Class ?
All citizens of an industrial society would rise or fall based on their own innate merits.
The tenets of naturalism were the only laws through which society should be governed.
The middle class was overly focused on its own comfort and consumption.
Land and natural resources should belong equally to all citizens.
C
Which of the following was not an element of realism?
social Darwinism
instrumentalism
naturalism
pragmatism
A
In what ways did writers, photographers, and visual artists begin to embrace more realistic subjects in their work? How were these responses to the advent of the industrial age and the rise of cities?
The growth of the industrial economy and the dramatic growth of cities created new, harsh realities that were often hidden from the public eye. Writers and artists, responding both to this fact and to the sentimentalism that characterized the writing and art of their predecessors, began to depict subjects that reflected the new truth. Photographers like Jacob Riis sought to present to the public the realities of working-class life and labor. Novelists began to portray true-to-life vignettes in their stories. Visual artists such as George Bellows, Edward Hopper, and Robert Henri formed the Ashcan School of Art, which depicted the often gritty realities of working-class city life, leisure, and entertainment.
Critical Thinking Questions
What triumphs did the late nineteenth century witness in the realms of industrial growth, urbanization, and technological innovation? What challenges did these developments pose for urban dwellers, workers, and recent immigrants? How did city officials and everyday citizens respond to these challenges?
What were the effects of urbanization on the working, middle, and elite classes of American society? Conversely, how did the different social classes and their activities change the scope, character, and use of urban spaces?
How do you think that different classes of city dwellers would have viewed the City Beautiful movement? What potential benefits and drawbacks of this new direction in urban planning might members of each class have cited?
How was Darwin’s work on the evolution of species exploited by proponents of the industrial age? Why might they have latched on to this idea in particular?
Historians often mine the arts for clues to the social, cultural, political, and intellectual shifts that characterized a given era. How do the many works of visual art, literature, and social philosophy that emerged from this period reflect the massive changes that were taking place? How were Americans—both those who created these works and those who read or viewed them—struggling to understand the new reality through art, literature, and scholarship?
Glossary
- instrumentalism
- a theory promoted by John Dewey, who believed that education was key to the search for the truth about ideals and institutions
- naturalism
- a theory of realism that states that the laws of nature and the natural world were the only relevant laws governing humanity
- pragmatism
- a doctrine supported by philosopher William James, which held that Americans needed to experiment and find the truth behind underlying institutions, religions, and ideas in American life, rather than accepting them on faith
- realism
- a collection of theories and ideas that sought to understand the underlying changes in the United States during the late nineteenth century


9.7: Evaluating Data as Sources
Evaluating data for relevance and credibility is just as important as evaluating any other source. As with other information sources with data there is never a 100% perfect source. You’ll have to make educated guesses (inferences) about whether the data are good enough for your purpose.
To evaluate data, you’ll need to find out how the data were collected. If the data are in another source, such as a book; web page; or newspaper, magazine, or research journal article, evaluate that source in the usual way. If the book or newspaper, magazine, or web page got the data from somewhere else, do the same evaluation of the source from which the book or article got the data. The article, book, or web page should cite where the data came from. If it doesn’t, then that is a black mark against using that data. The data in a research journal article are often the work of the authors of the article. But you’ll want to be sure they provide information about how they collected the data.
If the data are in a research journal article, read the entire article, including the section called Methodology, which tells how the data were collected. Then determine the data’s relevance to your research question by considering such questions as:
- Was the data collected recently enough?
- Is the data cross-sectional (based on information from people at any one time) or longitudinal (based on information from the same people over time)? If one is more appropriate for your research question than the other, is there information that you can still logically infer from this data?
- Were the types of people from whom the data was collected the same type of people your research question addresses? The more representative the study’s sample is of the group your research question addresses, the more confident you can be in using the data to make your argument in your final product.
- Was the data analysis done at the right level for your research question? For instance, it may have been done at the individual, family, business, state, or zip code level. But if that doesn’t relate to your research question, can you still logically make inferences that will help your argument? Here’s an example: Imagine that your research question asks whether participation in high school sports in Minneapolis schools is positively associated with enrolling in college. But the data you are evaluating is analyzed at the state level. So you have data about the whole state of Minnesota’s schools and not Minneapolis in particular. In this case, ask yourself whether there is still an inference you can make from the data.
To evaluate the credibility of the data in a research journal article you have already read, take the steps recommended in Evaluating Sources, plus consider these questions:
- Is the article in a peer-reviewed journal? Look at the journal’s instructions for authors, which are often located on the journal’s website, to see if it talks about peers reviewing the article and asking for changes (revisions) before publishing. If it is a peer-reviewed journal, consider that a plus for the article’s credibility. Being peer-reviewed doesn’t mean it’s perfect; just more likely to be credible.
- Do the authors discuss causation or correlation? Be wary of claims of causation; it is very difficult to determine a causal effect. While research studies often find relationships (correlation) between various variables in the data, this does not equal causation. For instance, let’s return to our example above: If the study of Minneapolis high school students’ sports participation showed a positive correlation between sports participation and college enrollment, the researcher cannot say that participation caused college enrollment. If it were designed to show cause and effect, the study would not have resulted in a correlation. Instead, it would have had to have been designed as an experiment or quasi-experiment, used different statistical analyses, and would have supported or not supported its hypotheses.


4.5: Spectra
Spectra are very important to astronomy because they can show the component wavelengths and specific elements present.
Continuous Spectrum
A continuous spectrum shows all wavelengths and is produced by hot, dense solid objects at high temperatures.
