Your colleague asks you to listen to them practice their speech practice and give them feedback.
Your teacher asks you to give feedback to another classmate about their speech.
Your boss asks, “What did you think about my speech?”
In each case, the person is looking to you to provide feedback. In this chapter, I will talk about how to assess the feedback situation, how to offer constructive criticism, and how to graciously receive criticism. Let’s start with how to ask for feedback and listen graciously.
Beta Testing with Friends
One useful way to test out your speech is with friends or trusted colleagues. In the world of fan fiction, writers send out a manuscript for beta testing. The idea is that a group of people with eyes on a text can provide helpful suggestions. Seek out those individuals in your life who will be beta testers for you. It is especially important for you to identify the purpose of your speech and what potential problems to look for. For example, are you wanting someone to proof your outline for formatting mistakes or are you wanting someone to tell you if your speech will work for the audience you have? You should seek out friends who will build you up, but you should also seek out those who will give you honest feedback.
Receiving Feedback
When you ask for feedback from others, receive their feedback as a gift. Someone is taking their time and giving it to you; someone is putting themselves out there and saying things that might cause discomfort, but they are doing it for you. Individuals vary on how they receive feedback and how comfortable they are with being evaluated. It can be hard, I know. Growth can be hard.
When receiving feedback, try doing the following:
Sit in a non-defensive posture. It is tempting to cross your arms and to tense up all your muscles when receiving oral feedback. Keep your body open and loose. Staying open helps them to feel like you really want their suggestions and closed arms can equal a closed mind — keep an open body.
Do not take feedback as a personal insult.
If the feedback is verbal, write down the suggestions (even if you disagree with the suggestions). Respect the other person’s opinions by writing them down. It makes them feel like they have been heard and you appreciate the feedback they are giving. (Writing the feedback down also helps you to not cross your arms defensively–see suggestion one– and it helps you remember the suggestions.
Do not take it as a personal insult. (I mean it).
Avoid the temptation to defend yourself. “I did it this way because…” or, “I thought it would be best to…” You already know why you did things the way you did. Interrupting them to tell them the reasons you did what you did comes off as defensive and reduces the likelihood they will give you all the feedback they have to offer. You already know what you were thinking and by telling them you haven’t advanced your situation. Use this time to learn what they are thinking.
Do not take it as a personal insult. (Really, this is so important).
Breathe. Most people feel stress when someone is giving them constructive criticism, breathe and relax so you can really listen.
Do not take it personally. Do not take it personally. Do not take it personally. I said it three times so it will stick. Since it is about your performance or your speech writing, it is hard not to feel criticism of your speech as a criticism of your person. Try to take criticism instead as someone caring enough about you to push you to grow.
Don’t be a Lynn. Yeah, that’s me, I’m the Lynn that you are
not
supposed to be. I remember being in high school debate my sophomore year and getting a grade from my teacher on my speech performance. She wrote, “slow down.” I can remember being so mad because this was proof that the “evil witch” hated me. It didn’t matter that I earned an A on that speech and that there were many positive things mentioned. It didn’t matter that she picked me to go to the best competitions. No, I was convinced that those two words meant she hated me. Now, I know to take advice as proof that someone cares and not as a criticism. It took me only twenty-plus years to get there. Don’t be a Lynn.
After Every Speech, Do a Self-Evaluation
Allison Shapira of Global Speaking suggests you do a self-evaluation after each speech:
What did I do well?
What didn’t I do so well?
What am I going to do differently next time?

(As a teacher, I do this every time I get student evaluations. I have a file of helpful suggestions from students. Yes, I even keep the critiques. I review the file every semester before I reteach the class. I think it makes me a better teacher. )
Constructive Criticism
There will be times when others look to you to read over their speech or listen to them practice and then give them constructive criticism. Constructive criticism is made up of two words: constructive–the building of something, and criticism–the giving of a critique. So constructive criticism is critiquing with the intention of building something. When we give others constructive criticism, our goal should be to help build them to be better speakers.
Give Them Help
Reagel and Reagle came up with a creative way to remember the goal of feedback, it should
HELP
:
Help
the speaker improve
Encourage
another speech
Lift
self-esteem
Provide
useful recommendations
Give Them a Sandwich
One way to give constructive criticism is to use the sandwich method. Say something positive, give feedback about something they can work on to improve, and then say something positive.
Ask Questions
Ask honest questions that can help lead them to solutions or ask questions to soften the sound of negative feedback:
“What did you mean by…”
“Have you considered?
For example:
“Have you considered the impact of showing such a gruesome photo on your slide?”
“Have you considered starting with a quote? ”
“Have you thought about whether the people in the back will be able to see your poster?”
“Have you thought about using a microphone so everyone can hear you?”
Different People, Different Types of Feedback
It is no surprise that people give and receive feedback differently. One person might take a feedback statement and be grateful for the corrections while the next person might take it as a complete insult. People are unique and I want to talk about a few of the most common differences. My goal here is not to put people in boxes or types, but rather to help you be more empathetic as a giver and receiver of feedback.
High and Low Self Monitors
Psychology researcher Mark Snyder identified people as being either high self-monitors or low self-monitors. High self-monitors typically try to fit in and play the role according to the context. They are about image, and they are motivated to fit in with their peers. They like to know what is expected, so they can adapt to the situation. Giving them useful feedback may mean pointing out how they can make changes in their message to meet the audience’s expectations. When giving feedback to high self-monitors, focus the feedback on how they can elevate their credibility in the eyes of the audience.
On the other hand, low self-monitors tend to be motivated to act based on their inner beliefs and values. They are motivated to be true to their sense of self and to above all– be genuine. When giving low self-monitors feedback, encourage them to be the best speaker they can be while being true to themselves. Focus on giving them feedback in a way that encourages them to harness their unique talents.
While you may not know exactly whether they are high or low self-monitor, I bet you have some idea of what motivates them. The more you can tailor your feedback to them, the more likely it is they will hear what you are saying. If you are curious about your type, you can take the quiz. You can have the person giving you feedback take the quiz as well. This can be a helpful exercise to think about how you give and receive feedback.
When you know your sickness
You’re halfway cured.
French saying
In the book,
The Culture Map,
a Dutch businessman is quoted as saying. “It is all a lot of hogwash. All that positive feedback just strikes us in the face and not in the least bit motivating.” People from different cultural groups have different feedback norms. As our society grows increasingly diverse, it is important to learn not just how to give good feedback, but to give feedback that demonstrates an awareness of how different cultures give and receive feedback.
Erin Meyer does international training to help business professionals understand differences and similarities and how to bridge the gap:
Managers in different parts of the world are conditioned to give feedback in drastically different ways. The Chinese manager learns never to criticize a colleague openly or in front of others, while the Dutch managers learns always to be honest and to give the message straight. Americans are trained to wrap positive messages around negative ones, while the French are trained to criticize passionately and provide positive feedback sparingly. Having a clear understanding of these differences and strategies for navigating them is crucial for leaders of cross-cultural teams.
Erin Meyer,
The Culture Map
Upgraders and Downgraders
Meyers identifies cultures as
Upgraders
and
Downgraders.
Upgraders use words or phrases to make negative feedback feel stronger. An upgrader might say, “this is
absolutely
inappropriate.” As you read this, see if you identify more as an upgrader or downgrader.
Upgraders say:
Absolutely–“That was absolutely shameless.”
Totally–“You totally missed the point.”
Strongly–” I strongly suggest that you…”
By contrast, downgraders use words to soften the criticism. A downgrader might say, “We are not quite there yet” or “This is just my opinion, but…”
Downgraders say:
“Kind of”
“Sort of”
“A little”
“Maybe”
“Slightly”
“This is just my opinion.”
When giving and receiving feedback across cultures, it is helpful to be aware of these differences so you can “hear” what they are really saying. Take for example this statement as a Dutch person complains about how Americans give feedback.
The problem is that we cant’ tell when the feedback is supposed to register to us as excellent, ok, or really poor. For a Dutchman, the word “excellent” is saved for a rare occasion and “okay” is…well, neutral. But with the Americans, the grid is different. “Excellent” is used all the time, “Okay” seems to mean, “not okay.” “Good” is only a mild complement. And when the message was intended to be bad, you can pretty much assume that, if an American is speaking and the listner is Dutch, the real meaning of the message will be lost all together.
Erin Meyer,
The Culture Map.
Nannette Ripmeester, Director of Expertise in Labour Mobility, illustrates these differences to her clients with a chart. This chart shows the differences between what the British say, what they mean, and what the Dutch understand. (This is a condensed version of her list.)

Perhaps you would think about…
I would suggest…
This is an order.
Do it or be prepared to justify yourself
Think about this idea and do it if you like it.
Please think about that some more
It’s a bad idea.
Don’t do it.
It’s a good idea, keep developing it.
I would suggest
Do it as I want you to
An open suggestion
An issue that worries me slightly
A great worry
A minor issue
A few issues that need to be addressed
A whole lot needs to be changed
2-3 issues need rewriting
Chances are as you read this list, you identified yourself in some of the statements and identified someone you know who is in the other list. Hopefully, this made you think about how personal style can be as different as cultural style. The big idea here is when you are giving and receiving feedback, it can be helpful to try to identify their communication style and adjust accordingly.
Politeness Strategies
As you already know, whenever you critique someone’s work, there is a potential to hurt their feelings. There are many factors that influence whether the feedback is helpful or hurtful. In communication, we use the term “face” to mean the sense of self a person projects. People can “take face” by creating a situation where someone looks bad to others or people can “lose face” by doing something that diminishes them in the eyes of others. Optimally, we want people to feel like they “gain-face” and feel encouraged. The way that you give feedback as well as the person’s natural tendencies will influence how “face” is affected.
When giving feedback, you should think about how your feedback takes or gives face. You also need to consider what is at stake for the other person.
How much you are willing to “take face” from someone may depend on the importance of the feedback. You will likely want to provide more suggestions for someone who is doing a career speech to get their dream job vs that same person doing a college speech worth minimal points. You will likely be more invested in helping a friend polish a speech to make it just right as opposed to someone you barely know.
Finally, the other thing influencing feedback is the power difference between people. You will likely give feedback differently to your little sister than you would to your boss. The status of the individuals and how important power is to them will impact how “face” is taken and given. For example, a high-power country like China would consider an open critique of a teacher, boss, or elder a huge insult, whereas someone from a low-power country, would be less offended. In any situation, you will be negotiating power, context, and the need to save face.
Taking all these factors into account, Brown and Levinson created Politeness Theory as a way to explain the different ways we give feedback to save face.
Bald on Record:
This type of feedback is very direct without concern for the person’s esteem face. This type of feedback is usually given if there is a small fix the speaker would feel strongly about.
Examples of bald on record feedback:
“Be sure you bold the headings.”
“Alphabetize the references.”
Positive Politeness:
In this type of feedback, you would build up the face or esteem of the other person. You would make them feel good before you make any suggestions. (It looks a lot like the sandwich method, hunh?)
