7.10: Learning Activities
Learning Activities
(Answers to “Learning Activities” can be found in the “Answer Key” at the end of the book. Answers to interactive activities are provided as immediate feedback.)
Visit the TeamSTEPPS® Instructor Manual: Specialty Scenarios for multiple AHRQ scenarios requiring application of TeamSTEPPS® to patient scenarios.
Scenario 1
Read this scenario and answer the following questions.
Jill and Neil are both nurses working the same shift. Jill is responsible for patients in Rooms 1–6, and Neil is responsible for patients in Rooms 7–12. Over the course of their shift, both nurses routinely visit their patients’ rooms to take vitals and deliver medication.
On one of his rounds, Neil attends to his patient in Room 8. He reads the chart and notices Jill’s initials signaling that she had already checked on this patient. A bit confused, he continues on to his next patient. After another hour goes by, Neil returns to Room 8 and again notices Jill’s initials on the chart. Neil thinks to himself, “What is she doing? I’ve got it covered. She’s checking my work. She must think I’m incompetent.” Neil decides to approach Jill and see what is going on.
Questions:
a) What might have been the initial cause of the conflict?
b) What kind of conflict did it turn into?
c) If Neil provides Jill with a copy of the room assignments, would that resolve the conflict?
Scenario 2
Read this scenario and answer the following questions.
A nursing team is having a routine meeting. One of the nurses, Stephen, is at the end of a 12-hour shift, and another nurse, Tanya, is just beginning hers. Tanya is a senior nurse in the unit with over ten years’ experience on this specific unit. Stephen is new to the unit with fewer than three years’ experience in nursing. Tanya has been asked to present information to the team about effective time management on the unit.  Tanya breaks from her presentation and asks, “Stephen, do you have anything to add?” Stephen replies, “No, I just don’t know why we need to talk about this again.” Tanya chooses to avoid engaging with Stephen further and finishes her presentation. Stephen continues to be disruptive throughout the presentation.
After the meeting concludes, Tanya approaches Stephen and asks if he had anything to add from the meeting. Stephen replies, “No, I don’t have anything. I just think we all know what the procedure is because we just learned it all during orientation training. Maybe if you don’t remember the training, you should take it again.” Tanya is shocked by his reply and quickly composes herself. She states, “Stephen, I have worked on this unit for over ten years. I was asked to present that information because there are current issues going on among the staff. Next time please respect my authority and listen to those who come before you.”
Questions:
a) What types of conflict are present?
b) What will need to happen to resolve this issue between Tanya and Stephen?
c) Take a moment to think about what your preferred approach to conflict may be. How might you adapt your approach to conflict when working with others?
Scenario 3
Read this scenario and answer the following questions.
Connie, the head nurse on Unit 7, is a respected member of the team. She has been working on this unit for a number of years and is seen by the other nurses as the “go to” person for questions and guidance. Connie is always thorough with patients and demonstrates excellence and quality in her work. Dr. Smith is a well-respected member of the medical profession and an expert in his field of medicine. He has a reputation for excellent bedside manner and is thorough in his approach with patients.
Connie is four hours into her 12-hour shift when she is approached by Dr. Smith. He asks, “Connie, why has the patient in Room 2 not received his blood pressure medication over the past few days? I was not notified about this!”
Connie, trying to find a quick solution, replies, “I didn’t know that patient had been missing medication. I’ll go check on it and get back to you.”
Dr. Smith is persistent, saying, “You don’t need to go check anything. I know this patient and should have been informed about the withholding of medication and the reasons why.”
Connie, again attempting to find a resolution, states, “Well, there must be some communication about this change somewhere . . .”
“There isn’t!” Dr. Smith interrupts.
Connie becomes upset and decides to leave the conversation after declaring, “Fine, if you know everything, then you figure it out; you’re the one with the medical degree, aren’t you?”
Connie makes her way to the nurses’ station where Dan and Elise are assessing charts and says to them, “You will not believe what Dr. Smith just said to me!”
Dan and Elise look shocked and ask “What?”
Connie explains, “Well, he thinks I don’t do my job, when really we nurses are the ones that keep this unit going. Who is he to question my ability to look after patients? I am the most knowledgeable person on this unit!”
Dan and Elise don’t know how to reply and decide to avoid the interaction with a simple, “Oh yeah.”
Meanwhile, Dr. Smith has made his way to the doctor’s lounge and finds his colleague Dr. Lee. Dr. Smith tells him, “That head nurse on Unit 7 is useless. She doesn’t know what she is doing and doesn’t understand that we must be informed about changes to our patients’ medications, does she?” Dr. Lee nods quickly and returns to reviewing his file.
About three hours later, Connie and Dr. Smith have each spoken to several people about the interaction. Connie bumps into another nurse, Jessie, one of her best friends. Connie pulls Jessie aside and says, “You will not believe what I just saw. I was going into the admin office to file my holiday requests and out of the corner of my eye, I saw Dr. Smith lurking around the corner, pretending to look at a chart, spying on me!”
“What?  That’s not safe,” replies Jessie.
Connie is relieved that Jessie is taking her side—it makes her feel as if she is not going crazy. “Yeah, I’m really getting worried about this. First, that creep is accusing me of not doing my job, and then he is wasting his time spying on me.  Maybe what I should do is file a complaint. That’ll show the ‘big man’ that he isn’t that big around here and maybe take him down a notch.”
When Jessie gives her an apprehensive look, Connie continues, “Oh Jessie, don’t worry. I have a perfect record and this won’t affect me. Even if it does, I will have done something good for all the other nurses around here. It is the principle of the matter at this point.”
Meanwhile, Dr. Smith has run into an old classmate of his, Dr. Drucker, and says, “Wow! You got hired! So happy to have you in the hospital. I do need to tell you though to be careful of the head nurse on Unit 7 . . .” Dr. Smith describes his troubles with Connie and adds, “She’s a real snitch! She makes trouble out of nothing. I was reviewing a chart down the corridor from the admin office, minding my own business and actually getting my work done, when I saw her slip into the admin office to squeal about me to the top bosses! This is something that needs to be watched! We can’t have people reporting doctors to administration over nothing. I think I’m going to write her up and get a black mark on that perfect record of hers. I will be fine—everyone knows I’m right. There will be consequences for her.”
Questions:
a) Using the scenario above, identify the stages of the conflict escalation.
b) What suggestions might you give to Dan, Elise, Jessie, Dr. Lee, and Dr. Drucker about how to respond to Connie and Dr. Smith?
c) How do you think this conflict will be resolved?


5.3: Statutory and Common Law
What you’ll learn to do: differentiate between statutory and common law
In this section you’ll learn about the differences between statutory and common law.
Learning Objectives
- Define common law
Statutory vs. Common Law
At both the federal and state levels, the law of the United States can be traced back to the common law system of English law, which was in force in the American colonies during the time of the Revolutionary War. Since then, U.S. law has diverged greatly from its English roots both in terms of substance and procedure. The main departure occurred when the United States ratified the Constitution in 1789. In effect, the Constitution and federal statutes and laws made in furtherance of the Constitution were established as “the supreme Law of the Land.” From that point on, the law of the land was no longer based on legal practices in England but became distinctly American and home grown. When the individual states ratified their state constitutions, the state legislatures obtained the power to establish state law, or the “Law of the State.” Together, this collection of federal and state laws constitutes something we often refer to as the “body of law.” This body of law governs the behavior of individuals, businesses, and even governments. Just like the human body, the “body of law” is comprised of multiple parts, each performing an individual function while simultaneously working together as a whole. In this section, we will examine two of the most fundamental types of laws, followed by nearly every nation in the world: statutory law and common law .
A statute is a law passed by a legislature; and statutory law is the body of law resulting from statutes. A statute—or the statutory law—may also be referred to as legislation . One of the benefits of statutory law is that whether it’s federal or state law, it’s a written law that you can locate and read at the law library or online. This is not true of common law, which is also known as “unwritten law, because it’s not collected in a single source.
Suppose you are headed over to a friend’s house to watch football on Sunday, and on your way you stop in at the local supermarket to buy some beer and pretzels for the gang. You carry your six-pack and snacks up to the counter to pay, and the clerk tells you that she’s sorry, but she can’t sell you the beer. At first you think it’s because she suspects you’re underage, but before you can show her your ID, she explains that she can’t sell alcohol before noon because (1) it’s Sunday and (2) you are in the State of North Carolina. Shocked, you think she’s joking until she refers you to the following NC Statute: N.C. General Statute 18B-1004(c) states, “It shall be unlawful to sell or consume alcoholic beverages on any licensed premises from the time at which sale or consumption must cease on Sunday morning until 12:00 noon on that day.” No amount of begging or pleading will get you the beer, because the owner of the supermarket knows that if she violates N.C. General Statute 18B-1004(c), the store’s ABC license could be revoked and its alcohol sales ended permanently . This is an example of statutory law.
However, when the federal and state constitutions were written, it wasn’t possible to anticipate and include every possible law in those documents. For instance, in 1789 there was no reason to write laws prohibiting people from operating motor vehicles while intoxicated, because there were no motor vehicles yet—people still rode horses. Instead, the Constitution made provisions for law to evolve as society evolved. In 1803, U.S. Supreme Court Chief Justice John Marshall stated that “[i]t is emphatically the province and duty of the Judicial Department to say what the law is.” This kind of judge-made law is c ommon law . Case law is developed by judges, courts, and similar tribunals, and, over time, the decisions in individual cases establish precedents for future cases. Precedent means that the decisions judges have made in earlier cases guide how future cases are decided. In common law systems, this principle is called stare decisis , and it has a binding effect on judges and courts: Stare decisis holds that cases should be decided according to consistent principled rules so that similar facts will yield similar results. If the court finds that the current dispute is fundamentally distinct from previous cases, judges have the authority and duty to make law by creating precedent. Thereafter, the new decision becomes precedent and will bind future courts.
William Frantz Elementary School, New Orleans, 1960. “After a Federal court ordered the desegregation of schools in the South, U.S. marshals escorted a young Black girl, Ruby Bridges, to school.”
In Brown v. Board of Education , the landmark case concerning racial segregation in U.S. public schools, the Supreme Court ultimately handed down a decision that established a new legal precedent. At the heart of the case was the contention that the separate school systems for blacks and whites were inherently unequal and thus violated the Equal Protection Clause of the Fourteenth Amendment to the U.S. Constitution.
After the case was reheard in 1953, Chief Justice Warren was able to persuade all of the justices to support a unanimous decision declaring segregation in public schools to be unconstitutional. On May 14, 1954, he delivered the opinion of the Court: “We conclude that in the field of public education the doctrine of ‘separate but equal’ has no place. Separate educational facilities are inherently unequal. . .”
Although it would be many years before all U.S. public schools were desegregated, the Supreme Court’s ruling in Brown was the legal turning point that paved the way for the this change. Under common law, the precedent it established was that separate educational facilities for different races are inherently unequal.


12.5: How can we teach "digital natives"?
