1.1: Tangle Basics
One of the aspects of knot theory that makes knots challenging is that knots must be understood in their wholeness: not on the basis of looking just at some of their crossings but looking at how all those crossings fit together to paint a global picture of the knot. Ultimately, the knot invariants we study later in the semester will help us to take this perspective. For now, though, we'll take the global questions out of the picture by slicing apart our knots and pinning down the cut ends, much like an entomologist might study a mounted butterfly. The objects we obtain for study in this process are called tangles , and the class of them that are easiest to study are the so-called rational tangles.
\(1.2.1\) Constructing Rational Tangles
We begin by thinking of rational tangles as the result of applying combinations of two geometric operations to an "empty" horizontal tangle:
- Twists of the rightmost strands, \(T\text{,}\) and
- Quarter-turn rotations of the entire tangle, \(R\text{.}\)
So for instance, we might describe a tangle as a "word" built from these letters such as \[\text { TTRTTTRTTTTRTRT. }\]
Ideally, what we'd like to find is a way to represent each of these operations as an operation happening to rational numbers and not just to rational tangles. In this process what we're hoping to obtain is:
- A group \(\Gamma\) of operations on tangles, and
- A (right-) group action of this group on the rational numbers, \(\rho: \Gamma \rightarrow \operatorname{Func}(\mathbb{Q} \rightarrow \mathbb{Q})\) .
The latter definition might be new to you, but we'll hopefully see by example that it's pretty natural. To make it precise:
Definition \(1.2.1\) Group action.
Let \(G\) be a group and \(S\) be a set. A (right) action of \(G\) on \(S\) is a function \[\rho: G \rightarrow\{\text { bijections } S \rightarrow S\}\]
that respects the group operation. That is, for all \(g,h \in G\) we have
- \(\bigl( \rho(g) \circ \rho(h) \bigr)(x) = \bigl(\rho(h\cdot g)\bigr)(x)\) and
- \(\bigl(\rho(g^{-1})\bigr)(x) = \bigl(\rho(g)\bigr)^{-1}(x).\)
In effect, what a group action does is assign to each
element
\(g\in G\) an invertible
function
\(\rho(g) : S \to S\) in such a way that the product of two elements \(h \cdot g \in G\) is assigned to the composition of the functions \(\rho(g) \circ \rho(h) \text{.}\) The reversal of this order is what makes this a right action, and permits us to read a product of group elements as acting from left to right, for example, (twist, then twist, then rotate, then twist, then rotate \[\begin{aligned}
T T R T R \\
\downarrow \\
r(t(r(t(t(x)))))
\end{aligned}\]
\(1.2.2\) How the Tangle Group Operates
Determining the structure of the tangle group will help us to understand what rational tangles can look like after all the twisting and rotating is done. We can observe how each operation interacts with itself:
- Iterated twists \(T, T^2, T^3, T^4, \ldots\) continue to add new crossings and change the tangle, so \(T\) is an element of infinite order in the tangle group.
- Iterated rotations \(R, R^2, R^3, R^4,\ldots\) undo themselves. It's easy to see that since \(R\) is a quarter-turn, \(R^4 = I\) will be the identity. What's more surprising, and will take until Section \(1.3\) to understand, is that for rational tangles, only two rotations are needed to return to the identity (\(R^2 = I\)). So \(R\) is an element of order 2 in the tangle group.
The final step is to understand the inverses of each of these basic operations ( generators ) in the tangle group. What's clear from point (2) above is that \(R\) is its own inverse. What's less clear is how to "un-twist a twist," i.e. how to construct an inverse twist using only more of the same twist (combined with possibly some rotations). The video above does the best justice to the observation that \[T \underbrace{R T R T R}_{T^{-1}}=I\]
and the discovery of this inverse completes our presentation of the tangle group.
Definition \(1.2.2\) Tangle group.
The tangle group \(\Gamma\text{,}\) defined as the set of all operations that carry one rational tangle into another, has the presentation \[\Gamma=\left\langle T, R \mid R^{2}=I=(T R)^{3}\right\rangle .\]
When we attempt to understand this group more concretely, we might try, as in the video above, to examine its action on the set of four vertices of the tangle (the four people holding the ropes). This serves to represent each tangle group operation by the permutation that it induces on the four people holding the ropes. But it doesn't tell the whole story because, for example, the permutation induced by the twist \(T\) has infinite order in the tangle group (where successive twists always compound on each other) but order two in the permutation group (which only looks at the fact that the two people holding the twisting ropes have returned back to their starting places after two twists).
\(1.2.3\) The Tangle Group Action on Rational Numbers
What we ultimately want when we determine an action of the tangle group on the rational numbers is that the above situation not happen: we need the functions operating on the rational numbers to reflect all the structure revealed in the presentation of Definition \(1.2.2\). So what we need are functions \(t : \mathbb{Q}\to\mathbb{Q}\) and \(r : \mathbb{Q}\to\mathbb{Q}\) which satisfy all of the following.
- \(t\) has infinite order, i.e. \(t^n(x) = t(t(t(\cdots t(x) \cdots )))\) must never equal the identity function \(i(x) = x\) unless \(n=0\text{.}\)
- \(r\) has order two, i.e. \(r^2(x) = r(r(x)) = i(x) = x\) for all \(x\in\mathbb{Q}\text{.}\)
- \((r\circ t)\) has order three, i.e. \((r\circ t)^3(x) = r(t(r(t(r(t(x)))))) = i(x)=x\) for all \(x \in\mathbb{Q}\text{.}\)
Beginning by positing that the simple "add one" function \(t(x) = x+1\) is a suitable choice to represent the infinite-order operation \(T\text{,}\) we then go in search of another function \(r\) which is an "involution" (i.e., it is equal to its own inverse function) and which also interacts with \(t\) in the manner specified by (3) above. We find that the opposite-reciprocal function miraculously fits the bill, and arrive at a right action of \(\Gamma\) on \(\mathbb{Q}\) defined by \[t(x)=x+1 \quad r(x)=-\frac{1}{x} .\]
The video above concludes with an explanation of how this permits us to compute a rational number, called the tangle number , for a rational tangle once we assign to the empty tangle the number 0.
\(1.2.4\) Exercises
Exercise \(<IP_ADDRESS>\).
Find an example of a knot diagram with an equal number of "over" crossings as "under" crossings, but the knot which it represents is not amphicheiral, i.e., \(K \neq -K\text{.}\)
Exercise \(<IP_ADDRESS>\).
(Adams, 2.12) Draw a sequence of pictures that show that the following two rational tangles are equivalent.
Figure 2.22 Two equivalent rational tangles.
Exercise \(<IP_ADDRESS>\).
Find an example of a 2-tangle which is not a rational tangle, and explain how you can tell that it is not rational. (Adams' text and the Kauffman/Lambropoulou paper are good places to start looking.)


A business plan is a twenty- to forty-page document that serves the dual role of being an internal road map for your team and an external sales tool for potential investors, customers, and partners. Your business plan describes the entirety of your venture: the problem you’re solving, your solution, the technology behind it, the size of your target market, the customers, the competition, your business model, team, financial needs, and exit strategy.
A business plan forces you to think things through early; it ensures you have well-defined venture goals. Clear goals help generate a clear path for you and your team to follow as you begin to implement your venture.
The elements of a business plan
Business plans vary widely, but most consist of:
- The executive summary
- The problem you’re solving or the need you’re filling
- Your solution
- Technology and IP
- Size of opportunity/market
- Customers and how you will reach them
- Competition
- Business model
- Team
- Financial needs
- Exit strategy
The executive summary is a 2- to 5-page section that summarizes the plan’s main points. In a few pages, the executive summary conveys the essence of the venture. It should contain only the key points from the important sections of the full plan.
The problem you’re solving or the need you’re filling is stated in an introductory section. Don’t shy away from aggressive terms and phrases in this section—you want to grab the reader’s attention.
Your solution , or how you will alleviate the pain, is stated in general terms.
Your technology and IP are described in more detail in the next section. Explain specifically how the technology works, but don’t overdo it; the explanation should be comprehensible to an intelligent layperson with some knowledge of the field. State whether the technology is yours or licensed, and, if so, from whom and under what conditions. Describe the status of your IP protection. What patents have been granted, applied for, will be applied for? Supply the patent numbers or the application numbers if you have them. State if you’re protected by other forms of IP.
Size of opportunity/market If you’re creating a new market it can be difficult to gauge its potential size, whereas if you’re introducing a better technology into an existing market, the estimate can be more accurate. Regardless, you and your potential investors need to feel comfortable that the potential market is large enough to sustain a profitable business.
Customers and how you will reach them Points to discuss include who your target customers are, your strategy for selling to them, what channels you will use, and when. Demonstrate an understanding of your target customers.
Competition Demonstrate knowledge of your competitors. Who are they? Are they selling the same or different technology? Who are the likely new entrants? Are they both domestic and foreign? What are their strengths and weaknesses?
Business model is a general outline of the way your company will make a profit. We go into more detail on business models later in this section.
Team This section should convince potential investors that they can trust your team with their money. Does your team:
- Have the knowledge, experience, diverse skill sets, integrity, drive, persistence, and passion required to make it happen, in spite of the adversity and obstacles that are likely to arise along the way?
- Understand its limitations? Are you willing to seek help and listen?
- Work with solid, experienced directors and advisors?
Although important, this section of the plan is only the first step in the convincing process. Personal interactions with investors and the due diligence process also play an important role.
Financial needs including the amount of money the venture is seeking and over what time frame; how the money will be used; the major assumptions involved; and when you will achieve cash flow break-even and profitability.
Exit strategy details how investors will get their money back (hopefully with a healthy return) and exit your company. Some exit strategies are Initial Public Offering (IPO), merger/acquisition, and buyout by a strategic partner
|
Nine questions every business plan should answer
From “How to Write a Great Business Plan” by William A. Sahlman. Harvard Business Review, July-August, 1997.
The real benefits of a business plan
Spitzer says that before actually sitting down to write the business plan for Zform, he hadn’t thought at all about marketing. He soon realized that he needed to do some research and identify the competition. “Don’t just jump into a new venture [without planning] because you will drown in the small details and won’t even realize that you’re way off base. A business plan forces you to think of all the details that you wouldn’t normally think about. When you write your plans down on paper, you have to be clear. The process of writing forces you to work out the details.” Spitzer emphasizes that there is no such thing as a final plan. “You need to accept and be comfortable with the fact that your business plan is a live document that will always be changing.” Silva suggests looking to others for advice. “Planning is essentially answering questions that have already been laid out for you by people that are experienced and know exactly what will hurt your venture if you don’t plan. The process of planning was ten times more valuable than the actual business plan itself.” Silva believes in learning from others’ mistakes. He regularly reads the “Postmortem” section of Game Developer magazine, which talks about games that have failed and why they have failed. Silva advises new venture seekers, “Talk to an industry veteran who has experienced failure, and ask why.“ Another viewpoint on business plans comes from Phyl Speser of Foresight Science & Technology. “Most folks will disagree with me, but I think [business plans] are highly overrated. Boeing never had a business plan until late in the last century. They knew what they did: they built airplanes. If you’re going out for venture capital money, you need a twenty+ page plan. If you’re not, you usually need a much smaller one. Essentially, you need as much of a plan as is necessary for your team or your investors. Otherwise, don’t waste the time.” |