Emission Line Spectrum
An Emission Line spectrum produces a spectrum with bright lines . It occurs from heating an object, causing the electrons in the object’s atoms/molecules to show at a specific wavelength.
Absorption Line Spectrum
An Absorption Line spectrum produces a spectrum with dark lines . It occurs when one object absorbs specific photons given off by a second object. For example, a light behind a cloud of gas absorbs photons at the cloud’s energy level while the rest of the photons pass through the cloud.


9.2.1: A Look Into Transcendentalism (Group Anthology Contribution 2019)
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- Robin DeRosa, Abby Goode et al.
- Plymouth State University/
In the mid-nineteenth century, a new literary and philosophical movement called Transcendentalism was born. It promoted the idea of living completely by one’s own motives and reality, ultimately ignoring those ideas held by others. This arose from the period before it known as Intellectualism which focused on the practice and exercise of the mind to gain reason.
Transcendentalism began in New England and is largely attributed to two prominent writers—Ralph Waldo Emerson and Henry David Thoreau (Khan). Other highly esteemed writers like Nathaniel Hawthorne, Henry Wadsworth Longfellow, Walt Whitman, and Margaret Fuller were also well-known transcendentalist writers. Popular writings of the movement include Walden by Henry David Thoreau, Leaves of Grass by Walt Whitman, or “Self Reliance” by Ralph Waldo Emerson. In addition to the individual work of each, many of the writers were connected through a club known as The Transcendental Club based in Cambridge, Massachusetts on the Harvard University campus. Here they compiled and spread the various tenets of Transcendentalism through a periodical titled The Dial (ushistory).
Transcendentalism celebrates the individual over society, believing that every person contains knowledge that transcends the physical realm. As such, transcendental thinkers encouraged people to form their own truths independent from those imposed on them by the church, society, and other scholars. The movement marked a shift back to nature as transcendentalists found simplicity and spirituality away from the pressures of civilization.
Ralph Waldo Emerson, author of “Self-Reliance” and other notable poems and essays, is widely regarded as the founder of the transcendental movement. Thus, he is responsible for many key shifts in the movement from start to finish, beginning when he left the Unitarian ministry in 1932 to become an essayist and orator, the event that launched the transcendental movement.
Four years later, Emerson anonymously published his groundbreaking book Nature , which many consider the defining text on Transcendentalism. Nature propelled Transcendentalism from its origins as a religious reform movement to one with philosophical, theological, political, and literary relevance.
Emerson contributed many important transcendental pieces, such as “The American Scholar,” his controversial Harvard Divinity School address, and some poetry. Perhaps his most famous text, “Self-Reliance” serves alongside Nature as a foundational piece in the transcendental movement (Richardson).
Published in 1841, “Self-Reliance” exemplifies a transcendental text and reinforces the elements central to the movement, such as nonconformity, free thought, and nature over society. Notions of God are intertwined with those of nature and man throughout Emerson’s essay, hinting at the earlier Unitarian reform teachings that served as the springboard for Transcendentalism as society’s relationship with God became increasingly individual. “Self-Reliance” urges people to adopt an individualistic lifestyle, where the ideas of the church and of known intellectuals do not influence one’s own beliefs. This essay is Emerson’s rallying cry to convince the broader public to adopt transcendental principles and live according to his principles of self-sufficiency and nonconformity (Carbone II).
The end of the practice of Transcendentalism might have been signaled by the decline of Brook Farm between 1844 and 1847. Brook Farm was the commune inhabited by some members of the Transcendental Club. Emerson himself chose not to live there, but visited occasionally . The decline was the result of members not having a clear vision of the views of the farm’s mission, fighting amongst members, and financial problems. The 1850s is when Transcendentalism lost its influence, which is likely due to the death of fundamental member Margaret Fuller. (history.com). Even though other figures of the practice continued to write and publish, Transcendentalists were no longer a functioning group.
Carbone II, Steven A. American Transcendentalism and Analysis of Emerson’s “Self-Reliance.” Inquiries Journal. http://www.inquiriesjournal.com/articles/1663/american-transcendentalism-and-analysis-of-ralph-waldo-emersons-self-reliance . Wednesday, October 2, 2019.
Getchell, Michelle. Transcendentalism. Khan Academy. https://www.khanacademy.org/humanities/us-history/the-early-republic/culture-and-reform/a/transcendentalism Tuesday, October 1, 2019.
History.com Editors. “Transcendentalism.” History.com , A&E Television Networks, 15 Nov. 2017, https://www.history.com/topics/19th-century/transcendentalism .
Richardson, Jr. Robert D. The Transcendetal Club . American Transcendentalism Web. https://archive.vcu.edu/english/engweb/transcendentalism/ideas/club.html . Wednesday October 2, 2019.
ushistory. Transcendentalism, An American Philosophy. U.S History Online Textbook. http://www.ushistory.org/us/26f.asp . Wednesday, October 2, 2019.


4.8: CRAAP Analysis
Describe the components of the CRAAP analysis process
There are several tools available when evaluating sources. Perhaps the most common and well known is the (well-titled) CRAAP Analysis. The University of Santa Cruz library has a full breakdown of the CRAAP method . Below is a summarized form; CRAAP stands for:
- Currency: How current is the source?
- Reliability: How important is the information, and has it been consistently presented?
- Authority: What is the source of the information?
- Accuracy: Judged against other sources (which themselves will need appropriate evaluation), how correct is the source?
- Purpose: What is the goal of the source—why was it created?