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While we often think of millennials – those born between 1981 and 1996 – as the first wired generation, the truth is that most millennials were teenagers when the iPhone was invented. Anyone born before those years is called a "digital immigrant". But today’s K-12 students, on the other hand, are true “digital natives” in the sense that they’ve never known a world without swiping and apps — and many were watching videos on their parents’ phones or tablets when they were toddlers.
This generation of students, Generation Z, has very different expectations for how they’ll learn, and teaching them requires a different approach as well. K-12 educators and administrators have to be very thoughtful and deliberate about how to best reach them.
We’ve spent a lot of time talking with educators and administrators about how to reach today’s students effectively. Based on what we’ve learned, here are five recommendations for educating the true digital natives of Gen Z.
1) Meet them where they are
Incorporate tools students are comfortable with and are already using every day for communication and entertainment. Many of these resources have education-specific versions available that are purposefully built to encourage self-directed learning.
For instance, Gen Z frequently streams video from sources such as YouTube and TikTok. According to the 2019 Speak Up survey from Project Tomorrow, a third of students in grades 6-12 use YouTube to help with their homework, and 49% say they use it to explore topics they’re interested in on their own. What’s more, 28% of middle schoolers have their own personal YouTube channel — and 7% are running a YouTube business! Educators can engage students more effectively by incorporating short video clips from well-vetted sources into their instruction and having students create their own videos to demonstrate their learning.
Besides YouTube, the most popular online platforms among teens are Instagram and Snapchat, according to Pew Research Center. Students like the positive reinforcement they get when their posts generate likes and comments on these social media platforms. An app called Seesaw is like Instagram for learning, turning school projects and assignments into social media that students share with the world. Instead of collecting likes and comments for posting selfies, students receive positive reinforcement on their schoolwork.
Note
Are You a Digital Immigrant?
- If you answer yes to any of these questions then you are a digital immigrant:
- Do you remember when telephones had a dial on them (after all, we still “dial” a phone number don’t we?) and were actually connected to the wall by a wire?
- Do you remember TV test patterns and antennas on the roof?
- Do you remember the electric typewriter?
- Do you remember carbon paper?
- Do you remember the Apple II, The TRS 80?
- Do you remember the IBM PC and the black and white Macintosh?
- Do you remember America Online, The Source, CompuServe or other online services, i.e., Mosaic or Netscape?
- Do you remember getting broadband in your home?
- Do you remember the Compact Disc (CD)? (Riedl, 2007)
2) Focus on equity and inclusion.
Make sure every student has equitable access to digital devices and learning opportunities. If you’re rolling out a 1:1 learning program but don’t yet have the resources to provide a device for every student, we recommend giving all students in a particular grade level a device. Rather than divvying up devices by class type, distributing devices by grade is a more equitable approach. If you establish a “bring your own device” policy, make sure you have extra devices to give to students who don’t have one of their own.
Consider how students will access technology not just at school, but also at home. The “homework gap,” or the disadvantage that students face in completing homework when they don’t have broadband access at home, is real. The 2019 Speak Up survey revealed that 13% of students struggle to complete homework because they lack home internet service. Lending students mobile hotspots, keeping the library open before and after school, and partnering with local businesses to give families discounted broadband can help solve this challenge. For more ideas, read Bridging the Digital Divide When Students Go Home
3) Teach and model effective digital citizenship.
Just because today’s students are digital natives doesn’t mean they understand how to behave safely and responsibly online. Teachers need to explicitly teach and model responsible and ethical digital behavior.
A Pew survey reveals that 59% of teens have directly experienced some form of cyberbullying. According to the 2019 Speak Up survey, only one in five girls and 36% of boys rate their digital citizenship skills as “advanced.”
While many schools address digital citizenship in an assembly or one-time workshop with students, these skills should be taught and reinforced throughout the core curriculum. Common Sense Media is one of many organizations with a free digital citizenship curriculum guide for schools.
4) Provide a safer online environment.
Cyberbullying takes a heavy toll on students. Of children currently experiencing a mental health problem, more than two-thirds (68%) say they experienced cyberbullying in the last year, according to The Children’s Society and Young Minds. The organization notes that a study by Birmingham University found that cyberbullying victims are twice as likely to attempt suicide or engage in self-harm as non-victims.
Aside from teaching responsible behavior, teachers need tools that can help keep students safer while they’re working online. Classroom management software that provides insight into students’ online activity provides an additional layer of defense. And now, a new partnership between LanSchool and Bark for Schools gives schools even more security. Bark’s research backed solution alerts administrators to activities that may indicate issues such as depression, cyberbullying, self-harm, and sexual content on students’ school accounts. Schools that register for Bark as part of their LanSchool relationship will also receive complimentary access to Bark’s image removal software, a powerful tool that helps prevent explicit images of minors from ever entering the digital space.
5) Turn them loose.
Teachers have to be willing to give up some degree of control in their classroom by letting students create, collaborate, and take charge of their own learning. Research shows students are more engaged, explore content in greater depth, and retain knowledge more effectively when they take ownership of their learning.
But this doesn’t mean giving up all control – teachers are still there to guide students and ensure a safe learning environment. Teachers must establish clear rules for their classrooms and hold students accountable for breaking them. In fact, research suggests that classroom management plays a critical role in student achievement, as Robert Marzano and his colleagues have indicated.
Classroom management software can help teachers effectively manage instruction for today’s digital natives, providing a safer digital environment that supports student-centered learning while empowering the teacher to guide on the side.
The true digital natives of Generation Z expect to learn differently than prior generations. By integrating the kinds of digital tools they’re familiar with, focusing on equity, teaching and modeling responsible digital citizenship, providing a safer online environment, and empowering them to take charge of their learning, schools can deeply engage Gen Z in their learning and achieve better outcomes.


23.1: External Female Genitals
(CC-BY-4.0, OpenStax, Human Anatomy )
The female reproductive system functions to produce gametes and reproductive hormones, just like the male reproductive system; however, it also has the additional task of supporting the developing fetus and delivering it to the outside world. Unlike its male counterpart, the female reproductive system is located primarily inside the pelvic cavity (Figure \(\PageIndex{1}\)). Recall that the ovaries are the female gonads. The gamete they produce is called an oocyte .
External Female Genitals
The external female reproductive structures are referred to collectively as the vulva (Figure \(\PageIndex{2}\)). The mons pubis is a pad of fat that is located at the anterior, over the pubic bone. After puberty, it becomes covered in pubic hair. The labia majora (labia = “lips”; majora = “larger”) are folds of hair-covered skin that begin just
posterior to the mons pubis. The thinner and more pigmented labia minora (labia = “lips”; minora = “smaller”) extend medial to the labia majora. Although they naturally vary in shape and size from woman to woman, the labia minora serve to protect the female urethra and the entrance to the female reproductive tract.
The superior, anterior portions of the labia minora come together to encircle the clitoris (or glans clitoris), an organ that originates from the same cells as the glans penis and has abundant nerves that make it important in sexual sensation and orgasm. The hymen is a thin membrane that sometimes partially covers the entrance to the vagina. An intact hymen cannot be used as an indication of “virginity”; even at birth, this is only a partial membrane, as menstrual fluid and other secretions must be able to exit the body, regardless of penile–vaginal intercourse. The vaginal opening is located between the opening of the urethra and the anus. It is flanked by outlets to the Bartholin’s glands (or greater vestibular glands).
Vagina
The vagina , shown at the bottom of \(\PageIndex{1}\) and \(\PageIndex{2}\), is a muscular canal (approximately 10 cm long) that serves as the entrance to the reproductive tract. It also serves as the exit from the uterus during menses and childbirth. The outer walls of the anterior and posterior vagina are formed into longitudinal columns, or ridges, and the superior portion of the vagina—called the fornix—meets the protruding uterine cervix. The walls of the vagina are lined with an outer, fibrous adventitia; a middle layer of smooth muscle; and an inner mucous membrane with transverse folds called rugae . Together, the middle and inner layers allow the expansion of the vagina to accommodate intercourse and childbirth. The thin, perforated hymen can partially surround the opening to the vaginal orifice. The hymen can be ruptured with strenuous physical exercise, penile–vaginal intercourse, and childbirth. The Bartholin’s glands and the lesser vestibular glands (located near the clitoris) secrete mucus, which keeps the vestibular area moist.
The vagina is home to a normal population of microorganisms that help to protect against infection by pathogenic bacteria, yeast, or other organisms that can enter the vagina. In a healthy woman, the most predominant type of vaginal bacteria is from the genus Lactobacillus . This family of beneficial bacterial flora secretes lactic acid, and thus protects the vagina by maintaining an acidic pH (below 4.5). Potential pathogens are less likely to survive in these acidic conditions. Lactic acid, in combination with other vaginal secretions, makes the vagina a self- cleansing organ. However, douching—or washing out the vagina with fluid—can disrupt the normal balance of healthy microorganisms, and actually increase a woman’s risk for infections and irritation. Indeed, the American College of Obstetricians and Gynecologists recommend that women do not douche, and that they allow the vagina to maintain its normal healthy population of protective microbial flora.


1.2: Percents
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In the 2004 vice-presidential debates, Edwards's claimed that US forces have suffered "90% of the coalition casualties" in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies "have taken almost 50 percent" of the casualties[1]. Who is correct? How can we make sense of these numbers?
Percent literally means “per 100,” or “parts per hundred.” When we write 40%, this is equivalent to the fraction \(\frac{40}{100}\) or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since \(\frac{80}{200}=\frac{10}{25}=\frac{40}{100}\).
Example 1
243 people out of 400 state that they like dogs. What percent is this?
Solution
\(\frac{243}{400}=0.6075=\frac{60.75}{100}\). This is 60.75%.
Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.
Example 2
Write each as a percent: a) \(\frac{1}{4}\) b) 0.02 c) 2.35
Solution
a) \(\frac{1}{4}=0.25=25 \%\) b) \(0.02=2 \%\) c) \(2.35=235 \%\)
Percents
If we have a part that is some percent of a whole , then
\(\text{percent }=\frac{\text { part }}{\text { whole }}\), or equivalently, \(\text{part }=\text{ percent} \cdot \text{whole}\)
To do the calculations, we write the percent as a decimal.
Example 3
The sales tax in a town is 9.4%. How much tax will you pay on a $140 purchase?
Solution
Here, $140 is the whole, and we want to find 9.4% of $140. We start by writing the percent as a decimal by moving the decimal point two places to the left (which is equivalent to dividing by 100). We can then compute:
\(\operatorname{tax}=0.094(140)=\$ 13.16\) in tax.
Example 4
In the news, you hear “tuition is expected to increase by 7% next year.” If tuition this year was $1200 per quarter, what will it be next year?
Solution
The tuition next year will be the current tuition plus an additional 7%, so it will be 107% of this year’s tuition:
\(\$ 1200(1.07)=\$ 1284\).
Alternatively, we could have first calculated 7% of $1200: \(\$ 1200(0.07)=\$ 84\).
Notice this is not the expected tuition for next year (we could only wish). Instead, this is the expected increase , so to calculate the expected tuition, we’ll need to add this change to the previous year’s tuition:
\(\$ 1200+\$ 84=\$ 1284\).