13.9: Non-Cash Activities
Learning Outcomes
- Explain the nature of non-cash activities
What business activities are considered non-cash activities? In order to prepare a cash flow statement, we need to understand which items on our income statement and balance sheet may not involve the transfer of cash, thus will not have a place on our statement of cash flows.
These non-cash activities may include depreciation and amortization, as well as obsolescence. Property, plant and equipment resides on the balance sheet. These items are taken on the income statement in small increments called depreciation or amortization.
If we purchase a new dump truck, we don’t take the entire purchase price as an expense when we purchase it. We put it as an asset on our balance sheet, and then take depreciation expense over the life of the dump truck.
These non-cash items need to be properly recorded on the income statement, but disregarded for the cash flow statement.
Learn More
Visit Accounting for Management’s website for some additional information about non-cash investing and financing activities to keep in mind as we work through the cash flow statement process.
Cash and cash equivalents are those items on the balance sheet that are liquid assets. Cash can be spent, so it is the most liquid of the assets.
Cash equivalents might include money market accounts, treasury bills or commercial paper. It is essentially a place to sit money, to make a return on it. Cash sitting in a checking account may get no interest, but a money market account will earn interest while it sits. In this way, a company can put its idle cash to work.
These cash equivalents are pretty easy to move to a checking account if the cash becomes needed, but getting interest rather than just letting it sit there is a smart step!
Practice Questions


1.6: Early Number Concepts and Number Sense
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- Janet Stramel
- Fort Hays State University
As a teacher of mathematics, remember these fundamental beliefs as you develop early number concepts in your students.
- Thinking about the problem, not just the answer, is what is most important.
- The teacher should be the facilitator, not the one with all the answers.
- Process is more important than product. Mathematics is not just “facts” to be memorized. The concepts children learn are of utmost importance; and then the memorization will come. Children must experience through play and using manipulatives and “reinvent” the concepts of mathematics in their own minds.
To help children develop early early number concepts, use activities that focus on verbal counting , one-to-one correspondence , cardinality , and subitizing .
Watch this video from Graham Fletcher on “The Progression of Early Number & Counting.”
Verbal counting is critical in developing quantitative thinking in young children. A focus on counting in context is one way to build a child’s mathematical abilities, therefore it is critical to count with children every day. Some activities include:
- Practice counting words and written numerals with pictures or representations of objects.
- Count aloud as a class.
- Use a number chart so children can see what number symbols look like visually.
For more activities related to counting, go to the K-5 Counting Center .
Suggested Activity:
To practice counting on, begin with a number and context the child recognizes, like four bear manipulatives. Add another bear and say, “five,” then add another bear and say, “six.” This helps students connect the number names with the increase in objects.
Help students see the patterns and make connections. Ask probing questions such as, “How do you know?” and “How did you figure that out?”
One-to-one correspondence is a developmental skill for young children that we take for granted. When a child touches each toy and says the number name out loud, “one, two, three…,” he/she demonstrates the ability to count with one-to-one correspondence. To develop this skill in young children, you can point to objects as you say the number word, or move each object as you say the number word out loud.
Suggested Activity:
To develop one-to-one correspondence, ask students to match two different types of objects together, such as counting bears and a die that shows five dots.
To be successful in mathematics, students must understand cardinality by counting various objects and hold that number “in their head.” A child who understands the cardinality concept will count a set once and not need to count it again.
To develop this skill, students need constant repetition of counting and teaching through modeling. Children learn how to count (matching counting words with objects) before they understand that the last word stated in a count indicates the amount of the set. Counting should be reinforced throughout the day, not just during “math time.”
Suggested Activity:
Show students a collection of counters. Ask, “How many are here?”
Other examples:
- Ask students to count the number of empty chairs in the classroom.
- Count the number of groups.
- Count the number of crayons in their desk.
Next in the progression of counting is subitizing , the ability to quickly identify the number of items in a group without counting the items. Developing the ability to subitize is critical to the success of early number concepts. Subitizing allows children to see groupings, and these groupings are foundational to addition, subtraction, multiplication and division.
Christina Tondevold (2019) explains: “It’s the same thing when kids are learning to read. When we first teach kids to learn to read, they do it in isolation. C-A-T. And then they put that together to make cat. It’s the same thing as 1, 2, 3, that’s 3. But eventually, we don’t want them saying “C-A-T,” and having to sound it all out the whole way. Like for catnap, you don’t want them to sound out “c-a-t-n-a-p.” You want them to see the chunks. Cat-nap.”
In mathematics, when adding 8 + 7, students may not immediately know the sum. But instead of counting on fingers, students can visualize these amounts. Students may see the 7 as 5 and 2 and the 8 as 5 and 3. They can then put together the 5 and 5 and the 2 and 3 to know 10 and 5 is 15.
Games are an excellent way to develop subitizing skills in young students; dice, dominoes, playing cards, etc. Watch the subitizing video , as groups of objects quickly flash on the screen.
Suggested Activity: Subitizing with Paper Plates
Hold up a paper plate with dots on it for 3 seconds and say, “How many dots did you see? What does the pattern look like?” Spend time discussing the configuration of the pattern and how many dots are on the plate. Then show the plate again so children can self-check. (Van de Walle, Karp, Bay-Williams, 2019)
The goal is for the children to recognize the shape of the dots on the plate and when held up, they will recognize that it’s a five or a 9 relatively quickly. You want the children to get past one to one counting of the dots and to recognize the number by the dot arrangement. Think of how you recognize the number on dice, you don’t count the pips but you know when you see a 5. This is what you want your children to learn.
Number Sense
Number sense is defined as a “good intuition about numbers and their relations. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989). When students consider the context of a problem, look at the numbers in a problem, and make a decision about which strategy would be most efficient in each particular problem, they have number sense. Number sense is the ability to think flexibly between a variety of strategies in context.
Building on subitizing, composing and decomposing numbers, using ten-frames, and the hundreds chart will help develop number sense in your students. When students have number sense, they gain computational fluency , are flexible in their thinking, and are able to choose the most efficient strategy to solve a problem.
Number sense is a foundational idea in mathematics; it encourages students to think flexibly and promotes confidence with numbers. According to Marilyn Burns (2007), “students come to understand that numbers are meaningful and outcomes are sensible and expected.” And “just as our understanding of phonemic awareness has revolutionized the teaching of beginning reading, the influence of number sense on early math development and more complex mathematical thinking carries implications for instruction” (Gersten & Chard, 1999).
Number sense is a skill that allows students to work with numbers fluently and efficiently. This includes understanding skills such as quantities, concepts of more and less, symbols represent quantities (8 means the same thing as eight), and making number comparisons (12 is greater than 9, and three is half of six). Students must
- Understand numbers, ways of representing numbers, relationships among numbers, and the number system,
- Understand the meanings of operations and how they related to one another, and
- Compute fluently and make reasonable estimates (NCTM, 2000).
Teachers must give students many opportunities to develop number sense through activities with physical objects, such as counters, blocks, or small toys. Most children need that concrete experience of physically manipulating groups of objects. After these essential experiences, a teacher can move to more abstract materials such as dot cards.
Ten-Frames
Another way to develop number sense in children is to use ten-frames. A “ten-frame” is a rectangle, separated into two rows, with 10 equal spaces. Here is a blank ten-frame from NRICH.
Ten-frames help students develop number sense, a sense of ten, and learn basic number facts. A ten-frame also gives students a visual of numbers, of composing and decomposing numbers, and is a good way to teach addition and subtraction within 10.
This Ten Frame applet from NCTM Illuminations is a fun way for students to practice counting and addition skills.
Hundreds Chart
A Hundreds Chart is essential in every K-2 classroom. Ask students what patterns they see on the Hundreds Chart. You can also begin to “hide” numbers on the Hundreds Chart and ask students to tell what that number is and how they know.
Part-Part-Whole Relationships
Teaching part-part-whole relationships is critical to understanding addition and subtraction concepts. The part-part-whole model shows that two parts make up a whole. The following examples show the value of part-part-whole relationships.
Example #1: Samantha has 4 red marbles and 3 blue stickers. How many stickers does Samantha have?
To use the part-part-whole model, think first about the unknown. Since we want to know how many stickers Samantha has in all, that is the unknown or the “whole.”
Example #2: Addison has 27 chocolate candies. 15 are red, and the rest are green. How many green candies does Addison have?
To use the part-part-whole model, think first about the unknown. Since we want to know how many of Addison’s candies are green, that is the unknown or the “part.”
Basic facts are any number or mathematical fact or idea that is instantly recognizable or retrievable without having to resort to a strategy or a calculation. Just knowing basic facts is not enough. Teachers must help students develop the ability to quickly and accurately understand the relationships between numbers. Students need to make sense of numbers as they find and make strategies for joining and separating quantities.
Students use properties of addition (the commutative and associative properties) to add whole numbers. They can then use other strategies such as making tens and near doubles. (Student do not need to know the names of the properties, but they should understand them.)
In the strategy “making tens,” students decompose numbers to find a 10. For example, when learning 8 + 5, students may decompose the 5 into 2 and 3, and then see that 8 + 2 is 10; therefore 10 + 3 = 13.
Giving children multiple opportunities to experience early number concepts and number sense is critical for early childhood teachers. Children must experience mathematics through play and manipulatives as they make sense of numbers in their own minds.