Lastly, Flinders University has an excellent commentary below on the use of Wikipedia. Depending on your organizational culture, Wikipedia might be sufficient as a source. This is hotly debated, but it all depends on the purpose of your writing. Not all of your business reports need to be lengthy and high caliber items supported by scholarly sources; you might find yourself writing a “quick” report within a day or two where your boss indicates that Wikipedia or some other introductory website (and encyclopedia maybe) is sufficient. It is important to point out that the report, while important, is not the end, it simply a means to making better decisions. Direction from your organization’s decision-makers is key. Flinder’s comments: [1]
A note on Wikipedia. While you certainly would not cite a Wikipedia article as a credible source in your essay, Wikipedia can be a great place to start to get an understanding of a topic. It may also lead you to relevant, high quality resources. Try looking at the references for a Wikipedia page (at the bottom of each Wikipedia page) and assess the quality of the references you find. Many Wikipedia entries will cite scholarly resources (including books and journal articles) in their references, which if appropriate, can then be cited in your essay.
If we return to Martha’s project, we can imagine her running each secondary source through a CRAAP analysis. Her sources are likely to be a mix of books, magazine articles, videos and other media. When considering what might make for a good source for Martha, we might use a reverse version the CRAAP analysis to illustrate her analysis:
Currency
How current is the source?
Martha is not likely to use anything older than 3 to 5 years. There is no hard-and-fast rule here, but civic politics, dynamic life, technology and other features of human experience are likely to render anything too much older than 5 years less helpful.
Reliability
How important is the information, and has it been consistently presented?
Martha is likely to read or watch materials that are professionally put together and have a linked look and feel to the material.
Authority
What is the source of the information?
Martha is likely to look at sources that have a reputation of doing work in the homeless community or have done good work in other similar areas of social work.
Accuracy
Judged against other sources (which themselves will need appropriate evaluation), how correct is the source?
The bottom line here is whether the material has been reviewed by other experts. For scholarly work, we refer to this as “peer-reviewed.” Clearly not all of Martha’s sources need to be peer-reviewed; however, the more of her sources that are, the better her information will be. Flinders University recommends asking the following questions: [2]
- Is it scholarly?
- Is the information supported by evidence?
- Is that evidence referenced by the source?
- Has the content been peer-reviewed or edited by a publisher?
- Can the information be verified by other literature on the same topic?
- Is the tone objective and impartial?
- Is it free from obvious errors such as spelling or grammar?
- Is it written by a scholar with expertise in the field?
Purpose
What is the goal of the source—why was it created?
This is where intention and bias are more clear. For professional scholarly work, you will often see a note at the end of the document indicating any funding or entities that supported the work. This is there to inform the reader of external influences on the material. A professional author will work to limit his or her bias, or they will use an alternative technique, which is to discuss their bias in their work, and make their agenda clear to the reader.
- Flinders University. "Evaluating your source." Web. flinders.libguides.com/evaluate. 18 June 2018.
- Ibid.


2.1: What is a Theory?
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- Erika Gutierrez, Janét Hund, Shaheen Johnson, Carlos Ramos, Lisette Rodriguez, & Joy Tsuhako
- Long Beach City College, Cerritos College, & Saddleback College via ASCCC Open Educational Resources Initiative (OERI)
Sociologists study social events, interactions, and patterns, and they develop a theory in an attempt to explain why things work as they do. In sociology, a theory is a way to explain different aspects of social interactions and social structures as well as to create a testable proposition, called a hypothesis, about society (Allan, 2006).
For example, although suicide is generally considered an individual phenomenon, Émile Durkheim was interested in studying the social factors that affect it. He studied social ties within a group, or social solidarity , and hypothesized that differences in suicide rates might be explained by religion-based differences. Durkheim gathered a large amount of data about Europeans who had ended their lives, and he did indeed find differences based on religion. Protestants were more likely to commit suicide than Catholics in Durkheim’s society, and his work supports the utility of theory in sociological research.
Theories vary in scope depending on the scale of the issues that they are meant to explain. Macro-level theories relate to large-scale issues and large groups of people, while micro-level theories look at very specific relationships between individuals or small groups. Grand theories attempt to explain large-scale relationships and answer fundamental questions such as why societies form and why they change. Sociological theory is constantly evolving and should never be considered complete. Classic sociological theories are still considered important and current, but new sociological theories build upon the work of their predecessors and add to them (Calhoun, 2002).
In sociology, a few theories provide broad perspectives that help explain many different aspects of social life, and these are called paradigms. Paradigms are philosophical and theoretical frameworks used within a discipline to formulate theories, generalizations, and the experiments performed in support of them. Students are encouraged to shift their paradigms by considering a variety of perspectives and content covered in this textbook. To test your ability to shift paradigms, what do you see in this picture? When was the picture taken, and where?
 Taken in 2001, this photo is of a residence in Taiwan; the swastika on the front door communicates it is a Buddhist household, welcoming visitors. Throughout Asia and Indigenous societies worldwide, the swastika symbolizes peace, happiness, love, and long life - not hatred, anti-semitism, racism, or violence which many associate with the swastika symbol, due to the atrocity of the Holocaust during German-occupied Europe from 1941-1945. Considering different paradigms is important to understand our vast human history and contemporary society.
Three paradigms have come to dominate sociological thinking, because they provide useful explanations: structural functionalism, conflict theory, and symbolic interactionism. These are discussed further in the next section as well as two more recent theoretical contributions: intersectionality and critical race theory.
Contributors and Attributions
- Ramos, Carlos. (Long Beach City College)
- Gutierrez, Erika. (Santiago Canyon College)
- Introduction to Sociology 2e (OpenStax) ( CC BY 4.0 )
Works Cited
- Allan, K. (2006). Contemporary Social and Sociological Theory: Visualizing Social Worlds . Thousand Oaks, CA: Pine Forge Press.