Examples of positive politeness feedback:
“You are so organized; this one little fix and it will be perfect.”
“I love the story you told, a few more details would really help me see the character.”
Negative Politeness:
The name of this type of feedback is a little misleading. It doesn’t mean you are negative. It means you acknowledge that getting feedback may make them feel negative. You would say things that acknowledge their discomfort. You might minimize the criticism so it doesn’t make them feel bad or find other ways to soften the blow of criticism.
Examples of negative politeness feedback:
“I know this critique might sound rough and I hope it helps, but I think you really need to work on the middle section.”
“This is just me making suggestions, but I would be able to understand more if your slide has a heading.”
I’m not an expert on this, but I think you might need to have a stronger thesis.”
“I see what you are trying to do here, but I think some of your audience members might not get it.”
Off Record:
When you give feedback that is off the record, you are hinting vaguely that they should make a change.
Examples of off the record feedback.
“How many sources are we supposed to have?” (Instead of saying, “You need to have more research”)
“I thought we were supposed to have slides with our speech, maybe I heard that wrong.”
“Are other people in the class dressing up?”
Avoidance:
Some people are afraid of giving feedback so they will avoid the situation altogether.
Try This
Avoid the three C’s
Criticize
Complain
Condemn
Perform the three R’s
Review
Reward
Recommend
From Westside Toastmasters
Giving Feedback During a Speech
When you are listening to someone speak, you are giving constant nonverbal feedback.  The way you listen lets the speaker know that you value them and what they are saying. It can be reassuring to the speaker to have people who are in the audience smiling and nodding.
Try this little experiment: If you have a speaker who is average or boring, lean in and listen intently. Don’t be insincere and cheesy, but rather try to be an earnest listener. You will find that when the speaker notices you paying attention, they will usually become less monotone and more engaging. The speaker affects the audience, and the audience affects the speaker.
Asking for Feedback During Your Speech
“It is not OK to say OK at the end of every sentence!” That is what my friend told me after one of my speeches. I wasn’t even aware I was saying “OK.” I asked my friend to raise a finger every time I said, “OK” in my speech. At first, it was distracting because he kept raising his fingers, but eventually, I figured out the way to get him to stop was to stop saying “OK.” Sometimes, it helps to get feedback after a speech, and sometimes it helps to get it during the speech. Having a friend put their hand to their ear to tell you to speak up can be really helpful. I once stood at the back of the room and tried to signal to my husband as he was speaking with the fly of his pants unzipped. I kept making the zipper gesture. He saw me gesturing and he could tell I wanted something, but he couldn’t understand what. He finally stopped lecturing and said, “Lynn, what do you want?” What was I to do? What would you do? I said, “Zip your pants.”
 Someone who will tell you if your pants are unzipped (preferably before you start) and will give you the signal to speak louder or to check your microphone. If you know that you tend to pace, lean on the podium, or say um’s, have them give you the signal.
Courage is what it takes to stand up and speak.
Courage is also what it takes to sit down and listen.
Winston Churchill
Former Prime Ministre of the United Kingdom
Assess the Situation
“How do I look?” I asked my husband before we go out on a date. “What do you think of my new dress?” Many of you know that this question can be a trap. Am I really wanting his opinion or am I wanting him to compliment me? A savvy partner will assess the situation and answer accordingly. The thought process might go
, “She already bought the dress, she is excited about trying out a new restaurant and has been talking about it all week. She has been getting ready for a long time– the right answer is clearly one that makes her feel good. She is not really wanting my advice, she is fishing for a compliment.”
Take this story to heart when thinking about giving speech feedback. Your boss spent three weeks preparing for this big speech with the clients. He told you he has not slept in two days because he wanted everything to be perfect. After the presentation, he asks you, “What did you think?” A savvy co-worker will read the situation and pick out the positive parts. There may be constructive criticism that needs to happen, but when the boss is in the moment of high adrenaline after laying it all on the line, you should read the situation and hold any negative feedback for another time.
“My honesty is not a gift that everyone wants,” this phrase was from one of my students who was writing a reflective essay about navigating being honest and being appropriate. Be savvy and read the situation to know when to give feedback and when to be quiet. You should also read the person. The more you adjust your feedback to the person’s personal style and to the situation, the more likely they are to use that feedback to improve.
Key Takeaways
Remember This!
Be open to the feedback of others, it can help you improve as a speaker.
When giving feedback to others consider the context, their needs, the impact on their esteem, and their culture.
Use the feedback sandwich as a model for giving constructive criticism.
Please share your feedback, suggestions, corrections, and ideas
I want to hear from you.
Do you have an activity to include?
Did you notice a typo that I should correct?
Are you planning to use this as a resource and do you want me to know about it?
Do you want to tell me something that really helped you?
Research suggests the following regarding giving students feedback on their speeches:
Discuss the purpose of feedback.
Discuss ways to view criticism.
Give smaller, low-stakes opportunities to receive speech feedback.
Give negative feedback privately.
Allow feedback to come from multiple sources: In a speech class, have other students offer speech feedback.
Phrase it as skill-building rather than deficiency. Instead of “poor eye contact” say “increasing eye contact will build a connection with your audience.”
Give as many positive comments as possible.
Develop a warm, positive relationship with your students so they sense goodwill when they receive feedback from you.
I asked numerous professionals to Beta Test (review) this book. The feedback I received had great examples of the politeness strategies mentioned in this chapter. For me, the review process was the living version of this chapter. I felt exposed and vulnerable (and still feel exposed letting you see these comments). Because I was open to criticism, I was able to correct my mistakes, become a better writer, and give you a book that represented the profession and not just my opinions.
Read these and see if you can figure out which politeness strategy is at work.
As a speaker, it will not be easy to be open to criticism, but doing so will allow you to grow. As an act of reflection, ask yourself, which feedback below would be the most palatable to you–the direct, the apologetic, the explanatory.
After posting five sentences of praise, this reviewer wrote: “The only things I would look at are the critical comments I am making below. I hope you don’t mind. I’m a teacher.
Is there an extra space in that highlighted area?
This sentence reads funny to me. Seems like you are taking away from your powerful point if you end with the phrase “a little.” Sorry, just saying.
I am an over-user of the word “that”. If you can eliminate it and the sentence still makes sense, I would go ahead and cut it.
I was curious about your choice to provide transcripts for some videos but not for others.
I realize you are totally confident with saying NEVER. I say NEVER and ALWAYS in class, but I am not sure I should. Maybe “Instead of” can sub for NEVER.
Review bulleted lists for punctuation consistency: Some lines have periods in the end; some don’t.
“It is no surprise then, that when researchers looked at 500 Ted talks.” I suggest the style “TED Talks,” as that’s how the TED org itself expresses its name, and TED was originally an acronym for Technology, Entertainment, and Design.
The content was spot on and have no critical-minded commentary. My comments and sharings would be just on formatting. Because I’m a Monica… (maybe you’re a Friends gal, too?)
There is a noticeable lack of diversity in the speakers featured.  This is a big problem. It makes white, male, English-first speakers the norm rather than an example of many. This would make me wary of using this text in my classes because typically this demographic of white, male, English-first speakers only represents a bare plurality of my students.
References
Brown, P., & Levinson, S. (1978). Universals in Language Usage: Politeness Phenomena. In E. Goody (Ed.),
Questions and Politeness: Strategies in Social Interaction
(pp. 56-310). Cambridge University Press.
King, P. E., Young, M. J., & Behnke, R. R. (2000). Public speaking performance improvement as a function of information processing in immediate and delayed feedback interventions.
Communication Education, 49,
365–374.
https://doi.org/10.1080/03634520009379224
Mehra, A., Kilduff, M. & Brass, D.J. (2001). The social networks of high and low self-monitors Implications for workplace performance.
Administrative Science Quarterly, 46
(1), 121-146.
https://doi.org/10.2307/2667127
Ripmeester, N. Rottier, B., & Bush, A. (2010). Separated by a common translation? How the Brits and the Dutch communicate.
Pediatric Pulmonology. 46(
4). 409-411.
https://doi.org/10.1002/ppul.21380
Smith, C.D. & King, P.E. (2007). Student feedback sensitivity and the efficacy of feedback interventions in public speaking performance improvement.
Communication Education 53
(3).
https://doi.org/10.1080/0363452042000265152


2.7.1: Overview
First-order logic is not the only system of logic of interest: there are many extensions and variations of first-order logic. A logic typically consists of the formal specification of a language, usually, but not always, a deductive system, and usually, but not always, an intended semantics. But the technical use of the term raises an obvious question: what do logics that are not first-order logic have to do with the word “logic,” used in the intuitive or philosophical sense? All of the systems described below are designed to model reasoning of some form or another; can we say what makes them logical?
No easy answers are forthcoming. The word “logic” is used in different ways and in different contexts, and the notion, like that of “truth,” has been analyzed from numerous philosophical stances. For example, one might take the goal of logical reasoning to be the determination of which statements are necessarily true, true a priori, true independent of the interpretation of the nonlogical terms, true by virtue of their form, or true by linguistic convention; and each of these conceptions requires a good deal of clarification. Even if one restricts one’s attention to the kind of logic used in mathematics, there is little agreement as to its scope. For example, in the Principia Mathematica , Russell and Whitehead tried to develop mathematics on the basis of logic, in the logicist tradition begun by Frege. Their system of logic was a form of higher-type logic similar to the one described below. In the end they were forced to introduce axioms which, by most standards, do not seem purely logical (notably, the axiom of infinity, and the axiom of reducibility), but one might nonetheless hold that some forms of higher-order reasoning should be accepted as logical. In contrast, Quine, whose ontology does not admit “propositions” as legitimate objects of discourse, argues that second-order and higher-order logic are really manifestations of set theory in sheep’s clothing; in other words, systems involving quantification over predicates are not purely logical.
For now, it is best to leave such philosophical issues for a rainy day, and simply think of the systems below as formal idealizations of various kinds of reasoning, logical or otherwise.


6.4.1: The Nature of Light (Answers)
Check Your Understanding
1.1. 2.1% (to two significant figures)
1.2. 15.1°
1.3. air to water, because the condition that the second medium must have a smaller index of refraction is not satisfied
1.4. 9.3 cm
1.5. \(\displaystyle AA'\) becomes longer, \(\displaystyle A'B'\) tilts further away from the surface, and the refracted ray tilts away from the normal.
1.6. also \(\displaystyle 90.0%\)
1.7. There will be only refraction but no reflection.
Conceptual Questions
1. model as a ray when devices are large compared to wavelength, as a wave when devices are comparable or small compared to wavelength
3. This fact simply proves that the speed of light is greater than that of sound. If one knows the distance to the location of the lightning and the speed of sound, one could, in principle, determine the speed of light from the data. In practice, because the speed of light is so great, the data would have to be known to impractically high precision.
