1.3: Thermometers and Temperature Scales
By the end of this section, you will be able to:
- Describe several different types of thermometers
- Convert temperatures between the Celsius, Fahrenheit, and Kelvin scales
Any physical property that depends consistently and reproducibly on temperature can be used as the basis of a thermometer. For example, volume increases with temperature for most substances. This property is the basis for the common alcohol thermometer and the original mercury thermometers. Other properties used to measure temperature include electrical resistance, color, and the emission of infrared radiation (Figure \(\PageIndex{1}\)).
Thermometers measure temperature according to well-defined scales of measurement. The three most common temperature scales are Fahrenheit, Celsius, and Kelvin. Temperature scales are created by identifying two reproducible temperatures. The freezing and boiling temperatures of water at standard atmospheric pressure are commonly used.
On the Celsius scale , the freezing point of water is \(0^oC\) and the boiling point is \(100^oC\). The unit of temperature on this scale is the degree Celsius (\(^oC\)). The Fahrenheit scale (still the most frequently used for common purposes in the United States) has the freezing point of water at \(32^o F\) and the boiling point a. \(212^oF\) Its unit is the degree Fahrenheit (\(^oF\)). You can see that 100 Celsius degrees span the same range as 180 Fahrenheit degrees. Thus, a temperature difference of one degree on the Celsius scale is 1.8 times as large as a difference of one degree on the Fahrenheit scale, or
\[\Delta T_F = \frac{9}{5}\Delta T_C. \nonumber \]
The definition of temperature in terms of molecular motion suggests that there should be a lowest possible temperature, where the average kinetic energy of molecules is zero (or the minimum allowed by quantum mechanics). Experiments confirm the existence of such a temperature, called absolute zero . An absolute temperature scale is one whose zero point is absolute zero. Such scales are convenient in science because several physical quantities, such as the volume of an ideal gas, are directly related to absolute temperature.
The Kelvin scale is the absolute temperature scale that is commonly used in science. The SI temperature unit is the kelvin , which is abbreviated K (not accompanied by a degree sign). Thus 0 K is absolute zero. The freezing and boiling points of water are 273.15 K and 373.15 K, respectively. Therefore, temperature differences are the same in units of kelvins and degrees Celsius, or \(\Delta T_C = \Delta T_K\).
The relationships between the three common temperature scales are shown in Figure \(\PageIndex{2}\). Temperatures on these scales can be converted using the equations in Table \(\PageIndex{1}\).
| To convert from… | Use this equation… |
|---|---|
| Celsius to Fahrenheit | \(T_F = \frac{9}{5}T_C + 32\) |
| Fahrenheit to Celsius | \(T_C = \frac{5}{9}(T_F - 32)\) |
| Celsius to Kelvin | \(T_K = T_C + 273.15\) |
| Kelvin to Celsius | \(T_C = T_K - 273.15\) |
| Fahrenheit to Kelvin | \(T_K = \frac{5}{9}(T_F - 32) + 273.15\) |
| Kelvin to Fahrenheit | \(T_F = \frac{9}{5}(T_K - 273.15) + 32\) |
To convert between Fahrenheit and Kelvin, convert to Celsius as an intermediate step.
“Room temperature” is generally defined in physics to be \(25^oC\). (a) What is room temperature in \(^oF\)? (b) What is it in K?
Strategy To answer these questions, all we need to do is choose the correct conversion equations and substitute the known values.
Solution
To convert from \(^oC\) to \(^oF\), use the equation
\[T_F = \dfrac{9}{5}T_C + 32. \nonumber \]
Substitute the known value into the equation and solve:
\[T_F = \dfrac{9}{5}(25^oC) + 32 = 77^oF. \nonumber \]
Similarly, we find that \(T_K = T_C + 273.15 = 298 \, K\).
The Kelvin scale is part of the SI system of units, so its actual definition is more complicated than the one given above. First, it is not defined in terms of the freezing and boiling points of water, but in terms of the triple point . The triple point is the unique combination of temperature and pressure at which ice, liquid water, and water vapor can coexist stably. As will be discussed in the section on phase changes, the coexistence is achieved by lowering the pressure and consequently the boiling point to reach the freezing point. The triple-point temperature is defined as 273.16 K. This definition has the advantage that although the freezing temperature and boiling temperature of water depend on pressure, there is only one triple-point temperature.
Second, even with two points on the scale defined, different thermometers give somewhat different results for other temperatures. Therefore, a standard thermometer is required. Metrologists (experts in the science of measurement) have chosen the constant-volume gas thermometer for this purpose. A vessel of constant volume filled with gas is subjected to temperature changes, and the measured temperature is proportional to the change in pressure. Using “TP” to represent the triple point,
\[T = \dfrac{p}{p_{TP}}T_{TP}. \nonumber \]
The results depend somewhat on the choice of gas, but the less dense the gas in the bulb, the better the results for different gases agree. If the results are extrapolated to zero density, the results agree quite well, with zero pressure corresponding to a temperature of absolute zero.
Constant-volume gas thermometers are big and come to equilibrium slowly, so they are used mostly as standards to calibrate other thermometers.


1.19: Centered lattices
When the unit cell does not reflect the symmetry of the lattice, it is usual in crystallography to refer to a 'conventional', non-primitive, crystallographic basis, a c , b c , c c instead of a primitive basis, a , b , c . This is done by adding lattice nodes at the center of the unit cell or at one or three faces. The vectors joining the origin of the unit cell to these additional nodes are called 'centering vectors'. In such a lattice a c , b c and c c with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors t ∈ L , t = t 1 a c + t 2 b c + t 3 c c ; with at least two of the coefficients t 1 , t 2 , t 3 being fractional. The table below gives the various types of centering vectors and the corresponding types of centering. Each one is described by a letter, called the Bravais letter, which is to be found in the Hermann-Mauguin symbol of a space group.
The 'multiplicity', m , of the centered cell is the number of lattice nodes per unit cell (see table).
The volume of the unit cell, V c = ( a c , b c , c c ) is given in terms of the volume of the primitive cell, V = ( a , b , c ), by:
V c = m V


6.14: Case Study 1- Adult Client
Initial Assessment Data
- Biographical data: Adult client
- Reason for seeking care: Vomiting and diarrhea
- History of presenting illness: Vomited x 4 daily x 3 days, diarrhea x 6 daily x 3 days, currently severely nauseated and dizzy, tolerating sips of clear fluid
- Past history: No medications, no illnesses
Think about the client data and try to answer the following questions.
- What infection control measures should the healthcare provider implement?
- What method should be used to measure temperature for this adult client?
- What pulse and blood pressure readings should be taken for this adult client?
Go to the next page to check your answers.


3.3.1: Distinguishing Circumference and Area
Lesson
Let's contrast circumference and area.
Exercise \(\PageIndex{1}\): Filling the Plate
About how many cheese puffs can fit on the plate in a single layer? Be prepared to explain your reasoning.
Exercise \(\PageIndex{2}\): Cart Sort: Circle Problems
Your teacher will give you cards with questions about circles.
- Sort the cards into two groups based on whether you would use the circumference or the area of the circle to answer the question. Pause here so your teacher can review your work.
- Your teacher will assign you a card to examine more closely. What additional information would you need in order to answer the question on your card?
- Estimate measurements for the circle on your card.
- Use your estimates to calculate the answer to the question.
Exercise \(\PageIndex{3}\): Visual Display of Circle Problem
In the previous activity you estimated the answer to a question about circles.
Create a visual display that includes:
- The question you were answering
- A diagram of a circle labeled with your estimated measurements
- Your thinking, organized so that others can follow it
- Your answer, expressed in terms of \(\pi\) and also expressed as a decimal approximation
Exercise \(\PageIndex{4}\): Analyzing Circle Claims
Here are two students’ answers for each question. Do you agree with either of them? Explain or show your reasoning.
1. How many feet are traveled by a person riding once around the merry-go-round?
- Clare says, “The radius of the merry-go-round is about 4 feet, so the distance around the edge is about \(8\pi\) feet.”
- Andre says, “The diameter of the merry-go-round is about 4 feet, so the distance around the edge is about \(4\pi\) feet.”
2. How much room is there to spread frosting on the cookie?
- Clare says “The radius of the cookie is about 3 centimeters, so the space for frosting is about \(6\pi\) cm 2 .”
- Andre says “The diameter of the cookie is about 3 inches, so the space for frosting is about \(2.25\pi\) in 2 .”
3. How far does the unicycle move when the wheel makes 5 full rotations?
- Clare says, “The diameter of the unicycle wheel is about 0.5 meters. In 5 complete rotations it will go about \(\frac{5}{2}\pi\) m 2 .”
- Andre says, “I agree with Clare's estimate of the diameter, but that means the unicycle will go about \(\frac{5}{4}\pi\) m.”
Are you ready for more?
A goat (point \(G\)) is tied with a 6-foot rope to the corner of a shed. The floor of the shed is a square whose sides are each 3 feet long. The shed is closed and the goat can't go inside. The space all around the shed is flat, grassy, and the goat can't reach any other structures or objects. What is the area over which the goat can roam?
Summary
Sometimes we need to find the circumference of a circle, and sometimes we need to find the area. Here are some examples of quantities related to the circumference of a circle:
- The length of a circular path.
- The distance a wheel will travel after one complete rotation.
- The length of a piece of rope coiled in a circle.
Here are some examples of quantities related to the area of a circle:
- The amount of land that is cultivated on a circular field.
- The amount of frosting needed to cover the top of a round cake.
- The number of tiles needed to cover a round table.
In both cases, the radius (or diameter) of the circle is all that is needed to make the calculation. The circumference of a circle with radius \(r\) is \(2\pi r\) while its area is \(\pi r^{2}\). The circumference is measured in linear units (such as cm, in, km) while the area is measured in square units (such as cm 2 , in 2 , km 2 ).
Practice
Exercise \(\PageIndex{5}\)
For each problem, decide whether the circumference of the circle or the area of the circle is most useful for finding a solution. Explain your reasoning.
- A car’s wheels spin at 1000 revolutions per minute. The diameter of the wheels is 23 inches. You want to know how fast the car is travelling.
- A circular kitchen table has a diameter of 60 inches. You want to know how much fabric is needed to cover the table top.
- A circular puzzle is 20 inches in diameter. All of the pieces are about the same size. You want to know about how many pieces there are in the puzzle.
- You want to know about how long it takes to walk around a circular pond.
Exercise \(\PageIndex{6}\)
The city of Paris, France is completely contained within an almost circular road that goes around the edge. Use the map with its scale to:
- Estimate the circumference of Paris.
- Estimate the area of Paris.
Exercise \(\PageIndex{7}\)
Here is a diagram of a softball field:
- About how long is the fence around the field?
- About how big is the outfield?
Exercise \(\PageIndex{8}\)
While in math class, Priya and Kiran come up with two ways of thinking about the proportional relationship shown in the table.
| \(x\) | \(y\) |
|---|---|
| \(2\) | ? |
| \(5\) | \(1750\) |
Both students agree that they can solve the equation \(5k=1750\) to find the constant of proportionality.
- Priya says, “I can solve this equation by dividing 1750 by 5.”
- Kiran says, “I can solve this equation by multiplying 1750 by \(\frac{1}{5}\).”
- What value of \(k\) would each student get using their own method?
- How are Priya and Kiran's approaches related?
- Explain how each student might approach solving the equation \(\frac{2}{3}k=50\).
(From Unit 2.2.2)