- Calhoun, C. (2002). Contemporary Sociological Theory . Hoboken, NJ: Wiley-Blackwell.


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1.1: Playwriting
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Playwriting is a terrifying and exhilarating endeavor and it is the basis of dramatic art. The audience came to hear a story, escape from their lives and problems, and be transformed by experiencing the struggles of another individual. The playwright is the creator of worlds, deity of conflict, and messenger of the essential truths human beings need to hear over and over throughout their lives. However, like with all things, with great power comes great responsibility.
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1.2: The Steps of the Writing Process
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An overview of the 15 Steps involved in writing a play.
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1.3: The Idea Store
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Some writers are idea factories and can look almost anywhere and find a new twist on a story or situation that is worth watching, but what do you do when you cannot find an idea? That is easy, you go to the Idea Store.  There will always be times where you are at a loss of ideas and so the Idea Store was created to help you get started on a concept for your story.
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1.4: Research
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Your level research is dependant upon the story you are telling. Some projects like biographies or historical dramas require vast amounts of research, while others require very little because you are creating an original world. You also can choose where to add research into your process, for some projects it may be in the beginning as you are gathering ideas and other times it may be once you have developed characters and have identified a setting and conflict.
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1.5: Theme
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A theme is an idea, a simple, clear, and powerful idea.  The more powerful and life changing the lesson, the more sacred the story. Human beings are incredible entities that extract meaning from stories that are not true. It is this ability that has allowed the arts and creativity to thrive.
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1.6: Conflict
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Conflict is the primary ingredient of drama and without it drama ceases to exist. Conflict is when two opposing forces collide. David and Goliath, Harry Potter and Voldemort, You and Your Future are all examples of opposing forces and these forces meet within the battlefield of story. Stories were invented to help us make sense of the world and find our place in it, and the world is full of conflict.
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1.7: Types of Story
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One element of the book that broadened my perspective of storytelling was his section on the 10 different types of stories. Once his observation was pointed out I started to see how each type of conflict could fit neatly in each story type and I feel that this knowledge is incredibly helpful to anyone entering the art of storytelling. The following is my understanding of Blake Snyder’s information but I encourage you to read the book for a deeper analysis.
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1.8: Setting
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The setting is the environment in which the conflict can take place. The conflict and setting will also contribute to the action (or things that happen) on stage. You can set your play in any location you choose and as many locations as you would like. You can have a single location or multiple locations however, there are pros and cons to each.
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1.9: Character and Story Archetypes
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There are many character and story archetypes yet each story has 5 essential archetypes. In some cases one character can be the vessel for several of the following types, yet these archetypes are always present in stories. The following are an amalgamation of many theories, yet the more you study story structure you will notice how each theory and perspective has many commonalities with its counterparts. The following is my attempt to combine them.
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1.10: Plot
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Plot is the series of events that are structured to take the hero on a transformative journey throughout the story. Stories have been around for centuries and throughout that time the structure of storytelling has become solidified and with that there are essential components or rules you must follow. There are always attempts to break the form and some are done quite well, but if you just follow the structure and deliver a strong and well crafted story your audience will be receptive and happy.
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1.11: Dialogue
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Dialogue is the primary tool used for storytelling in theater. Film uses imagery, novels use descriptions, comics use pictures, theater uses dialogue. As a playwright the audience will not read, hear, or see your stage directions, but they will hear every word of your dialogue. Dialogue and action are how you drive the plot, develop characters relationships, and tell your story. Dialogue is also used to reveal characters inner emotions, their past experiences, or vulnerability and frustrations.
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1.12: Creating Characters
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Creating original characters is one of the most fun and rewarding experiences you will have as an artist. You are bringing something into being through your imagination and audience members will live and see the world through this character’s eyes and experiences and often this fictional character will make a lasting impact on actual people’s lives.
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1.13: Scene
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A scene is a character’s attempt to solve a problem. At the beginning of your story you introduce the world of the play and the characters who inhabit it. You then need to establish a normal day for the characters in the world. At the end of the first scene you will need to introduce the major conflict of the play. Every additional scene is a character attempting to solve the problem or conflict and by failing to do so making their plight worse.
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1.14: Crafting the Plot
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Every story you tell will be creating a world where you present a problem and over the course of the story you solve it. In the process of solving the problem the hero is transformed and through the character’s journey the audience learns an important essential truth. However, in order to tell this story you need to first chart the course of events that will take place so that you can maximize the impact of your narrative.
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1.15: The Process
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Every playwright has a process unique to them and this process will begin to shape how you develop ideas. The process I am going to outline for you is a helpful guide to keep you moving towards completion. Tell your story, and along the way you will find that even though you will have plotted out where you wish to go with the story, it can change and you will find that as the author you discover the play more than create it.
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1.16: Writing Exercises
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7.1: Introduction to Matrices
In this section, you will learn to:
- Add and subtract matrices.
- Multiply a matrix by a scalar.
- Multiply two matrices.
A matrix is a 2-dimensional array of numbers arranged in rows and columns. Matrices provide a method of organizing, storing, and working with mathematical information. Matrices have an abundance of applications and use in the real world. Matrices provide a useful tool for working with models based on systems of linear equations. We’ll use matrices in sections 7.2, 7.3, and 7.4 to solve systems of linear equations with several variables in this chapter.
Matrices are used in encryption, which we will explore in section 7.5, and in economic modeling, explored in section 7.6. We use matrices again in Chapter 9, in optimization problems such as maximizing profit/revenue or minimizing cost. Matrices are used in business for scheduling, routing transportation and shipments, and managing inventory.