Try it Now 1
A TV originally priced at $799 is on sale for 30% off. There is then a 9.2% sales tax. Find the price after including the discount and sales tax.
- Answer
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The sale price is \(\$799(0.70) = \$559.30\). After tax, the price is \(\$559.30(1.092) = \$610.76\).
Example 5
The value of a car dropped from $7400 to $6800 over the last year. What percent decrease is this?
Solution
To compute the percent change, we first need to find the dollar value change: \(\$ 6800-\$ 7400=-\$ 600\) Often we will take the absolute value of this amount, which is called the absolute change : \(|-600|=600\).
Since we are computing the decrease relative to the starting value, we compute this percent out of \(\$7400\):
\(\frac{600}{7400}=0.081=8.1 \%\) decrease. This is called a relative change .
Absolute and Relative Change
Given two quantities,
Absolute change = \(|\text{ending quantity }-\text{ starting quantity}|\)
Relative change: \(\frac{\text { absolute change }}{\text { starting quantity }}\)
Absolute change has the same units as the original quantity.
Relative change gives a percent change.
The starting quantity is called the base of the percent change.
The base of a percent is very important. For example, while Nixon was president, it was argued that marijuana was a “gateway” drug, claiming that 80% of marijuana smokers went on to use harder drugs like cocaine. The problem is, this isn’t true. The true claim is that 80% of harder drug users first smoked marijuana. The difference is one of base: 80% of marijuana smokers using hard drugs, vs. 80% of hard drug users having smoked marijuana. These numbers are not equivalent. As it turns out, only one in 2,400 marijuana users actually go on to use harder drugs[2].
Example 6
There are about 75 QFC supermarkets in the U.S. Albertsons has about 215 stores. Compare the size of the two companies.
Solution
When we make comparisons, we must ask first whether an absolute or relative comparison. The absolute difference is \(215 – 75 = 140\). From this, we could say “Albertsons has 140 more stores than QFC.” However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful. There are two different relative changes we could calculate, depending on which store we use as the base:
Using QFC as the base, \(\frac{140}{75}=1.867\).
This tells us Albertsons is 186.7% larger than QFC.
Using Albertsons as the base, \(\frac{140}{215}=0.651\).
This tells us QFC is 65.1% smaller than Albertsons.
Notice both of these are showing percent differences . We could also calculate the size of Albertsons relative to QFC: , which tells us Albertsons is 2.867 times the size of QFC. Likewise, we could calculate the size of QFC relative to Albertsons: , which tells us that QFC is 34.9% of the size of Albertsons.
Example 7
Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started?
Solution
To answer this question, suppose the value started at $100. After one week, the value dropped by 60%:
\(\$ 100-\$ 100(0.60)=\$ 100-\$ 60=\$ 40\).
In the next week, notice that base of the percent has changed to the new value, $40. Computing the 75% increase:
\(\$ 40+\$ 40(0.75)=\$ 40+\$ 30=\$ 70\).
In the end, the stock is still $30 lower, or \(\frac{\$ 30}{\$ 100}=30 \%\) lower, valued than it started.
Try it Now 2
The U.S. federal debt at the end of 2001 was $5.77 trillion, and grew to $6.20 trillion by the end of 2002. At the end of 2005 it was $7.91 trillion, and grew to $8.45 trillion by the end of 2006[3]. Calculate the absolute and relative increase for 2001-2002 and 2005-2006. Which year saw a larger increase in federal debt?
- Answer
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2001-2002: Absolute change: $0.43 trillion. Relative change: 7.45%
2005-2006: Absolute change: $0.54 trillion. Relative change: 6.83%
2005-2006 saw a larger absolute increase, but a smaller relative increase.
Example 8
A Seattle Times article on high school graduation rates reported “The number of schools graduating 60 percent or fewer students in four years – sometimes referred to as “dropout factories” – decreased by 17 during that time period. The number of kids attending schools with such low graduation rates was cut in half.”
a) Is the “decrease by 17” number a useful comparison?
b) Considering the last sentence, can we conclude that the number of “dropout factories” was originally 34?
Solution
a) This number is hard to evaluate, since we have no basis for judging whether this is a larger or small change. If the number of “dropout factories” dropped from 20 to 3, that’d be a very significant change, but if the number dropped from 217 to 200, that’d be less of an improvement.
b) The last sentence provides relative change which helps put the first sentence in perspective. We can estimate that the number of “dropout factories” was probably previously around 34. However, it’s possible that students simply moved schools rather than the school improving, so that estimate might not be fully accurate.
Example 9
In the 2004 vice-presidential debates, Edwards's claimed that US forces have suffered "90% of the coalition casualties" in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies "have taken almost 50 percent" of the casualties. Who is correct?
Solution
Without more information, it is hard for us to judge who is correct, but we can easily conclude that these two percents are talking about different things, so one does not necessarily contradict the other. Edward’s claim was a percent with coalition forces as the base of the percent, while Cheney’s claim was a percent with both coalition and Iraqi security forces as the base of the percent. It turns out both statistics are in fact fairly accurate.
Try it Now 3
In the 2012 presidential elections, one candidate argued that “the president’s plan will cut $716 billion from Medicare, leading to fewer services for seniors,” while the other candidate rebuts that “our plan does not cut current spending and actually expands benefits for seniors, while implementing cost saving measures.” Are these claims in conflict, in agreement, or not comparable because they’re talking about different things?
- Answer
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Without more information, it is hard to judge these arguments. This is compounded by the complexity of Medicare. As it turns out, the $716 billion is not a cut in current spending, but a cut in future increases in spending, largely reducing future growth in health care payments. In this case, at least the numerical claims in both statements could be considered at least partially true. Here is one source of more information if you’re interested: http://factcheck.org/2012/08/a-campaign-full-of-mediscare/
We’ll wrap up our review of percents with a couple cautions. First, when talking about a change of quantities that are already measured in percents, we have to be careful in how we describe the change.
Example 10
A politician’s support increases from 40% of voters to 50% of voters. Describe the change.
Solution
We could describe this using an absolute change: \(|50 \%-40 \%|=10 \%\). Notice that since the original quantities were percents, this change also has the units of percent. In this case, it is best to describe this as an increase of 10 percentage points .
In contrast, we could compute the percent change: \(\frac{10 \%}{40 \%}=0.25=25 \%\) increase. This is the relative change, and we’d say the politician’s support has increased by 25%.
Lastly, a caution against averaging percents.
Example 11
A basketball player scores on 40% of 2-point field goal attempts, and on 30% of 3-point of field goal attempts. Find the player’s overall field goal percentage.
Solution
It is very tempting to average these values, and claim the overall average is 35%, but this is likely not correct, since most players make many more 2-point attempts than 3-point attempts. We don’t actually have enough information to answer the question. Suppose the player attempted 200 2-point field goals and 100 3-point field goals. Then they made 200(0.40) = 80 2-point shots and \(100(0.30) = 30\) 3-point shots. Overall, they made 110 shots out of 300, for a \(\frac{110}{300}=0.367=36.7 \%\) overall field goal percentage.
[1] www.factcheck.org/cheney_edwards_mangle_facts.html
[2] http://tvtropes.org/pmwiki/pmwiki.php/Main/LiesDamnedLiesAndStatistics
[3] www.whitehouse.gov/sites/defa...s/hist07z1.xls


10.1: Youth Crime
[1] Sensationalized media exposure in the 1990s facilitated the public’s fear of youth crime, which resulted in get tough legislation and a perceived need to “do something” about juvenile crime. [2] The juvenile court was criticized for its inability to control youth crime and, as a result, policies shifted from rehabilitation to punishment of juvenile offenders. [3] This punishment included an increase in the number of states that adopted new legislation or revised their previous statutes to facilitate the transfer of youthful offenders from juvenile court to criminal court to be tried as adults. [4]
-
Benekos, P., & Merlo, A. (2004).
Controversies in juvenile justice and delinquency.
Anderson Publishing.
↵
-
Myers, D.L. (2001).
Excluding violent youth from the juvenile court: The effectiveness of legislative waiver
.New York: LBF Scholarly Press.
↵
-
Feld, B.C. (2001). Race, youth violence, and the changing jurisprudence of waiver.
Behavioral Sciences & the Law, 19
(1), 3-22.
↵
-
Snyder, H. N., & Sickmund, M. (2006). Juvenile offenders and victims: 2006 National Report. Washington, DC:
U.S. Department of Justice, Office of Justice Programs, Office of Juvenile Justice and Delinquency Prevention
↵


8.4: Reading- Causes of Earthquakes
The following video explains the cause of earthquakes.
In an earthquake, the initial point where the rocks rupture in the crust is called the focus . The epicenter is the point on the land surface that is directly above the focus. In about 75% of earthquakes, the focus is in the top 10 to 15 kilometers (6 to 9 miles) of the crust. Shallow earthquakes cause the most damage because the focus is near where people live. However, it is the epicenter of an earthquake that is reported by scientists and the media (figure 1).
Watch this animation summarizing elastic rebound theory.
Tectonic earthquakes occur anywhere in the earth where there is sufficient stored elastic strain energy to drive fracture propagation along a fault plane . The sides of a fault move past each other smoothly and aseismically only if there are no irregularities or asperities along the fault surface that increase the frictional resistance. Most fault surfaces do have such asperities and this leads to a form of stick-slip behavior. Once the fault has locked, continued relative motion between the plates leads to increasing stress and therefore, stored strain energy in the volume around the fault surface. This continues until the stress has risen sufficiently to break through the asperity, suddenly allowing sliding over the locked portion of the fault, releasing the stored energy. [1]
This energy is released as a combination of radiated elastic strain seismic waves, frictional heating of the fault surface, and cracking of the rock, thus causing an earthquake. This process of gradual build-up of strain and stress punctuated by occasional sudden earthquake failure is referred to as the elastic-rebound theory. It is estimated that only 10 percent or less of an earthquake’s total energy is radiated as seismic energy. Most of the earthquake’s energy is used to power the earthquake fracture growth or is converted into heat generated by friction. Therefore, earthquakes lower the Earth’s available elastic potential energy and raise its temperature, though these changes are negligible compared to the conductive and convective flow of heat out from the Earth’s deep interior. [2]
Earthquake Fault Types
There are three main types of fault, all of which may cause an interplate earthquake: normal, reverse (thrust) and strike-slip. Normal and reverse faulting are examples of dip-slip, where the displacement along the fault is in the direction of dip and movement on them involves a vertical component. Normal faults occur mainly in areas where the crust is being extended such as a divergent boundary. Reverse faults occur in areas where the crust is being shortened such as at a convergent boundary. Strike-slip faults are steep structures where the two sides of the fault slip horizontally past each other; transform boundaries are a particular type of strike-slip fault. Many earthquakes are caused by movement on faults that have components of both dip-slip and strike-slip; this is known as oblique slip.