5.1: Key Variables in Planning
- List some key controllable and uncontrollable variables that should be considered when creating an overall retail strategy
Firms use strategic planning to develop comprehensive plans for how best to compete by taking a holistic view of the market and the factors that influence it. Decision-makers try to understand all the variables that shape the firm’s internal capabilities and resources relative to external opportunities and threats. In doing so, they work to find areas where they can differentiate, anticipate competitive response, and build sustainable competitive advantage.
Yet, not all variables are controllable. Factors like overall economic trends, government regulation, volatility of supply, demographic shifts, changes in cultural values, and technology can have profound impact upon strategic plans. Think again about the massive macro-factors influencing retail that we’ve already discussed—technology, wellness, labor. They all exist outside the control of the firm.
Previously, we’ve examined how consumers are using convenient digital interfaces and low-cost fulfillment to order products on-line and in-app for at-home delivery. This change in consumer behavior, empowered by advancements in technology, has an effect upon the retail industry, as well. Of course, forward-looking retail strategy should include a consideration of how this is shaping the shopping experience from search through fulfillment to service. But, what was the likelihood that strategists were incorporating these factors into their retail strategy 10 years ago? (Keep in mind that the first Apple iPhone® was introduced in 2007.)
What should this imply for strategic planning now, when artificial intelligence and machine learning are gaining wider-spread acceptance? Nothing? What does the future look like, if consumers can order items or subscribe for automatic fulfillment with in-home assistants like Amazon Alexa and Google Home? What if these machines “learn” consumption patterns and preferences, able to place orders without the consumer initiating them? In what profound and uncontrollable ways will shopping change?
You also read how massive demographic changes have elevated interest in health & wellness, especially around fresh, all-natural ,and organic foods. Consider what this has meant for product marketing and in-store assortment. What are the implications for retailers as these trends become even more mainstream? And, what are the implications for the supply chain, as the vast majority of agricultural output in the United States is not geared toward organic farming? This likely constraint of supply will influence prices. What will this mean for retailer and consumers? How can firms account for this opportunity and related risk in their strategic plans?
And, of course, we discussed labor. Cultural values have coalesced around resolving economic disparity, pressuring local, state, and federal governments to increase minimum wage. They’ve done so through regulation. And, while this has great benefit for associates, it puts incredible pressure on retailers. As you know, retailers operate with very low margins. So, what does this mean for strategic plans? How can retailers manage operations and service while also managing labor costs profitably? What could be the next area of activism or regulation?
As strategists and decision-makers within a firm try to focus internal capabilities & resources for a specific strategy, they must recognize that external variables can undermine their efforts. These uncontrollable variables, including economic trends, government regulation, volatility of supply, demographic shifts, changes in cultural values, and technology, can have profound impact upon strategic plans. Today’s opportunity can vanish very quickly.
Contributors and Attributions
- Key Variables in Planning. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution


12.1: Introduction
Learning Objectives
- Assess children or adolescents with mental health disorders, developmental disabilities, or disruptive behaviors
- Assess family dynamics and include family members in planning nursing care for children and adolescents
- Describe psychotherapy, behavioral therapy, and medical treatments for children or adolescents with mental health disorders, developmental disabilities, or disruptive behaviors
- Discuss conditional confidentiality and mandatory reporting related to caring for children and adolescents
- Apply the nursing process to children or adolescents with mental health disorders, developmental disabilities, or disruptive behaviors
- Identify community resources for children and adolescents with mental health disorders that enhance individual and family functioning
- Evaluate the effectiveness of interventions
Mental health disorders among children are described as “serious changes in the way children typically learn, behave, or handle their emotions, causing distress and problems getting through the day.” [1] One in six children in the United States aged 2 to 8 years (17.4%) has a diagnosed mental, behavioral, or developmental disorder. Behavior problems are more common among children aged 6 to 11 years. [2]
This chapter will discuss common mental health disorders and disabilities in children and adolescents, as well as interprofessional treatments and related nursing plans of care.
- Centers for Disease Control and Prevention. (2022, February 24). Data and statistics on children’s mental health. https://www.cdc.gov/childrensmentalhealth/data.html ↵
- Centers for Disease Control and Prevention. (2022, February 24). Data and statistics on children’s mental health. https://www.cdc.gov/childrensmentalhealth/data.html ↵


4.6: Immigration
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For Americans at the turn of the century, imperialism and immigration were two sides of the same coin. The involvement of American women with imperialist and anti-imperialist activity demonstrates how foreign policy concerns were brought home and became, in a sense, domesticated. It is also no coincidence that many of the women involved in both imperialist and anti-imperialist organizations were also concerned with the plight of new arrivals to the United States. Industrialization, imperialism, and immigration were all linked. Imperialism had at its core a desire for markets for American goods, and those goods were increasingly manufactured by immigrant labor. This sense of growing dependence on “others” as producers and consumers, along with doubts about their capability of assimilation into the mainstream of white, Protestant American society, caused a great deal of anxiety among native-born Americans.
Between 1870 and 1920, over twenty-five million immigrants arrived in the United States. This migration was largely a continuation of a process begun before the Civil War, though by the turn of the twentieth century, new groups such as Italians, Poles, and Eastern European Jews made up a larger percentage of the arrivals while Irish and German numbers began to dwindle.
Although the growing U.S. economy needed large numbers of immigrant workers for its factories and mills, many Americans reacted negatively to the arrival of so many immigrants. Nativists opposed mass immigration for various reasons. Some felt that the new arrivals were unfit for American democracy, and that Irish or Italian immigrants used violence or bribery to corrupt municipal governments. Others (often earlier immigrants themselves) worried that the arrival of even more immigrants would result in fewer jobs and lower wages. Such fears combined and resulted in anti-Chinese protests on the West Coast in the 1870s. Still others worried that immigrants brought with them radical ideas such as socialism and communism. These fears multiplied after the Chicago Haymarket affair in 1886, in which immigrants were accused of killing police officers in a bomb blast. 28
In September 1876, Franklin Benjamin Sanborn, a member of the Massachusetts Board of State Charities, gave an address in support of the introduction of regulatory federal immigration legislation at an interstate conference of charity officials in Saratoga, New York. Immigration might bring some benefits, but “it also introduces disease, ignorance, crime, pauperism and idleness.” Sanborn thus advocated federal action to stop “indiscriminate and unregulated immigration.” 29
Sanborn’s address was aimed at restricting only the immigration of paupers from Europe to the East Coast, but the idea of immigration restrictions was common across the United States in the late nineteenth century, when many variously feared that the influx of foreigners would undermine the racial, economic, and moral integrity of American society. From the 1870s to the 1920s, the federal government passed a series of laws limiting or discontinuing the immigration of particular groups, and the United States remained committed to regulating the kind of immigrants who would join American society. To critics, regulations legitimized racism, class bias, and ethnic prejudice as formal national policy.
The first move for federal immigration control came from California, where racial hostility toward Chinese immigrants had mounted since the midnineteenth century. In addition to accusing Chinese immigrants of racial inferiority and unfitness for American citizenship, opponents claimed that they were also economically and morally corrupting American society with cheap labor and immoral practices, such as prostitution. Immigration restriction was necessary for the “Caucasian race of California,” as one anti-Chinese politician declared, and for European Americans to “preserve and maintain their homes, their business, and their high social and moral position.” In 1875, the anti-Chinese crusade in California moved Congress to pass the Page Act, which banned the entry of convicted criminals, Asian laborers brought involuntarily, and women imported “for the purposes of prostitution,” a stricture designed chiefly to exclude Chinese women. Then, in May 1882, Congress suspended the immigration of all Chinese laborers with the Chinese Exclusion Act, making the Chinese the first immigrant group subject to admission restrictions on the basis of race. They became the first illegal immigrants. 30
On the other side of the country, Atlantic Seaboard states also facilitated the formation of federal immigration policy. Since the colonial period, East Coast states had regulated immigration through their own passenger laws, which prohibited the landing of destitute foreigners unless shipmasters prepaid certain amounts of money in the support of those passengers. State-level control of pauper immigration developed into federal policy in the early 1880s. In August 1882, Congress passed the Immigration Act, denying admission to people who were not able to support themselves and those, such as paupers, people with mental illnesses, or convicted criminals, who might otherwise threaten the security of the nation.
The category of excludable people expanded continuously after 1882. In 1885, in response to American workers’ complaints about cheap immigrant labor, Congress added foreign workers migrating under labor contracts with American employers to the list of excludable people. Six years later, the federal government included people who seemed likely to become wards of the state, people with contagious diseases, and polygamists, and made all groups of excludable people deportable. In 1903, those who would pose ideological threats to American republican democracy, such as anarchists and socialists, also became the subject of new immigration restrictions.
Many immigration critics were responding to the shifting demographics of American immigration. The center of immigrant-sending regions shifted from northern and western Europe to southern and eastern Europe and Asia. These “new immigrants” were poorer, spoke languages other than English, and were likely Catholic or Jewish. White Protestant Americans typically regarded them as inferior, and American immigration policy began to reflect more explicit prejudice than ever before. One restrictionist declared that these immigrants were “races with which the English-speaking people have never hitherto assimilated, and who are most alien to the great body of the people of the United States.” The increased immigration of people from southern and eastern Europe, such as Italians, Jews, Slavs, and Greeks, led directly to calls for tighter restrictive measures. In 1907, the immigration of Japanese laborers was practically suspended when the American and Japanese governments reached the so-called Gentlemen’s Agreement, according to which Japan would stop issuing passports to working-class emigrants. In its forty-two-volume report of 1911, the U.S. Immigration Commission highlighted the impossibility of incorporating these new immigrants into American society. The report highlighted their supposed innate inferiority, asserting that they were the causes of rising social problems in America, such as poverty, crime, prostitution, and political radicalism. 31
The assault against immigrants’ Catholicism provides an excellent example of the challenges immigrant groups faced in the United States. By 1900, Catholicism in the United States had grown dramatically in size and diversity, from 1 percent of the population a century earlier to the largest religious denomination in America (though still outnumbered by Protestants as a whole). As a result, Catholics in America faced two intertwined challenges: one external, related to Protestant anti-Catholicism, and the other internal, having to do with the challenges of assimilation.
Externally, the Church and its members remained an “outsider” religion in a nation that continued to see itself as culturally and religiously Protestant. Torrents of anti-Catholic literature and scandalous rumors maligned Catholics. Many Protestants doubted whether Catholics could ever make loyal Americans because they supposedly owed primary allegiance to the pope.
Internally, Catholics in America faced the question every immigrant group has had to answer: to what extent should they become more like native-born Americans? This question was particularly acute, as Catholics encompassed a variety of languages and customs. Beginning in the 1830s, Catholic immigration to the United States had exploded with the increasing arrival of Irish and German immigrants. Subsequent Catholic arrivals from Italy, Poland, and other Eastern European countries chafed at Irish dominance over the Church hierarchy. Mexican and Mexican American Catholics, whether recent immigrants or incorporated into the nation after the Mexican-American War, expressed similar frustrations. Could all these different Catholics remain part of the same Church?
Catholic clergy approached this situation from a variety of perspectives. Some bishops advocated rapid assimilation into the English-speaking mainstream. These “Americanists” advocated an end to “ethnic parishes”—the unofficial practice of permitting separate congregations for Poles, Italians, Germans, and so on—in the belief that such isolation only delayed immigrants’ entry into the American mainstream. They anticipated that the Catholic Church could thrive in a nation that espoused religious freedom, if only they assimilated. Meanwhile, however, more conservative clergy cautioned against assimilation. While they conceded that the United States had no official religion, they felt that Protestant notions of the separation of church and state and of licentious individual liberty posed a threat to the Catholic faith. They further saw ethnic parishes as an effective strategy protecting immigrant communities and worried that Protestants would use public schools to attack the Catholic faith. Eventually, the head of the Catholic Church, Pope Leo XIII, weighed in on the controversy. In 1899, he sent a special letter (an encyclical) to an archbishop in the United States. Leo reminded the Americanists that the Catholic Church was a unified global body and that American liberties did not give Catholics the freedom to alter church teachings. The Americanists denied any such intention, but the conservative clergy claimed that the pope had sided with them. Tension between Catholicism and American life, however, would continue well into the twentieth century. 32
The American encounter with Catholicism—and Catholicism’s encounter with America—testified to the tense relationship between native-born and foreign-born Americans, and to the larger ideas Americans used to situate themselves in a larger world, a world of empire and immigrants.