Just about any application that collects and manages data can apply matrices. The use of matrices has grown as the availability of data in many areas of life and business has increased. They are important tools for organizing data and solving problems in all fields of science (physics, chemistry, biology, genetics, meteorology, and economics).
Computer science analyzes diagrams of networks to understand how things are connected, such as relationships between people on a social website, relationships between results in line searches, and how people link from one website to another. The mathematics to work with network diagrams comprise the field of “graph theory”; it relies on matrices to organize the information in the graphs that diagram connections and associations in a network. For example, if someone uses Facebook, Linked In, or other social media sites, these sites use network graphs and matrices to organize their relationships with other users.
Introduction to Matrices
A matrix is a rectangular array of numbers. Matrices are useful in organizing and manipulating large amounts of data. In order to understand matrices, we will look at the following example.
Fine Furniture Company makes chairs and tables at its San Jose, Hayward, and Oakland factories. The total production, in hundreds, from the three factories for the years 2014 and 2015 is listed in the table below.
| 2014 | 2015 | |||
| CHAIRS | TABLES | CHAIRS | TABLES | |
| SAN JOSE | 30 | 18 | 36 | 20 |
| HAYWARD | 20 | 12 | 24 | 18 |
| OAKLAND | 16 | 10 | 20 | 12 |
- Represent the production for the years 2014 and 2015 as the matrices A and B.
- Find the difference in sales between the years 2014 and 2015.
- The company predicts that in the year 2020, the production at these factories will be double that of the year 2014. What will the production be for the year 2020?
Solution
a) The matrices are as follows:
\[A=\left[\begin{array}{ll}
30 & 18 \\
20 & 12 \\
16 & 10
\end{array}\right] \nonumber \]
\[\mathrm{B}=\left[\begin{array}{ll}
36 & 20 \\
24 & 18 \\
20 & 12
\end{array}\right] \nonumber \]
b) We are looking for the matrix \(B - A\). When two matrices have the same number of rows and columns, the matrices can be added or subtracted entry by entry. Therefore, we get
\[\mathrm{B}-\mathrm{A}=\left[\begin{array}{ll}
36-30 & 20-18 \\
24-20 & 18-12 \\
20-16 & 12-10
\end{array}\right]=\left[\begin{array}{ll}
6 & 2 \\
4 & 6 \\
4 & 2
\end{array}\right] \nonumber \]
c) We would like a matrix that is twice the matrix of 2014, i.e., \(2A\).
Whenever a matrix is multiplied by a number, each entry is multiplied by the number.
\[2 \mathrm{A}=2\left[\begin{array}{ll}
30 & 18 \\
20 & 12 \\
16 & 10
\end{array}\right]=\left[\begin{array}{ll}
60 & 36 \\
40 & 24 \\
32 & 20
\end{array}\right] \nonumber \]
Before we go any further, we need to familiarize ourselves with some terms that are associated with matrices. The numbers in a matrix are called the entries or the elements of a matrix.
Whenever we talk about a matrix, we need to know the size or the dimension of the matrix. The dimension of a matrix is the number of rows and columns it has. When we say a matrix is a “3 by 4 matrix”, we are saying that it has 3 rows and 4 columns. The rows are always mentioned first and the columns second. This means that a \(3 \times 4\) matrix does not have the same dimension as a \(4 \times 3\) matrix.
\[A=\left[\begin{array}{cccc}
1 & 4 & -2 & 0 \\
3 & -1 & 7 & 9 \\
6 & 2 & 0 & 5
\end{array}\right] \nonumber \]
\[B=\left[\begin{array}{ccc}
2 & 9 & 8 \\
-3 & 0 & 1 \\
6 & 5 & -2 \\
-4 & 7 & 8
\end{array}\right] \nonumber \]
Matrix \(A\) has dimensions \(3 \times 4\) and matrix \(B\) has dimensions \(4 \times 3\).
A matrix that has the same number of rows as columns is called a square matrix . A matrix with all entries zero is called a zero matrix. A square matrix with 1's along the main diagonal and zeros everywhere else, is called an identity matrix . When a matrix (size r x c) is multiplied by an identity matrix(size c x c), the matrix remains the same (size r x c).
\[I=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right] \nonumber \]
Matrix \(I\) is a \(3 \times 3\) identity matrix
A matrix with only one row is called a row matrix , and a matrix with only one column is called a column matrix . Two matrices are equal if they have the same size and the corresponding entries are equal.
The following operations can be performed on matrices: addition, subtraction, scalar multiplication, and matrix multiplication. The topics are defined in the sections below.
Matrix Addition and Subtraction
If two matrices are the same size, they can be added or subtracted. The operations are performed on corresponding entries.
Given the matrices \(A\), \(B\), \(C\) and \(D\), below
\[A=\left[\begin{array}{lll}
1 & 2 & 4 \\
2 & 3 & 1 \\
5 & 0 & 3
\end{array}\right] \quad B=\left[\begin{array}{lll}
2 & -1 & 3 \\
2 & 4 & 2 \\
3 & 6 & 1
\end{array}\right] \quad C=\left[\begin{array}{l}
4 \\
2 \\
3
\end{array}\right] \quad D=\left[\begin{array}{r}
-2 \\
-3 \\
4
\end{array}\right] \nonumber \]
Find, if possible.