Reverse faults, particularly those along convergent plate boundaries are associated with the most powerful earthquakes, megathrust earthquakes, including almost all of those of magnitude 8 or more. Strike-slip faults, particularly continental transforms, can produce major earthquakes up to about magnitude 8. Earthquakes associated with normal faults are generally less than magnitude 7. For every unit increase in magnitude, there is a roughly thirtyfold increase in the energy released.  An 8.6 magnitude earthquake releases the same amount of energy as 10,000 atomic bombs like those used in World War II.
This is so because the energy released in an earthquake, and thus its magnitude, is proportional to the area of the fault that ruptures2.0.CO;2." id="return-footnote-276-3" href="#footnote-276-3" aria-label="Footnote 3"> [3] and the stress drop. Therefore, the longer the length and the wider the width of the faulted area, the larger the resulting magnitude. The topmost, brittle part of the Earth’s crust, and the cool slabs of the tectonic plates that are descending down into the hot mantle, are the only parts of our planet which can store elastic energy and release it in fault ruptures. Rocks hotter than about 300 degrees Celsius flow in response to stress; they do not rupture in earthquakes. [4] The maximum observed lengths of ruptures and mapped faults (which may break in a single rupture) are approximately 1000 km. Examples are the earthquakes in Chile, 1960; Alaska, 1957; Sumatra, 2004, all in subduction zones. The longest earthquake ruptures on strike-slip faults, like the San Andreas Fault (1857, 1906), the North Anatolian Fault in Turkey (1939) and the Denali Fault in Alaska (2002), are about half to one third as long as the lengths along subducting plate margins, and those along normal faults are even shorter.
The most important parameter controlling the maximum earthquake magnitude on a fault is however not the maximum available length, but the available width because the latter varies by a factor of 20. Along converging plate margins, the dip angle of the rupture plane is very shallow, typically about 10 degrees. [5] Thus the width of the plane within the top brittle crust of the Earth can become 50 to 100 km (Japan, 2011; Alaska, 1964), making the most powerful earthquakes possible.
Strike-slip faults tend to be oriented near vertically, resulting in an approximate width of 10 km within the brittle crust, [6] thus earthquakes with magnitudes much larger than 8 are not possible. Maximum magnitudes along many normal faults are even more limited because many of them are located along spreading centers, as in Iceland, where the thickness of the brittle layer is only about 6 km. [7]
In addition, there exists a hierarchy of stress level in the three fault types. Thrust faults are generated by the highest, strike slip by intermediate, and normal faults by the lowest stress levels. [8] This can easily be understood by considering the direction of the greatest principal stress, the direction of the force that “pushes” the rock mass during the faulting. In the case of normal faults, the rock mass is pushed down in a vertical direction, thus the pushing force ( greatest principal stress) equals the weight of the rock mass itself. In the case of thrusting, the rock mass “escapes” in the direction of the least principal stress, namely upward, lifting the rock mass up, thus the overburden equals the least principal stress. Strike-slip faulting is intermediate between the other two types described above. This difference in stress regime in the three faulting environments can contribute to differences in stress drop during faulting, which contributes to differences in the radiated energy, regardless of fault dimensions.
Earthquakes away from Plate Boundaries
Where plate boundaries occur within the continental lithosphere, deformation is spread out over a much larger area than the plate boundary itself. In the case of the San Andreas fault continental transform, many earthquakes occur away from the plate boundary and are related to strains developed within the broader zone of deformation caused by major irregularities in the fault trace (e.g., the “Big bend” region). The Northridge earthquake was associated with movement on a blind thrust within such a zone. Another example is the strongly oblique convergent plate boundary between the Arabian and Eurasian plates where it runs through the northwestern part of the Zagros Mountains. The deformation associated with this plate boundary is partitioned into nearly pure thrust sense movements perpendicular to the boundary over a wide zone to the southwest and nearly pure strike-slip motion along the Main Recent Fault close to the actual plate boundary itself. This is demonstrated by earthquake focal mechanisms. [9]
All tectonic plates have internal stress fields caused by their interactions with neighboring plates and sedimentary loading or unloading (e.g. deglaciation). [10] These stresses may be sufficient to cause failure along existing fault planes, giving rise to intraplate earthquakes. [11]
Shallow-Focus and Deep-Focus Earthquakes
The majority of tectonic earthquakes originate at the ring of fire in depths not exceeding tens of kilometers. Earthquakes occurring at a depth of less than 70 km are classified as shallow-focus earthquakes, while those with a focal-depth between 70 and 300 km are commonly termed mid-focus or intermediate-depth earthquakes. In subduction zones, where older and colder oceanic crust descends beneath another tectonic plate, deep-focus earthquakes may occur at much greater depths (ranging from 300 up to 700 kilometers). [12]
These seismically active areas of subduction are known as Wadati–Benioff zones. Deep-focus earthquakes occur at a depth where the subducted lithosphere should no longer be brittle, due to the high temperature and pressure. A possible mechanism for the generation of deep-focus earthquakes is faulting caused by olivine undergoing a phase transition into a spinel structure. [13]
Earthquakes and Volcanic Activity
Earthquakes often occur in volcanic regions and are caused there, both by tectonic faults and the movement of magma in volcanoes. Such earthquakes can serve as an early warning of volcanic eruptions, as during the 1980 eruption of Mount St. Helens. [14] Earthquake swarms can serve as markers for the location of the flowing magma throughout the volcanoes. These swarms can be recorded by seismometers and tiltmeters (a device that measures ground slope) and used as sensors to predict imminent or upcoming eruptions. [15]
Rupture Dynamics
A tectonic earthquake begins by an initial rupture at a point on the fault surface, a process known as nucleation. The scale of the nucleation zone is uncertain, with some evidence, such as the rupture dimensions of the smallest earthquakes, suggesting that it is smaller than 100 m while other evidence, such as a slow component revealed by low-frequency spectra of some earthquakes, suggest that it is larger. The possibility that the nucleation involves some sort of preparation process is supported by the observation that about 40% of earthquakes are preceded by foreshocks. Once the rupture has initiated, it begins to propagate along the fault surface. The mechanics of this process are poorly understood, partly because it is difficult to recreate the high sliding velocities in a laboratory. Also the effects of strong ground motion make it very difficult to record information close to a nucleation zone. [16]
Rupture propagation is generally modeled using a fracture mechanics approach, likening the rupture to a propagating mixed mode shear crack. The rupture velocity is a function of the fracture energy in the volume around the crack tip, increasing with decreasing fracture energy. The velocity of rupture propagation is orders of magnitude faster than the displacement velocity across the fault. Earthquake ruptures typically propagate at velocities that are in the range 70–90% of the S-wave velocity, and this is independent of earthquake size. A small subset of earthquake ruptures appear to have propagated at speeds greater than the S-wave velocity. These supershear earthquakes have all been observed during large strike-slip events. The unusually wide zone of coseismic damage caused by the 2001 Kunlun earthquake has been attributed to the effects of the sonic boom developed in such earthquakes. Some earthquake ruptures travel at unusually low velocities and are referred to as slow earthquakes. A particularly dangerous form of slow earthquake is the tsunami earthquake, observed where the relatively low felt intensities, caused by the slow propagation speed of some great earthquakes, fail to alert the population of the neighboring coast, as in the 1896 Sanriku earthquake. [17]
Earthquake Clusters
Most earthquakes form part of a sequence, related to each other in terms of location and time. [18] Most earthquake clusters consist of small tremors that cause little to no damage, but there is a theory that earthquakes can recur in a regular pattern. [19]
Aftershocks
An aftershock is an earthquake that occurs after a previous earthquake, the mainshock. An aftershock is in the same region of the main shock but always of a smaller magnitude. If an aftershock is larger than the main shock, the aftershock is redesignated as the main shock and the original main shock is redesignated as a foreshock. Aftershocks are formed as the crust around the displaced fault plane adjusts to the effects of the main shock. [20]
Earthquake Swarms
Earthquake swarms are sequences of earthquakes striking in a specific area within a short period of time. They are different from earthquakes followed by a series of aftershocks by the fact that no single earthquake in the sequence is obviously the main shock, therefore none have notable higher magnitudes than the other. An example of an earthquake swarm is the 2004 activity at Yellowstone National Park. [21] In August 2012, a swarm of earthquakes shook Southern California’s Imperial Valley, showing the most recorded activity in the area since the 1970s. [22]
Sometimes a series of earthquakes occur in what has been called an earthquake storm, where the earthquakes strike a fault in clusters, each triggered by the shaking or stress redistribution of the previous earthquakes. Similar to aftershocks but on adjacent segments of fault, these storms occur over the course of years, and with some of the later earthquakes as damaging as the early ones. Such a pattern was observed in the sequence of about a dozen earthquakes that struck the North Anatolian Fault in Turkey in the 20th century and has been inferred for older anomalous clusters of large earthquakes in the Middle East. [23]
- Ohnaka, M. (2013). The Physics of Rock Failure and Earthquakes . Cambridge University Press. p. 148. ↵
- Spence, William; S. A. Sipkin; G. L. Choy (1989). " Measuring the Size of an Earthquake ." United States Geological Survey . Retrieved 2006-11-03 . ↵
- Wyss, M. (1979). "Estimating expectable maximum magnitude of earthquakes from fault dimensions". Geology 7 (7): 336–340 doi: 10.1130/0091-7613(1979)7<336:EMEMOE>2.0.CO;2 . ↵
- Sibson R. H. (1982) "Fault Zone Models, Heat Flow, and the Depth Distribution of Earthquakes in the Continental Crust of the United States," Bulletin of the Seismological Society of America , Vol 72, No. 1, pp. 151–163. See also Sibson, R. H. (2002) "Geology of the crustal earthquake source," International handbook of earthquake and engineering seismology , Volume 1, Part 1, page 455, eds. W.H.K. Lee, H. Kanamori, P.C. Jennings, and C. Kisslinger, Academic Press. ↵
- " Global Centroid Moment Tensor Catalog ." Globalcmt.org . Retrieved 2011-07-24 . ↵
- " Instrumental California Earthquake Catalog ." WGCEP. Retrieved 2011-07-24. ↵
- Hjaltadóttir S., 2010, "Use of relatively located microearthquakes to map fault patterns and estimate the thickness of the brittle crust in Southwest Iceland." See also " Reports and publications | Seismicity | Icelandic Meteorological office ." En.vedur.is. Retrieved 2011-07-24. ↵
- Schorlemmer, D.; Wiemer, S.; Wyss, M. (2005). "Variations in earthquake-size distribution across different stress regimes." Nature 437 (7058): 539–542. doi: 10.1038/nature04094 . ↵
- Talebian, M; Jackson, J (2004). "A reappraisal of earthquake focal mechanisms and active shortening in the Zagros mountains of Iran." Geophysical Journal International 156 (3): 506–526. doi: 10.1111/j.1365-246X.2004.02092.x . ↵
- Nettles, M.; Ekström, G. (May 2010). "Glacial Earthquakes in Greenland and Antarctica." Annual Review of Earth and Planetary Sciences 38 (1): 467–491. doi: 10.1146/annurev-earth-040809-152414 . ↵
- Noson, Qamar, and Thorsen (1988). Washington State Earthquake Hazards: Washington State Department of Natural Resources . Washington Division of Geology and Earth Resources Information Circular 85. ↵
- " M7.5 Northern Peru Earthquake of 26 September 2005 " (PDF). National Earthquake Information Center. 17 October 2005 . Retrieved 2008-08-01 . ↵
- Greene II, H. W.; Burnley, P. C. (October 26, 1989). "A new self-organizing mechanism for deep-focus earthquakes." Nature 341 (6244): 733–737. doi: 10.1038/341733a0 . ↵
- Foxworthy and Hill (1982). Volcanic Eruptions of 1980 at Mount St. Helens, The First 100 Days: USGS Professional Paper 1249 . ↵
- Watson, John; Watson, Kathie (January 7, 1998). " Volcanoes and Earthquakes ." United States Geological Survey . Retrieved May 9, 2009 . ↵
- National Research Council (U.S.). Committee on the Science of Earthquakes (2003). "5. Earthquake Physics and Fault-System Science." Living on an Active Earth: Perspectives on Earthquake Science . Washington D.C.: National Academies Press. p. 418. Retrieved 8 July 2010 . ↵
- Ibid. ↵
- "What are Aftershocks, Foreshocks, and Earthquake Clusters?" ↵
- " Repeating Earthquakes ." United States Geological Survey. January 29, 2009 . Retrieved May 11, 2009 . ↵
- "What are Aftershocks, Foreshocks, and Earthquake Clusters?" ↵
- " Earthquake Swarms at Yellowstone ." United States Geological Survey . Retrieved 2008-09-15 . ↵
- Duke, Alan. " Quake 'swarm' shakes Southern California ." CNN . Retrieved 27 August 2012 . ↵
- Amos Nur; Cline, Eric H. (2000). "Poseidon's Horses: Plate Tectonics and Earthquake Storms in the Late Bronze Age Aegean and Eastern Mediterranean" (PDF). Journal of Archaeological Science 27 (1): 43–63. See also " Earthquake Storms ." Horizon . 1 April 2003 . Retrieved 2007-05-02 . ↵
Contributors and Attributions
- 7.2: The Nature of Earthquakes. Provided by : CK-12. Located at : http://www.ck12.org/book/CK-12-Earth-Science-For-High-School/section/7.2/ . License : CC BY-NC: Attribution-NonCommercial
- Naturally occurring earthquakes. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Earthquake#Naturally_occurring_earthquakes . License : CC BY-SA: Attribution-ShareAlike
- Earthquakes 101. Authored by : National Geographic. Located at : https://youtu.be/VSgB1IWr6O4 . License : All Rights Reserved . License Terms : Standard YouTube License


21.2.3: Represent an Inequality as an Interval on a Number Line
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Learning Outcomes
- Graph and inequality on a number line.