11.5: John Newton - “Amazing Grace”
So far, we have examined compositions for use in Christian worship that are fixed in terms of their musical details. Two recordings of Sleepers, Wake , for example, might differ slightly in terms of tempo or timbre, but they will sound essentially the same. They will certainly contain all of the same pitches and rhythms, and will be similar in length. These are all characteristics of the classical tradition, in which the composer exercises a great deal of control over the musical work.
Next, however, we will examine an example of Christian worship music that has changed dramatically as it has been adopted and adapted by different religious communities. In fact, the only element of “Amazing Grace” to remain consistent throughout its lifetime has been the words, although all of the versions under consideration here also use the same melody—that which will be familiar to those who know the hymn. This stylistic flexibility is typical of music in the vernacular tradition, which permits the reinterpretation and transformation of musical compositions by individual performers.
No-one has ever performed Sleepers, Wake without being aware that it was composed by J.S. Bach, but people sing “Amazing Grace” every day without knowing who penned the words or music. Indeed, those are not easy questions to answer—the text, although initially written by John Newton (1725-1807), was later expanded by an anonymous author, while people still debate the identity of the tune’s composer. People who know this hymn are more likely to identify it with one of its great interpreters, such as Aretha Franklin, whose version we will encounter below.
Newton’s Life
The story of “Amazing Grace” begins with John Newton, who wrote six verses— some familiar from modern usage, some not—in 1772. Newton was a clergyman in the Church of England. He served as curate in the village of Olney, where his parishioners were largely impoverished and uneducated. Newton gained a reputation for impassioned preaching that spoke to the personal moral struggles of his congregants. Unlike other preachers, Newton willingly shared sins from his own past—and those were certainly in no short supply.
Newton took to the sea as a ship’s apprentice at the age of eleven, but was pressed into service with the Royal Navy after refusing to obey his captain’s orders. After deserting the Navy to visit a young lady, Polly Catlett, he was traded to a slave ship, where he developed a reputation for writing obscene songs and using language that shocked even sailors. Newton had renounced his Christian faith early in his seagoing career, but in 1748 a near-death experience aboard the ship Greyhound inspired him to reconsider his beliefs. He was further encouraged by his love for Polly, whom he married in 1750. All the same, it was many years (and another brush with death) before Newton reformed in any meaningful way, and he continued to work in the slave trade well into the 1750s. Newton began studying theology in 1756 and was finally successful in securing ordination and his position at Olney in 1764.
At Olney, Newton began writing hymns for his congregation to sing together at weekly prayer meetings. Newton’s hymns used simple language that was easily intelligible to his parishioners, and many of his texts were written in the first person. They focused on the confession of sins and the joys of salvation. Although the quality of his verse was criticized by some, Newton’s hymns became quite popular. They first appeared in print as part of the 1779 collection Olney Hymns , which included “Amazing Grace”:
Amazing grace! (how sweet the sound) That sav’d a wretch like me!
I once was lost, but now am found, Was blind, but now I see.
‘Twas grace that taught my heart to fear, And grace my fears reliev’d;
How precious did that grace appear The hour I first believ’d!
Thro’ many dangers, toils, and snares, I have already come;
‘Tis grace hath brought me safe thus far, And grace will lead me home.
The Lord has promis’d good to me, His word my hope secures;
He will my shield and portion be As long as life endures.
Yes, when this flesh and heart shall fail, And mortal life shall cease;
I shall possess, within the veil, A life of joy and peace.
The earth shall soon dissolve like snow, The sun forbear to shine;
But God, who call’d me here below, Will be forever mine.
Although it is known that Newton’s congregation used his “Amazing Grace” text beginning in 1773, we have no idea what tune or tunes the text was sung to. It certainly was not the tune we know today. Newton was not a composer, but he always intended for his devotional poems to be sung. Like other hymn text authors, he crafted his verses using specific patterns of syllables and rhymes so that they could be sung to preexisting melodies. This system of interchangeable texts and tunes meant that any hymn text could be sung to a variety of tunes and any tune could be supplied with a variety of texts. Only over time have specific texts and tunes in the hymn tradition come to be paired off, such that churchgoers expect a text—“Amazing Grace,” for example—always to be sung to the same melody.
“Amazing Grace”5 in the Sacred Harp Tradition
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“Amazing Grace” 5. Performance: Texas Sacred Harp Singers, Southwest Sacred Harp Singing Convention (2011) |
The pairing of “Amazing Grace” with its tune took place in 1835, when William Walker published a version in his hymnal Southern Harmony . The name of the tune was “New Britain” and it had already been in circulation for some time, although in the company of different texts. The authorship of “New Britain” is still contested. It was based in turn on two older melodies that first appeared in the 1829 hymnal Columbian Harmony . These tunes are unattributed, which indicates that they might be the work of hymnal compilers Charles H. Spilman and Benjamin Shaw. It is equally likely, however, that the tunes derive from folk tradition and might trace their origin to the British Isles. The “New Britain” tune also appears in an 1828 manuscript compiled by hymn composer Lucius Chapin, who has been proposed as yet another potential author. It is unlikely that a composer will ever be identified—if, indeed, “New Britain” was even the product of a single composer, which we have reason to doubt.
By the time William Walker published his version in 1835, the “Amazing Grace” text had already become very popular in the United States. It was widely used during the Second Great Awakening, which saw the staging of revivals across the country. These revivals featured charismatic preachers, who swayed crowds using emotionally charged speech punctuated with song. The verses of “Amazing Grace”—which embodied the personal, confessional approach to conversion favored by these preachers—were paired with simple refrains and sung to familiar tunes without the aid of hymnals.
Walker was not personally associated with the Second Great Awakening, but he made his living as a hymn publisher and singing teacher, and was therefore aware of trends in the world of Protestant worship music. Walker belonged to the American hymn tradition known as shape-note singing , which encompasses a unique form of notation, an approach to music education, and a compositional style.
The notation used in the shape-note tradition was first developed in late-18th century England for the purpose of simplifying the task of reading music. As its name suggests, shaped notation employs various shapes—each paired with a syllable—to represent the different pitches. In its original form, the system used only four shapes, even though a scale contains seven distinct pitches. The shapes were repeated in a way that maintained consistent patterns of intervals between shapes. If a singer could learn the intervallic distance between two shapes, they could easily sing music at sight. The syllables provide an additional tool for singers and also make it easy to teach the system. (The use of syllables to learn melodies actually dates back to ca. 1026, when the Italian monk Guido of Arezzo proposed a system that used six syllables—including the four later adapted by shape-note enthusiasts.)
Shaped notation was adopted in the United States as part of a movement to improve church music. During the second half of the 18th century, music lovers in New England began to lament the sorry state of American congregational singing. Churchgoers were largely illiterate, and they did not have access to hymnals or instruments. The most common method of hymn singing, known as lining out , required a songleader to call out a musical phrase, which the congregants would then repeat in unison. Activists wanted to restore four-part harmony to the worship service, but this would require music literacy and access to printed notation.
To fulfill both of these needs, entrepreneurs began to publish hymn books and offer singing schools. A singing master would travel around a region, spending two weeks at a time teaching anyone who was interested how to read using the shape- note system. Singing schools were usually hosted by a church, and classes would meet daily. These schools not only paved the way to better church music, but also provided much-needed entertainment to farming communities during the winter months and facilitated interactions between young men and women, who had few opportunities to encounter one another unchaperoned. The students would not only pay tuition to the singing master but would purchase hymn books—another important source of income.
The hymns composed in the American shape-note tradition were unique. Most of the composers—William Billings (1746-1800) and Daniel Read (1757-1836) were the most famous to emerge from the First New England School—were self- taught, and they gleefully rejected the conventions of European harmony and part writing. Their hymns contain unusual and harsh sounds (the result of breaking voice-leading rules), uneven phrases, and incomplete chordal harmonies (open fifths in place of triads). The shape-note composers also adhered to the ancient practice of putting the melody in the tenor part instead of the soprano—an approach, dating back to medieval church polyphony, that had already disappeared from European choral music.
The singing school movement was launched in New England, but by the early 19th century had come to be regarded as outdated. The North, after all, aspired to a cosmopolitan identity and was embarrassed by the primitive efforts of its shape- note composers. The movement, however, found a new home in the South, where rural singing masters flourished up until the Civil War. It was there that the most influential hymn books were published. William Walker’s Southern Harmony was among these, but without question the most important shape-note hymnal was The Sacred Harp (1844), published in Georgia by Benjamin Franklin White and Elisha J. King. Unlike other such hymnals, The Sacred Harp has remained popular into the present day, and enthusiasts around the world regularly gather to sing from its pages. As such, the tradition of shape-note singing is today most commonly referred to as Sacred Harp singing .
Participants in modern Sacred Harp sings adhere to a number of practices that originated in the singing school movement. To begin with, their purpose is not worship but music marking. Many Sacred Harp enthusiasts also profess Christian beliefs, and sings often begin and end with prayer, but Sacred Harp thrives because participants are committed to the music.
At a typical sing, participants sit in a formation known as a “hollow square,” with singers—seated according to vocal part—facing a central open space. Although the vocal parts are gendered in typical choirs, this is not the case in Sacred Harp: Women will frequently sing the tenor part up an octave, while men will sing the soprano part down an octave. There is no conductor. Instead, participants will take turns selecting hymns, which each will direct from the central position. The person who chooses the hymn will specify which verses are to be sung, and someone will provide the starting pitches for each part. Then the group will sing through the hymn once on the syllables before turning to the text.
The vocal style associated with Sacred Harp singing is also unusual. Participants do not tend to approach their task with nuance. Instead, each sings as loudly and exuberantly as they can, often accenting the rhythms with physical movement (something that is expressly prohibited in choral singing). The tempos are steady. Finally, there is never instrumental accompaniment.
“Amazing Grace,” paired with the tune “New Britain,” appears in both Southern Harmony and The Sacred Harp . The Southern Harmony version contains only three voice parts (soprano, tenor, and bass), but editions of The Sacred Harp eventually added an alto part. (Experienced singers might notice that the soprano part is very high. This notation, however, is not meant to be taken literally, and most groups will sing the hymn in a lower key.) The familiar melody is in the tenor in both versions, while the other parts provide harmony above and below. As a result, a performance of this version of the hymn strikes most listeners as familiar, but somehow strange.