5. Powder consists of many small particles with randomly oriented surfaces. This leads to diffuse reflection, reducing shine.
7. “toward” when increasing n (air to water, water to glass); “away” when decreasing n (glass to air)
9. A ray from a leg emerges from water after refraction. The observer in air perceives an apparent location for the source, as if a ray traveled in a straight line. See the dashed ray below.
The figure is illustration of the formation of the image of a leg under water, as seen by a viewer in the air above the water. A ray is shown leaving the leg and refracting at the water air interface. The refracted ray bends away from the normal. Extrapolating the refracted ray back into the water, the extrapolated ray is above the actual ray so that the image of the leg is above the actual leg and the leg appears shorter.
11. The gemstone becomes invisible when its index of refraction is the same, or at least similar to, the water surrounding it. Because diamond has a particularly high index of refraction, it can still sparkle as a result of total internal reflection, not invisible.
13. One can measure the critical angle by looking for the onset of total internal reflection as the angle of incidence is varied. Equation 1.5 can then be applied to compute the index of refraction.
15. In addition to total internal reflection, rays that refract into and out of diamond crystals are subject to dispersion due to varying values of n across the spectrum, resulting in a sparkling display of colors.
17. yes
19. No. Sound waves are not transverse waves.
21. Energy is absorbed into the filters.
23. Sunsets are viewed with light traveling straight from the Sun toward us. When blue light is scattered out of this path, the remaining red light dominates the overall appearance of the setting Sun.
25. The axis of polarization for the sunglasses has been rotated \(\displaystyle 90°\).
Problems
27. \(\displaystyle 2.99705×10^8m/s; 1.97×10^8m/s\)
29. ice at \(\displaystyle 0°C\)
31. 1.03 ns
33. 337 m
35. proof
37. proof
39. reflection, \(\displaystyle 70°\); refraction, \(\displaystyle 45°\)
41. \(\displaystyle 42°\)
43. 1.53
45. a. 2.9 m;
b. 1.4 m
47. a. \(\displaystyle 24.42°\);
b. \(\displaystyle 31.33°\)
49. \(\displaystyle 79.11°\)
51. a. 1.43, fluorite;
b. \(\displaystyle 44.2°\)
53. a. \(\displaystyle 48.2°\);
b. \(\displaystyle 27.3°\)
55. \(\displaystyle 46.5°\) for red, \(\displaystyle 46.0°\) for violet
57. a. \(\displaystyle 0.04°\);
b. 1.3 m
59. \(\displaystyle 72.8°\)
61. \(\displaystyle 53.5°\) for red, \(\displaystyle 55.2°\) for violet
63. 0.500
65. 0.125 or 1/8
67. \(\displaystyle 84.3°\)
69. \(\displaystyle 0.250I_0\)
71. a. 0.500;
b. 0.250;
c. 0.187
73. \(\displaystyle 67.54°\)
75. \(\displaystyle 53.1°\)
Additional Problems
77. 114 radian/s
79. 3.72 mm
81. \(\displaystyle 41.2°\)
83. a. 1.92. The gem is not a diamond (it is zircon).
b. \(\displaystyle 55.2°\)
85. a. 0.898;
b. We cannot have \(\displaystyle n<1.00\), since this would imply a speed greater than c .
c. The refracted angle is too big relative to the angle of incidence.
87. \(\displaystyle 0.707B_1\)
89. a. \(\displaystyle 1.69×10^{−2}°C/s\);
b. yes
Challenge Problems
91. First part: \(\displaystyle 88.6°\). The remainder depends on the complexity of the solution the reader constructs.
93. proof; 1.33
95. a. 0.750;
b. 0.563;
c. 1.33


10.7: Lattice Structures in Crystalline Solids
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- Last updated
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Learning Objectives
By the end of this section, you will be able to:
- Describe the arrangement of atoms and ions in crystalline structures
- Compute ionic radii using unit cell dimensions
- Explain the use of X-ray diffraction measurements in determining crystalline structures
Over 90% of naturally occurring and man-made solids are crystalline. Most solids form with a regular arrangement of their particles because the overall attractive interactions between particles are maximized, and the total intermolecular energy is minimized, when the particles pack in the most efficient manner. The regular arrangement at an atomic level is often reflected at a macroscopic level. In this module, we will explore some of the details about the structures of metallic and ionic crystalline solids, and learn how these structures are determined experimentally.
The Structures of Metals
We will begin our discussion of crystalline solids by considering elemental metals, which are relatively simple because each contains only one type of atom. A pure metal is a crystalline solid with metal atoms packed closely together in a repeating pattern. Some of the properties of metals in general, such as their malleability and ductility, are largely due to having identical atoms arranged in a regular pattern. The different properties of one metal compared to another partially depend on the sizes of their atoms and the specifics of their spatial arrangements. We will explore the similarities and differences of four of the most common metal crystal geometries in the sections that follow.
Unit Cells of Metals
The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell . The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions, as illustrated in Figure \(\PageIndex{1}\).
Let us begin our investigation of crystal lattice structure and unit cells with the most straightforward structure and the most basic unit cell. To visualize this, imagine taking a large number of identical spheres, such as tennis balls, and arranging them uniformly in a container. The simplest way to do this would be to make layers in which the spheres in one layer are directly above those in the layer below, as illustrated in Figure \(\PageIndex{2}\). This arrangement is called simple cubic structure , and the unit cell is called the simple cubic unit cell or primitive cubic unit cell.
In a simple cubic structure, the spheres are not packed as closely as they could be, and they only “fill” about 52% of the volume of the container. This is a relatively inefficient arrangement, and only one metal (polonium, Po) crystallizes in a simple cubic structure. As shown in Figure \(\PageIndex{3}\), a solid with this type of arrangement consists of planes (or layers) in which each atom contacts only the four nearest neighbors in its layer; one atom directly above it in the layer above; and one atom directly below it in the layer below. The number of other particles that each particle in a crystalline solid contacts is known as its coordination number . For a polonium atom in a simple cubic array, the coordination number is, therefore, six.
In a simple cubic lattice, the unit cell that repeats in all directions is a cube defined by the centers of eight atoms, as shown in Figure \(\PageIndex{4}\). Atoms at adjacent corners of this unit cell contact each other, so the edge length of this cell is equal to two atomic radii, or one atomic diameter. A cubic unit cell contains only the parts of these atoms that are within it. Since an atom at a corner of a simple cubic unit cell is contained by a total of eight unit cells, only one-eighth of that atom is within a specific unit cell. And since each simple cubic unit cell has one atom at each of its eight “corners,” there is \(8 \times \frac{1}{8}=1\) atom within one simple cubic unit cell.
Example \(\PageIndex{1}\): Calculation of Atomic Radius and Density for Metals, Part 1
The edge length of the unit cell of alpha polonium is 336 pm.
- Determine the radius of a polonium atom.
- Determine the density of alpha polonium.
Solution
Alpha polonium crystallizes in a simple cubic unit cell:
- Two adjacent Po atoms contact each other, so the edge length of this cell is equal to two Po atomic radii: \(l = 2r\). Therefore, the radius of Po is: \[r=\frac{l}{2}=\frac{336 pm }{2}=168 pm . \nonumber \]
-
Density is given by \(\text{density}=\frac{\text{mass}}{\text{volume}}\). The density of polonium can be found by determining the density of its unit cell (the mass contained within a unit cell divided by the volume of the unit cell). Since a Po unit cell contains one-eighth of a Po atom at each of its eight corners, a unit cell contains one Po atom.
The mass of a Po unit cell can be found by \[1 \text { Po unit cell } \times \frac{1 \text { Po atom }}{1 \text { Po unit cell }} \times \frac{1 mol Po }{6.022 \times 10^{23} \text { Po atoms }} \times \frac{208.998 g }{1 mol Po }=3.47 \times 10^{-22} g \nonumber \]
The volume of a Po unit cell can be found by: \[V=l^3=\left(336 \times 10^{-10} cm \right)^3=3.79 \times 10^{-23} cm^3 \nonumber \]
(Note that the edge length was converted from pm to cm to get the usual volume units for density.) Therefore, the density is \[\text { Po }=\frac{3.471 \times 10^{-22} g }{3.79 \times 10^{-23} cm^3}=9.16 g / cm^3 \nonumber \]
Exercise \(\PageIndex{1}\)
The edge length of the unit cell for nickel is 0.3524 nm. The density of Ni is 8.90 g/cm 3 . Does nickel crystallize in a simple cubic structure? Explain.
- Answer
-
No. If Ni was simple cubic, its density would be given by:
\[1 Ni \text { atom } \times \frac{1 mol Ni }{6.022 \times 10^{23} Ni \text { atoms }} \times \frac{58.693 g }{1 mol Ni }=9.746 \times 10^{-23} g \nonumber \]
\[V=l^3=\left(3.524 \times 10^{-8} cm \right)^3=4.376 \times 10^{-23} cm^3 \nonumber \]
Then the density of Ni would be\[=\frac{9.746 \times 10^{-23} g }{4.376 \times 10^{-23} cm^3}=2.23 g / cm^3 \nonumber \]
Since the actual density of Ni is not close to this, Ni does not form a simple cubic structure.
Most metal crystals are one of the four major types of unit cells. For now, we will focus on the three cubic unit cells: simple cubic (which we have already seen), body-centered cubic unit cell , and face-centered cubic unit cell —all of which are illustrated in Figure \(\PageIndex{5}\). (Note that there are actually seven different lattice systems, some of which have more than one type of lattice, for a total of 14 different types of unit cells. We leave the more complicated geometries for later in this module.)
Some metals crystallize in an arrangement that has a cubic unit cell with atoms at all of the corners and an atom in the center, as shown in Figure \(\PageIndex{6}\). This is called a body-centered cubic (BCC) solid . Atoms in the corners of a BCC unit cell do not contact each other but contact the atom in the center. A BCC unit cell contains two atoms: one-eighth of an atom at each of the eight corners atom from the corners (\(8 \times \frac{1}{8}=1\) atom from the corners) plus one atom from the center. Any atom in this structure touches four atoms in the layer above it and four atoms in the layer below it. Thus, an atom in a BCC structure has a coordination number of eight.
Atoms in BCC arrangements are much more efficiently packed than in a simple cubic structure, occupying about 68% of the total volume. Isomorphous metals with a BCC structure include K, Ba, Cr, Mo, W, and Fe at room temperature. (Elements or compounds that crystallize with the same structure are said to be isomorphous .)
Many other metals, such as aluminum, copper, and lead, crystallize in an arrangement that has a cubic unit cell with atoms at all of the corners and at the centers of each face, as illustrated in Figure \(\PageIndex{7}\). This arrangement is called a face-centered cubic (FCC) solid . A FCC unit cell contains four atoms: one-eighth of an atom at each of the eight corners atom from the corners (\(8 \times \frac{1}{8}=1\) atom from the corners) and one-half of an atom on each of the six faces atoms from the faces (\(6 \times \frac{1}{2}=3\) atom from the faces). The atoms at the corners touch the atoms in the centers of the adjacent faces along the face diagonals of the cube. Because the atoms are on identical lattice points, they have identical environments.