- Graph the complement on a number line for both continuous and discrete variables.
Inequalities come up frequently in statistics and it is often helpful to plot the inequality on the number line in order to visualize the inequality. This helps both for inequalities that involve real numbers and for inequalities that refer to just integer values. As an extension of this idea, we often want to look at the complement of an inequality, that is all numbers that make the inequality false. In this section we will look at examples that accomplish this task.
Sketching an Inequality on a number line where the possible values are real numbers.
There are four different inequalities: \(<,\:\le,\:>,\:\ge\). What makes this the most challenging is when they are expressed in words. Here are some of the words that are used for each:
- \(<\): "Less Than", "Smaller", "Lower", "Younger"
- \(\le\): "Less Than or Equal to", "At Most", "No More Than", "Not to Exceed"
- \(>\): "Greater Than", "Larger", "Higher", "Bigger", "Older", "More Than"
- \(\ge\): "Greater Than or Equal to", "At Least", "No Less than"
These are the most common words that correspond to the inequalities, but there are others that come up less frequently.
Example \(\PageIndex{1}\)
Graph the inequality: \(3<x\le5\) on a number line
Solution
First notice that the interval does not include the number 3, but does include the number 5. We can represent not including a number with an open circle and including a number with a closed circle. The number line representation of the inequality is shown below.
Example \(\PageIndex{2}\)
In statistics, we often want to find probabilities of an event being at least as large or no more than a given value. It helps to first plot the interval on a number line. Suppose you want to find the probability that you will have to wait in line for at least 4minutes. Sketch this inequality on a number line.
Solution
First, notice that "At Least" has the symbol \(\ge\). Thus, we have a closed circle on the number 4. There is no upper bound, so we draw a long arrow from 4 to the right of 4. The solution is shown below
Example \(\PageIndex{3}\)
Another main topic that comes up in statistics is confidence intervals. For example in recent poll to see the percent of Americans who think that Congress is doing a good job found that a 95% confidence interval had lower bound of 0.18 and an upper bound of 0.24. This can be written as [0.18,0,24]. Sketch this interval on the number line.
Solution
The first thing we need to do is decide on the tick marks to put on the number line. If we counted by 1's, then the interval of interest would be too small to stand out. Instead we will count by 0.1's. The number line is shown below.
Example \(\PageIndex{4}\)
Often in statistics, we deal with discrete variables. Most of the time this will mean that only whole number values can occur. For example, you want to find out the probability that a college student is taking at most three classes. Graph this on a number line.
Solution
First note that the outcomes can only be whole numbers. Second, note that "at most" means \(\le\). Thus the possible outcomes are: 0, 1, 2, and 3. The number line below displays these outcomes.
Graphing the Complement
In statistics, we often want to graph the complement of an interval. The complement means everything that is not in the interval.
Example \(\PageIndex{5}\)
Graph the complement of the interval [2,4).
Solution
Notice that the complement of numbers inside the interval between 2 and 4 is the numbers outside that interval. This will consist of the numbers to the left of 2 and to the right of 4. Since the number 2 is included in the original interval, it will not be included in the complement. Since the number 4 is not included in the original interval, it will be included in the complement. The complement is shown on the number line below.
Example \(\PageIndex{6}\)
Some calculators can only find probabilities for values less than a certain number. If we want the probability of an interval greater than a number, we need to use the complement. Suppose that you want to find the probability that a person will have traveled to more than two foreign countries in the last twelve months. Find the complement of this and graph it on a number line.
Solution
First notice that only whole numbers are possible since it does not make sense to go to a fractional number of countries. Second note that the lowest number that is more than 2 is 3. If 3 is included in the original list, then 3 will not be included in the complement. Thus, the highest number that is in the complement of "more than 2" is 2. The number line below shows the complement of more than 2.
Exercise
Suppose you want to find the probability that at least 4 people in your class have a last name that contains the letter "W". To make this calculation you will need to first find the complement of "at least 4". Sketch this complement on the number line.


6.1: A- Partial Wave Analysis
This Appendix describes the method of partial wave analysis , which can be used to solve 3D scattering problems with spherically symmetric scattering potentials. Such a potential, \(V(r)\) , depends only on the radial distance \(r = \sqrt{x^2 + y^2 + z^2}\) and not on direction. This typically describes a situation where a point particle or spherically-symmetric object sits at the coordinate origin, \(\mathbf{r} = 0\) , and is bombarded by incident particles.
A.1 Spherical Waves
We begin by considering “exterior” solutions to the Schrödinger wave equation. Far from the scatterer, where \(V(r) \rightarrow 0\) , the Schrödinger wave equation can be rearranged into
\[\Big(\nabla^2 + k^2\Big) \psi(\mathbf{r}) = 0,\;\;\;\mathrm{where}\;\;k = \sqrt{2mE/\hbar^2}.\]
This partial differential equation is called the Helmholtz equation . We emphasize that \(E\) plays the role of a tunable parameter, and is not an eigenvalue in an eigenproblem. For the moment, we will not specify the boundary conditions, and look instead for a general set of solutions for a given \(E\) (and hence \(k\) ).
In spherical coordinates \((r,\theta,\phi)\) , the Helmholtz equation has the explicit form
\[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\psi}{\partial\theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2\psi}{\partial\phi^2} + k^2\psi(r,\theta,\phi) = 0.\]
There is a standard procedure for solving this. The first step is to perform a separation of variables, and look for solutions of the form
\[\psi(r,\theta,\phi) = A(r) Y_{\ell m}(\theta,\phi),\]
where \(A(r)\) is a function to be determined and \(Y_{\ell m}(\theta,\phi)\) is a special function known as a spherical harmonic . Spherical harmonics are functions designed specifically to represent the angular dependence of solutions with definite angular momenta. In the context of quantum mechanics, \(\ell\) and \(m\) are the quantum numbers representing the total angular momentum and the \(z\) -component of the angular momentum. It can be shown that the indices \(\ell\) and \(m\) must be integers satisfying \(\ell \ge 0\) and \(-\ell\le m \le \ell\) , in order for \(Y_{\ell m}(\theta,\phi)\) to be periodic in \(\phi\) and regular at the poles of the spherical coordinate system.
After plugging in this form for \(\psi(r,\theta,\phi)\) , the Helmholtz equation reduces to the following ordinary differential equation, which is a variant of the Bessel equation:
\[\frac{d}{dr}\left(r^2\frac{dA}{dr}\right) + \Big[k^2r^2 - \ell(\ell+1)\Big] A(r) = 0, \;\;\;\ell \in \mathbb{Z}_0^+.\]
Note that \(m\) drops out of the equation. Thus, \(A(r)\) depends on \(\ell\) , but not on \(m\) .
The ordinary differential equation has two linearly independent real solutions,
\(j_\ell(kr)\)
and
\(y_\ell(kr)\)
, which are called
spherical Bessel functions
. Most scientific computing packages provide functions to calculate these; Scientific Python, for example, has
scipy.special.spherical_jn
and
scipy.special.spherical_yn
. Some spherical Bessel functions are plotted below:
Note that the \(y_\ell\) functions diverge at \(kr\rightarrow 0\) . This does not bother us, for we are interested in solutions defined in the exterior region, away from the coordinate origin.
For large values of the input, the spherical Bessel functions have the limiting forms
\[\begin{align} \begin{aligned}j_\ell(kr)\; &\overset{kr\rightarrow\infty}{\longrightarrow} \; \frac{\sin(kr-\frac{\ell\pi}{2})}{kr} \\ y_\ell(kr)\; &\overset{kr\rightarrow\infty}{\longrightarrow} \; - \frac{\cos(kr-\frac{\ell\pi}{2})}{kr}.\end{aligned}\end{align}\]
Since we are interested in incoming and outgoing spherical waves, it is convenient to define
\[h_\ell^\pm(kr) = j_\ell(kr) \pm i y_\ell(kr).\]
This complex function is called a spherical Hankel function of the first kind ( \(+\) ) or second kind ( \(-\) ). It solves the same differential equation, but has the limiting form
\[h_\ell^\pm(kr)\; \overset{kr\rightarrow\infty}{\longrightarrow} \; \pm \frac{\exp\left[\pm i(kr-\frac{\ell\pi}{2})\right]}{ikr}.\]
Using it, we can write down a solution to the Helmholtz equation, in the form
\[\Psi_{\ell m}^\pm(\mathbf{r}) = h_{\ell}^\pm(kr) \,Y_{\ell m}(\theta,\phi).\]
This describes a spherical wave that is outgoing ( \(+\) ) or incoming ( \(-\) ), and that has a definite angular momentum described by the quantum numbers \(\ell\) and \(m\) .