Our rendition of “Amazing Grace” was recorded at the Southwest Sacred Harp Singing Convention in McMahan, TX, in 2011. After an initial pass using the syllables, the participants sing the first four verses of Newton’s text.
“Amazing Grace” in the Southern Gospel Tradition6
Soon after the 1844 publication of The Sacred Harp , a rift emerged in the Southern hymn publishing community. In 1846, Jesse Aiken published The Christian Minstrel , which introduced a new shape-note system using seven shapes. This development seemed natural enough. As one advocate for the new system put it, “Would any parent having seven children, ever think of calling them by four names?” Aiken’s hymnal provoked a virulent debate. While some publishers refused to adopt the new system, others were won over. William Walker himself switched to seven-shape notation with his 1866 hymnal Christian Harmony . Eventually, the seven-shape system emerged victorious.
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6. “Amazing Grace”Performance: The Inspirations (1976) |
Other changes came upon the hymn publishing industry as well. By the late 19th century, the stark harmonies of The Sacred Harp had largely been eschewed in favor of more pleasing harmonies derived from popular music. Hymn composers moved away from the minor mode—which dominates the pages of The Sacred Harp —and began to write in a more cheerful vein. Composers also abandoned the archaic practice of placing the melody in the tenor voice, instead putting it in the soprano and supporting it with accompaniment in the lower parts. Finally, piano accompaniment was introduced in the early 20th century as churches gained access to instruments.
All of these characteristics describe the Southern gospel tradition, which continues to flourish. Southern gospel music is driven by hymn composers and publishers, who supply a steady diet of new hymns. These are sung at conventions, which attract vast numbers of singers eager to test their reading ability on unfamiliar music. At the same time, Southern gospel devotees enjoy singing old favorites, such as “Amazing Grace.” Although Southern gospel music does not belong to a single Christian denomination, it is closely associated with evangelical branches such as the Southern Baptist Convention.
In the early 20th century, Southern gospel publishers began sponsoring professional quartets to sing their music. These quartets—which were at first all- male—toured the singing conventions, where they would perform the latest hymns from a particular publisher. They also participated in revivals, gave concerts, sang on the radio, and released commercial recordings. It might be argued that Southern gospel transformed from a participatory musical tradition to a performative one, for today many people think of Southern gospel primarily as a commercial music genre. At the same time, collective singing of Southern gospel music continues to take place in churches and at singing conventions.
Our example was recorded by the Inspirations in 1976. The Inspirations follow solidly in the model established in the early 20th century by publishers’ quartets. The group was formed in 1964 when Martin Cook, a high school chemistry teacher in Bryson City, NC, began singing gospel music with four of his students. A couple years later they founded a gospel music festival, Singing in the Smokies, and by 1969 the Inspirations were a full-time professional group. A string of number-one gospel hits in the early 1970s cemented their reputation.
The Inspirations’ recording of “Amazing Grace” was created during a live performance in Warner Robbins, GA. This is important, both because the sounds of the audience contribute to the effectiveness of the recording and because the release of live albums is a meaningful practice in the gospel tradition. After all, this music is intended to have a profound and personal impact on the listener. The Inspirations describe themselves as “an enthusiastic, sincere, clean-cut group of fundamental conservative Christian gentlemen with a desire and an objective to witness to a needful and sinful world through the medium of Gospel Music.” They make music not to entertain an audience but to save it.
The first verse is sung by lead tenor Archie Watkins. As is typical of the genre, he sings the melody in the highest part of his range. This means that the top notes sound almost like cries. We can hear that he is straining to reach them—an effort that is emotionally compelling and communicates the sincerity of the message. This type of singing has roots in the tradition of secular Appalachian music- making, and can be heard in banjo songs and bluegrass. Watkins does not sing with a steady pulse, as we heard in the Sacred Harp rendition, but takes his time to express each individual word.
For the second verse, the ensemble enters to supply a wordless harmony, switching to text only for the final two words. The third verse is sung on text by the entire ensemble, although Watkins high melody still stands out from the texture. We also hear the low bass singing of Mike Holcomb—another characteristic feature of the Southern gospel music.
The third verse sung by the Inspirations, however, is not the third verse of Newton’s text:
When we’ve been there ten thousand years, Bright shining as the sun,
We’ve no less days to sing God’s praise, Than when we first begun.
In fact, this verse wasn’t written by Newton at all. It was first published in the 1790 Virginian hymnal A Collection of Sacred Ballads , in which it was incorporated into the much older hymn “Jerusalem, My Happy Home.” (In The Sacred Harp , the verse appears as part of “Jerusalem, My Happy Home” and “Ninety-Fifth Psalm,” but is not included in “Amazing Grace.”) The first person to associate this verse with “Amazing Grace” was Harriet Beecher Stowe, author of the 1852 anti-slavery novel Uncle Tom’s Cabin . In Stowe’s narrative, Uncle Tom sings three verses of “Amazing Grace”—two of Newton’s verses and the new one—to mark his moment of greatest spiritual need.
It is typical for favorite verses to appear in more than one hymn. This is the result of oral tradition. A certain verse—memorable for its imagery or message— sticks in a singer’s head. They then add the verse to another hymn, one that has a tune with the same pattern of strong and weak beats. It is believed that Stowe was drawing from African American oral tradition in particular when she included this verse in “Amazing Grace.” However, it was not included in a hymnal until the 1910 Coronation Hymns .
“Amazing Grace” in the Black Gospel Tradition7
Black gospel is not unrelated to Southern gospel. The two traditions share a great deal in common and have always influenced one another. In the period during which these traditions developed, however, American society was highly segregated—by Jim Crow laws in the South and by less visible means in the North. It is unsurprising, therefore, that a unique tradition of music, based in part on African American musical tradition, should have arisen among black worshippers.
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7. “Amazing Grace” Performance: Aretha Franklin (1972) |
The style of music making that would come to be known as black gospel emerged first in the Holiness churches of the early 20th century. These congregations traced their roots to the revival meetings of the Second Great Awakening, in which many African Americans had participated. They favored a highly expressive and emotional form of worship, in which individual congregants might be moved to speak or sing.
During the 1920s, musicians in Holiness churches began to incorporate popular influences into worship. These included styles of piano playing derived from ragtime, percussion and brass instruments associated with jazz, and blues- inspired vocals. Although some churchgoers were skeptical about the inclusion of secular sounds in worship, the music gained popularity with the success of early recordings by artists like Arizona Dranes, a blind pianist and singer who belonged to the Church of God in Christ.
In the 1930s, male quartets—much like those in the Southern gospel tradition— and choirs came to the fore as black gospel gained followers. Although the style borrowed from popular music, it maintained strict separation from the commercial world, and gospel singers who made secular recordings were often shunned by the church community. At the same time, black gospel exercised enormous influence on the development of popular music. This influence is especially evident in the singing styles that emerged in rhythm ‘n’ blues and, later, soul. It is no wonder that many successful gospel artists were attracted by the possibility of mainstream popular success.
Aretha Franklin (1942-2018) was among the most prominent of these crossover singers, although anxieties about secular music had largely subsided by the time her career took off. Franklin was the daughter of a Baptist minister, C.L. Franklin, and she first sang in the church. Her father was something of a celebrity, and Franklin got to know many of the great gospel singers while she was still a child. At the age of 12, she began accompanying her father on preaching tours, for which she would provide stirring music. At 18, however, she decided to leave gospel behind for a career in popular music. With her father’s support, she moved to New York City and signed with Columbia Records.
Franklin did not flourish at Columbia, and it was not until she moved to Atlantic Records in 1967 that her career took off. Her version of “Respect,” which topped the R&B and pop charts that year, was just the first in a chain of hit singles and albums. At the height of her success, however, Franklin returned to her roots and recorded a gospel album, the 1972 Amazing Grace . It was to be the top-selling album of her entire career.
Amazing Grace was recorded live over the course of two evening concerts at the New Temple Missionary Baptist Church in Los Angeles. Franklin was backed up by the Southern California Community Choir and the prominent gospel singer Reverend James Cleveland. The title track, “Amazing Grace,” is the longest on the double album, clocking in at nearly eleven minutes. (Our version, taken from the master recording, is over sixteen minutes long.) The selection was featured near the end of the first evening’s concert, and was without doubt the high point of the show.
As with the Inspirations’ recording discussed above, the fact that Franklin’s “Amazing Grace” was recorded live is important. The sounds of the audience help us to visualize the church setting and remind us that this is worship music. We don’t just hear the song—we also hear people being moved and transformed by the song. The shouts and applause of the listeners are integral to the performance. The audience noise also helps us to understand the motivations of the singer. Aretha is not just performing a song: She is expressing her deeply held beliefs in an intensely personal way. Finally, we can hear how the audience shapes the performance. As they respond to Franklin’s singing, they also inspire her to new heights of expression.
Franklin’s singing style is emblematic of the black gospel tradition. She in fact only sings two verses of the hymn: the first and third. However, she weaves them into an epic drama, full of twists and turns. Every vowel of the text is drawn out in a long melisma containing many notes, and there is never a sense of pulse. Although the first verse is elongated and intense, we soon find out that Franklin was only getting started. The emotional energy builds with the third verse, in which Franklin brightens her timbre as she moves to the top of her range. She also adds a great deal of new text, which gives the impression of having been improvised on the spot. With these added words, she personalizes the message and seems to confess her own sins. She speaks directly to the people in the room through music. In the final line of the verse, however, Franklin abandons the texts and insteads hums an additional iteration of the melody. By doing so, she draws down the emotional energy for the purpose of producing an even more dramatic final conclusion. Upon returning to the text of verse three, Franklin inaugurates a series of call and response passages between herself and the choir/audience. Again, the verse fails to conclude, and the choir enters with the third-line text. It seems almost as if they are trying to provide Franklin with the strength to finish the hymn. When Franklin returns with the first line of the verse, she builds her singing into shouts and cries before finally arriving “home.”
These three renditions of “Amazing Grace” are remarkably different. Variations in performing forces—choir vs male quartet vs solo female vocalist with piano accompaniment—do not even begin to explain the variations in aesthetic and emotional impact. Instead, we have to examine the vocal styles that are characteristic of each performing tradition—styles that in turn make sense only in the context of worship practices that are each unique to a time and place. In the end, we see how a hymn written centuries ago by a reformed slave trader can embody and express the spiritual values of diverse individuals and sects.