- \(A + B\)
- \(C - D\)
- \(A + D.\)
Solution
As we mentioned earlier, matrix addition and subtraction involve performing these operations entry by entry.
a) We add each element of \(A\) to the corresponding entry of \(B\).
\[A+B=\left[\begin{array}{lll}
1+2 & 2+-1 & 4+3 \\
2+2 & 3+4 & 1+2 \\
5+3 & 0+6 & 3+1
\end{array}\right] \nonumber \]
\[A+B=\left[\begin{array}{lll}
3 & 1 & 7 \\
4 & 7 & 3 \\
8 & 6 & 4
\end{array}\right] \nonumber \]
b) Just like the problem above, we perform the subtraction entry by entry.
\[\mathrm{C}-\mathrm{D}=\left[\begin{array}{c}
4-(-2) \\
2-(-3) \\
3 - 4
\end{array}\right] \nonumber \]
\[\mathrm{C}-\mathrm{D}=\left[\begin{array}{c}
6 \\
5 \\
-1
\end{array}\right] \nonumber \]
c) The sum \(A + D\) cannot be found because the two matrices have different sizes.
Note: Two matrices can only be added or subtracted if they have the same dimension.
Multiplying a Matrix by a Scalar
If a matrix is multiplied by a scalar, each entry is multiplied by that scalar. We can consider scalar multiplication as multiplying a number and a matrix to obtain a new matrix as the product.
Given the matrix \(A\) and \(C\) in the example above, find \(2A\) and \(- 3C\).
Solution
To find \(2A\), we multiply each entry of matrix \(A\) by 2, and to find \(-3C\), we multiply each entry of C by -3. The results are given below.
a) We multiply each entry of A by 2.
\[2 \mathrm{A}=\left[\begin{array}{ccc}
2·1 & 2·2 & 2·4 \\
2·2 & 2·3 & 2·1 \\
2·5 & 2·0 & 2·3
\end{array}\right] \nonumber \]
\[2 \mathrm{A}=\left[\begin{array}{ccc}
2 & 4 & 8 \\
4 & 6 & 2 \\
10 & 0 & 6
\end{array}\right] \nonumber \]
b) We multiply each entry of C by -3.
\[-3 C=\left[\begin{array}{c}
-3·4 \\
-3·2 \\
-3·3
\end{array}\right] \nonumber \]
\[-3 C=\left[\begin{array}{c}
-12 \\
-6 \\
-9
\end{array}\right] \nonumber \]
Multiplication of Two Matrices
Multiplying two matrices is not as easy as the addition, subtraction, or scalar multiplication of matrices. Because of its wide use in application problems, it is important that we learn it well. Therefore, we will try to learn the process in a step-by-step manner. We first begin by finding a product of a row matrix and a column matrix.
Find the product \(AB\), given
\[A=\left[\begin{array}{lll}
2 & 3 & 4
\end{array}\right] \nonumber \]
and
\[B=\left[\begin{array}{l}
a \\
b \\
c
\end{array}\right]. \nonumber \]
Solution
The product is a \(1 \times 1\) matrix whose entry is obtained by multiplying the corresponding entries and then forming the sum.
\[\begin{align*}
\mathrm{AB} &=\left[\begin{array}{lll}
2 & 3 & 4
\end{array}\right]\left[\begin{array}{l}
\mathrm{a} \\
\mathrm{b} \\
\mathrm{c}
\end{array}\right] \\[4pt] &=[2 \mathrm{a}+3 \mathrm{b}+4 \mathrm{c}]
\end{align*} \nonumber \]
Note that \(AB\) is a \(1 \times 1\) matrix, and its only entry is \(2a + 3b + 4c\).
Find the product \(AB\), given
\[A=\left[\begin{array}{lll}
2 & 3 & 4
\end{array}\right] \nonumber \]
and
\[B=\left[\begin{array}{l}
5 \\
6 \\
7
\end{array}\right] \nonumber \]
Solution
Again, we multiply the corresponding entries and add.
\[\begin{align*} \mathrm{AB} &=\left[\begin{array}{lll}
2 & 3 & 4
\end{array}\right]\left[\begin{array}{l}
5 \\
6 \\
7
\end{array}\right] \\[4pt] &=[2 \cdot 5+3 \cdot 6+4 \cdot 7]\\[4pt] &=[10+18+28]\\[4pt] &=[56] \end{align*} \nonumber \]
Note: In order for a product of a row matrix and a column matrix to exist, the number of entries in the row matrix must be the same as the number of entries in the column matrix.
Find the product AB, given
\[A=\left[\begin{array}{lll}
2 & 3 & 4
\end{array}\right] \nonumber \]
and
\[B=\left[\begin{array}{ll}
5 & 3 \\
6 & 4 \\
7 & 5
\end{array}\right]. \nonumber \]
Solution
We know how to multiply a row matrix by a column matrix. To find the product \(AB\), in this example, we will multiply the row matrix \(A\) to both the first and second columns of matrix \(B\), resulting in a \(1 \times 2\) matrix.
\[\mathrm{AB}=\left[\begin{array}{lll}
2 \cdot 5+3 \cdot 6+4 \cdot 7 & 2 \cdot 3+3 \cdot 4+4 \cdot 5
\end{array}\right]=\left[\begin{array}{ll}
56 & 38
\end{array}\right] \nonumber \]
We multiplied a \(1 \times 3\) matrix by a matrix whose size is \(3 \times 2\). So unlike addition and subtraction, it is possible to multiply two matrices with different dimensions, if the number of entries in the columns of the first matrix is the same as the number of entries in the rows of the second matrix.
Find the product \(AB\), given:
\[A=\left[\begin{array}{lll}
2 & 3 & 4 \\
1 & 2 & 3
\end{array}\right] \nonumber \]
and
\[\mathrm{B}=\left[\begin{array}{ll}
5 & 3 \\
6 & 4 \\
7 & 5
\end{array}\right] \nonumber \]
Solution
This time we are multiplying two rows of the matrix \(A\) with two columns of the matrix \(B\). Since the number of entries in each row of \(A\) is the same as the number of entries in each column of \(B\), the product is possible. We do exactly what we did in the last example. The only difference is that the matrix \(A\) has one more row.
