8: Tone
Tone
Creating drawings using tone has been an interest of mine since childhood. My mother was the colorist, so naturally I moved in a different direction focusing more on drawing in pencil and charcoal. Even in my professional career as a medical illustrator, I am known more for my carbon dust work than my color illustrations. Carbon dust is a specialized technique using mainly brushes and a mix of ground charcoal and soft, black Conté pencils. I will share some examples during this chapter to make certain educational points.
Historically, tonal studies have mainly been for the creation of larger works such as paintings, and frescoes. Since much of the artwork created prior to the 20 th century was commissioned by a patron, it tended to be for the patron’s enjoyment, to express power, to document history, to educate, for a gift, or for commemoration. Few drawings in black and white for created as an end unto itself. That changed during the age when artists began to separate from the patron and create art for the sake of creating art. This trend begins in the mid 1800s and continues today.
Creating drawings with a continuous tone allows for modeling of the form in a more natural way. No longer are we abstracting from the visible forms as in line drawing, but are observing the light as it falls on the subject creating highlights and shadows – form in three-dimensions. Most of these forms can be reduced down to geometric shapes like spheres, cylinders, cubes, pyramids, cones and so on. Mastering the fundamental qualities of light on these forms helps tremendously in drawing them in tone. Applying this knowledge to even the most difficult subjects, like the human figure, will help the student create the illusion of three-dimensionality on the two-dimensional plane of the paper.
The Importance of Light
- The quality of light on form is very important
- A one-point (like a lamp) light source will give predictable highlights and shadows
- A strong on point light (like a spotlight on a stage) will be dramatic!
- Try to arrange the object(s) to maximize the effect of light and shadow on the form
- Strong one-point lighting also creates the most contrast
- Strong one-point lighting may also show less number of tones
- Soft lighting often will reveal a higher number of tonal variations
- Sunlight is high quality, fluorescent lighting is low quality
- Creating a value scale is good practice!
Adjacent Values
One very important property of tone mentioned in the above figure is the relationship of value to all adjacent values. The student drawing below is excellent (Figure 1). So how can we take it further? We can do this by focusing on and changing around the Adjacent Values. Look at how the intensity of the light and three-dimensional qualities of the form change. In Figure 2, the focus now becomes the form and separating the value on the top plane from plane of the table surface, also the reflected light is darkened. Why? It is darkened because it should be darker than the highlight area catching the light directly (not reflected off another surface). Notice also that the tone added to the top surface is added as a gradation to further push contrast (Figure 3, #1). The darkened reflected light area is labeled #2 in Figure 3. The approach in Figure 2 does improve the overall drawing, however, an even better approach is Figure 4. Figure 4, #1 shows the background plane darkened creating the illusion of light on the flat top plane of the toilet paper roll. The reflected light is still darkened at Figure 4, #2, and the cast shadow from the toilet paper is darkened and developed further.
Adjacent Values have an effect on each other - they can create more contrast; make form appear more three-dimensional; create the illusion of light on an object and so on. Look at her drawing again, and then look at the effect created by changing values in specific areas.
Value Scale
- Create a scale that has even transitions from one value to the next
- Use a variety of media and mark-making
- Notice the First Intensity scale has the broadest range of values
- The Limit of Intensity has the most contrast
Tonal Study
- Draw basic shapes with one-point lighting
- Notice the first drawing has the broadest range of values
- The last drawing has the most contrast
Tonal Study with Sphere
- The forms have bright highlights
- The forms also have general highlights
- The forms have shadows
- The forms have cast shadows
- The forms have reflected light
Example of Contrast and Value Range
The figure above shows a very successful composition by an artist using black-and-white tones. Where do your eyes go when you look at this black-and-white painting? How does the artist use contrast and tone to direct your eyes to the most important part of the composition?
Answer:
The artist places the most value contrast at the woman's face and hair. Additionally, the top of the sofa with its dark wooden frame directs the eyes also to the face. The dark pillow to her right creates a line on the edge of her dress again leading us up to her face.  This is a great example of an artist carefully designing a composition that successfully directs the viewers eyes to the exact place where he/she intended for the eyes to land.
Also note the overall compositional form –it’s a pyramid shape which is very stable–a favorite design construct of Da Vinci.
Here the artist began by “mapping out” the drawing (see above figure)
The artist next filled in darker values, including the background –this creates the illusion of light on the form
Next, the artist continued working all over (not just in one area) to build up the drawing –by doing this, changes can easily be made during the process
The final art has strong lighting from the back (backlighting) and a soft “fill” light from the front
Tonal (Value) Studies
The next three figures show artists using tone to draw the human figure. The first dark drawing is by Rembrandt. It is very solid, almost like a sculpture. The next two charcoal drawings are by Pierre-Paul Prud’hon. Notice that Pru’dhon tones in the background of the paper in order to show light hitting the skin of the figure. He also uses white chalk in his second drawing of the female nude sitting by a fountain. The use of white chalk is possible in this case because he is using gray paper.
The lighting is very soft in the drawings (in charcoal) by Prud’hon. He is a French master artist from the 1800s. Notice also the use of reflected light in his drawings helping to reveal the roundness of the forms.
Use of Tone in a Landscape
- Notice the planes of the buildings have different values
- Also see the use of positive and negative shapes are revealed by tone, not line
- The branches of the trees are revealed by dark tones and the snow, by leaving the paper white
Use of Contrast in a Landscape
Use of Light to Show Depth and Shadows from Sunlight
Tone Assignment
Where to Begin?
If we look at the form above we see an image made up of values. This image of a rose is made out of a kneaded eraser by a student. Very creative. But the point is that we see soft light hitting the slightly curved planes of the sculpted petals. Each petal in turn has a gradient from the lightest area through the curve to the edge. Under each petal is a cast shadow. When you begin to break down a complex three-dimensional form like this logically, then you are not as overwhelmed by drawing it. Again, we see highlights, gradations, and cast shadows. There are the three major tonal characteristics of this form.
For this assignment, follow these directives:
- Choose a place inside to draw where you control the lighting
- A one-point light source (like a lamp) is ideal
- Determine your eye level
- Choose simple geometric forms like a ball, sphere or box
- Simple round organic forms may also be used like an apple, orange, or egg
- Try to arrange the object(s) to maximize the effect of light and shadow on the form
- Create a balanced composition by cropping and the placement of the cast shadow
- Possibly use “Rule of Thirds” to create balance
- Use softer pencils from HB up to 6B
- Hold the pencil loosely and relax
- Focus on values
- Repeat using charcoal
Here are student drawings in pencil and charcoal:
Below are student tonal studies using squares and organic forms:
Elevating your Drawings:
The above set of drawings show a student drawing that demonstrates upper left lighting. The sequence that follows shows adding a background value and additional highlights to create more of an illusion of light on the forms.


3.5: Energy Accounting with Non-Conservative Forces: Thermal Energy
Energy Accounting for Non-Conservative Forces
Okay, so we have seen how we can pull conservative forces due to macroscopic objects into a system and change the accounting to using the simpler potential energies, but what about the case of work done by an external macroscopic object in the case where the force is not conservative? As we have seen already in such cases, we can still account for the total energy if the internal energies are included in the tally. The schematic diagram is only a bit different from the case of the conservative force.
Figure 3.5.1 – Changing Energy Accounting for Non-Conservative Forces
In this case, the external non-conservative force does not allow for redefining the system with a potential energy. Instead, the transfer of energy can only be accounted-for in terms of the internal energies of the objects involved. The change of accounting from the work-energy theorem to the new system therefore looks like:
This rather abstract description is difficult to wrap one's mind around without a clear example, as gravity was for the conservative force case. So we'll demonstrate how this accounting works with the most common example of a non-conservative force – kinetic friction. We want to keep this as simple as possible, so we will consider the following scenario...
The system in this example consists of two books. They are sliding across a horizontal tabletop, but the tabletop is frictionless, so it exerts no horizontal force on the books. The tabletop does exert a vertical force (the normal force) upward on the books, but as the books do not displace vertically, this force does no work. The Earth also exerts a force (gravity) on the books, but again, as they do not displace vertically, this force does no work either. So despite the presence of the tabletop and the Earth's gravity, the two books really are an isolated system, with no external work done on them (we also assume no air resistance during their motion). Below is a diagram of what occurs between the two books: At first they are separate, and then book \(A\) slides on top of stationary book \(B\), the kinetic friction force between them causing book \(B\) to speed up and book \(A\) to slow down, until they eventually stop rubbing across each other and slide away together at the same speed.