Because the Helmholtz equation is linear, any linear combination of spherical waves, with various values of \((\ell,m)\) , is also a solution:
\[\psi(\mathbf{r}) = \sum_\pm \sum_{\ell = 0}^\infty \sum_{m = - \ell}^\ell c_{\ell m}^\pm \Psi_{\ell m}^\pm(\mathbf{r}).\]
It can be shown that the spherical waves form a complete solution basis. In other words, any solution to the 3D Helmholtz equation within the exterior region can be written in the above form, for some choice of complex coefficients \(c_{\ell m}^\pm\) . (By the way, these spherical waves are appropriately normalized so that the flux associated with each term is directly proportional to \(|c_{\ell m}^\pm|^2\) .)
A.2 The Scattering Matrix
For a given scattering problem, the exterior wavefunction is described by the complex numbers \(\{c_{\ell m}^+\}\) and \(\{c_{\ell m}^-\}\) . These two sets of coefficients cannot, however, be independent of each other. For fixed \(V(\mathbf{r})\) and \(E\) , suppose there is an incoming spherical wave with definite angular momentum, say \(c_{\ell m}^- = 1\) for some choice of \((\ell,m)\) . After striking the scatterer, the quantum particle bounces back out to infinity, and the outgoing wavefunction is some superposition of outgoing spherical waves with a variety of angular momenta, described by certain coefficients \(\{c_{\ell' m'}^+\}\) .
Thus, for each choice of incoming wave with definite angular momentum, there is a corresponding set of outgoing-wave coefficients. Since the Schrödinger wave equation is linear, the principle of superposition states that linear combinations of scattering solutions are also valid solutions—i.e., solutions to the Schrödinger wave equation for the same \(V(\mathbf{r})\) and \(E\) . So if we supply an arbitrary set of incoming coefficients \(\{c_{\ell m}^-\}\) , the outgoing coefficients must be determined by a linear relation of the form
\[c_{\ell m}^+ = \sum_{\ell'm'} S_{\ell m, \ell' m'} \;c_{\ell' m'}^-.\]
To make the notation cleaner, we rewrite this as
\[c_{\mu}^+ = \sum_{\nu} S_{\mu \nu} \, c_{\nu}^-,\]
where each \(\mu\) or \(\nu\) denotes a pair of angular momentum quantum numbers \((\ell,m)\) , called a scattering channel . The matrix \(S\) is called a scattering matrix . Knowing \(V(\mathbf{r})\) and \(E\) , we can calculate \(S\) , and knowing \(S\) we can determine the outgoing wavefunction produced by any set of incoming waves.
Thus far, we have not specified how the “incoming” and “outgoing” waves are related to the “incident” and “scattered” waves of a scattering experiment. We now consider an incident plane wave, \(\psi_i(\mathbf{r}) = \Psi_i \exp(i\mathbf{k}_i\cdot \mathbf{r})\) , where \(|\mathbf{k}_i| = k\) . This introduces an important complication: relative to the coordinate origin, a plane wave is neither purely “incoming” nor “outgoing”! In fact, there is a mathematical identity stating that
\[\begin{align} \begin{aligned}e^{i\mathbf{k}_i \cdot \mathbf{r}} &= \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell 4 \pi j_{\ell}(kr) e^{i\ell\pi/2} \, Y_{\ell m}^*(\hat{\mathbf{k}}_i) \, Y_{\ell m}(\hat{\mathbf{r}})\\ &= \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell 2 \pi \left[h_{\ell}^+(kr) + h_{\ell}^-(kr)\right] e^{i\ell\pi/2} \, Y_{\ell m}^*(\hat{\mathbf{k}}_i) \, Y_{\ell m}(\hat{\mathbf{r}}).\end{aligned} \label{plane_wave_decomp}\end{align}\]
Here, \(\hat{\mathbf{k}}_i\) denotes the angular components (in spherical coordinates) of the incident wave-vector \(\mathbf{k}_i\) , while \(\hat{\mathbf{r}}\) likewise denotes the angular components of the position vector \(\mathbf{r}\) . Equation \(\eqref{plane_wave_decomp}\) informs us that the incident plane wave can be decomposed into a superposition of incoming and outgoing spherical waves, with the wave coefficients
\[c^{\pm}_{i, \ell m} = 2 \pi e^{i\ell\pi/2} \, Y_{\ell m}^*(\hat{\mathbf{k}}_i)\; \Psi_i.\]
As described in Chapter 1, the total wavefunction in a scattering problem is the sum of the incident wavefunction \(\psi_i(\mathbf{r})\) and the scattered wavefunction \(\psi_s(\mathbf{r})\) . The latter must be a superposition of only outgoing spherical waves; let us denote the coefficients by \(c^+_{s,\ell m}\) . The scattering matrix relation can then be re-written as
\[\begin{align} c^+_\mu &= c^+_{i,\mu} + c^+_{s,\mu} = \sum_{\mu\nu} S_{\mu\nu} c^-_{i,\nu} \\ \Rightarrow \;\;\; c^+_{s,\ell m} &= 2 \pi \sum_{\ell' m'} \Big(S_{\ell m, \ell' m'} - \delta_{\ell \ell'}\delta_{mm'}\Big) e^{i\ell'\pi/2} \, Y_{\ell' m'}^*(\hat{\mathbf{k}}_i)\; \Psi_i.\end{align}\]
Using this, the scattered wavefunction can be written as
\[\begin{align}\begin{aligned}\psi_s(\mathbf{r}) &= \sum_{\ell m} c^+_{s,\ell m} h_{\ell}^+(kr) \, Y_{\ell m}(\hat{\mathbf{r}}) \\ &= \Psi_i \sum_{\ell m} \sum_{\ell' m'} 2 \pi \Big(S_{\ell m, \ell' m'} - \delta_{\ell \ell'}\delta_{mm'}\Big) e^{i\ell'\pi/2} \, Y_{\ell' m'}^*(\hat{\mathbf{k}}_i)\; h_{\ell}^+(kr) \, Y_{\ell m}(\hat{\mathbf{r}}).\end{aligned}\end{align}\]
Taking the large- \(r\) expansion of the spherical Hankel functions yields
\[\psi_s(\mathbf{r}) \overset{r\rightarrow\infty}{\longrightarrow} \, \Psi_i \frac{e^{ikr}}{r} \; \left[ \frac{2 \pi}{ik}\, \sum_{\ell m} \sum_{\ell' m'} \Big(S_{\ell m, \ell' m'} - \delta_{\ell \ell'}\delta_{mm'}\Big) \, e^{-i(\ell-\ell')\pi/2} \, Y_{\ell' m'}^*(\hat{\mathbf{k}}_i)\; Y_{\ell m}(\hat{\mathbf{r}})\right].\]
The quantity in square brackets is precisely what we call the scattering amplitude:
\[f(\mathbf{k}_i \rightarrow k\hat{\mathbf{r}}) = \frac{2 \pi}{ik}\, \sum_{\ell m} \sum_{\ell' m'} \Big(S_{\ell m, \ell' m'} - \delta_{\ell \ell'}\delta_{mm'}\Big) \, e^{-i(\ell-\ell')\pi/2} \, Y_{\ell' m'}^*(\hat{\mathbf{k}}_i)\; Y_{\ell m}(\hat{\mathbf{r}}).\]
A.3 Spherically Symmetric Scattering Potentials
Generally, the scattering matrix needs to be calculated numerically. The process is greatly simplified if the scattering potential is spherically symmetric, i.e. \(V(\mathbf{r}) = V(r)\) . In that case, angular momentum is conserved, so an incoming spherical wave with angular momentum quantum numbers \((\ell,m)\) must scatter exclusively into an outgoing spherical wave with the same \((\ell,m)\) . This means that the scattering matrix components have the form
\[S_{\ell m, \ell'm'} = s_{\ell m}\, \delta_{\ell\ell'}\, \delta_{mm'}.\]
The scattering amplitude then simplifies to
\[f(\mathbf{k}_i \rightarrow k\hat{\mathbf{r}}) = \frac{2 \pi}{ik}\, \sum_{\ell m} \Big(s_{\ell m} - 1\Big) \, Y_{\ell m}^*(\hat{\mathbf{k}}_i)\; Y_{\ell m}(\hat{\mathbf{r}}).\]
Our task is now to obtain the \(s_{\ell m}\) ’s. The procedure is very similar to what we already went through in Section A.1. The total wavefunction satisfies the Schrödinger wave equation, which can be written in spherical coordinates as
\[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\psi}{\partial\theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2\psi}{\partial\phi^2} + K^2(r) \psi(r,\theta,\phi) = 0,\]
where
\[K^2(r) = \sqrt{\frac{2m\big[E-V(r)\big]}{\hbar^2}}.\]
This is similar to the Helmholtz equation, but with the constant \(k^2\) replaced by a function \(K^2(r)\) . In scattering channel \((\ell,m)\) , the solution has the form
\[\psi(r,\theta,\phi) = A(r) \, Y_{\ell m}(\theta, \phi).\]
Upon substitution into the Schrödinger wave equation, we find that \(A(r)\) must satisfy
\[\frac{d}{dr}\left(r^2\frac{dA}{dr}\right) + \Big[K^2(r)\, r^2 - \ell(\ell+1)\Big] A(r) = 0, \;\;\;\ell \in \mathbb{Z}_0^+.\]
As before, the equation for \(A(r)\) does not involve \(m\) ; hence, the scattering matrix components do not depend on \(m\) , and can be written as simply
\[s_{\ell m} \,=\, s_\ell.\]
For any given \(V(r)\) , we can solve the second-order ordinary differential equation numerically by supplying two boundary conditions at \(r=0\) , integrating up to a large value of \(r\) , and matching to the exterior solution
\[A(r) \; \overset{r\rightarrow\infty}{\longrightarrow} \; c^-_\ell h^-_\ell(kr) + c^+_\ell h^+_\ell(kr) = c^-_\ell \Big(h^-_\ell(kr) + s_\ell h^+_\ell(kr)\Big).\]
The value of \(s_\ell\) can then be extracted.