Atoms in an FCC arrangement are packed as closely together as possible, with atoms occupying 74% of the volume. This structure is also called cubic closest packing (CCP) . In CCP, there are three repeating layers of hexagonally arranged atoms. Each atom contacts six atoms in its own layer, three in the layer above, and three in the layer below. In this arrangement, each atom touches 12 near neighbors, and therefore has a coordination number of 12. The fact that FCC and CCP arrangements are equivalent may not be immediately obvious, but why they are actually the same structure is illustrated in Figure \(\PageIndex{8}\).
Because closer packing maximizes the overall attractions between atoms and minimizes the total intermolecular energy, the atoms in most metals pack in this manner. We find two types of closest packing in simple metallic crystalline structures: CCP, which we have already encountered, and hexagonal closest packing (HCP) shown in Figure \(\PageIndex{9}\). Both consist of repeating layers of hexagonally arranged atoms . In both types, a second layer (B) is placed on the first layer (A) so that each atom in the second layer is in contact with three atoms in the first layer. The third layer is positioned in one of two ways. In HCP, atoms in the third layer are directly above atoms in the first layer (i.e., the third layer is also type A), and the stacking consists of alternating type A and type B close-packed layers (i.e., ABABAB⋯). In CCP, atoms in the third layer are not above atoms in either of the first two layers (i.e., the third layer is type C), and the stacking consists of alternating type A, type B, and type C close-packed layers (i.e., ABCABCABC⋯). About two–thirds of all metals crystallize in closest-packed arrays with coordination numbers of 12. Metals that crystallize in an HCP structure include Cd, Co, Li, Mg, Na, and Zn, and metals that crystallize in a CCP structure include Ag, Al, Ca, Cu, Ni, Pb, and Pt.
Example \(\PageIndex{2}\): Calculation of Atomic Radius and Density for Metals, Part 2
Calcium crystallizes in a face-centered cubic structure. The edge length of its unit cell is 558.8 pm.
- What is the atomic radius of Ca in this structure?
- Calculate the density of Ca.
Solution
(a) In an FCC structure, Ca atoms contact each other across the diagonal of the face, so the length of the diagonal is equal to four Ca atomic radii (d = 4 r ). Two adjacent edges and the diagonal of the face form a right triangle, with the length of each side equal to 558.8 pm and the length of the hypotenuse equal to four Ca atomic radii:
\[a^2+a^2=d^2 \longrightarrow(558.8 pm )^2+(558.5 pm )^2=(4 r)^2 \nonumber \]
Solving this gives
\[r=\sqrt{\frac{(558.8 pm )^2+(558.5 pm )^2}{16}}=197.6 pm \text { for a Ca radius. } \nonumber \]
(b) Density is given by \(\text { density }=\frac{\text { mass }}{\text { volume }}\). The density of calcium can be found by determining the density of its unit cell: for example, the mass contained within a unit cell divided by the volume of the unit cell. A face-centered Ca unit cell has one-eighth of an atom at each of the eight corners (\(8 \times \frac{1}{8}=1\) atom from the corners) and one-half of an atom on each of the six faces (\(6 \times \frac{1}{2}=3\) atom), for a total of four atoms in the unit cell.
The mass of the unit cell can be found by:
\[1 Ca \text { unit cell } \times \frac{4 Ca \text { atoms }}{1 Ca \text { unit cell }} \times \frac{1 mol Ca }{6.022 \times 10^{23} Ca \text { atoms }} \times \frac{40.078 g }{1 mol Ca }=2.662 \times 10^{-22} g \nonumber \]
The volume of a Ca unit cell can be found by:
\[V=a^3=\left(558.8 \times 10^{-10} cm \right)^3=1.745 \times 10^{-22} cm^3 \nonumber \]
(Note that the edge length was converted from pm to cm to get the usual volume units for density.)
Then, the density of Ca is
\[\frac{2.662 \times 10^{-22} g }{1.745 \times 10^{-22} cm^3}=1.53 g / cm^3 \nonumber \]
Exercise \(\PageIndex{2}\)
Silver crystallizes in an FCC structure. The edge length of its unit cell is 409 pm.
- What is the atomic radius of Ag in this structure?
- Calculate the density of Ag.
- Answer
-
- 144 pm
- 10.5 g/cm 3
In general, a unit cell is defined by the lengths of three axes ( a , b , and c ) and the angles ( α , β , and γ ) between them, as illustrated in Figure \(\PageIndex{10}\). The axes are defined as being the lengths between points in the space lattice. Consequently, unit cell axes join points with identical environments.
There are seven different lattice systems, some of which have more than one type of lattice, for a total of fourteen different unit cells, which have the shapes shown in Figure \(\PageIndex{11}\).
The Structures of Ionic Crystals
Ionic crystals consist of two or more different kinds of ions that usually have different sizes. The packing of these ions into a crystal structure is more complex than the packing of metal atoms that are the same size.
Most monatomic ions behave as charged spheres, and their attraction for ions of opposite charge is the same in every direction. Consequently, stable structures for ionic compounds result (1) when ions of one charge are surrounded by as many ions as possible of the opposite charge and (2) when the cations and anions are in contact with each other. Structures are determined by two principal factors: the relative sizes of the ions and the ratio of the numbers of positive and negative ions in the compound.
In simple ionic structures, we usually find the anions, which are normally larger than the cations, arranged in a closest-packed array. (As seen previously, additional electrons attracted to the same nucleus make anions larger and fewer electrons attracted to the same nucleus make cations smaller when compared to the atoms from which they are formed.) The smaller cations commonly occupy one of two types of holes (or interstices) remaining between the anions. The smaller of the holes is found between three anions in one plane and one anion in an adjacent plane.  The larger type of hole is found at the center of six anions (three in one layer and three in an adjacent layer) located at the corners of an octahedron; this is called an octahedral hole . Figure \(\PageIndex{12}\) illustrates both of these types of holes.
Depending on the relative sizes of the cations and anions, the cations of an ionic compound may occupy tetrahedral or octahedral holes, as illustrated in Figure \(\PageIndex{13}\). Relatively small cations occupy tetrahedral holes, and larger cations occupy octahedral holes. If the cations are too large to fit into the octahedral holes, the anions may adopt a more open structure, such as a simple cubic array. The larger cations can then occupy the larger cubic holes made possible by the more open spacing.
There are two tetrahedral holes for each anion in either an HCP or CCP array of anions. A compound that crystallizes in a closest-packed array of anions with cations in the tetrahedral holes can have a maximum cation:anion ratio of 2:1; all of the tetrahedral holes are filled at this ratio. Examples include Li 2 O, Na 2 O, Li 2 S, and Na 2 S. Compounds with a ratio of less than 2:1 may also crystallize in a closest-packed array of anions with cations in the tetrahedral holes, if the ionic sizes fit. In these compounds, however, some of the tetrahedral holes remain vacant.
Example \(\PageIndex{3}\): Occupancy of Tetrahedral Holes
Zinc sulfide is an important industrial source of zinc and is also used as a white pigment in paint. Zinc sulfide crystallizes with zinc ions occupying one-half of the tetrahedral holes in a closest-packed array of sulfide ions. What is the formula of zinc sulfide?
Solution
Because there are two tetrahedral holes per anion (sulfide ion) and one-half of these holes are occupied by zinc ions, there must be \(\frac{1}{2} \times 2\) or 1, zinc ion per sulfide ion. Thus, the formula is ZnS.
Exercise \(\PageIndex{3}\)
Lithium selenide can be described as a closest-packed array of selenide ions with lithium ions in all of the tetrahedral holes. What it the formula of lithium selenide?
- Answer
-
Li 2 Se
The ratio of octahedral holes to anions in either an HCP or CCP structure is 1:1. Thus, compounds with cations in octahedral holes in a closest-packed array of anions can have a maximum cation:anion ratio of 1:1. In NiO, MnS, NaCl, and KH, for example, all of the octahedral holes are filled. Ratios of less than 1:1 are observed when some of the octahedral holes remain empty.
Example \(\PageIndex{4}\): Stoichiometry of Ionic Compounds
Sapphire is aluminum oxide. Aluminum oxide crystallizes with aluminum ions in two-thirds of the octahedral holes in a closest-packed array of oxide ions. What is the formula of aluminum oxide?
Solution
Because there is one octahedral hole per anion (oxide ion) and only two-thirds of these holes are occupied, the ratio of aluminum to oxygen must be \(\frac{2}{3}:1\), which would give \(\ce{Al_{2/3}O}\). The simplest whole number ratio is 2:3, so the formula is \(\ce{Al2O3}\).
Exercise \(\PageIndex{4}\)
The white pigment titanium oxide crystallizes with titanium ions in one-half of the octahedral holes in a closest-packed array of oxide ions. What is the formula of titanium oxide?
- Answer
-
\(\ce{TiO2}\)
In a simple cubic array of anions, there is one cubic hole that can be occupied by a cation for each anion in the array. In CsCl, and in other compounds with the same structure, all of the cubic holes are occupied. Half of the cubic holes are occupied in SrH 2 , UO 2 , SrCl 2 , and CaF 2 .
Different types of ionic compounds often crystallize in the same structure when the relative sizes of their ions and their stoichiometries (the two principal features that determine structure) are similar.
Unit Cells of Ionic Compounds
Many ionic compounds crystallize with cubic unit cells, and we will use these compounds to describe the general features of ionic structures.
When an ionic compound is composed of cations and anions of similar size in a 1:1 ratio, it typically forms a simple cubic structure. Cesium chloride, CsCl, (illustrated in Figure \(\PageIndex{14}\)) is an example of this, with Cs + and Cl − having radii of 174 pm and 181 pm, respectively. We can think of this as chloride ions forming a simple cubic unit cell, with a cesium ion in the center; or as cesium ions forming a unit cell with a chloride ion in the center; or as simple cubic unit cells formed by Cs + ions overlapping unit cells formed by Cl − ions. Cesium ions and chloride ions touch along the body diagonals of the unit cells. One cesium ion and one chloride ion are present per unit cell, giving the l:l stoichiometry required by the formula for cesium chloride. Note that there is no lattice point in the center of the cell, and CsCl is not a BCC structure because a cesium ion is not identical to a chloride ion.
We have said that the location of lattice points is arbitrary. This is illustrated by an alternate description of the CsCl structure in which the lattice points are located in the centers of the cesium ions. In this description, the cesium ions are located on the lattice points at the corners of the cell, and the chloride ion is located at the center of the cell. The two unit cells are different, but they describe identical structures.