Figure 3.5.2 – Energy Accounting for Kinetic Friction
If we treat book \(B\) as a system, then the (non-conservative) friction force exerted on it by book \(A\) does external work, leading to a change in the kinetic energy of the system:
\[W_{AB}=+f\Delta x_B=\Delta KE_B=KE_{Bf}-KE_{Bo}=\frac{1}{2}m_Bv_f^2-0\]
If we instead treat both books as a system, then we need to include both of their kinetic energies and their internal energies, and we get the same result as is expressed in Equation 3.5.1 . The changes in the kinetic energies of the two books are equal to the works done on each, expressed in the diagram. There are two important things to note here. First, the change in the kinetic energy of book \(A\) is negative (it slows down), and this is evident from the work done on it by book \(B\), since the kinetic friction force is in the opposite direction of book \(A\)'s displacement. Second, the positive work done on book \(B\) (and therefore its kinetic energy change) is a smaller magnitude than the negative work done on book \(A\) (equal to its kinetic energy change), since they experience forces of equal magnitude (Newton's 3rd Law), and book \(A\) displaces farther than book \(B\) during the time that they are rubbing against each other. The upshot of these facts is that the quantity \(\Delta KE_B + \Delta KE_A\) is a negative number . This in turn requires that the changes of internal energy of the two books is positive – they collectively gain internal energy. This is manifested microscopically as an increase in the kinetic and potential energies of the particles in the books, a phenomenon we will address shortly.
We can't compute the change in internal energy of an individual book, but the combined increase in their internal energies can be expressed in terms of what is given in the diagram:
\[\Delta E_{int\;B}+ \Delta E_{int\;A}=-\left(\Delta KE_B + \Delta KE_A\right)=-f\Delta x_B+f\Delta x_A=f\left(\Delta x_A-\Delta x_B\right)\]
The quantity \(\Delta x_A-\Delta x_B\) is the distance that book \(A\) slides across book \(B\), and it is a positive number, which means that there is an increase in the internal energies of the two books. This makes sense, as it means that the amount of energy that goes into the internal energies of the books is a measure of the number of interactions between molecules at the surfaces of the books, and the distance that they slide across each other is directly related to the number of molecular interactions.
There is still much more we can do with this simple two-book model, and we will return to it in a future chapter.
Thermal Energy
There is one subset of the internal energy category that deserves special mention. In most macroscopic model calculations, the internal energy change that occurs in the system comes in the form of random motions of molecules in the participating objects. When two surfaces slide across each other with kinetic friction, the surface molecules push and pull against each other, and as they are bound to their respective objects, they react by vibrating. These vibrations are not in-sync – they are quite random, and the vibrations spread to their neighbor molecules as well. Such vibrating particles possess energy, both kinetic and potential (the latter due to the potential energy from the restoring Van der Waals forces), and this energy is within the objects themselves, so it is internal. This microscopic, random internal energy is so important that it is given its own name – thermal energy .
As is implied by the name, thermal energy is measured most easily through temperature – the two sliding books above become warmer. We will wait until Physics 9B to see how thermal energy and temperature are mathematically related to each other. For the purposes of this class, knowing that a change in temperature is an indication of a change in internal energy will be sufficient. One other important feature of thermal energy that we will not explain until Physics 9B is that with models like the sliding books, the conversion of some of a book's kinetic energy into thermal energy is a one-way trip. We would have a wait a long time if we ever wanted to see the two books cool off and spontaneously slide off each other. This comes from the fact that all (or a very large majority) of the randomly-moving momlecules would have to synchronize and move in the same direction at the same time, and this is improbable in the extreme.
Alert
While it will be some time before you encounter the idea of "heat" in Physics 9B, it is important to understand as early as possible that thermal energy is not the same as heat. Heat is not a form of internal energy.
To get some sense of how important thermal energy is to this discussion, it should be noted that it requires quite a contrived situation to produce an example in the mechanics of macroscopic objects where internal energy is not thermal energy. Indeed, for this reason, most textbooks don't even make the distinction, and instead go straight to the total energy conservation equation:
\[0=\Delta KE+\Delta U+\Delta E_{\text{thermal}}\]
Thermal energy can have different intrinsic properties for different physical systems. For example, for a system that is a gas, the particles don't interact very much, which means there is very little potential energy included in such a system's internal/thermal energy. For a solid, on the other hand, the molecules are bound to each other, so the energy is divided pretty equally between the potential energies of the particle interactions and the kinetic energies of their motions.
Summary of Energy Accounting
In the last two sections, we have talked about a dizzying number of ways that we can keep track of the energy accounting for a physical system, so let's get away from all the conceptual discussion and summarize what is most useful for solving problems. As a general rule, we will be dealing with systems involving macroscopic objects whose internal energies are all thermal. Aside from these thermal energies, all of the remaining energy will be mechanical. This means that what we will mainly need for any problem are two things: The equation for total energy conservation, and the ability to compute a change in thermal energy by performing the work integral for the non-conservative (usually kinetic friction) force:
\[0=\Delta KE + \Delta U +\Delta E_{thermal}\;,\;\;\;\;\;\Delta E_{thermal}=-W_f=-\int\limits_A^B f\cdot dl\]
Some important/useful notes:
- There could be more than one type of potential energy present at the same time (such as a spring and gravity), so keep in mind that the "\(\Delta U\)" in the formula above is a placeholder for all of them summed together.
- We are assuming here that the objects in the system are rigid and are not rotating, so that the kinetic energy for each object is simply \(\frac{1}{2}mv^2\). This allows us to not be concerned about the specifics of each object's center of mass motion, as every part of it is moving at the same speed. We will see how to deal with the kinetic energy associated with rotational motion in a future chapter.
- It is often helpful to break down the \(\Delta\)'s into "before" and "after", rewriting the energy conservation equation this way:
\[{KE}_{before}+U_{before} = {KE}_{after}+U_{after}+\Delta E_{thermal}\]
The interpretation of this equation is a simple one: The system begins with some total energy that is in the form of mechanical energy. Later, the distribution of this total energy has changed. The amount of kinetic and potential has changed, and assuming there was also a non-conservative force present, some of that starting energy has become thermal energy, internal to the objects. Naturally the system started with some thermal energy, and we could similarly divide that between the two sides of the equation, but this equation is used for specific calculations, and while it is possible to compute the thermal energy change using a work integral, it is not possible (in Physics 9A) to compute the thermal energy of an object directly.
A ball is launched straight up into the air with the apparatus shown below. The ball is pushed upward so that it compresses the spring, and is released from rest. It then travels around a frictionless half-circle track, at the bottom of which is a scale that measures the contact force the ball exerts on the track at that point.
- Analysis
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The first thing we note here is that if we ignore air resistance, there are no non-conservative forces present, which means that \(\Delta E_{thermal}=0\), and the mechanical energy of the system doesn't change from one moment to the next:
\[{KE}_{before}+U_{before} = {KE}_{after}+U_{after}\nonumber\]
Next, we can see that at various times in the motion of the ball, there are two different potential energies possible - one from the spring and one from gravity. We are told that the ball is launched by the spring, which means that the spring must have been compressed. If we call this amount of compression \(\Delta y\), then at the moment the ball is released, the system contains an amount of spring potential energy equal to:
\[U_{before}\left(spring\right)=\frac{1}{2}k{\Delta y}^2\nonumber\]
Determining the amount of gravitational potential energy stored in the system requires that we define a point of zero potential energy. Naturally we can choose anywhere we like as this zero-point, because ultimately only the change in the gravitational potential energy from "before" to "after" will matter, and that will be the same wherever we happen to call \(U_{grav}=0\). The diagram labels a height for the ball when it stops rising measured from the bottom of the track, so let's choose that position as zero gravity. Doing this, we can determine the gravitational potential energy of the ball just as it is launched. It's height above the bottom of the track is the radius of the track plus the amount the spring is compressed , so we have:
\[U_{before}\left(gravity\right)=mg\left(R+\Delta y\right)\nonumber\]
Clearly the ball starts at rest, so it starts with zero kinetic energy. This completes a "before" picture that we can use to solve a problem. There are a number of "after" times we might be asked about, most of them presumably after the ball has left contact with the spring. At such times, the ball will have a new height and a new speed. The resulting gravitational potential energy and kinetic energy can then be put into the "after" side of the equation, and one can then solve for unknowns.
One item that we have not accounted for is the presence of the scale. It doesn't contribute to the energy of the ball in any way, so why is it even mentioned? Well, the scale provides a clever way to measure the speed of the ball (and therefore its kinetic energy) at the bottom of the track. When the ball passes directly above the scale, it is moving in a circle, and the force exerted by the track on the ball at that point (which is measured by the scale) is straight up toward the center of the circle. The gravity force is straight down, so the net force on the ball is toward the center of the circle, which means its acceleration is centripetal. We therefore have:
\[F_{scale}-mg=m\frac{v_{bottom}^2}{R}\nonumber\]
So if we have the scale measurement (and obviously the mass of the ball), we can compute the kinetic energy of the ball at that bottom point from the radius of the track.