10.5: Juvenile Justice Process
[1] Some states with very specific and real gang problems devised targeted gang suppression laws and legislation, while other states did not. The fear of youth crime led states to create mandatory minimum legislation (like Measure 11 laws in Oregon), waiver and transfer laws, and zero tolerance policies.
[2] Due to the loose definitions of parens patrea and the court’s attempt to act in the best interest of the child, after World War II, the juvenile court was criticized for disregarding due process.
Due process refers to the procedural rights established in the Constitution, especially the Bill of Rights. It includes rights such as the right to legal counsel, right to call witnesses, and right to be notified of charges (which will be revisited in In re Gault ). The original juvenile court did not implement due process rights because it was intervening in the lives of youth for their own good, not in such a formalized adult way where they would need constitutional protections. However, because of the abuse of power, this changed in later decades.


7.4: Exercises
Encode
IXLOVEXMATH
using the cryptosystem in Example \(7.1\).
Decode
ZLOOA WKLVA EHARQ WKHA ILQDO
, which was encoded using the cryptosystem in Example \(7.1\).
Assuming that monoalphabetic code was used to encode the following secret message, what was the original message?
APHUO EGEHP PEXOV FKEUH CKVUE CHKVE APHUO EGEHU EXOVL EXDKT VGEFT EHFKE UHCKF TZEXO VEZDT TVKUE XOVKV ENOHK ZFTEH TEHKQ LEROF PVEHP PEXOV ERYKP GERYT GVKEG XDRTE RGAGA
What is the significance of this message in the history of cryptography?
What is the total number of possible monoalphabetic cryptosystems? How secure are such cryptosystems?
Prove that a \(2 \times 2\) matrix \(A\) with entries in \({\mathbb Z}_{26}\) is invertible if and only if \(\gcd( \det(A), 26 ) = 1\text{.}\)
Given the matrix
\[ A = \begin{pmatrix} 3 & 4 \\ 2 & 3 \end{pmatrix}\text{,} \nonumber \]
use the encryption function \(f({\mathbf p}) = A {\mathbf p} + {\mathbf b}\) to encode the message
CRYPTOLOGY
, where \({\mathbf b} = ( 2, 5)^\transpose\text{.}\) What is the decoding function?
Encrypt each of the following RSA messages \(x\) so that \(x\) is divided into blocks of integers of length \(2\text{;}\) that is, if \(x = 142528\text{,}\) encode \(14\text{,}\) \(25\text{,}\) and \(28\) separately.
- \(\displaystyle n = 3551, E = 629, x = 31\)
- \(\displaystyle n = 2257, E = 47, x = 23\)
- \(\displaystyle n = 120979, E = 13251, x = 142371\)
- \(\displaystyle n = 45629, E = 781, x = 231561\)
Compute the decoding key \(D\) for each of the encoding keys in Exercise \(7.4.7\).
Decrypt each of the following RSA messages \(y\text{.}\)
- \(\displaystyle n = 3551, D = 1997, y = 2791\)
- \(\displaystyle n = 5893, D = 81, y = 34\)
- \(\displaystyle n = 120979, D = 27331, y = 112135\)
- \(\displaystyle n = 79403, D = 671, y = 129381\)
For each of the following encryption keys \((n, E)\) in the RSA cryptosystem, compute \(D\text{.}\)
- \(\displaystyle (n, E) = (451, 231)\)
- \(\displaystyle (n, E) = (3053, 1921)\)
- \(\displaystyle (n, E) = (37986733, 12371)\)
- \(\displaystyle (n, E) =<PHONE_NUMBER>3, 34578451)\)
Encrypted messages are often divided into blocks of \(n\) letters. A message such as
THE WORLD WONDERS WHY
might be encrypted as
JIW OCFRJ LPOEVYQ IOC
but sent as
JIW OCF RJL POE VYQ IOC
. What are the advantages of using blocks of \(n\) letters?
Find integers \(n\text{,}\) \(E\text{,}\) and \(X\) such that
\[ X^E \equiv X \pmod{n}\text{.} \nonumber \]
Is this a potential problem in the RSA cryptosystem?
Every person in the class should construct an RSA cryptosystem using primes that are \(10\) to \(15\) digits long. Hand in \((n, E)\) and an encoded message. Keep \(D\) secret. See if you can break one another's codes.


4.1: Measurement and Scale - The Mole Concept
One of the important concepts to grasp in chemistry is scale. Atoms and molecules are very small. A single atom of hydrogen has a mass of about 1.67 x 10 -24 grams (that’s 0.00000000000000000000000167 grams). One cubic centimeter of water (one mL) contains about 3.3 x 10 22 water molecules (that’s 33 sextillion molecules). Because chemists routinely use numbers that are both incredibly small and incredibly large, unique units of measurement have been developed to simplify working with these numbers. As we learned in Chapter 1, the atomic mass unit (amu) helps us talk about the mass of atoms on a scale appropriate to atoms (one grams is about 600 sextillion amu). In this chapter, we will introduce the concept of a mole to help us talk about numbers of atoms on a scale appropriate to the size of a sample we could work with in a laboratory (for example, in grams). The mole is defined as the number of atoms contained in exactly 12 grams of carbon-12 (the isotope ). Chemist have measured this number and it turns out to be 6.0221415 x 10 23 . We can think of the term mole as a number, just like the word dozen represents the number 12. We will use the mole to represent this very large number (a chemist’s dozen ) and we will see that there is a special relationship between a mole of a pure substance and the mass of the substance measured in amu.
The origin of the mole concept is generally attributed to the Italian chemical physicist, Amadeo Avogadro. In 1811, Avogadro published an important article that drew a distinction between atoms and molecules (although these terms were not in use at the time). As part of his explanation of the behavior of gasses, he suggested that equal volumes of all gases at the same temperature and pressure contained the same number of molecules. This statement is referred to as Avogadro’s Hypothesis and today we commonly refer to the number of things in a mole, (6.0221415 x 10 23 ) as Avogadro’s number (this is rounded to 6.02 x 10 23 for most calculations). Because a mole can be thought of as a number , you can convert any number to moles by dividing that number by 6.02 x 10 23 . For example, at the time of this writing, the national debt of the United States is about 7.9 trillion dollars (7.9 x 10 12 dollars). This could be expressed as moles of dollars as shown below:
\[\left ( 7.9\times 10^{12}dollars \right )\left ( \frac{1\: mol\: dollars}{6.02\times 10^{23}dollars} \right )=1.3\times 10^{-11}\: mol\: dollars \nonumber \]
The ratio of any number to the number of things in a mole is often referred to as a mole fraction .
- It is estimated there are 7 x 10 22 stars in the universe. How many moles of stars is this?
- It is estimated there are 7.5 x 10 18 grains of sand on the earth. How many moles of sand grains is this?
- You have 0.0555 moles of jelly donuts. What number of donuts would that be?
- You drink a small bottle of drinking water that contains 13 moles of water. What is the number of molecules of water you drank?