When an ionic compound is composed of a 1:1 ratio of cations and anions that differ significantly in size, it typically crystallizes with an FCC unit cell, like that shown in Figure \(\PageIndex{15}\). Sodium chloride, NaCl, is an example of this, with Na + and Cl − having radii of 102 pm and 181 pm, respectively. We can think of this as chloride ions forming an FCC cell, with sodium ions located in the octahedral holes in the middle of the cell edges and in the center of the cell. The sodium and chloride ions touch each other along the cell edges. The unit cell contains four sodium ions and four chloride ions, giving the 1:1 stoichiometry required by the formula, NaCl.
The cubic form of zinc sulfide, zinc blende, also crystallizes in an FCC unit cell, as illustrated in Figure \(\PageIndex{16}\). This structure contains sulfide ions on the lattice points of an FCC lattice. (The arrangement of sulfide ions is identical to the arrangement of chloride ions in sodium chloride.) The radius of a zinc ion is only about 40% of the radius of a sulfide ion, so these small Zn 2 + ions are located in alternating tetrahedral holes, that is, in one half of the tetrahedral holes. There are four zinc ions and four sulfide ions in the unit cell, giving the empirical formula ZnS.
A calcium fluoride unit cell, like that shown in Figure \(\PageIndex{17}\), is also an FCC unit cell, but in this case, the cations are located on the lattice points; equivalent calcium ions are located on the lattice points of an FCC lattice. All of the tetrahedral sites in the FCC array of calcium ions are occupied by fluoride ions. There are four calcium ions and eight fluoride ions in a unit cell, giving a calcium:fluorine ratio of 1:2, as required by the chemical formula, CaF 2 . Close examination of Figure \(\PageIndex{17}\) will reveal a simple cubic array of fluoride ions with calcium ions in one half of the cubic holes. The structure cannot be described in terms of a space lattice of points on the fluoride ions because the fluoride ions do not all have identical environments. The orientation of the four calcium ions about the fluoride ions differs.
Calculation of Ionic Radii
If we know the edge length of a unit cell of an ionic compound and the position of the ions in the cell, we can calculate ionic radii for the ions in the compound if we make assumptions about individual ionic shapes and contacts.
Example \(\PageIndex{5}\): Calculation of Ionic Radii
The edge length of the unit cell of LiCl (NaCl-like structure, FCC) is 0.514 nm or 5.14 Å. Assuming that the lithium ion is small enough so that the chloride ions are in contact, as in Figure \(\PageIndex{15}\), calculate the ionic radius for the chloride ion. Note: The length unit angstrom, Å, is often used to represent atomic-scale dimensions and is equivalent to 10 −10 m.
Solution
On the face of a LiCl unit cell, chloride ions contact each other across the diagonal of the face:
Drawing a right triangle on the face of the unit cell, we see that the length of the diagonal is equal to four chloride radii (one radius from each corner chloride and one diameter—which equals two radii—from the chloride ion in the center of the face), so d = 4 r . From the Pythagorean theorem, we have:
\[ \nonumber \]a^2+a^2=d^2 \nonumber \]
which yields:
\[(0.514 nm )^2+(0.514 nm )^2=(4 r)^2=16 r^2 \nonumber \]
Solving this gives:
\[r=\sqrt{\frac{(0.514 nm )^2+(0.514 nm )^2}{16}}=0.182 nm (1.82 \AA) \text { for a } Cl^{-} \text {radius. } \nonumber \]
Exercise \(\PageIndex{1}\)
The edge length of the unit cell of KCl (NaCl-like structure, FCC) is 6.28 Å. Assuming anion-cation contact along the cell edge, calculate the radius of the potassium ion. The radius of the chloride ion is 1.82 Å.
- Answer
-
The radius of the potassium ion is 1.33 Å.
It is important to realize that values for ionic radii calculated from the edge lengths of unit cells depend on numerous assumptions, such as a perfect spherical shape for ions, which are approximations at best. Hence, such calculated values are themselves approximate and comparisons cannot be pushed too far. Nevertheless, this method has proved useful for calculating ionic radii from experimental measurements such as X-ray crystallographic determinations.
X-Ray Crystallography
The size of the unit cell and the arrangement of atoms in a crystal may be determined from measurements of the diffraction of X-rays by the crystal, termed X-ray crystallography . Diffraction is the change in the direction of travel experienced by an electromagnetic wave when it encounters a physical barrier whose dimensions are comparable to those of the wavelength of the light. X-rays are electromagnetic radiation with wavelengths about as long as the distance between neighboring atoms in crystals (on the order of a few Å).
When a beam of monochromatic X-rays strikes a crystal, its rays are scattered in all directions by the atoms within the crystal. When scattered waves traveling in the same direction encounter one another, they undergo interference , a process by which the waves combine to yield either an increase or a decrease in amplitude (intensity) depending upon the extent to which the combining waves’ maxima are separated (see Figure \(\PageIndex{18}\)).
When X-rays of a certain wavelength, \(λ\), are scattered by atoms in adjacent crystal planes separated by a distance, \(d\), they may undergo constructive interference when the difference between the distances traveled by the two waves prior to their combination is an integer factor, \(n\), of the wavelength. This condition is satisfied when the angle of the diffracted beam, \(θ\), is related to the wavelength and interatomic distance by the equation:
\[n \lambda=2 d \sin \theta \label{Bragg} \]
This relation is known as the Bragg equation in honor of W. H. Bragg , the English physicist who first explained this phenomenon. Figure \(\PageIndex{19}\): illustrates two examples of diffracted waves from the same two crystal planes. The figure on the left depicts waves diffracted at the Bragg angle, resulting in constructive interference, while that on the right shows diffraction and a different angle that does not satisfy the Bragg condition, resulting in destructive interference.
Link to Learning
Visit this site for more details on the Bragg equation and a simulator that allows you to explore the effect of each variable on the intensity of the diffracted wave.
An X-ray diffractometer, such as the one illustrated in Figure \(\PageIndex{20}\), may be used to measure the angles at which X-rays are diffracted when interacting with a crystal as described earlier. From such measurements, the Bragg equation may be used to compute distances between atoms as demonstrated in the following example exercise.
Example \(\PageIndex{6}\): Using the Bragg Equation
In a diffractometer, X-rays with a wavelength of 0.1315 nm were used to produce a diffraction pattern for copper. The first order diffraction ( n = 1) occurred at an angle θ = 25.25°. Determine the spacing between the diffracting planes in copper.
Solution
The distance between the planes is found by solving the Bragg equation (Equation \ref{Bragg}), \(nλ = 2d \sin θ\), for \(d\).
This gives:
\[d=\frac{n \lambda}{2 \sin \theta}=\frac{1(0.1315 nm )}{2 \sin \left(25.25^{\circ}\right)}=0.154 nm \nonumber \]
Exercise \(\PageIndex{6}\)
A crystal with spacing between planes equal to 0.394 nm diffracts X-rays with a wavelength of 0.147 nm. What is the angle for the first order diffraction?
- Answer
-
10.8°
Portrait of a Chemist: X-ray Crystallographer Rosalind Franklin
The discovery of the structure of DNA in 1953 by Francis Crick and James Watson is one of the great achievements in the history of science. They were awarded the 1962 Nobel Prize in Physiology or Medicine, along with Maurice Wilkins , who provided experimental proof of DNA’s structure. British chemist Rosalind Franklin made invaluable contributions to this monumental achievement through her work in measuring X-ray diffraction images of DNA. Early in her career, Franklin’s research on the structure of coals proved helpful to the British war effort. After shifting her focus to biological systems in the early 1950s, Franklin and doctoral student Raymond Gosling discovered that DNA consists of two forms: a long, thin fiber formed when wet (type “B”) and a short, wide fiber formed when dried (type “A”). Her X-ray diffraction images of DNA (Figure \(\PageIndex{21}\)) provided the crucial information that allowed Watson and Crick to confirm that DNA forms a double helix, and to determine details of its size and structure. Franklin also conducted pioneering research on viruses and the RNA that contains their genetic information, uncovering new information that radically changed the body of knowledge in the field. After developing ovarian cancer, Franklin continued to work until her death in 1958 at age 37. Among many posthumous recognitions of her work, the Chicago Medical School of Finch University of Health Sciences changed its name to the Rosalind Franklin University of Medicine and Science in 2004, and adopted an image of her famous X-ray diffraction image of DNA as its official university logo.


2.2: Dominated Strategies
Recall that in a zero-sum game, we know that one player's win is the other player's loss. Furthermore, we know we can rewrite any zero-sum game so that the player's payoffs are in the form \((a, -a)\text{.}\) Note, this works even if \(a\) is negative; in which case, \(-a\) is positive.
Consider the zero-sum game with payoff matrix in Table \(2.2.1\). Note that for simplicity our payoff matrix contains only the payoffs and not the strategy names; but Player 1 still chooses a row and Player 2 still chooses a column.
| Table \(2.2.1\): The Payoff Matrix for Example \(2.2.1\) | ||
|---|---|---|
| Player 2 | ||
| Player 1 | \((1, -1)\) | \((-0, 0)\) |
| \((-1, 1)\) | \((-2, 2)\) |
If we know we are playing a zero-sum game, then the use of ordered pairs seems somewhat redundant: If Player 1 wins \(1\), then we know that Player 2 must lose \(1\) (win \(-1\)). Thus, if we KNOW we are playing a zero-sum game, we can simplify our notation by just using Player 1's payoffs. The above matrix in Table \(2.2.2\) can be simplified as in Table \(2.2.2\).
| Table \(2.2.2\): The Payoff Matrix for Example \(2.2.1\) using Only Player 1's Payoffs. | ||
|---|---|---|
| Player 2 | ||
| Player 1 | \(1\) | \(0\) |
| \(-1\) | \(-2\) |
When simplifying, keep a few things in mind:
- You MUST know that the game is zero-sum.
- If it is not otherwise specified, the payoffs represent Player 1's payoffs.
-
You can always give a similar matrix representing Player 2's payoffs. However, due to (2), you should indicate that the matrix is for Player 2. For example, Player 2's payoff matrix would be given by Table \(2.2.3\).
Table \(2.2.3\): The Payoff Matrix for Example \(2.2.1\) using Only Player 2's Payoffs. Player 2 Player 1 \(-1\) \(0\) \(1\) \(2\) -
Both players can make strategy decisions by considering only Player 1's payoff matrix. (Why?) Just to test this out, by looking only at the matrix in Table 2.2.3 determine which strategy each player should choose.
Table \(2.2.4\): Player 1's Payoffs in Example \(2.2.1\). Player 2 Player 1 \(1\) \(0\) \(-1\) \(-2\)
In this last example, it should be clear that Player 1 is looking for rows which give her the largest payoff. This is nothing new. However, Player 2 is now looking for columns which give Player 1 the SMALLEST payoff. (Why?)
Now that we have simplified our notation for zero-sum games, let's try to find a way to determine the best strategy for each player.