A puck slides down a frictionless track, over a short horizontal rough (frictional) patch, around a frictionless loop-de-loop, and into an ideal spring fixed to a wall, where it bounces off and goes back the other way, returning through the loop-de-loop, over the rough patch, and back up the ramp.
- Analysis
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We clearly have several modes for energy to change form here. The puck will change gravitational potential energy as if moves vertically, it will convert energy from mechanical to thermal when it experiences kinetic friction on the rough patch, potential energy in the spring will change when the puck compresses it, and of course the kinetic energy of the puck can change throughout.
The only weird part here is the loop-de-loop. Like the ramp, it introduces an opportunity for the gravitational potential energy to change, but what else can we say about it? Well, in a previous 'Analyze This' example , we determined a condition for the puck going around the loop to remain in contact. We tend to assume that the puck will make it around the loop to get to the spring, but in fact it has to have at least enough speed at the top of the loop so that it the gravitational force is barely enough to maintain centripetal acceleration. In particular, we found that the minimum speed it must have at the top of the loop is:
\[v_{min}=\sqrt{gR}\nonumber\]
We can express this as a minimum kinetic energy that the puck must have at the top of the loop in order to maintain contact:
\[{KE}_{min}=\frac{1}{2}mv_{min}^2=\frac{1}{2}mgR\nonumber\]
Given that the gravitational potential energy at the top of the loop is greater than at the bottom by an amount \(mg\Delta y = 2mgR\), it means that to make it around the loop the puck must have a kinetic energy of at least \(\frac{5}{2}mgR\). If it came down the ramp from rest with a vertical drop of \( \frac{5}{2}R\), then that would be just enough to let it get around, but of course in this case, the starting point would need to be higher, since the puck loses some of it's kinetic energy to thermal energy thanks to work done by kinetic friction, which we address next.
If we know the coefficient of kinetic friction, then all we need is the normal force between the puck and the rough patch to get the force of kinetic friction, and as this force is constant, the work done by the kinetic friction force is easy to compute, and this energy goes into increasing the thermal energy of the system:
\[\Delta E_{thermal} = -W_f = -\int \vec{F}_f\cdot \vec{dl} = F_fx=\mu_kNx=\mu_kmgx\nonumber\]
[The last equality comes from the fact that the surface where the rough patch lies is horizontal, so the normal force equals the puck's weight.]
There is one last important point to make in this analysis. As many places as are available for energy to turn forms, it is not usually necessary to include all of them in the process of solving a problem. Here's an example... Suppose we want to know how high the puck goes back up the ramp after it bounces back one time. We could track its motions through the process, going around the loop and bouncing off the spring, but assuming we know that it has enough energy to traverse the loop-de-loop (something we will need to check, using the criterion found above), we simply set the "before" time as the moment the puck is released and the "after" time when it comes to rest, then all we need to do is include the thermal energy resulting from two trips across the rough patch:
\[{KE}_{before}+U_{before} = {KE}_{after}+U_{after} + \Delta E_{thermal}\;\;\;\Rightarrow\;\;\; 0+mgy_{before} = 0 + mgy_{after}+2\mu_kmgx\;\;\;\Rightarrow\;\;\;y=h-2\mu_kx\nonumber\]
A block slides along a horizontal frictionless surface until it runs into a spring at its equilibrium length. Just as it starts to compress the spring, the surface becomes rough.
- Analysis
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With everything occurring on a horizontal surface, gravitational potential energy does not play a role. It's not that it isn't present, but it never changes, so it will not appear in our equations. The block will change kinetic energy, and the spring will gain potential energy as it is compressed. In addition, as the spring is being compressed, there is work being done by the kinetic friction force, so this will result in a change of thermal energy for the system. Our energy accounting therefore comes to:
\[0=\Delta KE + \Delta U_{spring} +\Delta E_{thermal}\nonumber\]
If we assume the friction force remains constant along the rough portion of the horizontal surface, then the work done by the constant kinetic friction force is just the negative of that force multiplied by the distance of the slide across the surface. The surface is horizontal, so the normal force equals the weight of the block. This distance is also the compression of the spring, so putting it all together, we have:
\[\Delta E_{thermal}=-W_f=-\left(-F_f\Delta x\right)=\mu_kN\Delta x=\mu_kmg\Delta x\;\;\;\Rightarrow\;\;\;0=\frac{1}{2}m\left(v_f^2-v_o^2\right)+\left(\frac{1}{2}k{\Delta x}^2-0\right)+\mu_kmg\Delta x\nonumber\]
If the roughness of the surface (measured by the coefficient of kinetic friction) does not remain constant along the slide of the block, then the problem becomes a bit tougher. The normal force is the same, but \(\mu_k\) is now a function of position, complicating the work integral:
\[\Delta E_{thermal}=-W_f=\int\limits_0^{\Delta x}F_fdx = \int\limits_0^{\Delta x}\mu\left(x\right)Ndx=mg\int\limits_0^{\Delta x}\mu\left(x\right)dx\nonumber\]


2.6: The Pleiades and The Big Dipper
The Pleiades
An asterism is a pattern of stars within a constellation, or composed of a number of constellations. The Pleiades, an open star cluster in the constellation of Taurus, is easy to find, and often confused as the Little Dipper. The Greek Pleaides myth is that there were seven sisters who were the daughters of Atlas and Pleione, a sea nymph. Zeus placed the sisters in the sky to protect them.
The Japanese call the Pleiades Subaru , Hawaiians Makahiki or many little eyes, and the Navajo Dilyehe the planting stars which represent youth. (1)
The Big Dipper
Perhaps the best-known asterism, at least in the northern hemisphere, is the Big Dipper, part of the constellation of Ursa Major, the Big Bear. It is easy to recognize, and does look like a dipper. Other cultures have identified it as a plow, wagon, barbeque pit, and perhaps the most meaningful, the drinking gourd.
As escaped slaves were leaving the south during the Civil War, they traveled north at night to try to go undetected. They were given directions to “follow the drinking gourd.” Since the Big Dipper is a constellation near the north celestial pole and the pole star Polaris, they could easily find the asterism and walk towards it; their own GPS system. (1)
Modern Constellations
Today we recognize 88 modern constellations, adopted by the International Astronomical Union in 1928. At the same time, the International Astronomical Union adopted constellation boundaries, much like state boundaries within a country like the United States.
Perhaps one of the most-intriguing — and disturbing – modern constellation stories took place in 2013, when a 22-year old New Hampshire woman stabbed her father during a fight outside while trying to identify constellations.
Finding constellations depend on your location; the less lighting you have, the better. Under some city lighting conditions, it is difficult to even find the bright stars, nevertheless many of the fainter stars that make up a constellation’s stick figure outline. (1)
- Authored by : Florida State College at Jacksonville. License : CC BY: Attribution


4.3: Planning Your Degree Path
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Questions to consider:
- What types of college degrees or certifications can I pursue?
- What is the difference between majors and minors?
- How do preprofessional programs differ from other majors?
- Do some majors have special requirements beyond regular coursework?
To set goals for your academic and career path, you must first have an understanding of the options available for you to pursue and the requirements you will need to meet. The next section provides an overview of academic programs and college degrees that are common among many colleges and universities in the United States. Please note that each institution will have its own specific options and requirements, so the intention of this section is both to help you understand your opportunities and to familiarize you with language that colleges typically use to describe these opportunities. After reviewing this section, you should be better able to formulate specific questions to ask at your school or be better prepared to navigate and search your own college’s website.
Types of Degrees
Whereas in most states high school attendance through the 12th grade is mandatory, or compulsory , a college degree may be pursued voluntarily. There are fields that do not require a degree. Bookkeeping, computer repair, massage therapy, and childcare are all fields where certification programs—tracks to study a specific subject or career without need of a complete degree—may be enough.
However, many individuals will find that an associate’s or bachelor’s degree is a requirement to enter their desired career field. According to United States Census data published in 2017, more than one-third of the adult population in the country has completed at least a bachelor’s degree, so this may be the degree that is most familiar to you.
Not every job requires a bachelor’s degree, and some require even higher degrees or additional specialized certifications. As you develop your academic plan, it is important to research your field of interest to see what requirements might be necessary or most desirable.
| Require Associate’s Degrees | Require Bachelor’s Degrees | Require Additional Certifications | Require Graduate Degrees |
|---|---|---|---|
| Radiology Technician | Nurse | Public School Teacher | Lawyer |
| Dental Hygienist | Computer/Information Systems Manager | Accountant | College or University Professor |
| Web Developer | Airline Pilot | Financial Advisor | Pharmacist |
| Graphic Designer | Electrical Engineer | Marriage and Family Therapist | |
| Automotive Technician | Construction Manager | Occupational Therapist |
To distinguish between the types of degrees, it is useful to understand that courses are often assigned a number of credits, sometimes called semester hours as well. Credits relate to the calculated hours during a course that a student spends interacting with the instructor and/or the course material through class time, laboratory time, online discussions, homework, etc. Courses at all degree levels are typically assigned a value of one to six credits, although students often need to complete a developmental education course or two, often in English or math. These requirements, which cost as much as typical college courses but do not grant college credit, are meant to provide some basic information students may have missed in high school but that will be necessary to keep up in college-level coursework.