10.27: Introduction to Inference for a Difference in Two Population Means
What you’ll learn to do: Conduct a hypothesis test or construct a confidence interval to investigate a difference between two population means. Interpret results in context.
In the last section we learned about a hypothesis test for a population mean. We will build on this and learn to conduct a hypothesis test about a difference between two population means and state a conclusion in context under appropriate conditions. We will then construct a confidence interval to estimate a difference in two population means (when conditions are met) and interpret the confidence interval in context.
Contributors and Attributions
CC licensed content, Shared previously
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


5.2: Grammar
In this section
- You will learn about verbal nouns and how they are derived from verbs.
- You will use different connectors to make meaningful sentences.
Grammar (1)
Masdar (verbal noun/gerund)
المصدر is a verbal noun that is derived from the same root as the verb to imply the activity or action. It is treated as a noun in that it can be definite or indefinite depending on context. For example, it will be indefinite when it falls as the first part of Idafa. Consider the following sentence pairs:
|
والدتي تحب
شُرْب
القهوة في الصباح.

My mother likes to drink coffee in the morning |
|
أحب
مشاهدة
التليفزيون في المساء.
I like watching TV in the evening |
أحب
أن أشاهد
التليفزيون في المساء.
I like to watch TV in the evening |
|
بِنتي تريد
القراءة
في الليل.
My daughter wants reading at night |
بِنتي تريد
أن تقرأ
في الليل.
My daughter wants to read at night |
In the previous examples, you can see that the masdar form can equally be expressed b y the structure أن + verb. Note that using the definite article الـ is not necessarily a part of the gerund. It depends on context so you cannot use it in a case which you have an Idafa structure as in number 1 and 2. Otherwise, you must use الـ when you have examples like number 3 above.
Study the following list of verbs and gerunds. Listen and read the forms.
[table id=48 /]
Decide whether you should use the verb or the masdar form in each of the following sentences.
With your classmates. Find someone who: survey your classmates to find out who does or does not do these activities. Use the Masdar form in questions and answers.
- likes watching football or basketball.
- enjoys swimming in the winter.
- likes traveling in the summer.
- likes playing sports? Which sport?
- likes reading every day.
- likes eating in restaurants? Which restaurant?
- likes living in a big city.
- wants to obtain the Masters.
- likes to work in teaching.
- likes dancing? Which dancing?
Grammar (2)
Expressing purpose and result
- To express the purpose or reason of something you are doing, you can use the letter لـ and connect it with either a verb or a Masdar. Study the following examples:
|
نذهب إلى المكتبة لكتابة الواجب.
We go to the library for writing the homework |
نذهب إلى المكتبة لنكتب الواجب. We go to the library to write the homework |
|
أسافر إلى مصر لدراسة اللغة العربية
I travel to Egypt for studying Arabic |
أسافر إلى مصر لأدرس اللغة العربية I travel to Egypt to study Arabic |
|
صاحبتي تعمل في شركة للحصول على فلوس My friend works in a company for obtaining money |
صاحبتي تعمل في شركة لتحصل على فلوس My friend work in a company to obtain money |
- To express the consequence or result, you can use the connector لذلك which literally means “for this reason.” It implies the equivalent of the connector “so” in English. It is only followed by a complete sentence as in the following examples:
|
ليس عندي فصول يوم السبت ، لذلك لا أذهب إلى الجامعة. I do not have classes on Saturday, so I do not go to the university. |
|
|
لا نحب الأكل في المطاعم ، و لذلك نطبخ في البيت. We do not like eating in restaurants, that’s why we cook at home. |
|
|
أريد السفر إلى الشرق الأوسط ، لذلك أدرس اللغة العربية. I want traveling to the middle east, for this reason I study Arabic. |
Decide what connector to use to combine the sentences below.
Translate each of the following sentences in two ways: once using verb and once using Masdar.
Grammar (3)
Present tense case marking
The default case ending for the present tense is ُ as in يأكلُ – يشربُ – نحبُ .
This default case changes when the following precede the verb : أن – لن – لـ
Study these sentences and compare the case endings of verbs:
|
نحب أن نذهبَ إلى المكتبة كل يوم. We like to go to the library every day. |
نذهبُ إلى المكتبة كل يوم. We go to the library every day. |
(1) | |
|
لن أسافرَ إلى مصر في الصيف. I will not travel to Egypt in the summer. |
أسافرُ إلى مصر في الصيف. I travel to Egypt in the summer. |
(2) | |
|
صاحبتي تدرس الكمبيوتر لتعملَ في شركة مايكروسوفت. My friend studies computer to work in Microsoft company |
صاحبتي تعملُ في شركة مايكروسوفت. My friend work in Microsoft company. |
(3) |
With verb conjugations that end with ــن , you drop it after these connectors. Study the following examples and note what happens to the ending of verbs:
|
يا سارة! هل تحبين أن تخرجي في الويك إند؟ We like to go to the library every day. |
يا سارة! هل تخرجين في الويك إند؟ We go to the library every day. |
(4) | |
|
هل أنتم تستطيعون أن تسكنوا في مدينة كبيرة؟ Can you live in a big city? |
هل أنتم تسكنون في مدينة كبيرة؟ Do you live in a big city? |
(5) | |
|
أصحابي يريدون أن يلعبوا كرة القدم. My friends want to play soccer. |
أصحابي يلعبون كرة القدم. My friends play soccer. |
(6) |
What happens here is that the case marking change in these cases is achieved by dropping the final noun in the verb if that verb is preceded by أن- لن -لـ
Choose the appropriate word and structure to complete the following sentences.
Rewrite the following sentences using the verb and make necessary changes.
Key Takeaways
- You can use a variety of connectors to make a complex sentence.
- You can flexibly use different word forms to convey similar meanings.