Consider the zero-sum game given in Table \(2.2.5\).
| Table \(2.2.5\): Payoff Matrix for Example \(2.2.2\). | |||
|---|---|---|---|
| Player 2 | |||
| Player 1 | \(1\) | \(0\) | \(2\) |
| \(-1\) | \(-2\) | \(2\) |
Determine which row Player 1 should choose. Is there any situation in which Player 1 would choose the other row?
Consider the zero-sum game given in Table \(2.2.6\).
| Table \(2.2.6\): Payoff Matrix for Example \(2.2.3\). | |||
|---|---|---|---|
| Player 2 | |||
| Player 1 | \(1\) | \(0\) | \(2\) |
| \(-1\) | \(-2\) | \(3\) |
Determine which row Player 1 should choose. Is there any situation in which Player 1 would choose the other row?
In Example \(2.2.2\), no matter what Player 2 does, Player 1 would always choose Row 1, since every payoff in Row 1 is greater than or equal to the corresponding payoff in Row 2 (\(1\ge -1\text{,}\) \(0\ge -2\text{,}\) \(2\ge 2\)). In Example \(2.2.3\), this is not the case: if Player 2 were to choose Column 3, then Player 1 would prefer Row 2. In Example \(2.2.2\) we would say that Row 1 dominates Row 2.
A strategy \(X\) dominates a strategy \(Y\) if every entry for \(X\) is greater than or equal to the corresponding entry for \(Y\text{.}\) In this case, we say \(Y\) is dominated by \(X\text{.}\)
If strategy \(X\) dominates strategy \(Y\text{,}\) we can write \(X\succ Y\text{.}\)
In mathematical notation, let \(a_{ik}\) be the value in the \(i^{ \text{th}}\) row and \(k^{ \text{th}}\) column. Similarly, \(a_{jk}\) is the value in the \(j^{ \text{th}}\) row and \(k^{ th}\) column. The \(i^{ \text{th}}\) row dominates the \(j^{ \text{th}}\) row if \(a_{ik}\ge a_{jk}\) for all \(k\text{,}\) and \(a_{ik}> a_{jk}\) for at least one \(k\text{.}\)
This definition can also be used for Player 2: we consider columns instead of rows. If we are looking at Player 1's payoffs, then Player 2 prefers smaller payoffs. Thus one column \(X\) dominates another column \(Y\) if all the entries in \(X\) are smaller than or equal to the corresponding entries in \(Y\text{.}\)
Here is the great thing: we can always eliminate dominated strategies! (Why?) Thus, in Example \(2.2.2\), we can eliminate Row 2, as in Figure \(2.2.1\).
Now it is easy to see what Player 2 should do.
In Example \(2.2.3\), we cannot eliminate Row 2 since it is not dominated by Row 1. However, it should be clear that Column 2 dominates Column 3 (remember, Player 2 prefers SMALLER values). Thus we can eliminate Column 3 as in Figure \(2.2.2\).
AFTER eliminating Column 3, Row 1 dominates Row 2. Now, in Figure \(2.2.3\) we can eliminate Row 2.
Again, now it is easy to determine what each player should do.
Check that the strategy pairs we determined in Example \(2.2.2\) and Example \(2.2.3\) are, in fact, equilibrium pairs.
Now, look back at the election examples from Section 2.1.1 and apply the process of eliminating dominated strategies.
Use the idea of eliminating dominated strategies to determine what each player should do in the Arnold/Bainbridge examples in Table \(2.1.4\), Table \(2.1.5\), and Table \(2.1.6\). Do you get the same strategy pairs as you determined in the related exercises (Exercise \(2.1.1\), Exercise \(2.1.2\), Exercise \(2.1.3\))?
The next three exercises provide more practice in using dominated strategies to find equilibrium pairs.
Use the idea of eliminating dominated strategies to determine any equilibrium pairs in the zero-sum game given in Table \(2.2.7\). Note, since it is a zero-sum game we need only show Player 1's payoffs. Explain all the steps in your solution. If you are unable to find an equilibrium pair, explain what goes wrong.
| Table \(2.2.7\): Payoff Matrix for Exercise \(2.2.3\). | |||||
|---|---|---|---|---|---|
| Player 2 | |||||
| Player 1 | W | X | Y | Z | |
| A | \(1\) | \(0\) | \(0\) | \(10\) | |
| B | \(-1\) | \(0\) | \(-2\) | \(9\) | |
| C | \(1\) | \(1\) | \(1\) | \(8\) | |
| D | \(-2\) | \(0\) | \(0\) | \(7\) |
Determine any equilibrium pairs in the zero-sum game given in Table \(2.2.8\). Explain all the steps in your solution. If you are unable to find an equilibrium pair, explain what goes wrong.
| Table \(2.2.8\): Payoff Matrix for Exercise \(2.2.4\). | |||||
|---|---|---|---|---|---|
| Player 2 | |||||
| Player 1 | W | X | Y | Z | |
| A | \(1\) | \(2\) | \(3\) | \(4\) | |
| B | \(0\) | \(-1\) | \(-0\) | \(5\) | |
| C | \(-1\) | \(3\) | \(2\) | \(4\) | |
| D | \(0\) | \(1\) | \(-1\) | \(1\) |
Determine any equilibrium pairs in the zero-sum game given in Table \(2.2.9\). Explain all the steps in your solution. If you are unable to find an equilibrium pair, explain what goes wrong.
| Table \(2.2.9\): Payoff Matrix for Exercise \(2.2.5\). | |||||
|---|---|---|---|---|---|
| Player 2 | |||||
| Player 1 | W | X | Y | Z | |
| A | \(-2\) | \(0\) | \(3\) | \(20\) | |
| B | \(1\) | \(-2\) | \(-3\) | \(0\) | |
| C | \(10\) | \(-10\) | \(-1\) | \(1\) | |
| D | \(0\) | \(0\) | \(10\) | \(15\) |
Determine any equilibrium pairs in the zero-sum game given in Table \(2.2.10\). Explain all the steps in your solution. If you are unable to find an equilibrium pair, explain what goes wrong.
| Table \(2.2.10\): Payoff Matrix for Exercise \(2.2.6\). | |||||
|---|---|---|---|---|---|
| Player 2 | |||||
| Player 1 | W | X | Y | Z | |
| A | \(-2\) | \(0\) | \(3\) | \(20\) | |
| B | \(1\) | \(-2\) | \(-5\) | \(-3\) | |
| C | \(10\) | \(-10\) | \(-1\) | \(1\) | |
| D | \(0\) | \(0\) | \(10\) | \(8\) |
Chances are you had trouble determining an equilibrium pair for the game in Exercise \(2.2.10\). Does this mean there isn't an equilibrium pair? Not necessarily, but we are stuck if we try to use only the idea of eliminating dominated strategies. So we need a new method.
We might think of our next method as the “worst case scenario,” or “extremely defensive play.” The idea is that we want to assume our opponent is the best player to ever live. In fact, we might assume our opponent is telepathic. So no matter what we do, our opponent will always guess what we are going to choose.
Assume you are Player 1, and you are playing against this “infinitely smart” Player 2. Consider the game in Table \(2.2.7\). If you pick row A, what will Player 2 do? Player 2 will pick column X or Y. Try this for each of the rows. Which row is your best choice? If you pick A, you will get \(0\text{;}\) if you pick B, you will get \(-2\text{;}\) if you pick C, you will get \(1\text{;}\) and if you pick D you will get \(-2\text{.}\) Thus, your best choice is to choose C and get \(1\text{.}\) Now assume you are Player 2, and Player 1 is “infinitely smart.” Which column is your best choice? If you pick W, Player 1 will get \(1\) (you will get \(-1\)); if you pick X, Player 1 will get \(1\text{;}\) if you pick Y, Player 1 will get \(1\text{;}\) and if you pick Z, you will get \(10\text{.}\) Thus, you can choose W, X, or Y (since you want Player 1 to win the least) and get \(-1\text{.}\)
Using the method described in Example \(2.2.4\), determine what each player should do in the game in Table \(2.2.8\).
Using the method described in Example \(2.2.4\), determine what each player should do in the game in Table \(2.2.9\).
After working through a few examples can you describe more generally the process used in Example \(2.2.4\)? What is Player 1 looking for in each row? Then how does she choose which row to play? What is Player 2 looking for in each column? How does he choose which column to play?
Generalize the method described in Example \(2.2.4\). In other words, give a general rule for how Player 1 should determine his or her best move. Do the same for Player 2.
What do you notice about using this method on the games in Tables \(2.2.7\), Table \(2.2.8\), and Table \(2.2.9\)? Is the solution an equilibrium pair?
Now try this method on the elusive payoff matrix in Table \(2.2.10\). What should each player do? Do you think we get an equilibrium pair? Explain.
The strategies we found using the above method have a more official name. Player 1's strategy is called the maximin strategy. Player 1 is maximizing the minimum values from each row. Player 2's strategy is called the minimax strategy. Player 2 is minimizing the maximum values from each column. Notice, we can find the maximin and minimax strategies for any zero-sum game. But do our players always want to use these strategies? Will they always result in an equilibrium pair? The next five exercises explore these questions.
Let's consider another game matrix, given in Table \(2.2.11\). Explain why you cannot use dominated strategies to find an equilibrium pair. Do you think there is an equilibrium pair for this game (why or why not)?
| Table \(2.2.11\): Payoff Matrix for Exercise \(2.2.12\). | |||||
|---|---|---|---|---|---|
| Player 2 | |||||
| Player 1 | W | X | Y | Z | |
| A | \(-2\) | \(0\) | \(3\) | \(20\) | |
| B | \(1\) | \(2\) | \(-3\) | \(0\) | |
| C | \(10\) | \(-10\) | \(-1\) | \(1\) | |
| D | \(0\) | \(0\) | \(10\) | \(15\) |
If both players use the maximin/minimax strategy, what is the outcome of the game in Table \(2.2.11\)?
In the game in Table \(2.2.11\), if Player 1's opponent can guess that Player 1 will choose to use a maximin strategy, is Player 1 better off not using the maximin strategy?
Suppose both players initially decide to use the minimax/maximin strategy in the game in Table \(2.2.11\). Is Player 1 better off choosing a different strategy? If Player 2 guesses a change, is Player 2 better off changing strategies? Continue this line of reasoning for several iterations. What strategies do each of the players choose? Is at least one player always better off switching strategies? Can we conclude that the maximin/ minimax strategy does not lead to an equilibrium pair?
Compare your answers in Exercise \(2.2.15\) to what happens in Exercise \(2.2.3\), Exercise \(2.2.4\), and Exercise \(2.2.5\). Can you identify any key differences between the games in Exercise \(2.2.15\) and Exercise \(2.2.3\), Exercise \(2.2.4\), and Exercise \(2.2.5\).
Given a zero-sum matrix game, we can find equilibrium pairs (if they exist) by the “guess and check” method, by eliminating dominated strategies, and by looking for the minimax/maximin strategies. You should be able to apply all three methods and think about which method might be the most appropriate for a given matrix game. For example, although “guess and check” should always find an equilibrium point if it exists, it may be very tedious to apply to a really large matrix. The maximin/minimax method might be much faster.