The minimum or maximum number of credits required to graduate with different degrees varies by state or institution, but approximate minimum numbers of credits are explained below. Keep in mind that although a minimum number of credits must be completed to get a certain degree, total credits completed is not the only consideration for graduation—you must take your credits or courses in particular subjects indicated by your college.
To determine your best degree option, it might make sense to do some research to determine what kind of career you’re most interested in pursuing. Visit your campus career center to meet with a counselor to guide you through this process. These services are free to students—similar services can be pricey once you’ve graduated, so take advantage. There are other tools online you can investigate.
GET CONNECTED
These free, online self-assessments help you narrow down your choices.
- MyPlan identifies your motivations by having you rank different aspects of work, then creating a ranked list of different possible jobs.
- MAPP Test helps you determine what you love to do and what you don’t love to do and then creates a list of jobs that might be a good fit for you.
- The Career Cluster Interest Survey is a quick tool to let you create career clusters based on personal qualities and school subjects and activities you especially enjoy.
Associate’s Degrees
To enter an associate’s degree program, students must have a high school diploma or its equivalent. Associate’s degree programs may be intended to help students enter a technical career field, such as automotive technology, graphic design, or entry-level nursing in some states. Such technical programs may be considered an Associate of Applied Arts (AAA) or Associate of Applied Science (AAS) degrees, though there are other titles as well.
Other associate’s degree programs are intended to prepare a student with the necessary coursework to transfer into a bachelor’s degree program upon graduation. These transfer-focused programs usually require similar general education and foundational courses that a student would need in the first half of a bachelor’s degree program. Transfer-focused associate’s degrees may be called Associate of Arts (AA) or Associate of Science (AS), or other titles, depending on the focus of study.
An associate’s degree is typically awarded when a student has completed a minimum of 60 credits, approximately 20 courses, meeting the requirements of a specific degree. Some technical associate’s degrees, such as nursing, may require additional credits in order to meet requirements for special certifications. You may find that your college or university does not offer associate’s degrees. Most associate’s degrees are offered by community or junior colleges, or by career and technical colleges.
ACTIVITY
What to Ask
If you’re planning your associate’s degree, here are some specific questions you may want to research.
If you intend to enter a technical career that requires special certification:
- Does your college prepare you to take a certification exam, or will you be meeting those requirements through your courses?
- Does your college have any special internship or employment placement arrangements with employers to help you gain experience or get started in the field?
If you intend to transfer upon graduation:
- Is your college regionally accredited?
- Does your college have any special transfer agreements for guaranteed transfer of credits or perhaps for discounted tuition?
- Does your state have special transfer agreements or requirements that make it easier to transfer to colleges or universities within the same state?
Bachelor’s Degrees
When someone generally mentions “a college degree,” they are often referring to the bachelor’s degree, or baccalaureate degree. Because it takes four years of full-time attendance to complete a bachelor’s degree, this degree is also referred to as a “four-year degree.” Similar to an associate’s degree, to enter a bachelor’s degree program a student must have completed a high school diploma or its equivalent. Both associate’s degrees and bachelor’s degrees are considered undergraduate degrees , thus students working toward these degrees are often called undergraduates . A student with an associate’s degree may transfer that degree to meet some (usually half) of the requirements of a bachelor’s degree; however, completion of an associate’s degree is not necessary for entry into a bachelor’s degree program.
A bachelor’s degree is usually completed with a minimum of 120 credits, or approximately 40 courses. Some specialized degree programs may require more credits. (If an associate’s degree has been transferred, the number of credits from that degree usually counts toward the 120 credits. For example, if an associate’s degree was 60 credits, then a student must take 60 additional credits to achieve their bachelor’s degree.)
Bachelor of Arts (BA), Bachelor of Science (BS), Bachelor of Science in Nursing (BSN), and Bachelor of Fine Arts (BFA) are the most popular degree titles at this level and differ primarily in their focus on exploring a broader range of subject areas, as with a BA, versus focusing in more depth on a particular subject, as with a BS, BSN, or BFA. Regardless of whether a student is pursuing a BA, BS, BSN, or BFA, each of these programs requires a balance of credits or courses in different subject areas. In the United States, a bachelor’s degree comprises courses from three categories: general education courses, major courses, and electives . A fourth category of courses would be those required for a minor, which we will discuss in more detail in the section on majors and minors.
General Education
General education, also called core curriculum , is a group of courses that are often set as requirements by your state or by your individual college. These courses provide you with a foundation of knowledge across a breadth of fields and are also intended to help you further develop college-level critical-thinking and problem-solving abilities. You may be able to select courses from a general education menu of courses available at your institution. More than half of your bachelor’s degree program is likely made up of general education courses.
| English composition |
| Humanities courses that study our beliefs and the expression of our beliefs such as literature, philosophy, politics, art, or religious studies |
| Social science courses that study our behavior such as psychology, sociology, anthropology, or economics |
| Laboratory science courses such as biology, chemistry, physics, and environmental science |
| Mathematics |
| Technology or computer skills |
| Foreign language, diversity, or global studies courses that provide introduction to different cultures or global social issues and promote cultural awareness |
| College success or first-year experience courses that provide introduction to your specific institution, discuss college-level expectations and skills, and/or provide assistance with academic and career planning |
Major Courses
Major courses are courses in your field of interest and provide you with the foundational knowledge required for further study in that field or with the skills necessary to enter your career. Some schools may refer to these as career studies courses. Major courses often have a series of prerequisites , or courses that must be taken in sequence prior to other courses, starting with an introductory course and progressing into more depth. Major courses usually make up about a fourth or more of a bachelor’s degree (30 credits, or approximately 10 courses). A BS or BFA degree may require more major courses than a BA degree. Colleges and universities usually require students to select a major by the time they’ve completed 30 total credits.
Electives
Electives are free-choice courses. Though you may have a choice to select from a menu of options to meet general education and major requirements, electives are even less restricted. Some students may be able to take more electives than others due to their choice of major or if they are able to take courses that meet more than one requirement (for example, a sociology course may be both a major requirement and a general education social science course). Some colleges intentionally allow room for electives in a program to ensure that students, particularly those students who are undecided about their major, are able to explore different programs without exceeding the total number of credits required to graduate with a bachelor’s degree. In other cases, students may have taken all of their major courses and fulfilled their general education requirements but still need additional credits to fulfill the minimum to graduate. The additional courses taken to meet the total credit requirement (if necessary) are considered electives.
Graduate Degrees
According to United States Census data published in 2018, 13.1 percent of the U.S. adult population have completed advanced degrees. 3 Whereas associate’s and bachelor’s degrees are considered undergraduate degrees and require high school graduation for entry, advanced degrees called graduate degrees require prior completion of a bachelor’s degree. Some professions require graduate degrees as a minimum job qualification, such as pharmacists, lawyers, physical therapists, psychologists, and college professors. In other cases, students may be motivated to pursue a graduate degree to obtain a higher-level job or higher salary, or to be more competitive in their field. Some students are also interested in learning about some subject in greater depth than they did at the undergraduate level. Because graduate degrees do not include general education or free elective courses, they are very focused on career-specific knowledge and skills. Graduate degrees include master’s, doctoral, and professional degrees. Master’s degrees often require 30–60 credits and take one to two years of full-time attendance to complete. Some master’s degrees, like those for counselors, require supervised job experience as a component of the degree and therefore require more credits.
Doctorate and professional degrees are the highest level of advanced degrees. Approximately 3.5% of the U.S. adult population has completed a doctorate or professional degree. Very few careers require this level of education for entry, so fewer individuals pursue these degrees. Doctorates are offered in many subjects and primarily prepare students to become researchers in their field of study. This in-depth level of education often requires an additional 90–120 credits beyond the bachelor’s degree, and may or may not require a master’s degree prior to entry. (A master’s degree as an entry requirement may reduce the number of credits required to complete the doctoral degree.)
Professional degrees are a specific type of doctorate-level degree that focus on skills to be applied in a practical , or hands-on, career rather than as a researcher. The most common professional degrees are Doctor of Medicine (MD) for aspiring medical doctors, Juris Doctor (JD) for aspiring lawyers, Doctor of Pharmacy (PharmD) for aspiring pharmacists, and Doctor of Education (EdD) for aspiring school and college or university administrators. If the career you are pursuing requires a graduate degree, you should keep this end goal in mind as you plan for the timeline and finances required to meet your goals. You may also want to inquire about special agreements that your college or university may have to expedite admission into or completion of graduate degrees. For example, some universities offer 4+1 master’s programs , wherein students take both bachelor’s and master’s level courses during their last year as an undergraduate to accelerate the completion of both degrees.