7.8: Chapter 7 Review
Chapter 7 Review
- Answers
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1. T F Our core values have nothing to do with finding a rewarding career.
2. T F Effective career exploration includes academic and hands-on experiential
research.
3. T F Career development is a linear process leading to the selection of only
one career we hold for our entire professional lives.
4. T F An academic plan helps students complete their major requirements in the
most efficient manner.
5. T F You need to meet with your academic advisor only during your first
semester.
6. What are the personal aspects we need to analyze and explore in career development?
A. intelligence, abilities, thoughts, attitude
B. family, culture, education, experiences
C. values, personality traits, interests, skills
7. What is one example of doing hands-on, experiential career research?
A. reading a scholarly journal article
B. volunteering in a community organization
C. presenting a slideshow
8. Which of the following summarizes widely-accepted career development theories?
A. Life presents us with unexpected opportunities that we can use to advance our careers.
B. We should leave our career development to fate or chance.
C. Planning a direct career path is a guaranteed route to career satisfaction.
9. What are prerequisite classes?
A. classes designed to teach critical thinking and effective communication
B. classes required to be taken before other higher-level classes
C. classes teaching specific career-related skills
10. Who is the professional assigned to assist students with creating educational plans?
A. professor
B. admissions counselor
C. academic advisor
1) F 2) T 3) F 4) T 5) F 6) C 7) B 8) A 9) B 10) C


<IP_ADDRESS>: Articles
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Well, if you thought prepositions were “little” words, wait until we consider the part of speech called articles. Articles are similar to adjectives in that they modify nouns, but unlike adjectives, they don’t really describe a noun; they just identify a noun.
Articles are the smallest of the small but still serve an important function. We have three articles in the English language: "a, an" and "the".
"The" is the definite article, which means it refers to a specific noun in a group.
"A" or "an" is the indefinite article, which means it refers to any member of a group. You would use the indefinite article when you aren’t trying to distinguish a particular noun.
Whether you use "a" or "an" depends on the noun that follows it. In general, you would use "a" if the noun begins with a consonant and "an" if the noun begins with a vowel.
Then, as always in English grammar, there are a few tricky exceptions. If a noun begins with h, you should think about the sound it makes.
And if a noun starts with a vowel, but it makes a y sound, you should use "a" instead of "an".


8.1: Overview, Objectives, and Key Terms
Overview
This chapter examines interrogations, self-incriminating statements, and confessions. Interrogations, similar to other criminal investigation practices, are governed by rules and procedures to ensure due process and equal justice. The goal of interrogation is to obtain a statement of guilt from a guilty suspect or to prove the potential suspect's innocence in the crime. This chapter also examines how and why false confessions occur, and how to prevent them. Juvenile suspects are a unique population when conducting interview and interrogations. This chapter also explores how to properly interview juvenile suspects and the unique considerations officers must consider when conducting interrogations.
- Identify the difference between a confession and an incriminating statement.
- Explain the importance of Miranda, and interrogations.
- Identify the differences between questioning, interviewing, and interrogating.
- Explain how officers evaluate the outcome of interrogations.
- Explain the common reasons people may confess to a crime they did not commit.
- Identify the research on juvenile’s developmental and cognitive abilities and how this affect’s the way officers question juveniles.


7.6: Symptoms and signs of infection
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Symptoms are sensations in the body that only the person who is unwell can experience; for example, a headache, pain in the abdomen, blurred vision and nausea are all symptoms, because no one but the sufferer can experience them. Lay people, including children, can usually describe their symptoms accurately or report them if asked the right questions, but symptoms are subjective experiences that others cannot observe or verify.
By contrast, the signs of a disease are indicators of illness that other people can observe (e.g. a runny nose and frequent sneezing). Considering the symptoms and signs together may give enough information for a trained health care worker to make a diagnosis, i.e. identify the underlying cause of the illness and give it a definite name.


4.5: Summary
By fitting a Brownian motion model to phylogenetic comparative data, one can estimate the rate of evolution of a single character. In this chapter, I demonstrated three approaches to estimating that rate: PICs, maximum likelihood, and Bayesian MCMC. In the next chapter, we will discuss other models of evolution that can be fit to continuous characters on trees.
Footnotes
1: Throughout this chapter, when I say "rate" I will mean the Brownian motion parameter σ 2 . This is a little different from “traditional” estimates of evolutionary rate, like those estimated by paleontologists. For example, one might have measurements of trait in a series of fossils representing an evolutionary lineage sampled at different time periods. By calculating the amount of change over a given time interval, one can estimate an evolutionary rate. These rates can be expressed as Darwins (defined as the log-difference in trait values divided by time in years) or Haldanes (defined as the difference in trait values scaled by their standard deviations divided by time in generations). Both types of rates have been calculated from both fossil data and contemporary time-series data on evolution from both islands and lab experiments. Such rates best capture evolutionary trends, where the mean value of a trait is changing in a consistent way through time (for more information see review in Harmon 2014) . Rates estimated by Brownian motion are a different type of “rate”, and some care must be taken to compare the two (see, e.g., Gingerich 1983) .
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2: Note that there are more complicated optimization algorithms that are useful for more difficult problems in comparative methods. In the case presented here, where the surface is smooth and has a single peak, almost any algorithm will work.
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3:PICs are a transformation of the original data in which all information about the root state has been removed; our idea of what that root state might be has no effect on calculations using PICs. One can calculate the likelihood for the PIC REML method by assuming all of the standardized PICs are drawn from a normal distribution (eq. 4.5) with mean 0 and variance $\hat{\sigma}_{REML}^2$ (eq. 4.8). Alternatively, one can estimate the variance of the PICs directly, keeping in mind that one must use a mean of zero (eq. 4.4). These two methods give exactly the same results.
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References
Burnham, K. P., and D. R. Anderson. 2003. Model selection and multimodel inference: A practical information theoretic approach. Springer Science & Business Media.
Felsenstein, J. 2004. Inferring phylogenies. Sinauer Associates, Inc., Sunderland, MA.
Felsenstein, J. 1985. Phylogenies and the comparative method. Am. Nat. 125:1–15.
FitzJohn, R. G. 2012. Diversitree: Comparative phylogenetic analyses of diversification in R. Methods Ecol. Evol. 3:1084–1092.
Freckleton, R. P. 2012. Fast likelihood calculations for comparative analyses. Methods Ecol. Evol. 3:940–947.
Garland, T., Jr. 1992. Rate tests for phenotypic evolution using phylogenetically independent contrasts. Am. Nat. 140:509–519.
Gingerich, P. D. 1983. Rates of evolution: Effects of time and temporal scaling. Science 222:159–161.
Harmon, L. J. 2014. Macroevolutionary rates. in The Princeton guide to evolution. Princeton University Press.
Hill, J. E. 1974. A new family, genus and species of bat (mammalia, chiroptera) from thailand. British Museum (Natural History).
Ho, L. S. T., and C. Ané. 2014. A linear-time algorithm for Gaussian and non-Gaussian trait evolution models. Syst. Biol. 63:397–408.
Nocedal, J., and S. Wright. 2006. Numerical optimization. Springer Science & Business Media.
O’Meara, B. C., C. Ané, M. J. Sanderson, and P. C. Wainwright. 2006. Testing for different rates of continuous trait evolution using likelihood. Evolution 60:922–933.