22.1: Introduction
Ecological Succession of Bacteria in Milk
The communities within ecosystems develop over time, from very simple species assemblages, to complex, rich ecosystems. In this process, called succession, each succeeding species facilitates changes in environment which allow new species to come into the ecosystem. As the community becomes more and more complex, the biodiversity of the ecosystem also increases. Both biotic and abiotic processes can reset the succession process. That is, events cause by both the community itself, and outside events can return the community to an earlier succession state. The gradual changes in the community are both orderly and predictable in many ecosystems. The peak or most complex, advanced community that can develop in any abiotic environment is called the climax community. The picture below describes the developing communities as a series of steps, each of which can be driven against the succession process by disturbances:
QUESTION
- What types of events could “reset” a succession process? Name at least one biotic and one abiotic disturbance.
One example of a severe disturbance, reducing the land to bare ground, is the passage of a glacier. Though glaciers have not covered this part of North America for ten thousand years, there are parts of the continent that are even now becoming uncovered by receding glaciers. One area is the pacific northwest. From Juneau to Glacier Bay, many glaciers which have previously fallen directly into the ocean are now leaving bare soil which has not been exposed for more than fifty thousand years. Because the glaciers retreat very slowly, we can watch communities change across time in a single snap shot. Take the glaciers of Glacier Bay National Park, midway between Juneau and Anchorage, Alaska. The glaciers there have been retreating since the explorer Vancouver’s first expedition in 1794. Since then, the retreat has covered over 100 km, including new coastline, meadows and mountains:
QUESTION
- Explain how this retreat will result in different communities along the glacier’s retreat, though in similar environments. Hint: what is the difference between exposed soil at point A and point B.
As the glacier retreats, it leaves nutrient poor soil which can only support simple plants such as liverworts, lichens, and other primitive plants. As they photosynthesize and die, we see them enter the decomposer pathway and increase the quality of the soil for later plants. However, this slow glacier retreat is a unique situation. To set up an experiment to test our understanding of succession would require hundreds of years, longer than a scientists lifetime. However, some organisms and communities proceed at a much faster rate, within your own refrigerators. The process of milk decomposition from a community of bacteria can test the same processes and theories in a much more reasonable time frame. This substitution of a simpler and faster community for experimental purposes is called a ‘model’ system.
LICENSES AND ATTRIBUTIONS
CC LICENSED CONTENT, ORIGINAL
- Community Ecology Lab. Authored by : Dr. William Edwards. Provided by : Niagara University. Located at : Niagara.ed [www.niagara.edu] . License : CC BY: Attribution
CC LICENSED CONTENT, SHARED PREVIOUSLY
- Biology 102 Labs. Authored by : Lynette Hauser & Dr. James Holden. Provided by : Tidewater Community Colleg. Located at : [www.tcc.edu] . License : Public Domain: No Known Copyright


3.1: Introduction to Congruences
As we mentioned in the introduction, the theory of congruences was developed by Gauss at the beginning of the nineteenth century.
Let \(m\) be a positive integer. We say that \(a\) is congruent to \(b\) modulo \(m\) if \(m \mid (a-b)\) where \(a\) and \(b\) are integers, i.e. if \(a=b+km\) where \(k\in \mathbb{Z}\).
If \(a\) is congruent to \(b\) modulo \(m\), we write \(a\equiv b(mod\ m)\).
\(19\equiv 5 (mod \ 7)\). Similarly \(2k+1 \equiv 1 (mod\ 2)\) which means every odd number is congruent to 1 modulo 2.
There are many common properties between equations and congruences. Some properties are listed in the following theorem.
Theorem: Properties of Congruences
Let \(a, b, c\) and \(d\) denote integers. Let \(m\) be a positive integers. Then:
- If \(a \equiv b(mod \ m)\) , then \(b\equiv a (mod \ m)\).
- If \(a\equiv b(mod \ m)\) and \(b\equiv c(mod \ m)\), then \(a\equiv c (mod \ m)\).
- If \(a\equiv b(mod\ m)\) , then \(a+c \equiv b+c (mod \ m)\).
- If \(a\equiv b(mod\ m)\) , then \(a-c \equiv b-c (mod \ m)\).
- If \(a\equiv b(mod\ m)\) , then \(ac \equiv bc (mod \ m)\).
- If \(a\equiv b(mod\ m)\) , then \(ac \equiv bc (mod \ mc)\), for \(c>0\).
- If \(a\equiv b(mod\ m)\) and \(c \equiv d (mod \ m)\) then \(a+c \equiv (b+d) (mod \ m)\).
- If \(a\equiv b(mod\ m)\) and \(c \equiv d (mod \ m)\) then \(a-c \equiv (b-d) (mod \ m)\).
- If \(a\equiv b(mod\ m)\) and \(c \equiv d (mod \ m)\) then \(ac \equiv bd (mod \ m)\).
- If \(a \equiv b(mod \ m)\) , then \(m\mid (a-b)\). Thus there exists integer \(k\) such that \(a-b=mk\), this implies \(b-a=m(-k)\) and thus \(m\mid (b-a)\). Consequently \(b\equiv a (mod \ m)\).
- Since \(a\equiv b(mod \ m)\) , then \(m\mid (a-b)\). Also, \(b\equiv c(mod \ m)\), then \(m\mid (b-c)\). As a result, there exit two integers \(k\) and \(l\) such that \(a=b+mk\) and \(b=c+ml\), which imply that \(a=c+m(k+l)\) giving that \(a=c (mod \ m)\).
- Since \(a\equiv b (mod \ m)\) , then \(m \mid (a-b)\). So if we add and subtract \(c\) we get \[m\mid ((a+c)-(b+c))\] and as a result \[a+c\equiv b+c (mod \ m).\]
- Since \(a\equiv b (mod \ m)\) , then \(m \mid (a-b)\) so we can subtract and add \(c\) and we get \[m\mid ((a-c)-(b-c))\] and as a result \[a-c\equiv b-c (mod \ m).\]
- If \(a \equiv b(mod \ m)\) , then \(m\mid (a-b)\). Thus there exists integer \(k\) such that \(a-b=mk\) and as a result \(ac-bc=m(kc)\). Thus \[m\mid (ac-bc)\] and hence \[ac\equiv bc (mod \ m).\]
- If \(a \equiv b(mod \ m)\) , then \(m\mid (a-b)\). Thus there exists integer \(k\) such that \(a-b=mk\) and as a result \[ac-bc=mc(k).\] Thus \[mc\mid (ac-bc)\] and hence \[ac\equiv bc (mod \ mc).\]
- Since \(a\equiv b(mod \ m)\) , then \(m\mid (a-b)\). Also, \(c\equiv d(mod \ m)\), then \(m\mid (c-d)\). As a result, there exits two integers \(k\) and \(l\) such that \(a-b=mk\) and \(c-d=ml\). Note that \[(a-b)+(c-d)=(a+c)-(b+d)=m(k+l).\] As a result, \[m\mid ((a+c)-(b+d)),\] hence \[a+c\equiv b+d(mod \ m).\]
- If \(a=b+mk\) and \(c=d+ml\) where \(k\) and \(l\) are integers, then \[(a-b)-(c-d)=(a-c)-(b-d)=m(k-l).\] As a result, \[m\mid ((a-c)-(b-d)),\] hence \[a-c\equiv b-d(mod \ m).\]
- There exit two integers \(k\) and \(l\) such that \(a-b=mk\) and \(c-d=ml\) and thus \(ca-cb=m(ck)\) and \(bc-bd=m(bl)\). Note that \[(ca-cb)+(bc-bd)=ac-bd=m(kc-lb).\] As a result, \[m\mid (ac-bd),\]hence \[ac\equiv bd(mod \ m).\]
- Because \(14\equiv 8(mod\ 6)\) then \(8 \equiv 14 (mod\ 6)\).
- Because \(22\equiv 10(mod \ 6)\) and \(10 \equiv 4(mod \ 6)\). Notice that \(22\equiv 4(mod \ 6)\).
- Because \(50\equiv 20 (mod\ 15)\) , then \(50+5=55\equiv 20+5=25(mod\ 15)\).
- Because \(50\equiv 20 (mod\ 15)\) , then \(50-5=45\equiv 20-5=15(mod\ 15)\).
- Because \(19\equiv 16(mod3)\) , then \(2(19)=38\equiv 2(16)=32(mod \ 3).\)
- Because \(19\equiv 16(mod3)\) , then \(2(19)=38\equiv 2(16)=32(mod \ 2(3)=6).\)
- Because \(19\equiv 3 (mod \ 8)\) and \(17\equiv 9(mod \ 8)\), then \(19+17=36\equiv 3+9=12(mod \ 8)\).
- Because \(19\equiv 3 (mod \ 8)\) and \(17\equiv 9(mod \ 8)\), then \(19-17=2\equiv 3-9=-6(mod \ 8)\).
- Because \(19\equiv 3 (mod \ 8)\) and \(17\equiv 9(mod \ 8)\), then \(19(17)=323\equiv 3(9)=27(mod \ 8)\).
We now present a theorem that will show one difference between equations and congruences. In equations, if we divide both sides of the equation by a non-zero number, equality holds. While in congruences, it is not necessarily true. In other words, dividing both sides of the congruence by the same integer doesn’t preserve the congruence.
- If \(a,b, c\) and \(m\) are integers such that \(m>0\), \(d=(m,c)\) and \(ac\equiv bc(mod \ m)\), then \(a\equiv b (mod \ m/d)\).
- If \((m,c)=1\) then \(a=b(mod \ m)\) if \(ac\equiv bc(mod \ m)\).
Part 2 follows immediately from Part 1. For Part 1, if \(ac\equiv bc(mod \ m)\), then \[m\mid (ac-bc)=c(a-b).\] Hence there exists \(k\) such that \(c(a-b)=mk\). Dividing both sides by \(d\), we get \((c/d)(a-b)=k(m/d)\). Since \((m/d,c/d)=1\), it follows that \(m/d \mid (a-b)\). Hence \(a\equiv b (mod \ m/d)\).
\(38 \equiv 10 (mod\ 7)\). Since \((2,7)=1\) then \(19\equiv 5 (mod \ 7).\)
The following theorem combines several congruences of two numbers with different moduli.
If \[a\equiv b(mod \ m_1), a\equiv b(mod \ m_2),...,a\equiv b(mod \ m_t)\] where \(a,b,m_1,m_2,...,m_t\) are integers and \(m_1,m_2,...,m_t\) are positive, then \[a\equiv b(mod \ \langle m_1,m_2,...m_t\rangle)\]
Since \(a\equiv b (mod \ m_i)\) for all \(1\leq i\leq t\). Thus \(m_i \mid (a-b)\). As a result, \[\langle m_1,m_2,...,m_t\rangle \mid (a-b)\] (prove this as an exercise). Thus \[a\equiv b(mod \ \langle m_1,m_2,...m_t\rangle).\]
Exercises
- Determine whether 3 and 99 are congruent modulo 7 or not.