Other Post-Baccalaureate Credentials
Post-baccalaureate refers to structured learning experiences pursued after a bachelor’s degree is achieved. While some such activities are structured into graduate degrees as described in the sections above, other fields value continuing education credits, competency badges, and additional certifications. These post-baccalaureate credentials may need to be completed prior to entering a career field, may be obtained as an option to gain competitive advantage for hiring, or may be achieved during the course of an individual’s career to stay current, maintain qualification, or be promoted. To determine if your field requires post-baccalaureate credentials, you may want to speak with an established professional in that career, review the qualifications section of related job descriptions, or visit with a career counselor on your campus. In a world that changes as rapidly as ours, engaging in lifelong learning is advisable regardless of the specific requirements of any particular career choice.
ACTIVITY
Draft an Education Timeline
Use the Bureau of Labor Statistics online Occupational Outlook Handbook to search for occupations that interest you, and note the level of education that these jobs require. Refer to your college’s academic catalog (frequently located on the college’s website) or curriculum maps to see the suggested sequence of courses for majors that relate to your career or careers of interest.
Select three interesting jobs and sketch a timeline for each, starting with your first semester of undergraduate study (whether you’re getting an associate’s or a bachelor’s) and ending with the point when you will meet the minimum requirements to be qualified for that job. Keep in mind that the timeline of students attending full-time may differ from those who are attending part-time.
Majors and Minors
One of the most common questions an undergraduate college student will be asked is “What’s your major?” As we already noted, your major is only one part of your undergraduate (associate’s or bachelor’s) degree, but it is the part that most demonstrates your interests and possible future goals. At some point during your studies you will be asked to decide on, or declare , a major. You may also be able to select a minor or additional concentration. Whereas a major comprises approximately 10–12 courses of a bachelor’s degree program and is required, a minor is usually 5–8 courses, is often optional, and may count toward or contribute to exceeding the total number of credits required for graduation. Rather than take elective courses, some students will select courses that meet the requirements for a minor. When selecting a major and possibly a minor, you’ll want to consider how the knowledge and skills you gain through those fields of study prepare you for a particular career. Majors and minors can be complementary. For example, a major in business might be well-matched with a minor in a foreign language, thus allowing the student to pursue a career in business with a company that hires bilingual employees. It is important to research careers of interest to you when selecting your major and/or minor to determine what will best help you to meet your goals.
Preprofessional Programs
Some undergraduate degrees are specifically designed to prepare students to later pursue professional degrees (such as the MD or PharmD) at the graduate level. Such programs are called preprofessional programs , preprofessional majors, or preprofessional tracks. The most common preprofessional programs are premed, prelaw, and prepharmacy, but you may see other offerings. Although some preprofessional programs are structured as majors that you can declare, many preprofessional programs are a sequence of recommended courses and activities that a student can follow alongside a related major. While following a preprofessional program may not guarantee your admittance to a professional program, it does increase the likelihood of acceptance to and preparation for a graduate professional program.
Consider Loretta’s story as an example of how a student might be on a preprofessional track. Loretta has decided that she would like to become a medical doctor. She has declared biology as her major and is taking the courses required to graduate with a bachelor of science degree in biology. Her university does not have a premed major, but it does have a premed track. She informs her academic advisor of her career goals, and her advisor provides her information about the premed track. The premed track includes a list of courses that students should take to prepare for the medical school entrance exams, called MCATs. Some of these courses are biology courses that overlap with Loretta’s biology major, while others are higher-level chemistry courses that are not required for her major. She can take these chemistry courses, and any other premed-track courses, as her elective courses. The premed track at Loretta’s university includes opportunities to attend MCAT study workshops and premed student club meetings. It also provides recommendations for summer volunteering and internships that will strengthen Loretta’s resume and application to medical school following the completion of her bachelor’s degree.
Special Requirements of Majors
While preprofessional programs prepare students for entrance into graduate professional degree programs, some undergraduate majors involve special requirements beyond the usual courses and classroom experience to prepare students for entrance to their career. Such requirements provide students practical experience or prerequisites for licensure necessary for a particular job. When requirements are major-specific, it is often because the requirement is state-mandated for that job. Majors that often include state-mandated special requirements are education, social work, and nursing. Some colleges and universities may require all students to participate in additional experiences beyond their regular coursework. You will want to ask your college about details specific to your major or institution. In this section we will generally discuss four such special requirements and experiences: fieldwork and internships, clinicals, student teaching, and service learning.
Fieldwork and Internships
Fieldwork and internships may also be referred to as practicums and field experience. These requirements provide hands-on work experience in a career, or field . When fieldwork or internships are required for your major, such as with a social work major, it is often listed as a course requirement among your major requirements. In other words, you usually receive credits for your fieldwork as you would for a lab or lecture course. Your fieldwork instructor will likely ask you to reflect on and report on your experiences. They will likely confer with a supervisor at your fieldwork site , the place where you are working, to help assess your hands-on learning. Fieldwork and internships provide students with opportunities to practice the skills they’ve learned in the classroom while also introducing them to the values and culture of the organizations and communities in which they hope to be employed. It is important to note that fieldwork and internship experiences are often available to students even if they are not required for their major. You may want to inquire with your academic advisors, faculty, or career services office to determine what opportunities might be available for you to gain this type of experience in your field of interest.
Clinicals
Clinicals are a type of fieldwork specifically required of nursing students. Clinicals may take place in hospitals, nursing homes, or mental health facilities. They provide nursing students who are nearing the end of their degree programs with the opportunity to practice nursing skills that cannot be learned in a regular classroom. During clinicals, students will interact with real patients to conduct physical examinations, draw blood, administer medicine, and provide other care as necessary. Because of the risk to patients, students participating in clinicals are more closely supervised by experienced professionals than those in other types of fieldwork experiences. Thus, clinicals function very much like a real-world classroom and progress to more independent work through the semester. Before undertaking clinicals, nursing students will need to complete certain coursework and pass a physical examination and background check. Because clinicals are often much longer than a class meeting, students will need to work with staff from the program to plan their schedule. It may not be feasible to work at another job while completing clinicals, so if you must work while you’re in college, it’s important to discuss this with nursing staff or academic advisors and to plan ahead.
Student Teaching
Student teaching is a specific type of fieldwork undertaken by students who plan to teach at the preschool, elementary, or middle and high school levels. Education students are often required to complete a student teaching experience in order to obtain a teaching license in their state. Students must often complete core education coursework prior to student teaching and must complete a background check prior to placement in a school setting. During their student teaching experience, students are usually paired one-on-one with an experienced teacher and have the opportunity to observe that teacher, get to know the students, understand the classroom culture, and participate in lessons as a teaching assistant as needed or appropriate. Much like nursing clinicals, this highly supervised fieldwork experience usually progresses to more independent work when the student teacher is asked to deliver and reflect on a lesson plan of their own design. Keep in mind as you plan for student teaching that unlike other fieldwork experiences, student teaching is limited to fall or spring semesters and cannot be completed in the summer because most schools are closed during the summer terms. Also, it may not be feasible to work at another job while completing your student teaching experience, so if you must work while you’re in college, it’s important to discuss this with your program staff or academic advisors and to plan ahead.
Service Learning
While service learning may not be required of a specific major, you may see this special requirement for a course or as a general graduation requirement for your college or university. It’s also an excellent opportunity to try out something that interests you, something that could lead to or be part of your eventual career.
Service learning is very much like volunteering or community service. The purpose of service learning is to interact with and meet the needs of your local community. Service learning does differ from volunteering in that it is more structured to meet specific learning goals. For example, if you were engaging in service learning for an environmental science course, your activities would likely be focused on local environmental issues. Or, if you were engaging in service learning for a sociology course, you would likely be working with local community groups or organizations not only to assist these organizations, but also to observe how groups interact. Like fieldwork, service learning provides you an opportunity to observe and apply concepts learned in the classroom in a real-world setting. Students are often asked to reflect on their service learning activities in the context of what they’ve been learning in class, so if you’re engaged in service learning, be thinking about how the activities you do relate to what you’ve learned and know.
WHAT STUDENTS SAY
-
Does your major have any special requirements that must be completed outside of the classroom?
- Fieldwork, internships, and/or student teaching
- Clinicals
- Other (write in)
- None
-
While in college, which of the following do you think you are most likely to do?