We multiply the first row of the matrix \(A\) with the two columns of \(B\), one at a time, and then repeat the process with the second row of A. We get
\[\mathrm{AB}=\left[\begin{array}{lll}
2 & 3 & 4 \\
1 & 2 & 3
\end{array}\right]\left[\begin{array}{ll}
5 & 3 \\
6 & 4 \\
7 & 5
\end{array}\right]=\left[\begin{array}{ll}
2 \cdot 5+3 \cdot 6+4 \cdot 7 & 2 \cdot 3+3 \cdot 4+4 \cdot 5 \\
1 \cdot 5+2 \cdot 6+3 \cdot 7 & 1 \cdot 3+2 \cdot 4+3 \cdot 5
\end{array}\right] \nonumber \]
\[\mathrm{AB}=\left[\begin{array}{ll}
56 & 38 \\
38 & 26
\end{array}\right] \nonumber \]
Find, if possible:
- \(EF\)
- \(FE\)
- \(FH\)
- \(GH\)
- \(HG\)
\[\mathrm{E}=\left[\begin{array}{ll}
1 & 2 \\
4 & 2 \\
3 & 1
\end{array}\right] \quad \mathrm{F}=\left[\begin{array}{ll}
2 & -1 \\
3 & 2
\end{array}\right] \quad \mathrm{G}=\left[\begin{array}{lll}
4 & 1
\end{array}\right] \quad \mathrm{H}=\left[\begin{array}{l}
-3 \\
-1
\end{array}\right] \nonumber \]
Solution
a) To find \(EF\), we multiply the first row \(\left[\begin{array}{ll}
1 & 2
\end{array}\right]\)
of E with the columns \(\left[\begin{array}{l}
2 \\
3
\end{array}\right] \text { and }\left[\begin{array}{l}
1 \\
-2
\end{array}\right]\) of the matrix F, and then repeat the process by multiplying the other two rows of E with these columns of F. The result is as follows:
\[\mathrm{EF}=\left[\begin{array}{ll}
1 & 2 \\
4 & 2 \\
3 & 1
\end{array}\right]\left[\begin{array}{cc}
2 & -1 \\
3 & 2
\end{array}\right]=\left[\begin{array}{cc}
1 \cdot 2+2 \cdot 3 & 1 \cdot-1+2 \cdot 2 \\
4 \cdot 2+2 \cdot 3 & 4 \cdot-1+2 \cdot 2 \\
3 \cdot 2+1 \cdot 3 & 3 \cdot-1+1 \cdot 2
\end{array}\right]=\left[\begin{array}{cc}
8 & 3 \\
14 & 0 \\
9 & -1
\end{array}\right] \nonumber \]
b) Product \(FE\) is not possible because the matrix F has two entries in each row, while the matrix E has three entries in each column. In other words, the matrix F has two columns, while the matrix E has three rows.
c) \[\mathrm{FH}=\left[\begin{array}{cc}
2 & -1 \\
3 & 2
\end{array}\right]\left[\begin{array}{c}
-3 \\
-1
\end{array}\right]=\left[\begin{array}{c}
2 \cdot-3+-1 \cdot-1 \\
3 \cdot-3+2 \cdot-1
\end{array}\right]=\left[\begin{array}{c}
-5 \\
-11
\end{array}\right] \nonumber \]
d) \[\mathrm{GH}=\left[\begin{array}{ll}
4 & 1
\end{array}\right]\left[\begin{array}{l}
-3 \\
-1
\end{array}\right]=\left[\begin{array}{l}
4 \cdot-3+1 \cdot-1 \\
-1
\end{array}\right]=[-13] \nonumber \]
e) \[\mathrm{HG}=\left[\begin{array}{l}
-3 \\
-1
\end{array}\right]\left[\begin{array}{ll}
4 & 1
\end{array}\right]=\left[\begin{array}{ll}
-3 \cdot 4 & -3 \cdot 1 \\
-1 \cdot 4 & -1 \cdot 1
\end{array}\right]=\left[\begin{array}{cc}
-12 & -3 \\
-4 & -1
\end{array}\right] \nonumber \]
We summarize some important properties of matrix multiplication that we observed in the previous examples.
In order for product \(\bf{AB}\) to exist:
- the number of columns of \(\bf{A}\) must equal the number of rows of \(\bf{B}\)
- if matrix \(\bf{A}\) has dimension \(\bf{m \times n}\) and matrix \(\bf{B}\) has dimension \(\bf{n \times p}\) , then the product \(\bf{AB}\) will be a matrix with dimension \(\bf{m \times p}\) .
Matrix multiplication is not commutative: if both matrix products \(\bf{AB}\) and \(\bf{BA}\) exist, most of the time \(\bf{AB}\) will not equal \(\bf{BA}\) .