To simplify the problem even further, let the scattering potential take the form of a spherical potential well of radius \(R\) and depth \(U\) :
\[V(r) = \begin{cases}-U &\mathrm{for}\;\; r < R, \\ \;\;\; 0 & \mathrm{otherwise}.\end{cases}\]
We will take \(U > 0\) , so that the potential is attractive. (The interested reader can work through the repulsive case, \(U < 0\) . The process is almost the same as what is presented below, except that for some values of \(E\) , the wave inside the scatterer becomes evanescent.) Now, the Schrödinger wave equation in the interior region reduces to the Helmholtz equation, but with \(k\) replaced with
\[q = \sqrt{2m(E+U)/\hbar^2}.\]
Note that \(q \in \mathbb{R}^+\) for \(E > 0\) , since we have assumed that \(U > 0\) . The elementary solutions for \(A(r)\) in the interior region are \(j_\ell(qr)\) and \(y_\ell(qr)\) . However, we must exclude the latter, since they diverge at \(r = 0\) . (When we got to a similar point in Section A.1, we did not exclude the spherical Bessel functions of the second kind, because at the time we were concerned with solutions in the exterior region.) We thus arrive at a solution of the form
\[A(r) = \begin{cases} \alpha_\ell\, j_\ell(qr), & r \le R \\ c^-_\ell \Big(h^-_\ell(kr) + s_\ell h^+_\ell(kr)\Big) & r \ge R.\end{cases}\]
So far, the values of \(\alpha_\ell\) , \(c^-_\ell\) , and \(s_\ell\) remain unknown. To proceed, we match the wavefunction and its derivative at the boundary \(r = R\) :
\[\begin{align} \begin{aligned} \alpha_\ell\, j_\ell(qR) &= c^-_\ell \Big(h^-_\ell(kR) + s_\ell h^+_\ell(kR)\Big) \\ \alpha_\ell\, q j_\ell'(qR) &= c^-_\ell k \Big({h^-_\ell}'(kR) + s_\ell {h^+_\ell}'(kR)\Big).\end{aligned}\end{align}\]
Here, \(j_\ell'\) denotes the derivative of the spherical Bessel function, and likewise for \({h_\ell^\pm}'\) . Taking the ratio of these two equations eliminates \(\alpha_\ell\) and \(c_\ell^-\) :
\[\frac{q j_\ell'(qR)}{j_\ell(qR)} = k \frac{{h^-_\ell}'(kR) + s_\ell {h^+_\ell}'(kR)}{h^-_\ell(kR) + s_\ell h^+_\ell(kR)}.\]
With a bit of rearrangement, this becomes
\[s_\ell = - \frac{k{h_\ell^-}'(kR) j_\ell(qR) - qh_\ell^-(kR)j_\ell'(qR)}{k{h_\ell^+}'(kR) j_\ell(qR) - qh_\ell^+(kR)j_\ell'(qR)}.\]
The numerator and denominator are complex conjugates of one another, since \(j_\ell\) is real and \((h_\ell^+)^* = h_\ell^-\) . Hence,
\[s_\ell = e^{2i\delta_\ell}, \;\;\;\mathrm{where}\;\; \delta_\ell = \frac{\pi}{2} - \mathrm{arg}\!\left[k{h_\ell^+}'(kR) j_\ell(qR) - qh_\ell^+(kR)j_\ell'(qR)\right].\]
In other words, the scattering matrix component is a pure phase factor. This is actually a consequence of energy conservation. Since the scattering matrix does not couple different angular momentum channels (due to the spherical symmetry), the incoming flux and outgoing flux in each channel must be equal. Hence, the only thing the scattering potential can do is to shift the phase of the outgoing spherical wave component in each channel.
Once we find \(\delta_\ell\) , we can compute the scattering amplitude
\[f(\mathbf{k}_i\rightarrow k\hat{\mathbf{r}}) = \frac{2 \pi}{ik}\, \sum_{\ell =0}^\infty \big(e^{2i\delta_\ell} - 1\big) \, \sum_{m=-\ell}^\ell \,Y_{\ell m}^*(\hat{\mathbf{k}}_i)\; Y_{\ell m}(\hat{\mathbf{r}}).\]
This can be simplified with the aid of the following addition theorem for spherical harmonics:
\[P_\ell(\hat{\mathbf{r}}_1\cdot\hat{\mathbf{r}}_2) = \frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{\ell} Y_{\ell m}^*(\hat{\mathbf{r}}_1) Y_{\ell m}(\hat{\mathbf{r}}_2).\]
where \(P_\ell(\cdots)\) denotes a Legendre polynomial . We finally obtain
Definition: Legendre Polynomial
\[\begin{align}\begin{aligned}f(\mathbf{k}_i \rightarrow k\hat{\mathbf{r}}) &= \frac{1}{2ik}\, \sum_{\ell =0}^\infty \big(e^{2i\delta_\ell} - 1\big) \big(2\ell+1\big)\, P_{\ell}(\hat{\mathbf{k}}_i\cdot \hat{\mathbf{r}}) \\ \delta_\ell &= \frac{\pi}{2} - \mathrm{arg}\!\left[k{h_\ell^+}'(kR) \, j_\ell(qR) - qh_\ell^+(kR)\, j_\ell'(qR)\right] \\ k &= |\mathbf{k}_i| = \sqrt{2mE/\hbar^2}, \;\; q = \sqrt{2m(E+U)/\hbar^2}.\end{aligned}\end{align}\]
This result for the scattering amplitude depends upon two variables: (i) \(E\) , the particle energy (which is conserved), and (ii) \(\Delta \theta = \cos^{-1}(\hat{\mathbf{k}}_i\cdot \hat{\mathbf{r}})\) , the deflection angle (i.e., the angle between the direction of incidence and the direction into which the particle is scattered).


18: Electric Charge and Electric Field
This chapter begins the study of electromagnetic phenomena at a fundamental level. The next several chapters will cover static electricity, moving electricity, and magnetism—collectively known as electromagnetism. In this chapter, we begin with the study of electric phenomena due to charges that are at least temporarily stationary, called electrostatics, or static electricity.
-
- 18.0: Prelude to Electric Charge and Electric Field
- Franklin demonstrated a connection between lightning and static electricity. Sparks were drawn from a key hung on a kite string during an electrical storm. These sparks were like those produced by static electricity, such as the spark that jumps from your finger to a metal doorknob after you walk across a wool carpet.
-
- 18.1: Static Electricity and Charge - Conservation of Charge
- When various materials are rubbed together in controlled ways, certain combinations of materials always produce one type of charge on one material and the opposite type on the other. By convention, we call one type of charge “positive”, and the other type “negative.” E.g., when glass is rubbed with silk, the glass becomes positively charged and the silk negatively charged. Since the glass and silk have opposite charges, they attract one another like clothes that have rubbed together in a dryer.
-
- 18.2: Conductors and Insulators
- Some substances, such as metals and salty water, allow charges to move through them with relative ease. Some of the electrons in metals and similar conductors are not bound to individual atoms or sites in the material. These free electrons can move through the material much as air moves through loose sand.
-
- 18.3: Coulomb's Law
- Through the work of scientists in the late 18th century, the main features of the electrostatic force—the existence of two types of charge, the observation that like charges repel, unlike charges attract, and the decrease of force with distance—were eventually refined, and expressed as a mathematical formula. The mathematical formula for the electrostatic force is called Coulomb’s law after the French physicist Charles Coulomb.
-
- 18.4: Electric Field- Concept of a Field Revisited
- Contact forces, such as between a baseball and a bat, are explained on the small scale by the interaction of the charges in atoms and molecules in close proximity. They interact through forces that include the Coulomb force. Action at a distance is a force between objects that are not close enough for their atoms to “touch.” That is, they are separated by more than a few atomic diameters.
-
- 18.5: Electric Field Lines- Multiple Charges
- Drawings using lines to represent electric fields around charged objects are very useful in visualizing field strength and direction. Since the electric field has both magnitude and direction, it is a vector. Like all vectors, the electric field can be represented by an arrow that has length proportional to its magnitude and that points in the correct direction. (We have used arrows extensively to represent force vectors, for example.)
-
- 18.6: Electric Forces in Biology
- Classical electrostatics has an important role to play in modern molecular biology. Large molecules such as proteins, nucleic acids, and so on—so important to life—are usually electrically charged. DNA itself is highly charged; it is the electrostatic force that not only holds the molecule together but gives the molecule structure and strength.
-
- 18.7: Conductors and Electric Fields in Static Equilibrium
- Conductors contain free charges that move easily. When excess charge is placed on a conductor or the conductor is put into a static electric field, charges in the conductor quickly respond to reach a steady state called electrostatic equilibrium.
Thumbnail: This diagram describes the mechanisms of Coulomb's law; two equal (like) point charges repel each other, and two opposite charges attract each other, with an electrostatic force F which is directly proportional to the product of the magnitudes of each charge and inversely proportional to the square of the distance r between the charges. Regardless of attraction, repulsion, charges or distance, the magnitudes of the forces, |F| (absolute value), will always be equal. (CC-BY-3.0).


1.7: To the Instructor: Optional Assignments/Activities/Extra Credit Opportunities for Unit 1
Ice Breaker
Since this is the first week of the course, an “ice breaker” for face-to-face classes helps students get to know each other better while also providing an opportunity to introduce two mnemonic devices: remembering by using association and organizing information by using a graphic (in this case, a three-column chart).
- Direct students to create a chart that has three columns. In the first column, they will write their classmates’ names. Across from each name in the second a third columns, they will write a word then a picture or symbol for each classmate that will help them remember the names (a fishing pole for one who likes to fish, for example, a flower for a gardener, a hat for another who says he always wears hats). Explain that they will be filling it out for as many classmates as they can in whatever time is available for this activity.
- Demonstrate by filling out the first entry for yourself. A “word association” for me might be that I am tall, or that I am the instructor. I ask the students what kind of picture or symbol usually represents a teacher to them. Most often, it will be an apple.
- Give students an amount of time to get as many names down as they can. Afterwards, they can test themselves using this mnemonic creation by studying it however they want to, to help them remember names. Of course, they have also engaged in a good opportunity to find out who they will be learning and working with in a casual and enjoyable way–as has the instructor.
Syllabus Treasure Hunt
Many students will simply tuck a course syllabus away in their notebook. It’s a pretty dull document, after all, with information such as course standards, objectives, classroom rules and regulations, and the like. They might do little more than glance at it and transfer the instructor’s email address and phone number to their cell phone.
However, there is often not only vital but also very useful information in a syllabus: grading information, holidays/school closure lists, where they might access disability accommodations, campus emergency numbers, how to lodge a grievance and many more useful items. One way to not only help them realize that a syllabus contains vital data but also to practice annotation skills is to assign a treasure hunt.
I usually tell the students to annotate (highlight, underline, circle, make marginal notes) for the following three items:
- Information they especially want to remember (most annotate my contact info)
- Information they have a question or concern about
- Any additional information they would appreciate having on this handy resource
Here is a sample EL115 syllabus treasure hunt for a typical EL115 course syllabus at Lane.
Simple Notebook Organization Plan
An additional activity for a face-to-face class is to assign the students a simple notebook organization task (for in-class or for homework). Of course, some students may already have a lovely organizational plan. Here is my suggestion:
From the topmost page when you open the notebook, insert:
- Your assignment/to-do list(s)
- Course syllabus
- Perhaps a list of classmates names/contact information: sample classmate contact form.