- Internship
- Part-time job
- Full-time job
- Study Abroad
-
What has influenced your academic and career plan the most? (rank order)
- The advice of friends and/or family
- The practical, hands-on experiences I’ve had outside of the classroom
- The knowledge and skills I’ve learned in class
- The advice I’ve gotten from mentors, advisors, or college faculty
You can also take the anonymous What Students Say surveys to add your voice to this textbook. Your responses will be included in updates.
Students offered their views on these questions, and the results are displayed in the graphs below.
Does your major have any special requirements that must be completed outside of the classroom?
While in college, which of the following do you think you are most likely to do?
What has influenced your academic and career plan the most?
Footnotes
- 2 Minimum degree qualifications may vary by state.
- 3 United States Census Bureau. (2019, February 21). Number of People with Master’s and Doctoral Degrees Doubles Since 2000. Retrieved from: https://www.census.gov/library/stori...ince-2000.html


1.23: Book XXIII
PENELOPE EVENTUALLY RECOGNISES HER HUSBAND—EARLY IN THE MORNING ULYSSES, TELEMACHUS, EUMAEUS, AND PHILOETIUS LEAVE THE TOWN.
Euryclea now went upstairs laughing to tell her mistress that her dear husband had come home. Her aged knees became young again and her feet were nimble for joy as she went up to her mistress and bent over her head to speak to her. “Wake up Penelope, my dear child,” she exclaimed, “and see with your own eyes something that you have been wanting this long time past. Ulysses has at last indeed come home again, and has killed the suitors who were giving so much trouble in his house, eating up his estate and ill treating his son.”
“My good nurse,” answered Penelope, “you must be mad. The gods sometimes send some very sensible people out of their minds, and make foolish people become sensible.  Why should you thus mock me when I have trouble enough already—talking such nonsense, and waking me up out of a sweet sleep that had taken possession of my eyes and closed them? I have never slept so soundly from the day my poor husband went to that city with the ill-omened name. Go back again into the women’s room; if it had been any one else who had woke me up to bring me such absurd news I should have sent her away with a severe scolding. As it is your age shall protect you.”
“My dear child,” answered Euryclea, “I am not mocking you. It is quite true as I tell you that Ulysses is come home again. He was the stranger whom they all kept on treating so badly in the cloister. Telemachus knew all the time that he was come back, but kept his father’s secret that he might have his revenge on all these wicked people.”
Then Penelope sprang up from her couch, threw her arms round Euryclea, and wept for joy. “But my dear nurse,” said she, “explain this to me; if he has really come home as you say, how did he manage to overcome the wicked suitors single handed, seeing what a number of them there always were?”
“I was not there,” answered Euryclea, “and do not know; I only heard them groaning while they were being killed. We sat crouching and huddled up in a corner of the women’s room with the doors closed, till your son came to fetch me because his father sent him. Then I found Ulysses standing over the corpses that were lying on the ground all round him, one on top of the other. You would have enjoyed it if you could have seen him standing there all bespattered with blood and filth, and looking just like a lion. But the corpses are now all piled up in the gatehouse that is in the outer court, and Ulysses has lit a great fire to purify the house with sulphur. He has sent me to call you, so come with me that you may both be happy together after all; for now at last the desire of your heart has been fulfilled; your husband is come home to find both wife and son alive and well, and to take his revenge in his own house on the suitors who behaved so badly to him.”
“My dear nurse,” said Penelope, “do not exult too confidently over all this. You know how delighted every one would be to see Ulysses come home—more particularly myself, and the son who has been born to both of us; but what you tell me cannot be really true. It is some god who is angry with the suitors for their great wickedness, and has made an end of them; for they respected no man in the whole world, neither rich nor poor, who came near them, and they have come to a bad end in consequence of their iniquity; Ulysses is dead far away from the Achaean land; he will never return home again.”
Then nurse Euryclea said, “My child, what are you talking about? but you were all hard of belief and have made up your mind that your husband is never coming, although he is in the house and by his own fire side at this very moment. Besides I can give you another proof; when I was washing him I perceived the scar which the wild boar gave him, and I wanted to tell you about it, but in his wisdom he would not let me, and clapped his hands over my mouth; so come with me and I will make this bargain with you—if I am deceiving you, you may have me killed by the most cruel death you can think of.”
“My dear nurse,” said Penelope, “however wise you may be you can hardly fathom the counsels of the gods. Nevertheless, we will go in search of my son, that I may see the corpses of the suitors, and the man who has killed them.”
On this she came down from her upper room, and while doing so she considered whether she should keep at a distance from her husband and question him, or whether she should at once go up to him and embrace him. When, however, she had crossed the stone floor of the cloister, she sat down opposite Ulysses by the fire, against the wall at right angles 180 [to that by which she had entered], while Ulysses sat near one of the bearing-posts, looking upon the ground, and waiting to see what his brave wife would say to him when she saw him. For a long time she sat silent and as one lost in amazement. At one moment she looked him full in the face, but then again directly, she was misled by his shabby clothes and failed to recognise him, 181 till Telemachus began to reproach her and said:
“Mother—but you are so hard that I cannot call you by such a name—why do you keep away from my father in this way? Why do you not sit by his side and begin talking to him and asking him questions? No other woman could bear to keep away from her husband when he had come back to her after twenty years of absence, and after having gone through so much; but your heart always was as hard as a stone.”
Penelope answered, “My son, I am so lost in astonishment that I can find no words in which either to ask questions or to answer them. I cannot even look him straight in the face. Still, if he really is Ulysses come back to his own home again, we shall get to understand one another better by and by, for there are tokens with which we two are alone acquainted, and which are hidden from all others.”
Ulysses smiled at this, and said to Telemachus, “Let your mother put me to any proof she likes; she will make up her mind about it presently.  When one man has killed another—even though he was not one who would leave many friends to take up his quarrel—the man who has killed him must still say good bye to his friends and fly the country; whereas we have been killing the stay of a whole town, and all the picked youth of Ithaca. I would have you consider this matter.”
“Look to it yourself, father,” answered Telemachus, “for they say you are the wisest counsellor in the world, and that there is no other mortal man who can compare with you. We will follow you with right good will, nor shall you find us fail you in so far as our strength holds out.”
“I will say what I think will be best,” answered Ulysses. “First wash and put your shirts on; tell the maids also to go to their own room and dress; Phemius shall then strike up a dance tune on his lyre, so that if people outside hear, or any of the neighbours, or some one going along the street happens to notice it, they may think there is a wedding in the house, and no rumours about the death of the suitors will get about in the town, before we can escape to the woods upon my own land. Once there, we will settle which of the courses heaven vouchsafes us shall seem wisest.”
Thus did he speak, and they did even as he had said. First they washed and put their shirts on, while the women got ready. Then Phemius took his lyre and set them all longing for sweet song and stately dance. The house re-echoed with the sound of men and women dancing, and the people outside said, “I suppose the queen has been getting married at last. She ought to be ashamed of herself for not continuing to protect her husband’s property until he comes home.” 182
This was what they said, but they did not know what it was that had been happening. The upper servant Eurynome washed and anointed Ulysses in his own house and gave him a shirt and cloak, while Minerva made him look taller and stronger than before; she also made the hair grow thick on the top of his head, and flow down in curls like hyacinth blossoms; she glorified him about the head and shoulders just as a skilful workman who has studied art of all kinds under Vulcan or Minerva—and his work is full of beauty—enriches a piece of silver plate by gilding it. He came from the bath looking like one of the immortals, and sat down opposite his wife on the seat he had left. “My dear,” said he, “heaven has endowed you with a heart more unyielding than woman ever yet had. No other woman could bear to keep away from her husband when he had come back to her after twenty years of absence, and after having gone through so much. But come, nurse, get a bed ready for me; I will sleep alone, for this woman has a heart as hard as iron.”
“My dear,” answered Penelope, “I have no wish to set myself up, nor to depreciate you; but I am not struck by your appearance, for I very well remember what kind of a man you were when you set sail from Ithaca. Nevertheless, Euryclea, take his bed outside the bed chamber that he himself built. Bring the bed outside this room, and put bedding upon it with fleeces, good coverlets, and blankets.”
She said this to try him, but Ulysses was very angry and said, “Wife, I am much displeased at what you have just been saying. Who has been taking my bed from the place in which I left it? He must have found it a hard task, no matter how skilled a workman he was, unless some god came and helped him to shift it. There is no man living, however strong and in his prime, who could move it from its place, for it is a marvellous curiosity which I made with my very own hands. There was a young olive growing within the precincts of the house, in full vigour, and about as thick as a bearing-post. I built my room round this with strong walls of stone and a roof to cover them, and I made the doors strong and well-fitting. Then I cut off the top boughs of the olive tree and left the stump standing. This I dressed roughly from the root upwards and then worked with carpenter’s tools well and skilfully, straightening my work by drawing a line on the wood, and making it into a bed-prop. I then bored a hole down the middle, and made it the centre-post of my bed, at which I worked till I had finished it, inlaying it with gold and silver; after this I stretched a hide of crimson leather from one side of it to the other. So you see I know all about it, and I desire to learn whether it is still there, or whether any one has been removing it by cutting down the olive tree at its roots.”