- Show that if \(x\) is an odd integer, then \(x^2\equiv 1(mod \ 8)\)
- Show that if \(a,b, m\) and \(n\) are integers such that \(m\) and \(n\) are positive, \(n \mid m\) and \(a\equiv b (mod \ m)\), then \(a\equiv b (mod \ n).\)
- Show that if \(a_i\equiv b_i(mod \ m)\) for \(i=1,2,...,n\), where \(m\) is a positive integer and \(a_i,b_i\) are integers for \(j=1,2,...,n\), then \(\sum_{i=1}^na_i\equiv \sum_{i=1}^nb_i(mod \ m)\)
- For which \(n\) does the expression \(1+2+...+(n-1)\equiv 0(mod \ n)\) holds.
Contributors and Attributions
-
Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution ( CC BY ) license.


12.10.1: Types of Equipment
Patients may have a number of special types of equipment that have been prescribed for them to use in order to be as independent as possible. Home Health Aides/Personal Care Aides should always be sure they know how to use the equipment correctly prior to using it with a patient.
Bed Equipment
Patients may be using their own bed or they may have a prescribed hospital bed. Learn how to properly use a hospital bed by practicing using the buttons to raise and lower the bed prior to using it with a patient.
Mobility Equipment
Patients may have any number of equipment to help them with mobility. Learn which types have been prescribed to the patient and how to correctly use them. An important task of the HHA/PCA is to use and teach patients how to properly use mobility devices.
Types of Mobility Equipment Include:
- Canes
- Walkers
- Wheelchairs
- Transfer/sliding board
- Hydraulic Lift (e.g. Hoyer Lift)
- Electric Lift Chair
- Gait belts
- Braces, splints, slings
- Prosthetic limbs
- Special shoes or boots
Toileting Equipment
Patients may have special equipment to use which will assist them with toileting if they have problems with mobility. These types of equipment help to meet the patient’s elimination needs and maintain as much independence as possible. Remember to offer toileting at least every two hours, or more frequently if the patient requires. Use proper safety procedures for transferring a patient to a toilet or bedside commode.
Types of Toileting Equipment:
- Bedpan
- Urinal
- Bedside commode
- Catheters
Scales
One of the tasks of the Home Health Aide/Personal Care Aide may be to measure the patient’s weight on a regular basis. This helps to determine if the patient is maintaining an adequate weight. Some patients may be on a special diet to lose weight, while others are prescribed diets to help them gain weight. Patients with conditions such as kidney or heart failure may have to be weighed on a daily basis in order to measure if the patient is retaining fluid. Retaining fluid can lead to a worsening of their health condition.
Accuracy is important when weighing a patient. Always ensure the scale is balanced to zero before weighing a patient. When weighing a patient, Home Health Aides/Personal Care Aides should encourage them to empty their bladder and bowels first. Weigh them first thing in the morning, and while they are wearing light clothing, such as pajamas.
Depending on the patient’s mobility, they may have a bed scale, chair scale, balance scale, or bathroom scale. Learn how to properly use each type of scale. It is important to ensure the safety of the patient while weighing them. If a patient appears unsteady and unable to stand in order to be weighed, inform the supervisor.
Remember too that people may be sensitive about their weight. Never chastise (scold) or be judgmental toward a person about their weight. Maintain privacy while weighing a patient and do not disclose a patient’s weight to others outside of the healthcare team, unless the patient states it is okay.
Types of Scales:
- Bed scale (the patient is weighed while in bed)
- Chair scale (the patient sits while being weighed)
- Balance scale (the type found in a doctor’s office)
- Bathroom scale (digital or standard)
This section will provide instruction about how to properly weigh a patient, use a hydraulic lift, and a slide board. The Home Health Aide/Personal Care Aide should always seek training from their agency to ensure they can correctly complete any procedure.


Exercise \(\PageIndex{1}\): Notice and Wonder: Wrapping Paper
Kiran is wrapping this box of sports cards as a present for a friend.
What do you notice? What do you wonder?
Exercise \(\PageIndex{2}\): Building Prisms and Pyramids
Your teacher will give you a drawing of a polyhedron. You will draw its net and calculate its surface area.
- What polyhedron do you have?
- Study your polyhedron. Then, draw its net on graph paper. Use the side length of a grid square as the unit.
- Label each polygon on the net with a name or number.
- Find the surface area of your polyhedron. Show your thinking in an organized manner so that it can be followed by others.
Exercise \(\PageIndex{3}\): Comparing Boxes
Here are the nets of three cardboard boxes that are all rectangular prisms. The boxes will be packed with 1-centimeter cubes. All lengths are in centimeters.
- Compare the surface areas of the boxes. Which box will use the least cardboard? Show your reasoning.
- Now compare the volumes of these boxes in cubic centimeters. Which box will hold the most 1-centimeter cubes? Show your reasoning.
Are you ready for more?
Figure C shows a net of a cube. Draw a different net of a cube. Draw another one. And then another one. How many different nets can be drawn and assembled into a cube?
Summary
The surface area of a polyhedron is the sum of the areas of all of the faces. Because a net shows us all faces of a polyhedron at once, it can help us find the surface area. We can find the areas of all polygons in the net and add them.
A square pyramid has a square and four triangles for its faces. Its surface area is the sum of the areas of the square base and the four triangular faces:
\((6\cdot 6)+4\cdot (\frac{1}{2}\cdot 5\cdot 6)=96\)
The surface area of this square pyramid is 96 square units.
Glossary Entries
Definition: Base (of a Prism or Pyramid)
The word base can also refer to a face of a polyhedron.
A prism has two identical bases that are parallel. A pyramid has one base.
A prism or pyramid is named for the shape of its base.
Definition: Face
Each flat side of a polyhedron is called a face. For example, a cube has 6 faces, and they are all squares.
Definition: Net
A net is a two-dimensional figure that can be folded to make a polyhedron.
Here is a net for a cube.
Definition: Polyhedron
A polyhedron is a closed, three-dimensional shape with flat sides. When we have more than one polyhedron, we call them polyhedra.
Here are some drawings of polyhedra.
Definition: Prism
A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.
Here are some drawings of prisms.
Definition: Pyramid
A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.
Here are some drawings of pyramids.
Definition: Surface Area
The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.
For example, if the faces of a cube each have an area of 9 cm 2 , then the surface area of the cube is \(6\cdot 9\), or 54 cm 2 .
Practice
Exercise \(\PageIndex{4}\)
Jada drew a net for a polyhedron and calculated its surface area.
- What polyhedron can be assembled from this net?
- Jada made some mistakes in her area calculation. What were the mistakes?
- Find the surface area of the polyhedron. Show your reasoning.
Exercise \(\PageIndex{5}\)
A cereal box is 8 inches by 2 inches by 12 inches. What is its surface area? Show your reasoning. If you get stuck, consider drawing a sketch of the box or its net and labeling the edges with their measurements.
Exercise \(\PageIndex{6}\)
Twelve cubes are stacked to make this figure.
- What is its surface area?
- How would the surface area change if the top two cubes are removed?
(From Unit 1.5.1)
Exercise \(\PageIndex{7}\)
Here are two polyhedra and their nets. Label all edges in the net with the correct lengths.
Exercise \(\PageIndex{8}\)
- What three-dimensional figure can be assembled from the net?
- What is the surface area of the figure? (One grid square is 1 square unit.)
(From Unit 1.5.3)


  People who were unable to orgasm ( anorgasmic ) can learn to have orgasms, those with premature ejaculation can learn to control orgasm, those with vaginismus (vaginal spasms) can learn to relax those muscles. But there’s also a group of people who don’t respond to therapy informed by the four-phase model.
In 1977 psychotherapist Helen Singer Kaplan published Hypoactive Sexual Desire in an effort to build upon this model and address a missing piece in the previous work done by Masters and Johnson, desire. Reviewing treatment failures among patients, she found that the clients with the least successful outcomes were those who lacked interest in sex. Kaplan realized something important was entirely missing from the four-phase model, desire (Kaplan, 1977 p. 3). The biological model put forth by Masters and Johnson removed how our brain connects us to the body and the reason it may not have come into play in the studies done in a lab might make sense.
It seems like a glaring oversight in retrospect, but of course it was missing, people who come to a laboratory to masturbate for science don’t have to want sex before they begin; they just have to get aroused for the purpose of the experiment. Kaplan took the four-phase model out of the laboratory and adapted it to the lived experience of her clients. Her “triphasic” model of the sexual response cycle begins with desire, which she conceptualized as “interest in” or “appetite for” sex, much like hunger or thirst. The second phase is arousal, which combines excitement and plateau into one phase, and the third phase is orgasm (Nagoski 2015, p. 50).
Kaplan’s model of sexual response has served as the foundation for diagnostic criteria in the American Psychiatric Association’s Diagnostic and Statistical Manual . Low desire and desire discrepancy between partners are the most common reasons people seek sex therapy. Some also seek help for “hypersexuality,” where they feel their desire and behaviors are out of their control (Nagoski, 2015 p.51).  Circling back to the beginning of this chapter, the answer is there is not a right amount of desire, everybody is different.  If there is something that concerns you about your level of desire, it is always good to talk with a sexual health educator.
 Because we are all different, our desires are as well. Bottom line, there is no “right” way to be sexual. It is time to let go of feeling the need to be a certain way, look a certain way, want sex in a certain way and focus on what you really want.
Sara Paules describes the dual control of sexual response that is referenced by Emily Nagoski in her 2015 groundbreaking text that we have referred to throughout this chapter, Come As You Are. Nagoski asserts that we all develop these systems based on our lived experiences and we all vary. It can be broken into two parts that can be compared to a car:
- Sexual Excitation System (SES). The system that responds to any sexually relevant stimuli in the environment-something we see, smell, remember etc. It is the accelerator.
- Sexual Inhibition System (SIS). The system that responds to any sexually inhibiting stimuli-like your parent being in the next room. It is the brakes.
This system can be further broken down into two different SIS systems:
- Responds to fear of performance failure (erectile dysfunction, premature ejaculation, etc.)
- Responds to fear of performance consequences (STI transmission, unwanted pregnancy, social consequences).
Sexual arousal can be understood as two equally important processes:
- Having gradually increasing stimulation for the SES
- Removing anything that the SIS might respond to, whether it be physical or emotional
Sara Paules, MA, LPC from the Mindful Soul Center in Austin, TX has an adapted sexual temperament quiz based on the work of Emily Nagoski and other sex researcher studying sexual temperament. You can take the quiz here. These types of assessments are meant to inform, not judge your personal temperament. People vary in their responses and there is not a normal temperament type (Nagoski, 2015, Paules, n.d.).