2.2: Working with units
There are several rules when working with units that are not always obvious, and sometimes not correctly applied, even in standard textbooks. Errors with units have led to serious confusions and accidents like the collapse of a Mars orbiter and an aeroplane running out of fuel in mid-air. It is therefore important for every engineer to be able to correctly work with units and we repeat the most important rules here.
In mechanics, we often deal with scalar variables that represent physical quantities that have a measurement unit, like forces, velocities, or positions. Scalar variables consist of the product of two elements: a number and a measurement unit.
Force can be expressed as the product of a numerical value and the measurement unit newton:
\[F=3 \cdot \mathrm{N}=3 \mathrm{~N}. \nonumber\]
Following the ISO norm [5] and the "Red Book" [2], for quantities that have a measurement unit, the numerical value of a scalar variable \(Q\) that represents such quantities is denoted by \(\{Q\}\), and the measurement unit is denoted by \([Q]\), such that
\[Q=\{Q\} \cdot[Q] \tag{2.6} \label{2.6}\]
A mass \(m=3.5 \mathrm{~kg}\) has the numerical value \(\{m\}=3.5\) and the unit \([m]=\mathrm{kg}\). The same quantity value can be expressed as \(3500 \mathrm{~g}\). Then, the numerical value is \(\{m\}=3500\), and \([m]=\mathrm{g}\).
Note that square brackets only have meaning when placed around the physical quantity. For axes labels one commonly uses round brackets \([4,8]\) around the unit like for a time axis: \(t\) (s). Another option [1] is to use division: \(t / \mathrm{s}\). Generally, the letters of units are printed in roman (upright), those of scalar variables in italic.
For time \(t\),
- good use is: \(t=4 \mathrm{~s}, t=4(\mathrm{~s}), t / \mathrm{s}=4\), or \([t]=\mathrm{s}\),
- not good use is: \(\cancel{t \Rightarrow 4}\), or \(\cancel{t}\).
Besides paying attention to the units, also you need to make sure that if you have a vector on the left side of the \(=\) sign, you also need to have a vector on the right side.
From Equation 2.6 it can be seen that every scalar equation in mechanics can be split into two equations, one for the numerical values, and one for the units. For example, an equation like \(F=m a\), with \(F=1 \mathrm{~N}, m=1 \mathrm{~kg}\) and \(a=1 \mathrm{~m} / \mathrm{s}^{2}\) can be split into:
\[\begin{align} F & =m \cdot a \tag{2.7} \label{2.7}\\[4pt] \{F\}[F] & =\{m\}[m] \cdot\{a\}[a] \tag{2.8} \label{2.8}\\[4pt] \{F\} & =\{m\} \cdot\{a\} \rightarrow 1=1 \cdot 1 \tag{2.9} \label{2.9}\\[4pt] {[F] } & =[m] \cdot[a] \rightarrow \mathrm{N}=\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2} \tag{2.10} \label{2.10}\end{align}\]
When making a calculation it is important that in the end result both the numerical values and the units are correct since obviously, in equations, the units on both sides of the equal sign need to be identical, otherwise the equality can never hold. Normally, the units automatically work out well if you work with SI units. In some cases, one has to deal with equations like \(a=c_{1} t^{2}\). In that case the unit of the constant \(c_{1}\) is not given, but can be obtained from the equation \([a]=\left[c_{1}\right]\left[t^{2}\right]\), which can be written as \(\mathrm{m} / \mathrm{s}^{2}=\left[c_{1}\right] \cdot \mathrm{s}^{2}\), from which it follows that the unit of \(c_{1}\) is: \(\left[c_{1}\right]=\mathrm{m} / \mathrm{s}^{4}\).
A general advice is to work as long as possible with variables, and only to substitute numbers and units in the final solution. Units almost always help as a part of plausibility checks for calculations. Moreover, writing the numerical value and unit in each step of the derivation is much more work, and errors in numerical values are harder to find than in equations.
In this textbook we mostly work with the International System of Units, also called SI units, the SI system, or metric units, which are the world’s most widely used system of measurement. If you have to work with non-SI units, as
a habit first convert it to the SI unit as a first step to prevent problems later on. Often this conversion is just a matter of multiplying the non-SI unit value by a constant factor, e.g. \(1.0 \mathrm{inch}=1.0 \mathrm{inch} \times 0.0254 \mathrm{~m} / \mathrm{inch}\), however in some cases the conversion is more complex, like when converting temperature in Fahrenheit to degrees Kelvin. Preferably also work in the base SI unit, remove any factors related to prefixes (like converting \(\mathrm{mm}\) to \(\mathrm{m}\) by dividing by a factor 1000) before starting calculations, using scientific notation (e.g. \(3.02 \times 10^{8}\) ) for very large or small values.
There are 7 base SI units, and the only base SI units relevant in this textbook are \(\mathrm{m}, \mathrm{s}\) and \(\mathrm{kg}\). There are 22 derived SI units of which we use 6 in this textbook: the \(\mathrm{rad}=\mathrm{m} / \mathrm{m}, \mathrm{Hz}=1 / \mathrm{s}, \mathrm{N}=\mathrm{kg} \mathrm{m} / \mathrm{s}^{2}, \mathrm{~Pa}=\mathrm{N} / \mathrm{m}^{2}, \mathrm{~J}=\mathrm{Nm}\), and \(\mathrm{W}=\mathrm{J} / \mathrm{s}\). The unit to use for a certain quantity or parameter can often be determined by evaluating known dynamics equations. As an example, since \(x\) has unit \(\mathrm{m}\) and \(t\) has unit \(\mathrm{s}\), we can use the equation for acceleration \(a=\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}\) to determine that acceleration \(a\) has the unit \(\mathrm{m} / \mathrm{s}^{2}\) (note how the square sign \({ }^{2}\) is placed behind the \(t\) but before the \(x\) in the second derivative notation). Similarly, one obtains from \(F=m \times a\) that \(1 \mathrm{~N}=1 \mathrm{~kg} \times 1 \mathrm{~m} / \mathrm{s}^{2}=1 \mathrm{~kg} \mathrm{~m} / \mathrm{s}^{2}\). This method can often help to find the right unit by using a known equation, such that units don’t have to be memorised. Keep in mind that the equations in this textbook only work correctly if one uses SI units or units derived from SI units (like \(\mathrm{N}\) or W).
In some cases the scalar variable represents a unitless quantity. Examples of unitless quantities are integer numbers (for example the number of wheels under a car) or ratios of quantities with the same unit (for example the ratio of two distances). Most mathematical functions, like trigonometric or logarithmic functions, only accept unitless quantities. For example there is no mathematical way to calculate \(\tan (1 \mathrm{~m})\). So, always make sure only unitless numbers or ratios are used. E.g. \(\log \left(x_{2} / x_{1}\right)\) is fine, but \(\log \left(x_{2}\right)\) cannot be calculated if \(\left[x_{2}\right]=\mathrm{m}\). It is important to note that additions and subtractions can only be performed between scalar quantities of the same unit, while multiplications, divisions and powers can be taken on quantities of different units as long as the operation is also applied to the respective units.
A special case are angles, even though they usually have degrees \({ }^{\circ}\) or rad behind them, these angle designations are actually unitless numbers, which measure e.g. the ratio of the length of a circular arc divided by the length of the perimeter of a full circle. The degree or radian symbol only indicates how the trigonometric function should be applied. When not otherwise specified in this textbook we work with radians. Since angles are unitless, most mathematical functions can accept angles as an argument.
When dealing with vectors, one also needs to specify the direction of the vector using one or more unit vectors, so for specifying a vector one uses
the product of a number, a measurement unit and at least one unit vector. Subtraction and addition is only possible for vectors with the same unit, scalar products are possible as long as the units are also multiplied, so for instance \(\overrightarrow{\boldsymbol{v}} \times t=\Delta \overrightarrow{\boldsymbol{x}}\) is a correct equation, because \(\mathrm{m} / \mathrm{s} \times \mathrm{s}=\mathrm{m}\) and because quantities on both sides of the \(=\) sign are vectors.
Expressions and equations may contain mixtures of quantities given with their numerical value and quantities given as variables. In all cases one needs to ensure that the units on both sides of the equal sign \(=\) are equal, and that one cannot equate a scalar to a vector or matrix. If a vector quantity evaluates to zero one we use the zero vector \(\overrightarrow{\mathbf{0}}\), which is a vector with magnitude zero and no direction.
For a particle of mass \(m=3 \mathrm{~kg}\) having acceleration vector \(\overrightarrow{\boldsymbol{a}}=1 \mathrm{~m} / \mathrm{s}^{2} \hat{\boldsymbol{i}}\), the resultant force vector \(\overrightarrow{\boldsymbol{F}}\) on the particle can be calculated as:
Correct is: \(\overrightarrow{\boldsymbol{F}}=m \overrightarrow{\boldsymbol{a}}=3 \mathrm{~kg} \times 1 \mathrm{~m} / \mathrm{s}^{2} \hat{\boldsymbol{\imath}}=3 \mathrm{~kg} \mathrm{~m} / \mathrm{s}^{2} \hat{\boldsymbol{\imath}}=3 \mathrm{~N} \hat{\boldsymbol{\imath}}\).
Incorrect is: \(\overrightarrow{\boldsymbol{F}}=\overrightarrow{3} \cdot \boldsymbol{a} \mathbb{N}\), because the vector \(\overrightarrow{\boldsymbol{a}}\) has unit \(\mathrm{m} / \mathrm{s}^{2}\) such that the units on both side of the equal sign are different.
A particle performs a harmonic oscillation, such that its location coordinate \(x\) is a function of amplitude \(x_{0}\), angular frequency \(\Omega\), and time \(t\) :
\[x=x_{0} \sin (\Omega t) \tag{2.11} \label{2.11}\]
Fill in suitable units for \(x, x_{0}, \Omega\), and \(t\) and check the equation for consistency.
Which of these expressions make correct use of units, for mass \(m\), time \(t\), force \(F\), angular velocity \(\omega\), and angle \(\theta\) ? Correct the mistakes.
A. \(\quad m=20\)
B. \(\quad m=3 \mathrm{~N}\)
C. \(\quad F=3 \cdot m[\mathrm{~N}]\)
D. \(\quad \mathrm{F}=20 N\)
E. \([m]=\mathrm{kg}\)
F. \(\quad m /[m]=20\)
G. \(\quad t=3 \mathrm{~s}\)
H. \(\quad \omega=2 \mathrm{rad} / \mathrm{s}\)
I. \(\quad \theta=\omega \cdot t=2 \cdot t[\mathrm{~s}]\)
J. \(\quad\{m\}=20\)
K. \(\quad \omega=t^{2}\)
L. \(\quad \omega=1 \mathrm{rad} / \mathrm{s} \cdot t^{2}\)
M. \(\quad F(t)=\cos (1 \mathrm{rad} / \mathrm{s} \cdot t) \mathrm{N}\)
N. \(\quad F(t)=3 e^{-t} \mathrm{~N}\)
O. \(\quad F(t)=\ln \left(t^{2} / \mathrm{s}^{2}\right) \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2}\)
\[\begin{align} & \text { Exemplary solution } \\[4pt] & \qquad \begin{array}{l} m \neq 20 \\[4pt] m \neq 3 \mathrm{~N} \\[4pt] F \neq 3 \cdot m[\mathrm{~N}] \\[4pt] \\[4pt] \\[4pt] \mathrm{F} \neq 20 \mathrm{~N} \\[4pt] \\[4pt] {[m]=\mathrm{kg}} \\[4pt] m /[\mathrm{m}]=20 \\[4pt] t=3 \mathrm{~s} \\[4pt] \omega=2 \mathrm{rad} / \mathrm{s} \\[4pt] \theta=\omega \cdot t \neq 2 \cdot t[\mathrm{~s}] \\[4pt] \{m\}=20 \\[4pt] \omega \neq t^{2} \\[4pt] \omega \neq 1 \mathrm{rad} / \mathrm{s} \cdot t^{2} \\[4pt] F(t)=\cos (1 \mathrm{rad} / \mathrm{s} \cdot t) \mathrm{N} \\[4pt] F(t) \neq 3 e^{-t} \mathrm{~N} \\[4pt] F(t)=\ln \left(t^{2} / \mathrm{s}^{2}\right) \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2} \end{array} \end{align}\]
unit omitted; \(m=20 \mathrm{~kg}\)
wrong unit for mass quantity; \(m=3 \mathrm{~kg}\) unit omitted and unit inserted with wrongly used square brackets; \(F=\) \(3 \mathrm{~m} / \mathrm{s}^{2} \cdot m\)
roman variable, italic unit symbol; \(F=\) \(20 \mathrm{~N}\)
correct!
correct!
correct!
correct!
wrong unit and \([\mathrm{s}]\) is wrong; \(\theta=2 \mathrm{rad} / \mathrm{s} \cdot t\) correct!
unit mismatch; e.g. \(\omega=1 \mathrm{rad} / \mathrm{s}^{3} \cdot t^{2}\)
unit mismatch; e.g. \(\omega=1 \mathrm{rad} / \mathrm{s}^{3} \cdot t^{2}\)
correct !
exponent should not have units; e.g. \(F(t)=e^{-t / \mathrm{s}} \mathrm{N}\)
correct !