When she heard the sure proofs Ulysses now gave her, she fairly broke down. She flew weeping to his side, flung her arms about his neck, and kissed him. “Do not be angry with me Ulysses,” she cried, “you, who are the wisest of mankind. We have suffered, both of us. Heaven has denied us the happiness of spending our youth, and of growing old, together; do not then be aggrieved or take it amiss that I did not embrace you thus as soon as I saw you. I have been shuddering all the time through fear that someone might come here and deceive me with a lying story; for there are many very wicked people going about. Jove’s daughter Helen would never have yielded herself to a man from a foreign country, if she had known that the sons of Achaeans would come after her and bring her back. Heaven put it in her heart to do wrong, and she gave no thought to that sin, which has been the source of all our sorrows. Now, however, that you have convinced me by showing that you know all about our bed (which no human being has ever seen but you and I and a single maidservant, the daughter of Actor, who was given me by my father on my marriage, and who keeps the doors of our room) hard of belief though I have been I can mistrust no longer.”
Then Ulysses in his turn melted, and wept as he clasped his dear and faithful wife to his bosom. As the sight of land is welcome to men who are swimming towards the shore, when Neptune has wrecked their ship with the fury of his winds and waves; a few alone reach the land, and these, covered with brine, are thankful when they find themselves on firm ground and out of danger—even so was her husband welcome to her as she looked upon him, and she could not tear her two fair arms from about his neck. Indeed they would have gone on indulging their sorrow till rosy-fingered morn appeared, had not Minerva determined otherwise, and held night back in the far west, while she would not suffer Dawn to leave Oceanus, nor to yoke the two steeds Lampus and Phaethon that bear her onward to break the day upon mankind.
At last, however, Ulysses said, “Wife, we have not yet reached the end of our troubles. I have an unknown amount of toil still to undergo. It is long and difficult, but I must go through with it, for thus the shade of Teiresias prophesied concerning me, on the day when I went down into Hades to ask about my return and that of my companions.  But as heaven has put it in your mind to speak of it, tell me about the task that lies before you. I shall have to hear about it later, so it is better that I should be told at once.”
“My dear,” answered Ulysses, “why should you press me to tell you? Still, I will not conceal it from you, though you will not like it. I do not like it myself, for Teiresias bade me travel far and wide, carrying an oar, till I came to a country where the people have never heard of the sea, and do not even mix salt with their food. They know nothing about ships, nor oars that are as the wings of a ship. He gave me this certain token which I will not hide from you. He said that a wayfarer should meet me and ask me whether it was a winnowing shovel that I had on my shoulder. On this, I was to fix my oar in the ground and sacrifice a ram, a bull, and a boar to Neptune; after which I was to go home and offer hecatombs to all the gods in heaven, one after the other. As for myself, he said that death should come to me from the sea, and that my life should ebb away very gently when I was full of years and peace of mind, and my people should bless me. All this, he said, should surely come to pass.”
And Penelope said, “If the gods are going to vouchsafe you a happier time in your old age, you may hope then to have some respite from misfortune.”
Thus did they converse. Meanwhile Eurynome and the nurse took torches and made the bed ready with soft coverlets; as soon as they had laid them, the nurse went back into the house to go to her rest, leaving the bed chamber woman Eurynome 183 to show Ulysses and Penelope to bed by torch light. When she had conducted them to their room she went back, and they then came joyfully to the rites of their own old bed.  They then laid themselves down to sleep in the cloisters.
When Ulysses and Penelope had had their fill of love they fell talking with one another.  Ulysses in his turn told her what he had suffered, and how much trouble he had himself given to other people. He told her everything, and she was so delighted to listen that she never went to sleep till he had ended his whole story.
He began with his victory over the Cicons, and how he thence reached the fertile land of the Lotus-eaters. He told her all about the Cyclops and how he had punished him for having so ruthlessly eaten his brave comrades; how he then went on to Aeolus, who received him hospitably and furthered him on his way, but even so he was not to reach home, for to his great grief a hurricane carried him out to sea again; how he went on to the Laestrygonian city Telepylos, where the people destroyed all his ships with their crews, save himself and his own ship only. Then he told of cunning Circe and her craft, and how he sailed to the chill house of Hades, to consult the ghost of the Theban prophet Teiresias, and how he saw his old comrades in arms, and his mother who bore him and brought him up when he was a child; how he then heard the wondrous singing of the Sirens, and went on to the wandering rocks and terrible Charybdis and to Scylla, whom no man had ever yet passed in safety; how his men then ate the cattle of the sun-god, and how Jove therefore struck the ship with his thunderbolts, so that all his men perished together, himself alone being left alive; how at last he reached the Ogygian island and the nymph Calypso, who kept him there in a cave, and fed him, and wanted him to marry her, in which case she intended making him immortal so that he should never grow old, but she could not persuade him to let her do so; and how after much suffering he had found his way to the Phaeacians, who had treated him as though he had been a god, and sent him back in a ship to his own country after having given him gold, bronze, and raiment in great abundance. This was the last thing about which he told her, for here a deep sleep took hold upon him and eased the burden of his sorrows.
Then Minerva bethought her of another matter. When she deemed that Ulysses had had both of his wife and of repose, she bade gold-enthroned Dawn rise out of Oceanus that she might shed light upon mankind. On this, Ulysses rose from his comfortable bed and said to Penelope, “Wife, we have both of us had our full share of troubles, you, here, in lamenting my absence, and I in being prevented from getting home though I was longing all the time to do so. Now, however, that we have at last come together, take care of the property that is in the house. As for the sheep and goats which the wicked suitors have eaten, I will take many myself by force from other people, and will compel the Achaeans to make good the rest till they shall have filled all my yards. I am now going to the wooded lands out in the country to see my father who has so long been grieved on my account, and to yourself I will give these instructions, though you have little need of them.  See nobody and ask no questions.” 185
As he spoke he girded on his armour. Then he roused Telemachus, Philoetius, and Eumaeus, and told them all to put on their armour also. This they did, and armed themselves. When they had done so, they opened the gates and sallied forth, Ulysses leading the way. It was now daylight, but Minerva nevertheless concealed them in darkness and led them quickly out of the town.


1.105: Symmetry operation
A symmetry operation is an isometry , i . e . a transformation under which two objects, or two configurations or an object, are brought to coincide. A symmetry operation is a Euclidean mapping: to each point of the first configuration there corresponds a point of the second configuration, the distances between two points are kept by the transformation, as are the angles.
The two configurations/objects can be either congruent or enantiomorphous. Correspondingly, the symmetry operations are classed into two kinds:
- symmetry operations of first kind : they relate congruent objects and consist of translations, rotations and screw rotations;
- symmetry operations of second kind : they relate enantiomorphous objects and consist of inversion, reflections, rotoinversions, and glide reflections. There exist a 1:1 correspondence between rotoinversion and rotoreflections: the latter are more used in Schoenflies notation, whereas rotoinversions are preferred in Hermann-Mauguin notation.
A symmetry operation is performed about a symmetry element.


5.3: The Importance of Sexuality
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Sexuality is important to us because it represents an activity that is a rite of passage into adulthood, because it is very pleasurable, and because it reinforces our roles and aspirations as males and females.
Samuel and Cynthia Janus published The Janus Report on Sexual Behavior in 1993. They studied a scientific sample of 2,765 men and women and reported some general trends in U.S. sexual practices and patterns. They found that age-based estimates indicate a great deal of similarity in sexual frequency between age groups with 2-3 sexual encounters per week.
Sex is a minor (yet important) part of our daily time allocation. People with a sexual partner available have sex about 3 times per week, taking about 25 minutes per experience. That means about 75 minutes per week or 3,900 minutes per year is spent having sex. Divide 3,900 by 60 minutes, and it equals about 65 hours per year having sex. At first glance that sounds like a great deal of time allocation, but keep in mind that in comparison, most of us spend most of our lives doing nonsexual things.
Consider these estimates: if the average person sleeps about eight hours in a 24 hour period, works 8.5 hours, eats 1.5 hours, commutes .5 hour, watches TV for three hours, and spends about 2.5 hours for miscellaneous activities, then compared to routine non-sexual activities, sexual intercourse comprises a relatively small portion of our time.
Relatively speaking sexual intercourse is a passive part of the average person’s life accounting for only 65 yearly hours of involvement per year. Many people refrain from regular sexual intercourse until their twenties and are less likely to participate in it if they are not married than are married people.  The average would be much lower if those categories of people were averaged into the equation.