- Your text or other official materials (downloaded text, if it is to be downloaded, any additional packets that might be required)
- 12 notebook dividers (one for the handouts, hand backs, scored/graded assignments for each of the 10 weeks of the course; and two for students’ choice. Some of my students prefer to keep all quiz/assessment type tasks in one place, for example. Some of them keep paper in another for note taking or other tasks.
Campus-wide virtual, or Online, website Treasure Hunt
This can be created as an actual physical search around campus for student services, or a website hunt. For our students at Lane, here is a sample online hunt: Lane Community College Students Services, campus treasure hunt.
Optional Extra Credit Opportunities and/or Grading Elements for Unit 1
Grades for this course as taught are determined primarily by the rubrics for the rubric for-take-home mid-term and for the take-home final portfolio projects assigned at the end of Unit 3 and Unit 6, respectively. This instructor teaches this as a face-to-face class presented on a Course Management System (MOODLE, at Lane), and includes points for attendance, as well as some miscellaneous homework assignments, mostly lesson exericises. However, this OER can be used for online or hybrid classes as well.
Here are some suggestions for additional course components and grading elements:
- The Glossary of Terms in the back is good information for weekly quizzes, mid-term tests, and/or final exams.
- For exercise 5.2 (re: the Brain dominance Theory), students could act out their responses.


4.3: Tab Stops
Learning Objectives
- Set tab stops.
You may recall from Alignment, Justification, and Indentation that hitting the Tab key indents a paragraph. Each hit of the Tab key inserted a tab space . By default, a tab space stretches to fit the next 1/2 inch interval on the page. Tab stops control the width of a tab space so that it can be greater or less than 1/2 an inch.
By changing the width of tab spaces, you can align text with tab stops rather than the edge of a column, page, or margin. This can be useful for aligning text different ways in the same line.
There are several kinds of tab stops, but the ones we will focus on in this course are the left tab, the center tab, and the right tab.
| Aligns text left along the tab stop line | Aligns text centrally to the tab stop line | Aligns text right along the tab stop line |
There are two main methods for setting tab stops: the ruler and the tab dialog box.
Tab Stops Using the Ruler
First, make the ruler visible (if it isn’t visible already). Go to View>Show and check the Ruler box.
The rulers display the width of the page in inches. The shaded-in gray spaces on the ruler represent the margins. This screenshot shows margins of one inch.
To add a tab stop, select the paragraph or paragraphs the tab stop will be added to. Select the type of tab stop by clicking the tab selector icon in the top-left corner until it shows the appropriate icon ( , , ). The tab selector button is located at the top of the vertical ruler and to the left of the horizontal ruler.
(Note: there are more tab options, such as decimal tab, bar tab, first-line indent, and hanging indent. Feel free to experiment with these later.)
Once the type of tab stop you want is selected, click anywhere along the horizontal ruler to place the tab stop.
The tab stop is represented by the icon in the ruler. In the screenshot above, the right tab is marked with a green circle. Notice that the lines below are right-aligned with the tab stop, with the exception of the last line. The last line is not aligned because the line is too long for the text to fit with the right-align.
To move the tab stop, click and drag the tab icon along the ruler.
The selected paragraphs moved with the tab stop. (And notice that the last line is aligned now that there is enough space for the text.)
To remove a tab stop, click the icon and drag it off of the ruler.
Practice Question
Tab Stops Using the Tab Dialog Box
You can also add and move tab stops using the tab dialog box. To reach it, go to the Paragraph dialog box in the Home tab.
Then click the Tabs button at the bottom-left corner of the Paragraph dialog box.
You have now reached the Tab dialog box.
Choose which type of tab stop you want in the Alignment area (Left, Center, Right, Decimal, or Bar). Type the distance of the tab stop in inches, then click Set or OK to apply that tab stop to whichever lines are selected.
Use Clear to remove tab stops. You can also still click and drag the tab stop along the ruler to change its location or drag the tab stop off the ruler to delete it.
Leader
The leader is a filler for the tab space. Instead of leaving the tab space blank, you can fill it with periods or other characters. This is particularly useful for a table of contents, but it can have other applications too. To choose a leader, select one of the options under the Leader section of the Tab dialog box.
Optional: Bar and Decimal Tabs
Two of the other tab stop types are the bar and decimal tabs.
The bar tab icon is a straight vertical line (|). Bar tabs do not align text. Instead, a bar tab adds a straight vertical line at the point of the bar tab. The line will be the length of the selected lines.
The decimal tab icon is . Decimal tabs align text based on the placement of periods (or rather, decimal points). This is useful when lining up numbers, especially dollar amounts.
| Side-by-Side Comparison of Decimal Tab Alignment | |
|---|---|
Notice that when a decimal tab is applied, the decimal points will line up regardless of what is on either side of the decimal point.
Contributors and Attributions
- Tab Stops. Authored by : Lumen Learning. License : CC BY: Attribution
- Lower Right. Authored by : Travis Avery. Located at : https://thenounproject.com/term/lower-right/546558/ . License : CC BY: Attribution
- Lower Left. Authored by : Travis Avery. Located at : thenounproject.com/term/lower-left/546557/. License : CC BY: Attribution
- Vertical Line. Authored by : Hector. Located at : https://thenounproject.com/cstudent11235/collection/vertical-line/?i=656405 . License : CC BY: Attribution . License Terms : Used in modifying Travis Avery's work to make new icon


11: Comparing Two Means
In the previous chapter we covered the situation when your outcome variable is nominal scale and your predictor variable 184 is also nominal scale. Lots of real world situations have that character, and so you’ll find that chi-square tests in particular are quite widely used. However, you’re much more likely to find yourself in a situation where your outcome variable is interval scale or higher, and what you’re interested in is whether the average value of the outcome variable is higher in one group or another. For instance, a psychologist might want to know if anxiety levels are higher among parents than non-parents, or if working memory capacity is reduced by listening to music (relative to not listening to music). In a medical context, we might want to know if a new drug increases or decreases blood pressure.  In other words, we want to compare the means of the two groups.
The standard answer to the problem of comparing means is to use a t-test, of which there are several varieties depending on exactly what question you want to solve. As a consequence, the majority of this chapter focuses on different types of t-test: one sample t-tests are discussed in Section 13.2, independent samples t-tests are discussed in Sections 13.3 and 13.4, and paired samples t-tests are discussed in Section 13.5. After that, we’ll talk a bit about Cohen’s d, which is the standard measure of effect size for a t-test (Section 13.8). The later sections of the chapter focus on the assumptions of the t-tests, and possible remedies if they are violated. However, before discussing any of these useful things, we’ll start with a discussion of the z-test.


5.5: Module 31 Competency Test No.6 Open Book
Module 31 Competency Test No.6 Open Book
Learning Outcomes
When you have completed this module, you will be able to:
- Within a three hour time limit, complete a written exam and the lab exercises without the aid of a key.
The AutoCAD 3D book was written with competency based modules. What that means is that you have not completed each module until you have mastered it. The Competency Test module contains multiple choice questions and a comprehensive lab exercise to test your mastery of the set of modules that you completed. There are no answers or keys supplied in a Competency Test module since it is meant to be checked by your instructor. If there are any parts of this module that you have trouble completing, you should go back and reread the module or modules containing the information that you are having trouble with. If necessary, redo as many lab exercises required until you fully understand the material.
If you are completing this book:
- Without the aid of an instructor, complete the written test and the lab exercise.
- In a classroom with an instructor, the instructor will give instructions on what to do after this module has been completed.
Multiple Choice Questions
Select the BEST answer.
-
Which three terms, of a 3D Solid, must the user be able to identify and select when using the SOLIDEDIT command?
- bodies, faces and solids
- faces, edges and regions
- bodies, edges and wireframes
- bodies, faces and edges
- edges, solids and regions
-
What visual style must be current when the SOLVIEW command is used to create a 2D multiview(s) of a solid model?
- Realistic
- 2D Wireframe
- Hidden
- 3D Wireframe
- Any visual style
-
Which one of the following item(s) must be set in the current drawing before the SOLDRAW and SOLPROF commands are used? Select the best answer.
- The Hidden linetype is loaded.
- The current visual style is 2D Wireframe.
- The current visual style is 3D Wireframe.
- A and B
- A and C
-
What command is used to find out if two or more solids occupy the same 3D space?
- SAMESPACE
- SUBTRACT
- INTERFERE
- UNION
- EXPLODE
-
Which SOLIDEDIT command option is used to divide two non-touching solid models back into individual solid models that have been joined with the UNION command?
- Separate
- Extrude
- Imprint
- Shell
- Check
-
Which SOLIDEDIT command option is used to attach a 2D object onto an existing solid model?
- Imprint
- Extrude
- Separate
- Shell
- Check
-
What two commands are used to a create 2D viewport(s), complete with hidden lines, of a solid model?
- SOLVIEW and MSPACE
- SOLVIEW and SOLDRAW
- MVIEW and SOLVIEW
- SOLPROF and MS
- MVIEW and PSPACE
-
Which SOLIDEDIT command option is used to create a hollow cavity, with a specified wall thickness, in a solid model?
- Imprint
- Extrude
- Separate
- Shell
- Check
-
What linetype must be loaded into the current drawing when using the SOLVIEW and SOLPROF commands?
- hidden
- center
- continuous
- dashed
- There is no specific linetype that has to be loaded.
-
Which one of the following terms does the SOLIDEDIT command use when it prompts the user to select a solid model?
- Model
- Face
- Body
- Solid
- Edge
Lab Exercise 31-1
Time Allowed: 3 Hours
| Drawing Name | Template | Units |
|---|---|---|
| AutoCAD 3D Lab 31-1 | 3D Layout English | Inches |
Step 1
Start a new drawing using the template shown above. Save the drawing with the name: AutoCAD 3D Lab 31-1 .
Step 2
Draw the polylines on layer: Pline .
Step 3
On layer: Solid 1 , draw a solid model of the object shown in the figures. (Figure Step 3A and 3B)
Step 4
On layout: Module Layout C , use the SOLVIEW command to create the three viewports of the model.
Step 5
On layer: VPORT, using the MVIEW command to create viewport. Set the view, in the viewport, to SE Isometric and the visual style Realistic.
Step 6
Set the scale of all four viewports to 1.25 and lock their display. (Figure Step 6)
Step 7
Use the SOLDRAW command to change the three multiviews views to 2D views.
Step 8
Set the current space to Paper and fill in the titleblock. (Figure Step 8)
Step 9
On layout: Module Layout D , using the SOLVIEW and SOLDRAW commands, create the three multiviews of the model.
Step 10
On layer: VPORT, using the MVIEW command, create two viewports. Set the visual style to Realistic and orbit them to match the figure the best you can by eye.
Step 11
Set the scale to 2:1 for the multiviews and 1.5:1 for the isometric views. Lock their display.
Step 12
Change the color of the layers so that the object lines are red and the hidden lines are blue. (Hint: See Module 30, WORKALONG: Creating 2D Drawings from a Solid Model, Step 16)
Step 13
Turn layer VPORT off.
Step 14
Set the current space to Paper and fill in the titleblock. (Figure Step 14)
Step 15
Save and close the drawing.