16.6: Maximal and Prime Ideals
In this particular section we are especially interested in certain ideals of commutative rings. These ideals give us special types of factor rings. More specifically, we would like to characterize those ideals \(I\) of a commutative ring \(R\) such that \(R/I\) is an integral domain or a field.
A proper ideal \(M\) of a ring \(R\) is a maximal ideal of \(R\) if the ideal \(M\) is not a proper subset of any ideal of \(R\) except \(R\) itself. That is, \(M\) is a maximal ideal if for any ideal \(I\) properly containing \(M\text{,}\) \(I = R\text{.}\) The following theorem completely characterizes maximal ideals for commutative rings with identity in terms of their corresponding factor rings.
Theorem \(16.35\)
Let \(R\) be a commutative ring with identity and \(M\) an ideal in \(R\text{.}\) Then \(M\) is a maximal ideal of \(R\) if and only if \(R/M\) is a field.
- Proof
-
Let \(M\) be a maximal ideal in \(R\text{.}\) If \(R\) is a commutative ring, then \(R/M\) must also be a commutative ring. Clearly, \(1 + M\) acts as an identity for \(R/M\text{.}\) We must also show that every nonzero element in \(R/M\) has an inverse. If \(a + M\) is a nonzero element in \(R/M\text{,}\) then \(a \notin M\text{.}\) Define \(I\) to be the set \(\{ ra + m : r \in R \text{ and } m \in M \}\text{.}\) We will show that \(I\) is an ideal in \(R\text{.}\) The set \(I\) is nonempty since \(0a+0=0\) is in \(I\text{.}\) If \(r_1 a + m_1\) and \(r_2 a + m_2\) are two elements in \(I\text{,}\) then
\[ (r_1 a + m_1) - ( r_2 a + m_2) = (r_1 - r_2)a + (m_1 - m_2) \nonumber \]is in \(I\text{.}\) Also, for any \(r \in R\) it is true that \(rI \subset I\text{;}\) hence, \(I\) is closed under multiplication and satisfies the necessary conditions to be an ideal. Therefore, by Proposition \(16.10\) and the definition of an ideal, \(I\) is an ideal properly containing \(M\text{.}\) Since \(M\) is a maximal ideal, \(I=R\text{;}\) consequently, by the definition of \(I\) there must be an \(m\) in \(M\) and an element \(b\) in \(R\) such that \(1=ab+m\text{.}\) Therefore,
\[ 1 + M = ab + M = ba + M = (a+M)(b+M)\text{.} \nonumber \]Conversely, suppose that \(M\) is an ideal and \(R/M\) is a field. Since \(R/M\) is a field, it must contain at least two elements: \(0 + M = M\) and \(1 + M\text{.}\) Hence, \(M\) is a proper ideal of \(R\text{.}\) Let \(I\) be any ideal properly containing \(M\text{.}\) We need to show that \(I = R\text{.}\) Choose \(a\) in \(I\) but not in \(M\text{.}\) Since \(a+ M\) is a nonzero element in a field, there exists an element \(b +M\) in \(R/M\) such that \((a+M)(b+M) = ab + M = 1+M\text{.}\) Consequently, there exists an element \(m \in M\) such that \(ab + m = 1\) and \(1\) is in \(I\text{.}\) Therefore, \(r1 =r \in I\) for all \(r \in R\text{.}\) Consequently, \(I = R\text{.}\)
Example \(16.36\)
Let \(p{\mathbb Z}\) be an ideal in \({\mathbb Z}\text{,}\) where \(p\) is prime.
Solution
Then \(p{\mathbb Z}\) is a maximal ideal since \({\mathbb Z}/ p {\mathbb Z} \cong {\mathbb Z}_p\) is a field.
A proper ideal \(P\) in a commutative ring \(R\) is called a prime ideal if whenever \(ab \in P\text{,}\) then either \(a \in P\) or \(b \in P\text{.}\) 6
Example \(16.37\)
It is easy to check that the set \(P = \{ 0, 2, 4, 6, 8, 10 \}\) is an ideal in \({\mathbb Z}_{12}\text{.}\)
Solution
This ideal is prime. In fact, it is a maximal ideal.
Proposition \(16.38\)
Let \(R\) be a commutative ring with identity \(1\text{,}\) where \(1 \neq 0\text{.}\) Then \(P\) is a prime ideal in \(R\) if and only if \(R/P\) is an integral domain.
- Proof
-
First let us assume that \(P\) is an ideal in \(R\) and \(R/P\) is an integral domain. Suppose that \(ab \in P\text{.}\) If \(a + P\) and \(b + P\) are two elements of \(R/P\) such that \((a + P)(b + P) = 0 + P = P\text{,}\) then either \(a + P = P\) or \(b + P = P\text{.}\) This means that either \(a\) is in \(P\) or \(b\) is in \(P\text{,}\) which shows that \(P\) must be prime.
Conversely, suppose that \(P\) is prime and
\[ (a + P)(b + P) = ab + P = 0 + P = P\text{.} \nonumber \]Then \(ab \in P\text{.}\) If \(a \notin P\text{,}\) then \(b\) must be in \(P\) by the definition of a prime ideal; hence, \(b + P = 0 + P\) and \(R/P\) is an integral domain.
Example \(16.39\)
Every ideal in \({\mathbb Z}\) is of the form \(n {\mathbb Z}\text{.}\) The factor ring \({\mathbb Z} / n{\mathbb Z} \cong {\mathbb Z}_n\) is an integral domain only when \(n\) is prime. It is actually a field.
Solution
Hence, the nonzero prime ideals in \({\mathbb Z}\) are the ideals \(p{\mathbb Z}\text{,}\) where \(p\) is prime. This example really justifies the use of the word “prime” in our definition of prime ideals.
Since every field is an integral domain, we have the following corollary.
Corollary \(16.40\)
Every maximal ideal in a commutative ring with identity is also a prime ideal
Historical Note
Amalie Emmy Noether, one of the outstanding mathematicians of the twentieth century, was born in Erlangen, Germany in 1882. She was the daughter of Max Noether (1844–1921), a distinguished mathematician at the University of Erlangen. Together with Paul Gordon (1837–1912), Emmy Noether's father strongly influenced her early education. She entered the University of Erlangen at the age of 18. Although women had been admitted to universities in England, France, and Italy for decades, there was great resistance to their presence at universities in Germany. Noether was one of only two women among the university's 986 students. After completing her doctorate under Gordon in 1907, she continued to do research at Erlangen, occasionally lecturing when her father was ill.
Noether went to Göttingen to study in 1916. David Hilbert and Felix Klein tried unsuccessfully to secure her an appointment at Göttingen. Some of the faculty objected to women lecturers, saying, “What will our soldiers think when they return to the university and are expected to learn at the feet of a woman?” Hilbert, annoyed at the question, responded, “Meine Herren, I do not see that the sex of a candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bathhouse.” At the end of World War I, attitudes changed and conditions greatly improved for women. After Noether passed her habilitation examination in 1919, she was given a title and was paid a small sum for her lectures.
In 1922, Noether became a Privatdozent at Göttingen. Over the next 11 years she used axiomatic methods to develop an abstract theory of rings and ideals. Though she was not good at lecturing, Noether was an inspiring teacher. One of her many students was B. L. van der Waerden, author of the first text treating abstract algebra from a modern point of view. Some of the other mathematicians Noether influenced or closely worked with were Alexandroff, Artin, Brauer, Courant, Hasse, Hopf, Pontryagin, von Neumann, and Weyl. One of the high points of her career was an invitation to address the International Congress of Mathematicians in Zurich in 1932. In spite of all the recognition she received from her colleagues, Noether's abilities were never recognized as they should have been during her lifetime. She was never promoted to full professor by the Prussian academic bureaucracy.
In 1933, Noether, who was Jewish, was banned from participation in all academic activities in Germany. She emigrated to the United States, took a position at Bryn Mawr College, and became a member of the Institute for Advanced Study at Princeton. Noether died suddenly on April 14, 1935. After her death she was eulogized by such notable scientists as Albert Einstein.