2.10: Chapter 2 Formula Review
2.2 Measures of the Location of the Data
\(i=\left(\frac{k}{100}\right)(n+1)\)
where \(i\) = the ranking or position of a data value,
\(k\) = the \(k\)th percentile,
\(n\) = total number of data.
Expression for finding the percentile of a data value: \(\left(\frac{x+0.5 y}{n}\right)(100)\)
where \(x\) = the number of values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile,
\(y\) = the number of data values equal to the data value for which you want to find the percentile,
\(n\) = total number of data
2.3 Measures of the Center of the Data
\(\mu=\frac{\sum f m}{\sum f}\) Where \(f\) = interval frequencies and \(m\) = interval midpoints.
The arithmetic mean for a sample (denoted by \(\overline{x}\)) is \(\overline{x}=\frac{\text { Sum of all values in the sample }}{\text { Number of values in the sample }}\)
The arithmetic mean for a population (denoted by μ ) is \({\mu}=\frac{\text { Sum of all values in the population }}{\text { Number of values in the population }}\)
2.7 Measures of the Spread of the Data
\(s_{x}=\sqrt{\frac{\sum f m^{2}}{n}-\overline{x}^{2}} \text { where } \) \(\begin{array}{l}{s_{x}=\text { sample standard deviation }} \\ {\overline{x}=\text { sample mean }}\end{array}\)
Formulas for Sample Standard Deviation \(s=\sqrt{\frac{\Sigma({x}_i-\overline{x})^{2}}{n-1}} \text { or } s=\sqrt{\frac{\Sigma {f}_i({x}_i - \overline{x})^{2}}{n-1}} \)
For the sample standard deviation, the denominator is n - 1, that is the sample size - 1.
Formulas for Population Standard Deviation \({\sigma}=\sqrt{\frac{\Sigma({x}_i - \mu)^{2}}{N}} \text { or } \sigma=\sqrt{\frac{\Sigma {f}_i({x}_i - \mu)^{2}}{N}} \)
For the population standard deviation, the denominator is N , the number of items in the population.


9.5: Practice
Express the shaded portion of each figure as both a fraction and as a decimal. Justify your answers.
Which number is bigger: 0.135 or 0.14? Justify your answer.
Arrange the digits 1, 2, 3, and 4 in the boxes to create the smallest possible sum. Use each digit exactly once. Justify that your answer is as small as possible.
\[\frac{\Box}{\Box} + \frac{\Box}{\Box} \nonumber \]
Arrange the digits 1, 2, 3, and 4 in the boxes to create the smallest possible (positive) difference. Use each digit exactly once. Justify that your answer is as small as possible.
\[\frac{\Box}{\Box} - \frac{\Box}{\Box} \nonumber \]
Use the “Dots & Boxes” model to show that \(\frac{1}{9} = 0. \bar{1}\). Then use this fact to answer these questions and justify your answers.
- What fraction is given by \(0. \bar{2}\)?
- What fraction is given by \(0. \bar{5}\)?
- What fraction is given by \(0. \bar{6}\)?
- What fraction is given by \(0. \bar{8}\)?
- What fraction is given by \(0. \bar{9}\)?
In this problem, you will focus on the calculation
\[170 \times \Box \ldotp \nonumber \]
Your goal is to get a product that is close to 200.
- Will you multiply 170 by a number greater or less than 1? Greater or less than 2? Justify your answers.
- Suppose you can use only one decimal place. Fill in the box with a number that gets as close to 200 as possible.
- Suppose you can use only two decimal places. Fill in the box with a number that gets as close to 200 as possible.
- Suppose you can use only three decimal places. Fill in the box with a number that gets as close to 200 as possible.
Do each computation below without using a calculator. Explain your thinking.
- $(23 \times 0.1) + (0.001 \times 55) \ldotp$
- $18.45 \div (0.63 \div 0.7) \ldotp$
- $22.65 - (0.03 \cdot 10) \ldotp$
Without actually calculating anything (just use your number sense!), order x, y, and z from smallest to largest. Explain your ordering.
\[\begin{split} x &= 0.07 + 0.000001 \\ y &= 0.07 \times 0.000001 \\ z &= 0.07 \div 0.000001 \end{split} \nonumber \]
For each question below, choose the correct calculation and explain your choice. Then estimate the answer (don’t calculate it exactly) and explain why your estimate is a good one.
- A large pizza has eight slices and costs $15.95. How much does each slice of pizza cost? Should you calculate 15.95 × 8 or 15.95 ÷ 8?
- There are 2.54 centimeters in an inch. A standard sheet of notebook paper is \(8 \frac{1}{2}\) inches wide and 11 inches long. How many centimeters wide is the page? Should you calculate 8.5 × 2.54 or 11 × 2.54 or 8.5 ÷ 2.54 or 11 ÷ 2.54?
- In a model train set, 1.38 inches represents one foot in real life. The height of One World Trade Center in New York City is 1776 feet. How tall would a scale model of the building be? Should you calculate 1776 × 1.38 or 1776 ÷ 1.38?
- Eight-tenths of a jumprope is 1.75 meters long. How long is the whole rope? Should you calculate 0.8 × 1.75 or 0.8 ÷ 1.75 or 1.75 ÷ 0.8?
Kaimi had no money at all when he cashed his paycheck. As he left the bank, he bought a piece of candy for a nickel from a machine. Later, he realized that the money in his pocket was equal to twice his paycheck. After a quick calculation, he figured out what happened: the teller accidentally switched the dollars and cents. How much was Kaimi supposed to be paid, and what did the teller give him? Justify your answer.
Here are the rules to a card game. Read the rules carefully and then answer the questions below.
- Each player starts with 10 points. The goal is to score as close to 100 points as possible without going over.
- On your turn: draw two cards, which will each have a decimal number on them. Using estimation (no computation), you can choose to multiply or divide your current score by one of the decimal numbers.
- After you decide, compute your new score exactly using a calculator. If your new score is over 100, you lose. If not, the other player takes a turn.
 If you do, the other player gets one more turn. Then, the player with the score that is closest to 100 without going over wins the game.
Here are the questions:
- On your turn, your score is 50. You draw the cards 0.2 and 1.75. Remember that your choices are: $\begin{split} &\text{divide by}\; 0.2 \qquad \text{multiply by}\; 0.2 \\ &\text{divide by}\; 1.75 \qquad \text{multiply by}\; 1.75 \ldotp \end{split}$What is your best move and why?
- On your turn, your score is 88. You draw 1.3 and 0.6. What is your best move and why?
- Your partner has a score of 57, and your score is 89. On her turn, your partner draws 0.8 and 1.8. She says she wants to end the game.  If you both make the best possible move, who will win the game? Justify your answer.


2.3: Recommended Immunization Schedules in Canada
The immunization schedule in each Canadian province and territory tells health professionals which vaccines should be administered at what time. Here is a link to the immunization schedule for infants and children in each province and territory: https://www.canada.ca/en/public-health/services/provincial-territorial-immunization-information/provincial-territorial-routine-vaccination-programs-infants-children.html .
The immunization schedule includes the recommended timing of all vaccine doses in order to be considered fully protected. There are many determinants that influence the schedule including: burden of disease in the province or territory, effectiveness and safety of vaccines, acceptability, ease of implementation, equity, and cost. When individuals are not on the routine schedule, the likelihood of missing doses and having incomplete vaccinations is increased. It also puts the individual at risk for getting the disease because they are not fully vaccinated. Catch-up schedules are available. If vaccination records are not available, assume the person is unvaccinated and use a catch-up schedule based on current age.
Accelerated immunization schedules are also available under necessary conditions such as outbreaks and unplanned travel. The interval between recommended doses can also often be lessened; however, health professionals should consult the Canadian Immunization Guide for vaccine-specific minimum dose intervals.
Points of Consideration
Being vaccinated on time is ideal. For people who start their vaccines later, health professionals can use a ‘ catch-up’ schedule which depends on the person’s age. For people with an unknown vaccine history (e.g. newcomers, refugees), the best practice is to treat the person as if they are unimmunized and begin a new immunization schedule.
Timing and Spacing of Vaccines
If vaccines are given before the recommended age or with less than the minimal spacing between doses, adequate immune response may not be achieved. Modifications to the recommended schedule may be necessary because of missed appointments or illness; however, for the most part, disruptions in the schedule do not necessitate restarting a vaccine series.
Community Immunity / Herd Immunity
By keeping up to date with vaccines, we can prevent the spread of disease to those who may be more likely to contract them, such as:
- babies.
- young children.
- pregnant people.
- people who are severely immunosuppressed.
Recall the discussion of community immunity, also known as herd immunity, from Chapter 1 , in which we learned that in general, more than 90% of the population needs to be immune in order to protect those who are not eligible to be vaccinated (shown in the image below).