3.2: Hazard Communication Safety Data Sheets
Hazard Communication Safety Data Sheets
The Hazard Communication Standard (HCS) requires chemical manufacturers, distributors, or importers to provide Safety Data Sheets (SDSs) to communicate the hazards of hazardous chemical products. As of June 1, 2015, the HCS requires new SDSs to be in a uniform format, and include the section numbers, the headings, and associated information under the headings below:
Section 1, Identification includes product identifier; manufacturer or distributor name, address, phone number; emergency phone number; recommended use; restrictions on use.
Section 2, Hazard(s) identification includes all hazards regarding the chemical; required label elements.
Section 3, Composition/information on ingredients includes information on chemical ingredients; trade secret claims.
Section 4, First-aid measures includes important symptoms/ effects, acute, delayed; required treatment.
Section 5, Fire-fighting measures lists suitable extinguishing techniques, equipment; chemical hazards from fire.
Section 6, Accidental release measures lists emergency procedures; protective equipment; proper methods of containment and cleanup.
Section 7, Handling and storage lists precautions for safe handling and storage, including incompatibilities.
Section 8, Exposure controls/personal protection lists OSHA's Permissible Exposure Limits (PELs); Threshold Limit Values (TLVs); appropriate engineering controls; personal protective equipment (PPE).
Section 9, Physical and chemical properties lists the chemical's characteristics.
Section 10, Stability and reactivity lists chemical stability and possibility of hazardous reactions.
Section 11, Toxicological information includes routes of exposure; related symptoms, acute and chronic effects; numerical measures of toxicity.
Section 12, Ecological information* Section 13, Disposal considerations* Section 14, Transport information* Section 15, Regulatory information*
Section 16, Other information includes the date of preparation or last revision.
*Note: Since other Agencies regulate this information, OSHA will not be enforcing Sections 12 through 15(29 CFR 1910.1200(g)(2)).
Employers must ensure that SDSs are readily accessible to employees.
See Appendix D of 1910.1200 for a detailed description of SDS contents.
Chemical manufacturers, importers and distributors must ensure that employers are provided copies of the SDS sheets with their initial shipment and with the first shipment after the SDS has been changed.
Maintaining SDS Sheets
The employer must maintain SDS sheets in the workplace for each of the hazardous chemicals on site. SDS sheets must be readily accessible, during each work shift, to employees when they are in their work area. Electronic access, such as computer databases, microfiche and other alternatives to maintaining paper copies of the SDS sheets are permitted, as long as no barriers to immediate employee access in each workplace are created by such options. In any event, SDS sheets must be readily accessible during each work shift to all employees.
If employees are required to travel between workplaces during a shift, SDS sheets are permitted to be kept at the primary workplace facility. SDS sheets may be kept in any form, provided they contain all of the required information as listed above.
Training
Employers provide employees with effective training on the hazardous chemicals in their workplaces at the time of their initial assignment, and whenever new hazardous chemicals are introduced into the workplace.
The required training shall cover at least the following topics:
- The requirements of the Hazard Communication Standard.
- Operations in the employee's area where hazardous chemicals are present.
- The location of the written hazard communication program and SDS sheets.
- Methods and operations to detect the presence of hazardous chemicals.
- The physical & health hazards associated with the chemicals in their work areas.
- PPE available to the employees and other control measures they can use to protect themselves.
- Details of the employer's hazard communication program, including how to use SDS sheets and proper PPE.
Process Safety Management
Unexpected releases of toxic, reactive, or flammable liquids and gases in processes involving highly hazardous chemicals have been reported for many years, in various industries using chemicals with such properties. Regardless of the industry that uses these highly hazardous chemicals, there is a potential for an accidental release any time they are not properly controlled, creating the possibility of disaster.
To help ensure safe and healthful workplaces, OSHA has issued the Process Safety Management of Highly Hazardous Chemicals standard (29 CFR 1910.119), which contains requirements for the management of hazards associated with processes using highly hazardous chemicals.
Process safety management (PSM) is addressed in specific standards for the general and construction industries. OSHA's standard emphasizes the management of hazards associated with highly hazardous chemicals and establishes a comprehensive management program that integrates technologies, procedures, and management practices.


10.3: Electromagnetic fields
Learning Objectives
- Explain electromagnetic field tensor
The electric field
Section 10.1 showed that relativity requires magnetic forces to exist, and section 10.2 gave us a peek at what this implies about or how electric and magnetic fields transform. To understand this on a more general basis, let’s explicitly list some assumptions about the electric field and see how they lead to the existence and properties of a magnetic field:
- Definition of the electric field : In the frame of reference of an inertial observer \(o\), take some standard, charged test particle, release it at rest, and observe the force \(F_o\) (section 4.5) acting on the particle. (The timelike component of this force vanishes.) Then the electric field three-vector \(E\) in frame \(o\) is defined by \(F_o = qE\), where we fix our system of units by taking some arbitrary value for the charge \(q\) of the test particle.
- Definition of electric charge : For charges other than the standard test charge, we take Gauss’s law to be our definition of electric charge.
- Charge is Lorentz invariant (section 1.3).
- Fields must have transformation laws (section 10.2). Many times already in our study of relativity, we’ve followed the strategy of taking a Galilean vector and trying to redefine it as a four-dimensional vector in relativity. Let’s try to do this with the electric field. Then we would have no other obvious thing to try than to change its definition to \(F = qE\), where \(F = ma\) is the relativistic force vector (section 4.5), so that the electric field three-vector was just the spacelike part of \(E\). Because \(a\cdot v = 0\) for a material particle, this would imply that \(E\) was orthogonal to o for any observer o. But this is impossible, since then a spacetime displacement vector s along the direction of \(E\) would be a vector of simultaneity for all observers, and we know that this isn’t possible in relativity.
The magnetic field
Our situation is very similar to the one encountered in section 9.1, where we found that knowledge of the charge density in one frame was insufficient to tell us the charge density in other frames. There was missing information, which turned out to be the current density. The problems we’ve encountered in defining the transformation properties of the electric field suggest a similar “missing-information” situation, and it seems likely that the missing information is the magnetic field. How should we modify the assumptions above to allow for the existence of a magnetic field in addition to the electric one? What properties could this additional field have? How would we define or measure it?
One way of imagining a new type of field would be if, in addition to charge \(q\), particles had some other characteristic, call it \(r\), and there was then be some entirely separate field defined by their action on a particle with this “r-ness.” But going down this road leads us to unrelated phenomena such as the the strong nuclear interaction.
The electromagnetic field tensor
The nature of the contradiction arrived at in the above section is such that our additional field is closely linked to the electric one, and therefore we expect it to act on charge, not on \(r\)-ness. Without inventing something new like \(r\)-ness, the only other available property of the test particle is its state of motion, characterized by its velocity vector \(v\). Now the simplest rule we could imagine for determining the force on a test particle would be a linear one, which would look like matrix multiplication:
\[F = q\mathcal {F}v\]
or in index notation,
\[F^a = q\mathcal {F}^a\: _b \: v^b\]
Although the form \(\mathcal {F}^a\: _b\) with one upper and one lower index occurs naturally in this expression, we’ll find it more convenient from now on to work with the upper-upper form \(\mathcal {F}^{ab }\). \(\mathcal {F}\) would be \(4 × 4\), so it would have \(16\) elements:
\[\begin{pmatrix} \mathcal {F}^{tt} & \mathcal {F}^{tx} & \mathcal {F}^{ty} & \mathcal {F}^{tz}\\ \mathcal {F}^{xt} & \mathcal {F}^{xx} & \mathcal {F}^{xy} & \mathcal {F}^{xz}\\ \mathcal {F}^{yt} & \mathcal {F}^{yx} & \mathcal {F}^{yy} & \mathcal {F}^{yz}\\ \mathcal {F}^{zt} & \mathcal {F}^{zx} & \mathcal {F}^{zy} & \mathcal {F}^{zz} \end{pmatrix}\]
Presumably these \(16\) numbers would encode the information about the electric field, as well as some additional information about the field or fields we were missing.
But these are not \(16\) numbers that we can choose freely and independently. For example, consider a charged particle that is instantaneously at rest in a certain observer’s frame, with \(v = (1, 0, 0, 0)\). (In this situation, the four-force equals the force measured by the observer.) The work done by a force is positive if the force is in the same direction as the motion, negative if in the opposite direction, and zero if there is no motion. Therefore the power \(P = dW/ dt\) in this example should be zero. Power is the timelike component of the force vector, which forces us to take \(\mathcal {F}^{tt} = 0\).
More generally, consider the kinematical constraint \(a\cdot v = 0\). When we require \(a\cdot v = 0\) for any \(v\), not just this one, we end up with the constraint that \(F\) must be antisymmetric, meaning that when we transpose it, the result is another matrix that looks just like the original one, but with all the signs flipped:
\[\begin{pmatrix} 0 & \mathcal {F}^{tx} & \mathcal {F}^{ty} & \mathcal {F}^{tz}\\ -\mathcal {F}^{tx} & 0 & \mathcal {F}^{xy} & \mathcal {F}^{xz}\\ -\mathcal {F}^{ty} & -\mathcal {F}^{xy} & 0 & \mathcal {F}^{yz}\\ -\mathcal {F}^{tz} & -\mathcal {F}^{xz} & -\mathcal {F}^{yz} & 0 \end{pmatrix}\]
Each element equals minus the corresponding element across the main diagonal from it, and antisymmetry also requires that the main diagonal itself be zero. In terms of the concept of degrees of freedom introduced in section 3.5, we are down to \(6\) degrees of freedom rather than \(16\). We now relabel the elements of the matrix and follow up with a justification of the relabeling. The result is the following rank-\(2\) tensor:
\[\mathcal {F}^{\mu \nu}=\begin{pmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{pmatrix}\]
We’ll call this the electromagnetic field tensor. The labeling of the left column simply expresses the definition of the electric field, which is expressed in terms of the velocity \(\vec{v}=(1,0,0,0)\) of a particle at rest. The top row then follows from antisymmetry. For an arbitrary velocity vector, writing out the matrix multiplication \(\mathcal {F}^{\mu} = q\mathcal {F}^{\mu}\: _{\nu} \: v^\nu\) results in expressions such as \(F^x=\gamma q(E_x+u_yB_z-u_zB_y)\) (problem , p.~). Taking into account the difference of a factor of \(\gamma\) between the four-force and the force measured by an observer, we end up with the familiar Lorentz force law,
\[\vec{F}_{o} = q(\vec{E}+\vec{u}\times\vec{B})\]
where \(\vec{B}\) is the magnetic field. This is expressed in units where \(c=1\), so that the electric and magnetic field have the same units. In units with \(c\ne 1\), the magnetic components of the electromagnetic field matrix should be multiplied by \(c\).
Thus starting only from the assumptions above, we deduce that the electric field must be accompanied by a magnetic field.
Example \(\PageIndex{1}\): Parity properties of E and B
In Example 9.2.3, we saw that under the parity transformation \((t, x, y, z) → (t, -x, -y, -z)\), any rank-\(2\) tensor expressed in Minkowski coordinates changes the signs of its components according to the same rule:
\[\begin{pmatrix} \text{no flip} & \text{flip} & \text{flip} & \text{flip}\\ \text{flip} & \text{no flip} & \text{no flip} & \text{no flip}\\ \text{flip} & \text{no flip} & \text{no flip} & \text{no flip}\\ \text{flip} & \text{no flip} & \text{no flip} & \text{no flip}\\ \end{pmatrix}\]
Since this holds for the electromagnetic field tensor \(F\), we find that under parity, \(E → -E\) and \(B → B\). For example, a capacitor seen in a mirror has its electric field pointing the opposite way, but there is no change in the magnetic field of a current loop, since the location of each current element is flipped to the other side of the loop, but its direction of flow is also reversed, so that the picture as a whole remains unchanged.
What about gravity?
A funny puzzle pops up if we go back and think about the assumptions above that went into all this. Those assumptions were so general that it almost seems as though the only possible behavior for fields is the behavior of electric and magnetic fields. But other fields do behave differently. How did the assumptions fail in the case of gravity, for example? Gauss’s law (assumption 2) certainly holds for gravity. But the source of gravitational fields isn’t charge, it’s mass-energy, and mass-energy isn’t a Lorentz invariant, contrary to assumption 3. Furthermore, assumption 1 entailed that our field could be defined in terms of forces measured by an inertial observer, but for an inertial observer gravity doesn’t exist (section 5.2).


10.E: Emotion and Motivation (Exercises)
10.1: Motivation
Review Questions
Q1
Need for ________ refers to maintaining positive relationships with others.
- achievement
- affiliation
- intimacy
- power
Q2
________ proposed the hierarchy of needs.
- William James
- David McClelland
- Abraham Maslow
- Albert Bandura
Q3
________ is an individual’s belief in her capability to complete some task.
- physiological needs
- self-esteem
- self-actualization
- self-efficacy
Q4
Carl mows the yard of his elderly neighbor each week for \(\$20\). What type of motivation is this?
- extrinsic
- intrinsic
- drive
- biological
Critical Thinking Questions
Q5
How might someone espousing an arousal theory of motivation explain visiting an amusement park?
Q6
Schools often use concrete rewards to increase adaptive behaviors. How might this be a disadvantage for students intrinsically motivated to learn? What are educational implications of the potential for concrete rewards to diminish intrinsic motivation for a given task?
Personal Application Question
Q7
Can you think of recent examples of how Maslow’s hierarchy of needs might have affected your behavior in some way?
Solution
S1
B
S2
C
S3
D
S4
A
S5
The idea of optimal levels of arousal is similar to a drive theory of motivation. Presumably, we all seek to maintain some intermediate level of arousal. If we are underaroused, we are bored. If we are overaroused, we experience stress. The rides at an amusement park would provide higher arousal (however, we would hope that these rides don’t actually pose significant threats to personal safety that would lead to a state of panic) to push us toward our own optimal level of arousal. Individuals at the park would choose different rides based on their specific arousal thresholds; for example, one person might find a simple water ride optimally arousing and an extreme roller coaster overarousing, while others would find the extreme roller coaster optimally arousing.
S6
We would expect to see a shift from learning for the sake of learning to learning to earn some reward. This would undermine the foundation upon which traditional institutions of higher education are built. For a student motivated by extrinsic rewards, dependence on those may pose issues later in life (post-school) when there are not typically extrinsic rewards for learning.
10.2: Hunger and Eating
Review Questions
Q1
According to your reading, nearly ________ of the adult population in the United States can be classified as obese.
- one half
- one third
- one fourth
- one fifth
Q2
________ is a chemical messenger secreted by fat cells that acts as an appetite suppressant.
- orexin
- angiotensin
- leptin
- ghrelin
Q3
________ is characterized by episodes of binge eating followed by attempts to compensate for the excessive amount of food that was consumed.
- Prader-Willi syndrome
- morbid obesity
- anorexia nervosa
- bulimia nervosa
Q4
In order to be classified as morbidly obese, an adult must have a BMI of ________.
- less than \(25\)
- \(25-29.9\)
- \(30-39.9\)
- \(40\) or more
Critical Thinking Questions
Q5
The index that is often used to classify people as being underweight, normal weight, overweight, obese, or morbidly obese is called BMI. Given that BMI is calculated solely on weight and height, how could it be misleading?
Q6
As indicated in this section, Caucasian women from industrialized, Western cultures tend to be at the highest risk for eating disorders like anorexia and bulimia nervosa. Why might this be?
Personal Application Question
Q7
Think about popular television programs on the air right now. What do the women in these programs look like? What do the men look like? What kinds of messages do you think the media is sending about men and women in our society?
Solution
S1
B
S2
C
S3
D
S4
D
S5
Using BMI as a sole metric can actually be misleading because people who have large amounts of lean muscle mass can actually be characterized as being overweight or obese based on their height and weight. Weight versus height is a somewhat crude measurement as it doesn’t distinguish the amount of body weight that comes from lean versus fatty tissue.
S6
These disorders are closely associated with sociocultural emphasis on a thin-ideal that is often portrayed in media. Given that non-Caucasians are under-represented in popular media in the West and that the thin-ideal is more heavily emphasized for women, this particular group is most vulnerable.
10.3: Sexual Behavior
Review Questions
Q1
Animal research suggests that in male rats the ________ is critical for the ability to engage in sexual behavior, but not for the motivation to do so.
- nucleus accumbens
- amygdala
- medial preoptic area of the hypothalamus
- hippocampus
Q2
During the ________ phase of the sexual response cycle, individuals experience rhythmic contractions of the pelvis that are accompanied by uterine contractions in women and ejaculation in men.
- excitement
- plateau
- orgasm
- resolution
Q3
Which of the following findings was not a result of the Kinsey study?
- Sexual desire and sexual ability can be separate functions.
- Females enjoy sex as much as males.
- Homosexual behavior is fairly common.
- Masturbation has no adverse consequences.
Q4
If someone is uncomfortable identifying with the gender normally associated with their biological sex, then he could be classified as experiencing ________.
- homosexuality
- bisexuality
- heterosexuality
- gender dysphoria
Critical Thinking Questions
Q5
While much research has been conducted on how an individual develops a given sexual orientation, many people question the validity of this research citing that the participants used may not be representative. Why do you think this might be a legitimate concern?
Q6
There is no reliable scientific evidence that gay conversion therapy actually works. What kinds of evidence would you need to see in order to be convinced by someone arguing that she had successfully converted her sexual orientation?
Personal Application Question
Q7
Issues related to sexual orientation have been at the forefront of the current political landscape. What do you think about current debates on legalizing same-sex marriage?
Solution
S1
C
S2
C
S3
A
S4
D
S5
Given the stigma associated with being non-heterosexual, participants who openly identify as homosexual or bisexual in research projects may not be entirely representative of the non-heterosexual population as a whole.
S6
Answers will vary, but it should be indicated that something more than self-reports of successful conversion would be necessary to support such a claim. Longitudinal, objective demonstrations of a real switch in both erotic attraction and the actual behavior in which the individual engaged would need to be presented in addition to assurances that this type of therapy was safe.
10.4: Emotion
Review Questions
Q1
Individuals suffering from posttraumatic stress disorder have been shown to have reduced volumes of the ________.
- amygdala
- hippocampus
- hypothalamus
- thalamus
Q2
According to the ________ theory of emotion, emotional experiences arise from physiological arousal.
- James-Lange
- Cannon-Bard
- Schachter-Singer two-factor
- Darwinian
Q3
Which of the following is not one of the seven universal emotions described in this chapter?
- contempt
- disgust
- melancholy
- anger
Q4
Which of the following theories of emotion would suggest that polygraphs should be quite accurate at differentiating one emotion from another?
- Cannon-Bard theory
- James-Lange theory
- Schachter-Singer two-factor theory
- Darwinian theory
Critical Thinking Questions
Q5
Imagine you find a venomous snake crawling up your leg just after taking a drug that prevented sympathetic nervous system activation. What would the James-Lange theory predict about your experience?
Q6
Why can we not make causal claims regarding the relationship between the volume of the hippocampus and PTSD?
Personal Application Question
Q7
Think about times in your life when you have been absolutely elated (e.g., perhaps your school’s basketball team just won a closely contested ballgame for the national championship) and very fearful (e.g., you are about to give a speech in your public speaking class to a roomful of 100 strangers). How would you describe how your arousal manifested itself physically? Were there marked differences in physiological arousal associated with each emotional state?
Solution
S1
B
S2
A
S3
C
S4
B
S5
The James-Lange theory would predict that I would not feel fear because I haven’t had the physiological arousal necessary to induce that emotional state.
S6
The research that exists is correlational in nature. It could be the case that reduced hippocampal volume predisposes people to develop PTSD or the decreased volume could result from PTSD. Causal claims can only be made when performing an experiment.
Contributors and Attributions
-
Rose M. Spielman with many significant contributors. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>.Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


13.20: Discussion- Moving Columns and Fields
You can view the transcript for “Access 2016 Field Order and Size” here (opens in new window).
Discussion Questions
- If there are a few different ways to rearrange fields, which way do you prefer and why?
- What are some common mistakes made to move fields or records in an Access table?
- What do you consider the best view (Design View, Datasheet View) for rearranging fields? Why?
Share your opinions below and respond to two of your classmates’ thoughts.
Contributors and Attributions
CC licensed content, Original
- Discussion: Moving Columns and Fields. Authored by : Sherri Pendleton. Provided by : Lumen Learning. License : CC BY: Attribution
All rights reserved content
- Access 2016 - Field Order and Size - How To Move the Column and Rearrange Fields Name in MS Database. Authored by : Professor Adam Morgan. Located at : https://youtu.be/VyZ234TaxcI . License : All Rights Reserved . License Terms : Standard YouTube License