2.4: Measures of the Center of the Data
The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean (average) and the median . To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. Technically this is the arithmetic mean. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts meaning an equal number of observations on each side. The weight of 25 people are below this weight and 25 people are heavier than this weight. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.
Note
The words “mean” and “average” are often used interchangeably. The substitution of one word for the other is common practice. The technical term is “arithmetic mean” and “average” is technically a center location. However, in practice among non-statisticians, “average" is commonly accepted for “arithmetic mean.”
When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an x with a bar over it (pronounced “\(x\) bar”): \(\overline x\).
The Greek letter \(\mu\) (pronounced "mew") represents the population mean . One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.
To see that both ways of calculating the mean are the same, consider the sample:
1; 1; 1; 2; 2; 3; 4; 4; 4; 4; 4
\[\overline{x}=\frac{1+1+1+2+2+3+4+4+4+4+4}{11}=2.7\nonumber\]
\[\overline{x}=\frac{3(1)+2(2)+1(3)+5(4)}{11}=2.7\nonumber\]
In the second calculation, the frequencies are 3, 2, 1, and 5.
You can quickly find the location of the median by using the expression \(\frac{n+1}{2}\).
The letter \(n\) is the total number of data values in the sample. If \(n\) is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If \(n\) is an even number, the median is equal to the two middle values added together and divided by two after the data has been ordered. For example, if the total number of data values is 97, then \(\frac{n+1}{2}=\frac{97+1}{2}=49\). The median is the 49 th value in the ordered data. If the total number of data values is 100, then \(\frac{n+1}{2}=\frac{100+1}{2}=50.5\). The median occurs midway between the 50 th and 51 st values. The location of the median and the value of the median are not the same. The upper case letter \(M\) is often used to represent the median. The next example illustrates the location of the median and the value of the median.
Example \(\PageIndex{1}\)
AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):
3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47;
Calculate the mean and the median.
-
The calculation for the mean is:
\(\overline{x}=\frac{[3+4+(8)(2)+10+11+12+13+14+(15)(2)+\ldots+35+37+40+(44)(2)+47]}{40}=23.6\)
To find the median, \(M\), first use the formula for the location. The location is:
\(\frac{n+1}{2}=\frac{40+1}{2}=20.5\)
Starting at the smallest value, the median is located between the 20 th and 21 st values (the two 24s):
\(3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47;\)\(M=\frac{24+24}{2}=24\)
Answer
Example \(\PageIndex{2}\)
Suppose that in a small town of 50 people, one person earns $5,000,000 per year and the other 49 each earn $30,000. Which is the better measure of the "center": the mean or the median?
-
\(\overline{x}=\frac{5,000,000+49(30,000)}{50}=129,400\)
\(M = 30,000\)
(There are 49 people who earn $30,000 and one person who earns $5,000,000.)
The median is a better measure of the "center" than the mean because 49 of the values are 30,000 and one is 5,000,000. The 5,000,000 is an outlier. The 30,000 gives us a better sense of the middle of the data.
Answer
Another measure of the center is the mode. The mode is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal.
Example \(\PageIndex{3}\)
Statistics exam scores for 20 students are as follows:
50; 53; 59; 59; 63; 63; 72; 72; 72; 72; 72; 76; 78; 81; 83; 84; 84; 84; 90; 93
Find the mode.
-
The most frequent score is 72, which occurs five times. Mode = 72.
Answer
Example \(\PageIndex{4}\)
Five real estate exam scores are 430, 430, 480, 480, 495. The data set is bimodal because the scores 430 and 480 each occur twice.
When is the mode the best measure of the "center"? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing.
Note
The mode can be calculated for qualitative data as well as for quantitative data. For example, if the data set is: red, red, red, green, green, yellow, purple, black, blue, the mode is red.
Calculating the Arithmetic Mean of Grouped Frequency Tables
When only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean: mean = \(\frac{\text { data sum }}{\text { number of data values }}\) We simply need to modify the definition to fit within the restrictions of a frequency table.
Since we do not know the individual data values we can instead find the midpoint of each interval. The midpoint is \(\frac{\text { lower boundary+upper boundary}}{2}\). We can now modify the mean definition to be \(\textbf{Mean of Frequency Table}=\frac{\sum f m}{\sum f}\) where f = the frequency of the interval and m = the midpoint of the interval.
Example \(\PageIndex{5}\)
A frequency table displaying professor Blount’s last statistic test is shown. Find the best estimate of the class mean.
|
Grade interval |
Number of students |
|---|---|
|
50–56.5 |
1 |
|
56.5–62.5 |
0 |
|
62.5–68.5 |
4 |
|
68.5–74.5 |
4 |
|
74.5–80.5 |
2 |
|
80.5–86.5 |
3 |
|
86.5–92.5 |
4 |
|
92.5–98.5 |
1 |
-
Find the midpoints for all intervals
Grade interval
Midpoint
50–56.5
53.25
56.5–62.5
59.5
62.5–68.5
65.5
68.5–74.5
71.5
74.5–80.5
77.5
80.5–86.5
83.5
86.5–92.5
89.5
92.5–98.5
95.5
Table \(\PageIndex{2}\)
-
Calculate the sum of the product of each interval frequency and midpoint. \(\sum f m\) \(53.25(1)+59.5(0)+65.5(4)+71.5(4)+77.5(2)+83.5(3)+89.5(4)+95.5(1)=1460.25\)
-
\(\mu=\frac{\sum f m}{\sum f}=\frac{1460.25}{19}=76.86\)
-
Answer
Exercise \(\PageIndex{1}\)
Maris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data:
|
Hours teenagers spend on video games |
Number of teenagers |
|---|---|
|
0–3.5 |
3 |
|
3.5–7.5 |
7 |
|
7.5–11.5 |
12 |
|
11.5–15.5 |
7 |
|
15.5–19.5 |
9 |
What is the best estimate for the mean number of hours spent playing video games?


11.5: Data Analysis Charts
By the end of this section, you will be able to:
- Determine the most appropriate chart for a given dataset or analysis
- Create a chart template
- Use the Quick Analysis tool to add sparklines to a table
At some point in your career, you might need to present an analysis of data to an audience of peers or management. Your information must appear clear and professional and will likely include graphs and charts of your analysis. Recall that the theme and color choices of these elements can be critical to developing a professional presentation that will effectively and clearly communicate your intended message (see Preparing Presentations for a review of design strategies and tips).
You learned how to create graphs and charts in PivotTables. Some of those skills, such as using legends, adding gridlines, editing axis titles and labels, and including chart titles, will make for a professional presentation of the data. This section will discuss several types of charts and graphs and when they might be appropriate to use based on your dataset.
With data visuals, it is key that you select the type of chart or graph that works best for a particular dataset and makes the information easy to understand. You cannot simply select a graph or a chart based on its appearance. All types of graphs and charts are not appropriate for every dataset. As you decide which chart or graph to use, consider the type and level of data that you have (Table 11.3). You should also consider what kind of message you are trying to convey. Are you trying to inform, compare, show trends over time, or reveal relationships? Your purpose can dictate what type of chart you will use.
| Purpose | Appropriate Charts/Graphs |
|---|---|
| Comparisons | Bar chart, pie chart, stacked bar chart, clustered column chart |
| Changes | Line graph, area chart, timeline |
| Relationships | Scatter plot, histogram, combo column/line chart |
Link to Learning
Selecting the right type of chart to represent your data is not always easy. How you present your data is incredibly important. You need to always be mindful that you are accurately presenting your data visually in a way that tells the story objectively, with no apparent bias or intention. Visit this link to read strategies for choosing the correct chart to visualize your data.
Clustered Column Chart
A clustered column chart is simply a column chart that includes multiple variables to show the comparison between the groups. A column chart compares one variable against another variable. For example, you might put the sales region on the horizontal axis and the total sales on the vertical axis. A clustered column would have that, but it would include the same data for multiple years of data. The clustered column chart is great for showing comparisons (Figure 11.85). Using the WorldCorp data, you can create a clustered column chart to see the comparisons between the quantity sold for each product type over time. A drawback of the clustered column chart is that it is not appropriate for large amounts of data. If you have more than five sets of data, the chart can look too busy and overwhelm your audience. The clustered column chart also works better with a 2D display rather than a 3D display. The 3D display can look too busy and can make it difficult to interpret the information easily.
Combination Chart
A combination chart combines two different chart types in one visual, allowing you to display data at different levels on the same chart. You can use combination charts to understand and examine comparisons between groups and the relationship between different variables. With this example, a combination chart can be created to show the FOB $ for each month based on the quantity sold for each of the two product types. With the combination chart, you determine which chart is appropriate for each data series you are including. Figure 11.86 uses a column chart for quantity sold and a line graph for FOB $. Combination charts are often easier to interpret as 2D than as 3D. Consider using contrasting colors to make the plotted series more visible.
Stacked Column Chart
The stacked column chart compares data like the clustered column chart but displays it in a different format. It shows the relationship between groups using the same scales. Instead of the columns being next to each other, the data is stacked, one series on top of the other, in a single column. A stacked column chart can be used to show part-to-whole relationships. Using contrasting colors will help make the distinction between the groups more visible. Using one of the 3D options for displaying a stacked column chart can give it a professional appearance that is easy to interpret (Figure 11.87). This stacked column chart shows WorldCorp’s sales agents’ FOB $ during a three-week period.
Bar Chart
A bar chart is a horizontal column chart. It is sometimes used to communicate complex datasets and can be easier to interpret in certain situations. Bar charts are effective for displaying negative and positive numbers together. They also work well for charts with long text labels along the y -axis because the labels are more readable when placed horizontally (Figure 11.88). You can create clustered and stacked bar charts to display multiple series in a single chart. Use a simple layout and contrasting colors for the most professional look. Like a clustered column chart, bar charts work best with smaller datasets. Too many bars on the chart can make it unreadable. Use a bar chart when you have ten or fewer categories to display. You can use WorldCorp data to construct a bar chart of the volume sold for four different sizes of televisions.
Line Chart
A line chart is used to show changes over time. It is appropriate for trend analysis and graphing continuous datasets. Line charts are not appropriate for categorical data. Be careful of the scaling of the axis so as to not overemphasize or deemphasize the trend in either the vertical or horizontal directions. Keep the chart to four lines or fewer for the most effective presentation of the material, and use thicker, solid lines rather than thin lines or dotted/dashed lines. Be sure to include a legend if you are plotting more than one series of data (Figure 11.89). Also consider including data markers on the lines to make interpretation clearer. The 3D line chart is difficult to interpret even with a single line; the 2D line chart presents data more clearly. You can examine the changes in sales for Home Stereo Systems at WorldCorp for the month of March with a line chart. Adding gridlines or data labels can help your audience quickly see values at certain points in time.
Earlier, you learned the importance of accurately representing data in a chart to prevent misinterpretation of the information. When you are presenting charts and graphs as opposed to the actual data, special care should be taken to make sure the data representation is valid. By adjusting the scaling on either axis, you can drastically change the interpretation of the information presented in a chart. Take a look at this article on how mistakes have been made in representing data to learn more. Now select one chart represented in the preceding link and re-create the chart so that it is not misleading, and then answer the following questions:
- What was the major issue(s) with the selected chart?
- What are some of the incorrect conclusions drawn from the chart?
- What adjustments did you make? How does it more accurately reflect the information?
Scatter Chart
A scatter chart or scatter plot shows the relationship between two variables. It is often used as part of a regression analysis, and it easily shows the direction of the slope of the line (Figure 11.90). A positive slope indicates a positive correlation between the variables, and a negative slope indicates a negative correlation. A positive slope is when the two variables move in the same direction. The chart of data will have a line that moves from the lower-left to the upper-right corner. A negative slope means the variables move in opposite directions. The chart of a negative slope will start in the upper-left corner and move down toward the lower-right corner of the chart. If the chart is relatively flat, meaning a near 0 slope, then there is little relationship between the two variables. The scatter plot for FOB $ versus Quantity Sold show a slight positive (upward) trend in the data.
A scatter chart can show if a relationship exists. It does not provide a quantitative measure of the relationship between the two variables; it simply shows the estimated relationship. In a scatter plot, the data points are not connected. Instead, it uses the data points only and then, if desired, you can add in a best fit regression line to examine the information more closely (Figure 11.91). If the data points are clustered around the best fit regression line, the relationship is strong. If the data points are scattered and show little to no discernible pattern around the fitted regression line, the relationship is weak. When you add the trendline to the previous scatter plot, you can see that the data points are not clustered around that trendline. Instead, they are quite spread out. This is a visual indication that the variables (FOB $ and Quantity Sold) are not strongly connected to each other.
Histogram
A histogram represents a single variable at various levels/ranges for that variable. With a histogram, a dataset is condensed into categories for a quick analysis of the distribution of the data (Figure 11.92). It is the visual representation of a histogram table. Histograms can represent discrete or continuous data, such as the distribution of customers. They are used frequently for a quick assessment of a dataset and to visually convey large datasets in an easy-to-understand format. They readily show outliers or gaps in information. The process of creating histograms was covered in more detail in Histograms.
Stock Chart
Stock charts are used to represent market data to show the highs and lows for stocks. They can also be used in quality control to show the range of data on a certain day or shift for a production process. To create a stock chart , you will need a date or time reference and the high values and low values for that date or time. The horizontal axis will be the span of time and the vertical axis will be dollars, in the case of a stock market chart. It could be another measure if you are using the stock chart for quality control purposes (Figure 11.93).
Chart Template
If you are creating the same type of chart on a regular basis, creating a chart template will save time. You can save features, themes, colors, and formats as a template that you can use with other data files. To begin, create the chart as you would normally create it. Add the desired customization to the chart. This could include formatting the chart with company colors or including the company logo as a watermark in the chart. When you are finished, right-click on the chart you just created and formatted (Figure 11.94). Choose Save As Template. You will then be prompted to enter a name for the chart template you just created (Figure 11.95). Consider a name for the template that will help identify the template’s use. Make sure the type is listed as Chart Template Files. Input a name and click Save.
To access the saved template, go to the Insert tab and choose Recommended Charts (Figure 11.96). Then, select All Charts and Templates. The saved template should be listed, and you can select that template to apply to the new dataset. Click on the template, then click OK, and Excel creates the chart based on your new dataset.
Quick Analysis Tool
The Quick Analysis tool provides a shortcut to analyzing a dataset with a group of options located in one place for ease of use. You can use these options to add formatting, to automatically insert totals and subtotals, or to add sparklines to your Data Table. There are three ways to access the Quick Analysis tool. If you highlight the Data Table you are interested in analyzing, the Quick Analysis tool icon will appear in the lower right of the selection (Figure 11.97). You can also use Ctrl+Q or right-click on a cell inside the dataset and select Quick Analysis from the menu that appears.
When you access the tool, a menu of options appears (Figure 11.98). Choose the appropriate tab and click on the option you want to use.
Formatting
The Formatting options in the Quick Analysis tool will insert conditional formatting based on the criteria you select. You can also access these options through the Home tab in the Styles command group. Select the drop-down menu next to the Conditional Formatting button. Conditional Formatting provided more detail on using conditional formatting for data. For formatting graphs and charts, there are a few options to choose from under the Conditional Formatting menu , and you can stack the formatting options upon each other. For example, you can choose to have both the Data Bars and Icon Set displayed in your table at the same time. You can add data bars into the selected cells, and they will appear on top of the values and will be appropriately sized for the data in that cell (Figure 11.99). You can format the bars through the Conditional Formatting menu on the Home tab. To remove the data bars, click on Clear Format on the far right of the Format tab in the Quick Analysis tools. You can also clear the formats by choosing the Clear Rules option in the Conditional Format menu on the Home tab.
The Color Scale option adds colors to the cells based on the values relative to each other. There are several different color scales you can choose. For example, one color scale uses blues to represent the higher values and reds to represent the lower values in the dataset (Figure 11.100). Again, to remove the colors, click on Clear Formats in the Quick Analysis tools or Clear Rules under Conditional Formatting.
Icon sets are a more visual way to see differences between the values in a dataset (Figure 11.101). The default icon is an arrow that is either red, yellow, or green based on the value in the cell. The icon can be changed by accessing the Conditional Formatting menu on the Home tab.
The Greater Than option in the Quick Analysis tools will highlight data that is greater than the value you set (Figure 11.102). You set this value in the input window, choose the color scheme for the highlighted cells that meet the criteria, and then click OK. Using the default color scheme, the cells that are greater than the value you set will be highlighted with a light red fill and a dark red text.
The final option in the Formatting tab of the Quick Analysis tools is the Top 10% tool. This tool highlights the values that are in the top 10 percent in the dataset. The default is to highlight those with light red fill and dark red text. This can be changed as desired (Figure 11.103).
Charts
The Quick Analysis tools give you the option to quickly insert a chart of the data (Figure 11.104) using the Charts tab. The chart will be inserted directly on the worksheet using the default color schemes. After you have created the chart, you can choose to customize it to meet your needs using the skills you have already learned. You can access charts not listed specifically by clicking on More Charts.
Totals
The Totals tab in the Quick Analysis tool gives you several options for summarizing your dataset. You can access the same options through the Formulas tab, but the process is much simpler through the Quick Analysis tool (Figure 11.105a). You can include the total of data (sum), average, count, percent of total, and running total. These options are available for both columns and rows. To access the additional formulas, click on the small arrow to the right of the window. With any of the options, the result of the formula will be included in your Data Table as another row or column (Figure 11.105b).
Sparklines
One way to show trends in your data is to use a sparkline , a small chart inserted into a cell next to your data that shows changes over time. There are three options in the Sparklines tab: Line, Column, and Win/Loss (Figure 11.106). These can also be accessed through the Insert menu in the Sparklines command group.
By default, sparklines are inserted at the end of each row. Once the sparkline is inserted, you have access to the Sparkline formatting tab. Here, you can format the sparkline’s style, change the input data, or change the type of sparkline. You can also add markers to the sparkline to show the highest and lowest points. In Figure 11.107, the Line sparkline was inserted. With just a quick glance, you can see that Antonio had the most sales in week two but did not do as well in week three, whereas Izabelle and James improved consistently.
Link to Learning
Marimekko charts are a way to visually examine market share. They are used often in consulting and finance companies. You can also include different segments within an industry. Read this article to find out more about this special type of chart and how to create it in Excel.


Use sketching techniques to produce orthographic sketches of the following figures. For the purposes of these exercises, do not be concerned with dimensions. Concentrate on producing good, dark outlines, good circular shapes, and correctly drawn hidden and center lines.
Sketch one orthographic view of the object shown pictorially in Figure \(\PageIndex{1}\). Remember that the holes in the gasket are circular (not elliptical as they appear in the three-dimensional sketch shown). Add a title block with the following:
Title: Gasket
Sk. No.: D-1/031
Sk. by: (your name)
Date: (today’s date)
Sketch two orthographic views (front view and top view), in the approximate scale of the object shown pictorially in Figure \(\PageIndex{2}\). Remember to place the views correctly and make sure that all hidden lines are clearly shown. Add a title block with the following:


Learning Objectives
After reading this chapter you should be able to:
-
Define verbal communication and explain its main characteristics.
-
Understand the three qualities of symbols.
-
Describe the rules governing verbal communication.
-
Explain the differences between written and spoken communication.
-
Describe the functions of verbal communication.
-
-
3.1: Nonverbal Communication Overview
-
Like verbal communication, nonverbal communication is essential in our everyday interactions. Remember that verbal and nonverbal communication are the two primary channels we study in the field of Communication. While nonverbal and verbal communications have many similar functions, nonverbal communication has its own set of functions for helping us communicate with each other.
-
-
3.2: Defining Nonverbal Communication
-
Like verbal communication, we use nonverbal communication to share meaning with others. Just as there are many definitions for verbal communication, there are also many ways to define nonverbal communication, let’s look at a few.
-
-
3.3: Differences Between Verbal and Nonverbal Communication
-
There are four fundamental differences between verbal and nonverbal communication.
-
-
3.4: Types of Nonverbal Communication
-
There are many types of nonverbal communication including kinesics, haptics, appearance, objects, artifacts, proxemics, our environment, chronemics, paralanguage, and silence. These types of nonverbal communication help us share meanings in our interactions.
-
-
3.5: Functions of Nonverbal Communication
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You learned that we use verbal communication to express ideas, emotions, experiences, thoughts, objects, and people. But what functions does nonverbal communication serve as we communicate (Blumer)? Even though it’s not through words, nonverbal communication serves many functions to help us communicate meanings with one another more effectively.
-
-
3.6: Nonverbal Communication Summary
-
-
-
3.7: Nonverbal Communication References
-
Thumbnail :pixabay.com/photos/sea-shaka...sture-1886481/


15.2: Why It Matters- Managing Process
Why explain how operations management contributes to organizational success?
Operations management is just what it sounds like: managing the operations of the business. The role of an operations manager is broad and encompasses multiple operational areas. While other employees can focus on a specialized area of operation (for example, finance and marketing), an operations manager wears many hats and does a bit of everything. While the term may be unfamiliar, you have probably already seen operations management in action—it even played a role in creating the beautiful turkey dinner in the photo below.
What does it take to make Thanksgiving dinner happen? Planning, scheduling, technology, logistics, supply chain management, quality assurance—all the aspects of operations management. As an overview of this module, let’s take Turkey Day as an example:
- Planning . Turkey dinner for fifteen people doesn’t just happen. It takes careful planning and possibly delegating tasks and duties to others. Who sets the table? Who brings the green bean casserole? What time should everyone arrive?
- Scheduling . A turkey can take up to six hours to cook, and if you have only one oven, you’ll need to schedule what time the bird goes in and comes out so the rolls and sweet potatoes get a turn in the oven.
- Technology . You’ll need an oven (or maybe a high-tech turkey fryer) and any number of cooking gadgets. Even the humble thermometer counts as technology. Unless you’re preparing a raw, paleo Thanksgiving dinner, technology will be essential.
- Logistics . Fifteen people won’t all fit around your current table. What should you do? Seat the children at a card table? Rent a larger table?  Logistics, logistics.
- Supply Chain Management. Aunt Sue is bringing pies, Bob is responsible for rolls, and Margaret is bringing the green beans. The host has to secure a fresh turkey before they sell out at the grocery store. If you live in the South, you’ll want to call the local fisherman and reserve some oysters for the oyster dressing. All are important components of the supply chain—leave one out, and you’ll miss a dish.
- Quality Assurance . Anyone who cooks knows you must taste, season, and taste again to ensure the food is up to snuff. Quality assurance might also include asking Jean to bring drinks and flowers because she’s a terrible cook.
Any undertaking that involves coordinating effort, tasks, and resources can be considered operations management. In this module, you’ll learn how operations management works in both manufacturing and service industries—in short, you’ll see how others get their turkey on the table.


What you’ll learn to do: examine common learning disabilities and other factors related to education during middle childhood
Across the world, by the time a child is entering middle childhood, they are being educated in some form or fashion.
That said, what children learn within that formal education program varies greatly across cultures. Further, most programs are set-up for typically developing children, but they may not be set-up to handle children who are accelerated learners or children with learning disabilities. In this section, we’ll take a look at some of these educational differences and developments, as well as struggles and learning difficulties during middle childhood.
Contributors and Attributions
CC licensed content, Original
-
Introduction to Educational Issues during Middle Childhood.
Authored by
: Stephanie Loalada for Lumen Learning.
Provided by
: Lumen Learning.
License
:
CC BY: Attribution
CC licensed content, Shared previously


2.19: Gramática- El verbo gustar + sustantivos
-
- Last updated
- Save as PDF
- Erica Brown, Alejandra Escudero, María Cristina Montoya, & Elizabeth Small
- SUNY Oneonta via OER SUNY
Objetivos
- Recognize the correct pronouns and correct forms of gustar to say what things you and others like and don’t like
In Unit 4 we saw how to say that I or you like to do something, using me/te gusta + infinitive. Of course, it’s possible to say that you like things, too, just a little more complicated.
The Spanish equivalent of “I like” is me gusta , which literally means “it pleases me”. To indicate whether someone else likes something, you change the indirect object pronoun so the thing is pleasing to that person:
| forms for different people liking a singular thing | ||
| persona singular | persona plural | |
| primera (1a) |
me gusta ( I like ) |
nos gusta ( We like ) |
| segunda (2a) | te gusta ( You like ) | os gusta ( You all like ) |
| tercera (3a) | le gusta ( He/She likes ) | les gusta ( They like ) |
When you use the verb gustar, the verb form you choose will depend on whether what you like is a singular noun, a plural noun, or a verb.
The verb form gusta is always in the singular when the noun that is liked or disliked is singular, because *it* is pleasing to the person:
- (A mí) me gusta la casa. ( I like the house. Literally: The house pleases me. )
- (A ti) te gusta la salsa picante. ( You—informal singular—like hot sauce. )
- (A usted) le gusta la comida. ( You—formal singular—like the food. )
- (A él) le gusta el libro. ( He likes the book. )
- (A ellas) les gusta el coche. ( They like the car. )
- (A nosotros) nos gusta el café. ( We like coffee. )
What if plural things are pleasing to you? The verb form gustan is always in the plural when the noun is plural or there are two nouns, because *they* please the person:
- (A mí) me gustan las casas. ( I like the houses. Literally: The houses please me. )
- (A ti) te gustan las manzanas. ( You—informal singular—like the apples. )
- (A usted) le gustan las comidas. ( You—formal singular—like the foods. )
- (A él) le gustan los libros. ( He likes the books. )
- (A ellas) les gustan los coches. ( They like the cars. )
- (A nosotros) nos gustan los animales. ( We like animals. )
As we saw in Unit 4, the verb form gusta is always in the singular when it is followed by a verb, even if there are many verbs listed, because *it* pleases the person *to do* the actions:
- (A mí) me gusta cantar mucho. ( I like to sing a lot. Literally: It pleases me a lot to sing. )
- (A tí) te gusta recibir y escribir cartas. ( You–informal singular– like to receive and write letters. )
- (A usted) le gusta escuchar la música. ( You–formal singular– like to listen to music. )
- (A él) le gusta comer y bailar. ( He likes to eat and dance. )
- (A ellas) les gusta aprender el español. ( They like to learn Spanish. )
- (A nosotros) nos gusta correr. ( We like to run. )
Note: In all of the above examples, the sentences are written with a prepositional phrase ( a mí, a ti, etc.) at the beginning of the sentence. This is optional and most Spanish-speakers will omit this phrase, but they may use it for emphasis in much the way we might emphasize in English by stressing a word or putting it in bold in writing: “ You don’t like beets? Well, I do!” Another use is for clarification in the third person. Since Le gusta comer y bailar could have several translations (“He likes to eat and dance” or “She likes to eat and dance” or “You <formal> like to eat and dance” or “<Any named person> likes to eat and dance”), the clarifying phrase is used more often with le or les .
Some other words that work the same way as gustar:
- Doler ( to hurt; literally: to be painful ) — Me duelen los pies. ( My feet hurt / are hurting me. )
- Encantar ( to love; literally: to be enchanting ) — A los mexicanos les encantan los dramas coreanos. ( Mexicans love Korean dramas. )
- Molestar ( to mind; literally: to be irritating, bothersome ) — ¿Le molesta la música? No, estoy bien. ( Do you mind the music? No, I’m fine. )
Contributors and Attributions
- Gramu00e1tica: El verbo gustar + sustantivos. Authored by : SUNY Oneonta with Lumen Learning. Provided by : SUNY Oneonta. License : CC BY: Attribution


9.7B: Chewing Muscles
Mastication, or chewing, involves the adduction and lateral motions of the jaw bone. It is controlled by four muscles of the face.
- Differentiate between the actions of the masseter and the temporalis muscles in chewing
Key Points
- The masseter elevates the jaw, closing the mouth.
- The temporalis elevates and retracts the jaw.
- The lateral pterygoid is the only muscle of mastication that actively opens the jaw. Unilateral action of a lateral pterygoid produces lateral movement in the jaw, usually performed in concert with the medial pterygoids.
- The medial pterygoid elevates and closes the jaw, contributes to protrusion of the mandible, and assists in mastication.
Key Terms
- medial pterygoid : A muscle of mastication with two heads. It lies inferiorly to the medial pterygoid.
- temporalis : A broad muscle that fans out to cover much of the temporal bone on the side of the skull.
- lateral pterygoid : A muscle of mastication with two heads. It lies superiorly to the medial pterygoid.
- masseter : The large muscle which raises the lower jaw, and assists in mastication.
Mastication, or the act of chewing, involves adduction and lateral motion of the jaw bone. It is controlled by four bilateral muscles in the face. The lower jaw, or mandible, connects to the temporal bone of the skull via the temporomandibular joint, which allows movement in all planes.
Location of the masseter muscle : The masseter muscle spans the opening of the mouth.
-
Masseter
: The most powerful muscle of mastication. It is quadrangular in shape and split into two regions, deep and superficial. It covers the other muscles of mastication.
- Attachments: The superficial region originates from the skull below the eye socket, while the deep part originates from the skull above the jaw. Both parts attach to the jaw.
- Actions: Elevates the jaw.
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Temporalis
: A broad muscle that fans out to cover much of the temporal bone on the side of the skull.
- Attachments: The temporalis muscle has a wide, fan-shaped origin on the side of the skull and condenses into a tendon which attaches to the jaw.
- Actions: Elevates and retracts the jaw.
-
Lateral Pterygoid
: The lateral pterygoid muscle has a triangular shape with two head, superior and inferior. It is the major protractor (opener) of the jaw.
- Attachments: Both heads originate adjacent to the eye socket and converge into a tendon which attaches to the jaw.
- Actions: Together, the lateral pterygoids protract the jaw, working independently to produce lateral movement.
-
Medial Pterygoid
: The medial pterygoid muscle has a quadrangular shape with two heads, deep and superficial. It is located inferior to the lateral pterygoid.
- Attachments: The superficial head originates from the front of the skull and the deep head attaches to the skull adjacent to the eye socket. Both heads attach to the jaw.
- Actions: Elevates the jaw and assists in the production of lateral movement.
Location of the temporalis muscle and the lateral pterygoid: (a) Highlighted in orange, the temporalis muscle is a broad muscle extending from zygomatic bone. (b) Arrows indicate the location of the lateral pterygoid, highlighted with the medial pterygoid in orange.
KEY MOVEMENTS
- Elevation of the Jaw: Produced by the masseter, temporalis and medial pterygoid.
- Depression of the Jaw: Produced by the lateral pterygoid, assisted by the digastric, mylohyoid and geniohyoid muscles found in the neck.
- Protraction of the Jaw: Produced by the lateral pterygoid.
- Retraction of the Jaw: Produced by the temporalis.
- Lateral Movement of the Jaw: Produced by the lateral and medial pterygoid.