Given matrices \(R\), \(S\), and \(T\) below, find \(2RS - 3ST\).
\[R=\left[\begin{array}{lll}
1 & 0 & 2 \\
2 & 1 & 5 \\
2 & 3 & 1
\end{array}\right] \quad S=\left[\begin{array}{lll}
0 & -1 & 2 \\
3 & 1 & 0 \\
4 & 2 & 1
\end{array}\right] \quad T=\left[\begin{array}{lll}
-2 & 3 & 0 \\
-3 & 2 & 2 \\
-1 & 1 & 0
\end{array}\right] \nonumber \]
Solution
We multiply the matrices R and S.
\begin{aligned}
&\mathrm{RS}=\left[\begin{array}{ccc}
8 & 3 & 4 \\
23 & 9 & 9 \\
13 & 3 & 5
\end{array}\right]\\
&\begin{array}{l}
2 \mathrm{RS}=2\left[\begin{array}{ccc}
8 & 3 & 4 \\
23 & 9 & 9 \\
13 & 3 & 5
\end{array}\right]=\left[\begin{array}{ccc}
16 & 6 & 8 \\
46 & 18 & 18 \\
26 & 6 & 10
\end{array}\right] \\
\mathrm{ST}=\left[\begin{array}{ccc}
1 & 0 & -2 \\
-9 & 11 & 2 \\
-15 & 17 & 4
\end{array}\right] \\
3 \mathrm{ST}=3\left[\begin{array}{ccc}
1 & 0 & -2 \\
-9 & 11 & 2 \\
-15 & 17 & 4
\end{array}\right]=\left[\begin{array}{ccc}
3 & 0 & -6 \\
-27 & 33 & 6 \\
-45 & 51 & 12
\end{array}\right]
\end{array}
\end{aligned}
Thus
\[2 \mathrm{RS}-3 \mathrm{ST}=\left[\begin{array}{ccc}
16 & 6 & 8 \\
46 & 18 & 18 \\
26 & 6 & 10
\end{array}\right]-\left[\begin{array}{ccc}
3 & 0 & -6 \\
-27 & 33 & 6 \\
-45 & 51 & 12
\end{array}\right]=\left[\begin{array}{ccc}
13 & 6 & 14 \\
73 & -15 & 12 \\
71 & -45 & -2
\end{array}\right] \nonumber \]
Find \(F^2\) given matrix
\[\mathrm{F}=\left[\begin{array}{ll}
2 & -1 \\
3 & 2
\end{array}\right] \nonumber \]
Solution
\(F^2\) is found by multiplying matrix \(F\) by itself, using matrix multiplication.
\[\mathrm{F}^{2}=\left[\begin{array}{cc}
2 & -1 \\
3 & 2
\end{array}\right]\left[\begin{array}{cc}
2 & -1 \\
3 & 2
\end{array}\right]=\left[\begin{array}{cc}
2 \cdot 2+(-1) \cdot 3 & 2 \cdot(-1)+(-1) \cdot 2 \\
3 \cdot 2+2 \cdot 3 & 3 \cdot(-1)+2 \cdot 2
\end{array}\right]=\left[\begin{array}{cc}
1 & -4 \\
12 & 1
\end{array}\right] \nonumber \]
Note that \(F^2\) is not found by squaring each entry of matrix \(F\). The process of raising a matrix to a power, such as finding \(F^2\), is only possible if the matrix is a square matrix .
USING MATRICES TO REPRESENT A SYSTEM OF LINEAR EQUATIONS
In this chapter, we will be using matrices to solve linear systems. In section 7.4, we will be asked to express linear systems as the matrix equation \(\bf{AX = B}\), where \(A\), \(X\), and \(B\) are matrices.
- Matrix \(A\) is called the coefficient matrix.
- Matrix \(X\) is a matrix with 1 column that contains the variables.
- Matrix \(B\) is a matrix with 1 column that contains the constants.
Verify that the system of two linear equations with two unknowns:
\begin{array}{l}
a x+b y=h \\
c x+d y=k
\end{array}
can be written as \(AX = B\), where
\[A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right] \quad X=\left[\begin{array}{l}
x \\
y
\end{array}\right] \text { and } B=\left[\begin{array}{l}
h \\
k
\end{array}\right] \nonumber \]
Solution
If we multiply the matrices \(A\) and \(X\), we get
\[A X=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]=\left[\begin{array}{c}
a x+b y \\
c x+d y
\end{array}\right] \nonumber \]
If \(AX = B\) then
\[\left[\begin{array}{l}
a x+b y \\
c x+d y
\end{array}\right]=\left[\begin{array}{l}
h \\
k
\end{array}\right] \nonumber \]
If two matrices are equal, then their corresponding entries are equal. It follows that
\begin{array}{l}
a x+b y=h \\
c x+d y=k
\end{array}
Express the following system as a matrix equation in the form \(AX = B\).
\begin{array}{l}
2 x+3 y-4 z=5 \\
3 x+4 y-5 z=6 \\
5 x \quad-6 z=7
\end{array}
Solution
This system of equations can be expressed in the form \(AX = B\) as shown below.
\[\left[\begin{array}{ccc}
2 & 3 & -4 \\
3 & 4 & -5 \\
5 & 0 & -6
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
5 \\
6 \\
7
\end{array}\right] \nonumber \]


16.3: Pedagogy
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The term pedagogy refers to the holistic nature of educators’ professional practice (especially those aspects that involve building and nurturing relationships), decision-making, and teaching and learning. When educators establish respectful relationships with children and families, they are able to work together to develop programs and experiences which are relevant to children and build on individual and group interests. These experiences create possibilities for children’s own ideas and activities, allowing them to celebrate their own interests and friendships and express themselves in different ways.
Educators’ professional judgements are central to their active role in facilitating children’s learning. In making professional judgements, they weave together their:
They also draw on their creativity, imagination and insight to help them improvise and adjust their practice to suit the time, place and context of learning.
Different theories about childhood inform approaches to children’s learning and development. School-age care educators draw upon a range of perspectives in their work. Drawing on a range of perspectives and theories can challenge traditional ways of seeing children, facilitating learning, and encourage educators, as individuals and with colleagues, to: