A dialectical approach to taking notes sounds much more complicated than it is.
Dialectic just means a dialogue
—a discussion between two (or more) voices trying to figure something out. Whenever you read new material, particularly material that is challenging in some way, it can be helpful to take dialectic notes to create clear spaces for organizing different sets of thoughts.
Start by drawing a
vertical line
down the middle of a fresh sheet of paper to make two long columns—leave some space at the bottom of the page.
Table 2.5
provides an overview of what you should put in the left and right column. It’s a good idea to leave space at the bottom of the page (or on the back) for additional notes about this piece or cite the source.
Table 2.5:
Dialectic note-taking
|
Left column – main ideas
|
Right column – your response
|
|
This column is a straightforward representation of the main ideas in the text you are reading. For example:
-
Also, take a look at
Table 2.6
for an example of dialectic notetaking, which demonstrates how to document the main ideas of a text and your comments.
Table 2.6:
Example of dialectic note-taking
|
Main ideas
|
My comments
|
-
main blood pressure (BP) methods: manual and automatic
-
both arms (measurements should be within 10 mm Hg)
-
sitting position with feet flat on floor, “bare arm at heart level” and resting for 5 minutes before measurement
-
accurate cuff size based on the person’s arm: “width of cuff 40% of the person’s arm circumference … length of cuff’s bladder is 80-100% of the person’s arm circumference” (see video)
-
palpate brachial artery firmly (2 cm medially from bicep tendon and 2-3 cm above antecubital fossa)
-
place cuff over bare arm with artery marker aligned with the artery
|
-
will need to figure out where to place the client’s arm (e.g., on a table)
-
how much pressure does “firm pressure” involve?
-
I have had my BP taken and the healthcare provider took it over my clothing. How come?
-
talk with teacher about how to pronounce “sphygmomanometer”?
-
>need to re-watch video when I get a chance
-
does it hurt to have your BP taken?
|
Source:
Lapum, J., Verkuyl, M., Garcia, W., St-Amant, O., & Tan, A. (2018). Vital sign measurement across the lifespan – 1
st
Canadian edition. Retrieved from:
https://pressbooks.library.ryerson.ca/vitalsign/
Once you have this set of dialectic notes, there are several ways you can use them. For example:
-
These notes can help you contribute to class discussion about this piece and the topics it addresses.
-
Significant questions you encountered while reading are already written down and collected in one place so you don’t have to sift through the reading again to find them.
-
Your observations and thoughts about the piece are already organized, which can help you see patterns and connections within those observations. Finding these connections can be a strong starting point for written assignments.
-
If you are asked to respond to this piece in writing, these notes can serve as a reference point as you develop a draft. They can give you new ideas if you get stuck and help keep the original connections you saw when reading fresh in your mind as you respond more formally to that reading.
Activities: Check Your Understanding
The original version of this chapter contained H5P content. You may want to remove or replace this element.
The original version of this chapter contained H5P content. You may want to remove or replace this element.
|


2.2: The Influence of Algorithms
“We live in an era of ambient information. Amidst the daily flood of digital news, memes, opinion, advertising, and propaganda, there is rising concern about how popular platforms, and the algorithms they increasingly employ, may influence our lives, deepen divisions in society, and foment polarization, extremism, and distrust” (Head et al. 1).
Positive & Negative
As companies, governments, and other organizations continue to collect and analyze massive amounts of our data, the use of algorithms has become pervasive. In fact, many are referring to this period as the “Age of Algorithms” or the “Algorithm Era,” and researchers are considering the significant impacts that these tools may have—both positive and negative.
There is no doubt that algorithms can be useful and help to improve our lives. For example, it certainly saves a lot of time and frustration to be able to pull up a map on our phones and instantly determine the fastest way to reach our destination. However, as technology and social media scholar danah boyd has noted, “the same technology can be used to empower people…or harm them. It all depends on who is using the information to what ends” (qtd. in Rainie and Anderson). Eni Mustafaraj, Assistant Professor of Computer Science at Wellesley College, similarly notes that “if we want more people in the world to have access to the total human knowledge accessible on the Internet, we need algorithms. However, what we need to object against are the values driving the companies that own these algorithms” (qtd. in Head et al. 42).
Concerns
What happens when algorithms are used to predict when college students are “cheating” on a test, or to predict who should be hired for a job or who should get a loan, or to decide the type of information we see in our social media newsfeeds, or to calculate credit scores, or even to predict criminal behavior and determine prison sentences? Are Google search results really an unbiased presentation of the best available information on a research question? How do algorithms impact our perception of a research topic, or of our own realities?
One of the most prominent examples in recent history is the Facebook-Cambridge Analytica data scandal, in which the personal profile data of at least 87 million Facebook users was harvested by a political consulting firm working for the Trump campaign. In this case, Cambridge Analytica used algorithms to develop psychographic profiles in order to more effectively direct its messaging and influence voter behavior in the 2016 U.S. Presidential election.
Zoom in or read the text version of the infographic
Sources
Head, Alison J., Barbara Fister, and Margy MacMillan. “ Information Literacy in the Age of Algorithms .” Project Information Literacy , 15 Jan. 2020. Licensed under CC BY-NC-SA 4.0
Image: “Big Data is Watching You” by ev is in the Public Domain, CC0
Image: “Facebook Concept Binary” by geralt on Pixabay
Infographic: “ Life in the Age of Algorithms ” by Jessica Yurkofsky for Project Information Literacy is licensed under CC BY-NC-SA 4.0
Rainie, Lee, and Janna Anderson. “ Code Dependent: Pros and Cons of the Algorithm Age .” Pew Research Center , 8 Feb. 2017.


9.6.6: Chapter Test
A student consumed 10 cans of their favorite soda last week. If each can contained 39 g of sugar, how many milligrams of sugar did they consume?
An athlete ran a total distance of 154 km in one month while training for a 5K fun run. How many meters did they run?
A doctor recommends their patient drinks 5 liters of water on days of vigorous exercise. If the patient drinks 3,400 mL, how much more do they need to drink?
A garden walkway has the dimensions shown. How many square meters of brick are needed to cover the walkway?
A studio apartment has a length of 1,200 cm and a width that one half of the length. What is the area of the apartment in meters?
A living room is 7 m long and 8 m wide. Carpet costs $9.50 per square meter. How much will it cost to carpet the room?
Arif mixed two cartons of orange juice that each contain 600 cL, two bottles of cranberry juice that each contain 1,250 mL, and one 2-liter bottle of soda water to make punch for a party. How much punch, in liters, did Arif make?
A fish tank has the dimensions \(40\,{\text{cm}}\, \times \,20\,{\text{cm}}\, \times \,25\,{\text{cm}}\). What is the capacity of the fish tank in liters?
A recipe for chili makes 5.4 L of chili. If a restaurant serves chili in 300 mL bowls, how many bowls of chili can they serve?
If a person weighs 76 kg but loses 3 kilograms, how much does the person weigh in grams?
A car weighs approximately 2 metric tons (mt). If a used car dealer has 222,000 kg of cars in their lot, how many cars are in the lot?
A restaurant has 126 kg of rice. If each serving contains 125 g, how many servings will they make?
A boiling pan of water is 212 °F. As the water cools the temperature drops 12 °C every 3 minutes. How many minutes until the temperature of the water reaches 28 °C?
On May 1 the temperature was 75 °F. On June 1 the temperature was 28 °C. Which day was warmer?
For each log placed on a fire, the temperature increases 45 °C. How many logs are needed for the campfire to increase 315 °C?


4.7E: Fitting Exponential Models to Data (Exercises)
section 4.7 exercise
Graph each function on a semi-log scale, then find a formula for the linearized function in the form \(\log \left(f\left(x\right)\right)=mx+b\).
1. \(f\left(x\right)=4\left(1.3\right)^{x}\)
2. \(f\left(x\right)=2\left(1.5\right)^{x}\)
3. \(f\left(x\right)=10\left(0.2\right)^{x}\)
4. \(f\left(x\right)=30\left(0.7\right)^{x}\)
The graph below is on a semi-log scale, as indicated. Find a formula for the exponential function \(y(x)\).
5. 6.
7. 8.
Use regression to find an exponential function that best fits the data given.
13. Total expenditures (in billions of dollars) in the US for nursing home care are shown below. Use regression to find an exponential function that models the data. What does the model predict expenditures will be in 2015?
| Year | 1990 | 1995 | 2000 | 2003 | 2005 | 2008 |
| Expenditure | 53 | 74 | 95 | 110 | 121 | 138 |
14. Light intensity as it passes through water decreases exponentially with depth. The data below shows the light intensity (in lumens) at various depths. Use regression to find an function that models the data. What does the model predict the intensity will be at 25 feet?
| Depth (ft) | 3 | 6 | 9 | 12 | 15 | 18 |
| Lumen | 11.5 | 8.6 | 6.7 | 5.2 | 3.8 | 2.9 |
15. The average price of electricity (in cents per kilowatt hour) from 1990 through 2008 is given below. Determine if a linear or exponential model better fits the data, and use the better model to predict the price of electricity in 2014.
| Year | 1986 | 1988 | 1990 | 1995 | 1997 | 2000 | 2002 | 2004 | 2006 | 2008 |
| Cost | 7.83 | 8.21 | 8.38 | 8.36 | 8.26 | 8.24 | 8.44 | 8.95 | 10.40 | 11.26 |
16. The average cost of a loaf of white bread from 1986 through 2008 is given below. Determine if a linear or exponential model better fits the data, and use the better model to predict the price of a loaf of bread in 2016.
| Year | 1986 | 1988 | 1990 | 1995 | 1997 | 2000 | 2002 | 2004 | 2006 | 2008 |
| Cost | 0.57 | 0.66 | 0.70 | 0.84 | 0.88 | 0.99 | 1.03 | 0.97 | 1.14 | 1.42 |
- Answer
-
1. \(\text{log} (f(x)) = \text{log} (1.3)x + \text{log} (4)\)
3. \(\text{log} (f(x)) = \text{log} (0.2) x + 1\)
5. \(y = e^{\dfrac{1}{2}x - 1} = e^{-1} e^{\dfrac{1}{2}x} \approx 0.368 (1.6487)^x\)
7. \(y = 10^{-x - 2}. = 10^{-2} 10^{-1x} = 0.01 (0.1)^x\)
9. \(y = 776.682 (1.426)^x\)
11. \(y = 731.92(0.738)^x\)
13. Expenditures are approximately $205
15. \(y = 7.599(1.016)^x\) \(r = 0.83064\), \(y = 0,1493x + 7.4893\), \(r = 0.81713\). Using the better function, we predict electricity will be 11.157 cents per kwh


2.7: Personal Pronouns with Verb "To be"
Verb “To be”
Instructor Video: Verb “to be”
| Singular pronouns and verb “to be.” | Plural pronouns and verb “to be.” | ||||
| Hindi | Transliteration | English | Hindi | Transliteration | English |
| मैं __हूँ | main __hoon | I am __. | हम __हैं | hum __hain | We are __. |
| तू __है | tu __hai | You are __. | |||
| तुम __हो | tum __ho | You are __. | तुम __हो | tum __ho | You are __. |
| आप __हैं | aap __hain | You are __. | आप __हैं | aap __hain | You are __. |
| यह/वह __है | ye/wo __hai | S/he, This/That is __. | ये/वे __हैं | ye/we __hain | They, These/Those are __. |
Table 3. List of Personal pronouns with the forms of verb होना “to be.”
Summary
| Verb “to be” | Pronouns | Examples |
| हूँ hoon | मैं | मैं अध्यापक हूँ | main adhyaapak hoon . I am a teacher. |
| हो ho | तुम | तुम छात्र हो | tum chhaatr ho . You are a student |
| है hai | तू, यह, वह (Singular nouns) |
तू लड़का है |
tu laRakaa hai
.
यह किताब है | ye kitaab hai . This is a book. वह मेज़ है | wo mez hai . That is a table. पीटर दोस्त है | Peter dost hai . Peter is a friend. |
| हैं hain | हम, आप, ये, वे (Plural nouns/honorific) |
हम छात्र हैं |
hum chhaatr hain
. We are students.
आप अच्छे हैं | aap achchhe hain . You are good. ये/वे किसान हैं | ye/we kisaan hain . They are farmers. ये सब्ज़ियाँ हैं | ye sabziyaan hain . These are vegetables. वे फल हैं | we phal hain . Those are fruits. फ़िलिप जी अध्यापक हैं | Philip ji adhyaapak hain . Philip ji is my teacher, |
Table 4. List of verb होना “to be,” with personal pronouns.
Positive Sentence and the Verb होना “to be”
Unlike English, the standard word order of Hindi is: Subject-Object-Verb (SOV).
Structure 1:
| English | Subject | Verb | Object |
| This | is | a pen | |
| Hindi | Subject | Object | Verb |
| This | a pen | is | |
| यह | एक कलम | है | |
Negative Sentence in Hindi and the Verb होना “to be”
Structure 2:
| English | Subject | Ver b | Negative | Object |
| This | is | not | a pen | |
| Hindi | Subject | Object | Negative | Verb |
| This | a pen | not | is | |
| यह | कलम | नहीं | है | |
Yes/No question and the Verb होना “to be”
In English, we move auxiliary verb at the beginning of a sentence to form a yes/no question. Whereas, in Hindi, the word क्या kyaa is used at the beginning of a sentence to form a yes/no question. Please note that here the word क्या kyaa does not mean “what.
Structure 3:
| English | Auxiliary verb | Subject | Object | |
| Is | this | a pen | ||
| Hindi | क्या kyaa | Subject | Object | Verb |
| Kyaa | this | a pen | is | |
| क्या | यह | कलम | है ? |
Yes/No question + Negative and the Verb होना “to be”
Structure 4:
| English | Auxiliary verb | (negative) | Subject | (Negative) | Object |
| Is | (not) | this | (not) | a pen | |
| Isn’t this a pen? / Is this not a pen? | |||||
| Hindi | क्या kyaa | Subject | Object | Negative | Auxiliary Verb |
| kyaa | this | a pen | not | is | |
| क्या | यह | कलम | नहीं | है ? |
Activities
Now as you know how to change a simple Hindi sentence into Negative, yes/no question, and yes/no question + negative, please fill the following table as instructed.
| Positive | Negative | Yes/No Question | Yes/No Question + Negative |
| मैं अध्यापक हूँ | | |||
| हम दोस्त नहीं हैं | | |||
| क्या वे छात्र हैं? | |||
| क्या आप हिन्दुस्तानी हैं ? | |||
| तू भाई है | | |||
| तुम छात्रा नहीं हो | | |||
| क्या यह आलू है ? | |||
| क्या वह टमाटर नहीं है ? |
Please fill in the blanks with suitable object to form five sentences with each pronoun given below. You can use the vocabulary list in this chapter or dictionary for any help that you need to complete this activity.
| Subject | Object | Verb “to be.” | English |
| मैं | अध्यापक | हूँ | | I am a teacher. |
| मैं | |||
| मैं | |||
| मैं | |||
| मैं | |||
| हम | दोस्त | हैं | | We are friends. |
| हम | |||
| हम | |||
| हम | |||
| हम | |||
| तू | होशियार | है | | You are smart. |
| तू | |||
| तू | |||
| तू | |||
| तू | |||
| तुम | लड़के | हो | | You are a boy. |
| तुम | |||
| तुम | |||
| तुम | |||
| तुम | |||
| आप | अच्छी | हैं | | You (female) are good. |
| आप | |||
| आप | |||
| आप | |||
| आप | |||
| यह | किताब | है | | This is a book. |
| यह | |||
| यह | |||
| यह | |||
| यह | |||
| वह | आपका तौलिया | है | | That is your towel. |
| वह | |||
| वह | |||
| वह | |||
| वह | |||
| ये | सब्जियां | हैं | | These are vegetables. |
| ये | |||
| ये | |||
| ये | |||
| ये | |||
| ये | |||
| वे | क्या | हैं ? | What are those? |
| वे | |||
| वे | |||
| वे | |||
| वे |
Test Yourself
https://openbooks.lib.msu.edu/ll151/?p=770
धन्यवाद !


13.1: Grounded Theory
-
- Last updated
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How can you analyse a vast set of qualitative data acquired through participant observation, in-depth interviews, focus groups, narratives of audio/video recordings, or secondary documents? One of these techniques for analysing text data is grounded theory —an inductive technique of interpreting recorded data about a social phenomenon to build theories about that phenomenon. The technique was developed by Glaser and Strauss (1967) in their method of constant comparative analysis of grounded theory research, and further refined by Strauss and Corbin (1990) to further illustrate specific coding techniques—a process of classifying and categorising text data segments into a set of codes (concepts), categories (constructs), and relationships. The interpretations are ‘grounded in’ (or based on) observed empirical data, hence the name. To ensure that the theory is based solely on observed evidence, the grounded theory approach requires that researchers suspend any pre-existing theoretical expectations or biases before data analysis, and let the data dictate the formulation of the theory.
Strauss and Corbin (1998) describe three coding techniques for analysing text data: open, axial, and selective. Open coding is a process aimed at identifying concepts or key ideas that are hidden within textual data, which are potentially related to the phenomenon of interest. The researcher examines the raw textual data line by line to identify discrete events, incidents, ideas, actions, perceptions, and interactions of relevance that are coded as concepts (hence called in vivo codes ). Each concept is linked to specific portions of the text (coding unit) for later validation. Some concepts may be simple, clear, and unambiguous, while others may be complex, ambiguous, and viewed differently by different participants. The coding unit may vary with the concepts being extracted. Simple concepts such as ‘organisational size’ may include just a few words of text, while complex ones such as ‘organizational mission’ may span several pages. Concepts can be named using the researcher’s own naming convention, or standardised labels taken from the research literature. Once a basic set of concepts are identified, these concepts can then be used to code the remainder of the data, while simultaneously looking for new concepts and refining old concepts. While coding, it is important to identify the recognisable characteristics of each concept, such as its size, colour, or level—e.g., high or low—so that similar concepts can be grouped together later . This coding technique is called ‘open’ because the researcher is open to and actively seeking new concepts relevant to the phenomenon of interest.
Next, similar concepts are grouped into higher order categories . While concepts may be context-specific, categories tend to be broad and generalisable, and ultimately evolve into constructs in a grounded theory. Categories are needed to reduce the amount of concepts the researcher must work with and to build a ‘big picture’ of the issues salient to understanding a social phenomenon. Categorisation can be done in phases, by combining concepts into subcategories, and then subcategories into higher order categories. Constructs from the existing literature can be used to name these categories, particularly if the goal of the research is to extend current theories. However, caution must be taken while using existing constructs, as such constructs may bring with them commonly held beliefs and biases. For each category, its characteristics (or properties) and the dimensions of each characteristic should be identified. The dimension represents a value of a characteristic along a continuum. For example, a ‘communication media’ category may have a characteristic called ‘speed’, which can be dimensionalised as fast, medium, or slow . Such categorisation helps differentiate between different kinds of communication media, and enables researchers to identify patterns in the data, such as which communication media is used for which types of tasks.
The second phase of grounded theory is axial coding , where the categories and subcategories are assembled into causal relationships or hypotheses that can tentatively explain the phenomenon of interest. Although distinct from open coding, axial coding can be performed simultaneously with open coding. The relationships between categories may be clearly evident in the data, or may be more subtle and implicit. In the latter instance, researchers may use a coding scheme (often called a ‘coding paradigm’, but different from the paradigms discussed in Chapter 3) to understand which categories represent conditions (the circumstances in which the phenomenon is embedded), actions/interactions (the responses of individuals to events under these conditions), and consequences (the outcomes of actions/interactions). As conditions, actions/interactions, and consequences are identified, theoretical propositions start to emerge, and researchers can start explaining why a phenomenon occurs, under what conditions, and with what consequences.
The third and final phase of grounded theory is selective coding , which involves identifying a central category or a core variable, and systematically and logically relating this central category to other categories. The central category can evolve from existing categories or can be a higher order category that subsumes previously coded categories. New data is selectively sampled to validate the central category, and its relationships to other categories—i.e., the tentative theory. Selective coding limits the range of analysis, and makes it move fast. At the same time, the coder must watch out for other categories that may emerge from the new data that could be related to the phenomenon of interest (open coding), which may lead to further refinement of the initial theory. Hence, open, axial, and selective coding may proceed simultaneously. Coding of new data and theory refinement continues until theoretical saturation is reached—i.e., when additional data does not yield any marginal change in the core categories or the relationships.
The ‘constant comparison’ process implies continuous rearrangement, aggregation, and refinement of categories, relationships, and interpretations based on increasing depth of understanding, and an iterative interplay of four stages of activities: comparing incidents/texts assigned to each category to validate the category), integrating categories and their properties, delimiting the theory by focusing on the core concepts and ignoring less relevant concepts, and writing theory using techniques like memoing, storylining, and diagramming. Having a central category does not necessarily mean that all other categories can be integrated nicely around it. In order to identify key categories that are conditions, action/interactions, and consequences of the core category, Strauss and Corbin (1990) recommend several integration techniques, such as storylining, memoing, or concept mapping, which are discussed here. In storylining , categories and relationships are used to explicate and/or refine a story of the observed phenomenon. Memos are theorised write-ups of ideas about substantive concepts and their theoretically coded relationships as they evolve during ground theory analysis, and are important tools to keep track of and refine ideas that develop during the analysis. Memoing is the process of using these memos to discover patterns and relationships between categories using two-by-two tables, diagrams, or figures, or other illustrative displays. Concept mapping is a graphical representation of concepts and relationships between those concepts—e.g., using boxes and arrows. The major concepts are typically laid out on one or more sheets of paper, blackboards, or using graphical software programs, linked to each other using arrows, and readjusted to best fit the observed data.
After a grounded theory is generated, it must be refined for internal consistency and logic. Researchers must ensure that the central construct has the stated characteristics and dimensions, and if not, the data analysis may be repeated. Researcher must then ensure that the characteristics and dimensions of all categories show variation. For example, if behaviour frequency is one such category, then the data must provide evidence of both frequent performers and infrequent performers of the focal behaviour. Finally, the theory must be validated by comparing it with raw data. If the theory contradicts with observed evidence, the coding process may need to be repeated to reconcile such contradictions or unexplained variations.


9: Sorting Algorithms
Unit Introduction
A sorting algorithm is an algorithm that puts elements of a collection such as array or list in a certain order. The most used orders are numerical order and lexicographical order. Efficient sorting is important to optimizing the use of other algorithms, such as in search and merge algorithms that require sorted collections to work correctly. It is also often useful for normalization of data and for producing human-readable output. More precisely, the output must satisfy two conditions (i) the output is in non-decreasing order (each element is no smaller than the previous element according to the desired total order) and (ii) the output is a permutation, or reordering, of the input.
The Sorting algorithms used in computer science are often classified by:
- Computational complexity (worst, average and best case behavior) in terms of the size of the list (n) - For typical sorting algorithms good behavior is O(n log n) and bad behavior is O(n2). Ideal behavior for a sort is O(n). Sort algorithms which only use an abstract key comparison operation often require at least O(n log n) comparisons on average.
- Memory usage (and use of other computer resources) - In particular, some sorting algorithms are “in place”, such that little memory is needed beyond the items being sorted, while others need to create auxiliary locations for data to be temporarily stored.
- Stability - Stable sorting algorithms maintain the relative order of records with equal keys (i.e. values). That is, a sorting algorithm is stable if whenever there are two records R and S with the same key and with R appearing before S in the original list, R will appear before S in the sorted list.
- Whether or not is a comparison sort - A comparison sort examines the data only by comparing two elements with a comparison operator.
- General sort method – whether uses insertion, exchange, selection, merging, etc.
Table \(\PageIndex{1}\) provides a summary of some sorting algorithms and their classification. When equal elements are indistinguishable, such as with integers, stability is not an issue. However, assume that the following pairs of numbers are to be sorted by their first coordinate: (4, 1) (3, 1) (3, 7) (5, 6). In this case, two different results are possible, that is, one which maintains the relative order of records with equal keys, and one which does not as shown below:
(3, 1) (3, 7) (4, 1) (5, 6) (order maintained)
(3, 7) (3, 1) (4, 1) (5, 6) (order changed)
Unstable sorting algorithms may change the relative order of records with equal keys, but stable sorting algorithms never do so. Unstable sorting algorithms can be specially implemented to be stable. One way of doing this is to artificially extend the key comparison, so that comparisons between two objects with otherwise equal keys are decided using the order of the entries in the original data order as a tie-breaker. Remembering this order, however, often involves an additional space cost.
Table \(\PageIndex{1}\), presents a classification summary of the sorting algorithms considered in this unit. In the table, n is the number of records to be sorted. The columns “Best”, “Average”, and “Worst” give the time complexity in each case. “Memory” denotes the amount of auxiliary storage needed beyond that used by the collection list itself. “Cmp” indicates whether the sort is a comparison sort.
Table \(\PageIndex{1}\): A Sorting Algorithms Classification
|
Name |
Best |
Average |
Worst |
Memory |
Stable |
Cmp |
Method |
|---|---|---|---|---|---|---|---|
|
Selection sort |
O(n2) |
O(n2) |
O(n2) |
O(1) |
No |
Yes |
Selection |
|
Insertion sort |
O(n) |
— |
O(n2) |
O(1) |
Yes |
Yes |
Insertion |
|
Binary tree sort |
O(nlog(n)) |
— |
O(nlog(n)) |
O(1) |
Yes |
Yes |
Insertion |
|
Merge sort |
O(nlog(n)) |
— |
O(nlog(n)) |
O(n) |
Yes |
Yes |
Merging |
|
Heap sort |
O(nlog(n)) |
— |
O(nlog(n)) |
O(1) |
No |
Yes |
Selection |
|
Quick sort |
O(nlog(n)) |
O(nlog(n)) |
O(n2) |
O(log n) |
No |
Yes |
Partitioning |
Unit Objectives
Upon completion of this unit you should be able to:
- Demonstrate understanding of various sorting algorithm
- Select best sorting method based on performance requirement
- Apply sorting algorithms in problem solving
Key Terms
Sorting : A process that puts elements of a collection such as array or list in a certain order


1.2.2: Reading - The Real Process of Science
The process of science , as represented here, is the opposite of “cookbook” (to see the full complexity of the process, roll your mouse over each element). In contrast to the linear steps of the simplified scientific method, this process is non-linear:
Click this link for interactive image: https://undsci.berkeley.edu/understanding-science-101/how-science-works/the-real-process-of-science/
The process of science is iterative.
Science circles back on itself so that useful ideas are built upon and used to learn even more about the natural world . This often means that successive investigations of a topic lead back to the same question, but at deeper and deeper levels. Let’s begin with the basic question of how biological inheritance works. In the mid-1800s, Gregor Mendel showed that inheritance is particulate — that information is passed along in discrete packets that cannot be diluted. In the early 1900s, Walter Sutton and Theodor Boveri (among others) helped show that those particles of inheritance, today known as genes, were located on chromosomes. Experiments by Frederick Griffith, Oswald Avery, and many others soon elaborated on this understanding by showing that it was the DNA in chromosomes which carries genetic information. And then in 1953, James Watson and Francis Crick, again aided by the ideas of many others and using data collected by Rosalind Franklin, provided an even more detailed understanding of inheritance by outlining the molecular structure of DNA. Still later in the 1960s, Marshall Nirenberg, Heinrich Matthaei, and others built upon this work to unravel the molecular code that allows DNA to encode proteins. And it doesn’t stop there. Biologists have continued to deepen and extend our understanding of genes, how they are controlled, how patterns of control themselves are inherited, and how they produce the physical traits that pass from generation to generation.
The process of science is not predetermined.
Any point in the process leads to many possible next steps, and where that next step leads could be a surprise. For example, instead of leading to a conclusion about tectonic movement, testing an idea about plate tectonics could lead to an observation of an unexpected rock layer. And that rock layer could trigger an interest in marine extinctions, which could spark a question about the dinosaur extinction — which might take the investigator off in an entirely new direction.
At first this process might seem overwhelming. And it is, a bit. Even within the scope of a single investigation, science may involve many different people engaged in all sorts of different activities in different orders and at different points in time — science is simply much more dynamic, flexible, unpredictable, and rich than many textbooks represent it as. But don’t panic! The scientific process may be complex, but the details are less important than the big picture…
Source: https://undsci.berkeley.edu/understanding-science-101/how-science-works/the-real-process-of-science/
(Content and media owned or made by the UC Museum of Paleontology (UCMP) for the Understanding Science (US) website are licensed under an Attribution-NonCommercial-ShareAlike 4.0 (BY-NC-SA 4.0) Creative Commons License )


6.5: Conductance
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Conductance, like resistance, is a property of devices. Specifically:
Conductance \(G\) (\(\Omega^{-1}\) or \(S\)) is the reciprocal of resistance \(R\).
Therefore, conductance depends on both the conductivity of the materials used in the device, as well as the geometry of the device.
A natural question to ask is, why do we require the concept of conductance, if it simply the reciprocal of resistance? The short answer is that the concept of conductance is not required; or, rather, we need only resistance or conductance and not both. Nevertheless, the concept appears in engineering analysis for two reasons:
- Conductance is sometimes considered to be a more intuitive description of the underlying physics in cases where the applied voltage is considered to be the independent “stimulus” and current is considered to be the response. This is why conductance appears in the lumped element model for transmission lines (Section 3.4), for example.
- Characterization in terms of conductance may be preferred when considering the behavior of devices in parallel, since the conductance of a parallel combination is simply the sum of the conductances of the devices.
Example \(\PageIndex{1}\): Conductance of a coaxial structure
Let us now determine the conductance of a structure consisting of coaxially-arranged conductors separated by a lossy dielectric, as shown in Figure \(\PageIndex{1}\). The conductance per unit length \(G'\) (i.e., S/m) of this structure is of interest in determining the characteristic impedance of coaxial transmission line, as addressed in Sections 3.4 and Section 3.10.
Figure \(\PageIndex{1}\): Determining the conductance of a structure consisting of coaxially-arranged conductors separated by a lossy dielectric.For our present purposes, we may model the structure as two concentric perfectly-conducting cylinders of radii \(a\) and \(b\), separated by a lossy dielectric having conductivity \(\sigma_s\). We place the \(+z\) axis along the common axis of the concentric cylinders so that the cylinders may be described as constant-coordinate surfaces \(\rho=a\) and \(\rho=b\).
There are at least 2 ways to solve this problem. One method is to follow the procedure that was used to find the capacitance of this structure in Section 5.24. Adapting that approach to the present problem, one would assume a potential difference \(V\) between the conductors, from that determine the resulting electric field intensity \({\bf E}\), and then using Ohm’s Law for Electromagnetics (Section 6.3) determine the density \({\bf J}=\sigma_s {\bf E}\) of the current that leaks directly between conductors. From this, one is able to determine the total leakage current \(I\), and subsequently the conductance \(G\triangleq I/V\). Although highly recommended as an exercise for the student, in this section we take an alternative approach so as to demonstrate that there are a variety of approaches available for such problems.
The method we shall use below is as follows:
- Assume a leakage current \(I\) between the conductors
- Determine the associated current density \({\bf J}\), which is possible using only geometrical considerations
- Determine the associated electric field intensity \({\bf E}\) using \({\bf J}/\sigma_s\)
- Integrate \({\bf E}\) over a path between the conductors to get \(V\). Then, as before, conductance \(G\triangleq I/V\).
The current \(I\) is defined as shown in Figure \(\PageIndex{1}\), with reference direction according to the engineering convention that positive current flows out of the positive terminal of a source. The associated current density must flow in the same direction, and the circular symmetry of the problem therefore constrains \({\bf J}\) to have the form
\[{\bf J} = \hat{\bf \rho}\frac{I}{A} \nonumber \]
where \(A\) is the area through which \(I\) flows. In other words, current flows radially outward from the inner conductor to the outer conductor, with density that diminishes inversely with the area through which the total current flows. (It may be helpful to view \({\bf J}\) as a flux density and \(I\) as a flux, as noted in Section 6.2.) This area is simply circumference \(2\pi\rho\) times length \(l\), so
\[{\bf J} = \hat{\bf \rho}\frac{I}{2\pi\rho l} \nonumber \]
which exhibits the correct units of A/m\(^2\).
Now from Ohm’s Law for Electromagnetics we find the electric field within the structure is
\[{\bf E} = \frac{\bf J}{\sigma_s} = \hat{\bf \rho}\frac{I}{2\pi\rho l \sigma_s} \nonumber \]
Next we get \(V\) using (Section 5.8)
\[V = -\int_{\mathcal C}{ {\bf E} \cdot d{\bf l} }\nonumber \]
where \(\mathcal{C}\) is any path from the negatively-charged outer conductor to the positively-charged inner conductor. Since this can be any such path (Section 5.9), we should choose the simplest one. The simplest path is the one that traverses a radial of constant \(\phi\) and \(z\).  Also note that this is expression depends only on materials (through \(\sigma_s\)) and geometry (through \(l\), \(a\), and \(b\)). Notably it does not depend on current or voltage, which would imply non-linear behavior.
To make the connection back to lumped-element transmission line model parameters (Section 3.4 and Section 3.10), we simply divide by \(l\) to get the per-length parameter:
\[\boxed{ G' = \frac{2\pi \sigma_s}{\ln\left(b/a\right)} } \label{m0105_eGp} \]
Example \(\PageIndex{2}\): Conductance of RG-59 coaxial cable
RG-59 coaxial cable consists of an inner conductor having radius \(0.292\) mm, an outer conductor having radius \(1.855\) mm, and a polyethylene spacing material exhibiting conductivity of about \(5.9\times 10^{-5}\) S/m. Estimate the conductance per length of RG-59.
Solution
From the problem statement, \(a=0.292\) mm, \(b=1.855\) mm, and \(\sigma_s \cong 5.9\times 10^{-5}\) S/m. Using Equation \ref{m0105_eGp}, we find \(G'\cong 200~\mu\)S/m.


9.1: Introduction
Replication begins at one or more origins of replication along DNA, where helicase enzymes catalyze unwinding of the double helix. DNA unwinding creates replicating bubbles, or replicons , with replication forks at either end. Making a new DNA strand starts with making an RNA primer with RNA nucleotides and primase enzymes. DNA nucleotides are then added to the 3’-ends of primers by one a DNA polymerase . Later, other DNA polymerases catalyze removal of the RNA primers and replacement of the hydrolyzed ribonucleotides with new deoxyribonucleotides. Finally, DNA ligases stitch together the fragments of new DNA synthesized at the replication forks. This complex mechanism is common to the replication of ‘naked’ prokaryotic DNA and of chromatinencased eukaryotic DNA, and must therefore have arisen early in the evolution of replication biochemistry. In this chapter, we look at the details of replication and the differences in detail between prokaryotic and eukaryotic replication that arise because of differences in DNA packing. As with any complex process with many moving parts, replication is error-prone. Therefore, we will also look at how the overall fidelity of replication relies mechanisms of DNA repair that target specific kinds of replication mistakes, or mutations . At the same time, lest we think that uncorrected errors in replication are always a bad thing, they usually do not have bad outcomes. Instead, they leave behind the very mutations that allow natural selection and the evolution of diversity .
Learning Objectives
- Explain how Cairns interpreted his theta (\(\Theta \)) images.
- Compare and contrast the activities of enzymes required for replication.
- Describe the order of events at an origin of replication and at each replication fork.
- Compare unidirectional and bidirectional DNA synthesis from an origin of replication.
- Outline the basic synthesis and proofreading functions of DNA polymerase.
- Identify the major players and their roles in the initiation of replication.
- Explain how Okazaki’s experimental results were not entirely consistent with how both strands of DNA replicate
- List the major molecular players (enzymes, etc.) that elongate a growing DNA strand.
- List the non-enzymatic players in replication and describe their functions.
- Describe how the structure of telomerase enables proper replication.
- Compare the activities of topoisomerases 1 and 2.
- Explain the reasoning behind the hypothesis of processive replication.
- Compare and contrast the impacts of germline and somatic mutations.
- Describe common forms of DNA damage.
- List enzymes of replication that were adapted to tasks of DNA repair.
- Explain why a DNA glycosylase is useful in DNA repair.
- Explain why the connection between 'breast cancer genes' and DNA repair.


4.3: Reading- What Is Weathering?
The footprints that astronauts left on the Moon will be there forever. Why? This is because the Moon has no atmosphere and, as a result, has no weathering. Weathering is one of the forces on Earth that destroy rocks and landforms. Without weathering, geologic features would build up but would be less likely to break down.
What is Weathering?
Weathering is the process that changes solid rock into sediments. Sediments were described in the Rocks chapter. With weathering, rock is disintegrated. It breaks into pieces.
Once these sediments are separated from the rocks, erosion is the process that moves the sediments. Erosion is the next chapter’s topic. The four forces of erosion are water, wind, glaciers, and gravity.
- Water is responsible for most erosion. Water can move most sizes of sediments, depending on the strength of the force.
- Wind moves sand-sized and smaller pieces of rock through the air.
- Glaciers move all sizes of sediments, from extremely large boulders to the tiniest fragments.
- Gravity moves broken pieces of rock, large or small, downslope.
While plate tectonics forces work to build huge mountains and other landscapes, the forces of weathering gradually wear those rocks and landscapes away. Together with erosion, tall mountains turn into hills and even plains. The Appalachian Mountains along the east coast of North America were once as tall as the Himalayas.
No human being can watch for millions of years as mountains are built, nor can anyone watch as those same mountains gradually are worn away. But imagine a new sidewalk or road. The new road is smooth and even. Over hundreds of years, it will completely disappear, but what happens over one year? What changes would you see (figure 1)? What forces of weathering wear down that road, or rocks or mountains over time?
Follow this link to view some animations of different types of weathering processes .
Contributors and Attributions
- 9.1: Weathering. Provided by : CK-12. Located at : http://www.ck12.org/book/CK-12-Earth-Science-For-High-School/section/9.1/ . License : CC BY-NC: Attribution-NonCommercial


5.6: Moving Knife
A somewhat different approach to continuous fair division is called the moving knife procedure.
In this method, applied to a cake,
- A referee starts moving a knife from left to right across a cake.
- As soon as any player feels the piece to the left of the knife is worth a fair share, they shout “STOP.” The referee then cuts the cake at the current knife position and the player who called stop gets the piece to the left of the knife.
- This procedure continues until there is only one player left. The player left gets the remaining cake.
Suppose that our four salespeople from above decided to use this approach to divide Washington. Rather than move the “knife” from left to right, they decide to move it from top to bottom.
Solution
The referee starts moving a line down a map of the state. Henry is the first to call STOP when the knife is at the position shown, giving him the portion of the state above the line.
Marjo is the next to call STOP when the knife is at the position shown, giving her the second portion of the state.
Bob is the next to call STOP, leaving Beth with the southernmost portion of the state
While this method guarantees a fair division, it clearly results in some potentially silly divisions in a case like this. The method is probably better suited to situations like dividing an actual cake.


Skip to main content
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Krawczyk,Daniel (2018). Reasoning: The Neuroscience of How We Think. Elsevier.
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Goldstein, E. Bruce (2005). Cognitive Psychology - Connecting, Mind Research, and Everyday Experience. Thomson Wadsworth.
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Marie T. Banich (1997). Neuropsychology. The neural bases of Mental Function. Houghton Mifflin.
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Wilson, Robert A.& Keil, Frank C. (1999). The MIT Encyclopedia of the Cognitive Sciences. Massachusetts: Bradford Book.
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Ward, Jamie (2006). The Student's Guide To Cognitive Science. Psychology Press.
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Levitin, D. J.(2002). Foundations of Cognitive Psychology.
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Schmalhofer, Franz. Slides from the course: Cognitive Psychology and Neuropsychology, Summer Term 2006/2007, University of Osnabrueck


13.6.4: Chapter Review
Section 13.1 Math and Art
A person’s length from their belly button to the floor is 25 in. What is the person’s approximate height rounded to the nearest inch?
The spiral seeds on a fruit mirror the Fibonacci sequence. If a row of the spiral contains 34 leaves, how many leaves would be found on the next larger spiral of seeds?
The seeds on a sunflower mirror the Fibonacci sequence. If a circular layer on the sunflower contains 89 seeds, approximately how many seeds would be found on the next larger circular layer?
A term of the Fibonacci sequence is 17,711 and the previous term is 10,946. What is the following term of the sequence?
The 35th term of the Fibonacci sequence is 9,227,465. Use the approximate value of \(\mathit{ϕ}\) of 1.618 to estimate the 34th term. Round your answer to the nearest whole number.
A frame has dimensions of 16 in by 24 in. Calculate the ratio of the sides rounded to the nearest tenth and determine if the size approximates a golden rectangle.
Section 13.2 Math and the Environment
An average toilet uses 5 gal of water per flush. A high-efficiency toilet uses about 1.25 gal per flush. How much water would a household save in a month (30 days) if the toilet was flushed 6 times a day?
When washing dishes, leaving the faucet running utilizes about 15 gal every 5 minutes, whereas filling the sink and turning the faucet off except to rinse the dishes uses about 6 gal for washing a dishpan-sized load of dishes. How much water is saved by not leaving the faucet running if it takes 10 minutes to wash a dishpan-sized load of dishes each day for a year containing 30 days in each month?
Leaving the water running when washing your hands consumes about 4 gal of water, whereas turning the water off when lathering reduces the water used to 1 gal. How much water is saved in an apartment of 4 people in 6 months (180 days) if each person averages washing their hands twice a day?
How long would it take for an energy-saving light bulb to consume 1 kW of power if the bulb is rated at 2.5 W? Round your answer to the nearest hour.
How long would it take for a 60 W light bulb to consume 1 kW of power? Round your answer to the nearest hour.
A portable fan uses about 40 W. How many kilowatt-hours are needed to run the fan during a 10-day trip if the fan is run for an average of 3.5 hours a day?
Section 13.3 Math and Medicine
In a 2% solution of lidocaine, how many milligrams are there per milliliter?
You are prescribed a nebulizer solution to aid with asthma. If the solution is 0.7% w/v, what volume is needed to deliver a 7 mg dose?
A prescription calls for a patient to receive 17 mg of a drug to be taken in oral form. If the pills are available in 3.4 mg, how many pills will the patient need daily?
A prescription calls for a patient to receive 9 mg daily of a drug to be taken daily for 10 days. If the pills are available in 2.25 mg, how many pills are needed to fill the prescription?
A patient is prescribed 2 mg/kg of a drug to be delivered daily intramuscularly, divided into three doses. If the patient weighs 30 kg, how many milligrams of the drug is needed for each dose?
A patient is prescribed 0.75 mg/kg of a drug to be delivered intramuscularly once per day. If the drug is available in 2.5 mg/mL and the patient weighs 63 kg, how many milliliters of the drug should be given per day?
Section 13.4 Math and Music
Select the most representative sound registering 50 dB: airport gate noise, light rain on a car, dirt bike racing.
Select the most representative sound registering 100 dB: city traffic when in a car with windows rolled up, children playing on a school playground, professional sporting event such as football.
What frequency range can teenagers typically hear but adults over 20 are often not able to hear?
What is three whole steps up from G # ?
What is one half-step down from F?
What is an octave above D 3 ?
Given that the frequency of \(\mathrm{A}_4=440 \mathrm{~Hz}\), what is the frequency of \(\mathrm{A}_6\) ?
Section 13.5 Math and Sports
For the following exercises, compute how many games would be played in the style of tournament and number of teams given in each question. Assume all tournaments are single round-round or single-elimination.
How many games would be played with 10 teams using a round-robin tournament?
How many games would be played with 25 teams using a round-robin tournament?
How many games would be played with 12 teams using a round-robin tournament?
How many games would be played with 4 teams using a single-elimination tournament?
How many games would be played with 9 teams using a single-elimination tournament?


7.3: Capacitors in Series and in Parallel
By the end of this section, you will be able to:
- Explain how to determine the equivalent capacitance of capacitors in series and in parallel combinations
- Compute the potential difference across the plates and the charge on the plates for a capacitor in a network and determine the net capacitance of a network of capacitors
Several capacitors can be connected together to be used in a variety of applications. Multiple connections of capacitors behave as a single equivalent capacitor. The total capacitance of this equivalent single capacitor depends both on the individual capacitors and how they are connected. Capacitors can be arranged in two simple and common types of connections, known as series and parallel , for which we can easily calculate the total capacitance. These two basic combinations, series and parallel, can also be used as part of more complex connections.
The Series Combination of Capacitors
Figure \(\PageIndex{1}\) illustrates a series combination of three capacitors, arranged in a row within the circuit. As for any capacitor, the capacitance of the combination is related to both charge and voltage :
\[ C=\dfrac{Q}{V}.\]
When this series combination is connected to a battery with voltage V , each of the capacitors acquires an identical charge Q . To explain, first note that the charge on the plate connected to the positive terminal of the battery is \(+Q\) and the charge on the plate connected to the negative terminal is \(-Q\). Charges are then induced on the other plates so that the sum of the charges on all plates, and the sum of charges on any pair of capacitor plates, is zero. However, the potential drop \(V_1 = Q/C_1\) on one capacitor may be different from the potential drop \(V_2 = Q/C_2\) on another capacitor, because, generally, the capacitors may have different capacitances. The series combination of two or three capacitors resembles a single capacitor with a smaller capacitance. Generally, any number of capacitors connected in series is equivalent to one capacitor whose capacitance (called the equivalent capacitance ) is smaller than the smallest of the capacitances in the series combination. Charge on this equivalent capacitor is the same as the charge on any capacitor in a series combination: That is, all capacitors of a series combination have the same charge . This occurs due to the conservation of charge in the circuit. When a charge Q in a series circuit is removed from a plate of the first capacitor (which we denote as \(-Q\)), it must be placed on a plate of the second capacitor (which we denote as \(+Q\)), and so on.
We can find an expression for the total (equivalent) capacitance by considering the voltages across the individual capacitors. The potentials across capacitors 1, 2, and 3 are, respectively, \(V_1 = Q/C_1\), \(V_2 = Q/C_2\), and \(V_3 = Q/C_3\). These potentials must sum up to the voltage of the battery, giving the following potential balance:
\[V = V_1 + V_2 + V_3.\]
Potential \(V\) is measured across an equivalent capacitor that holds charge \(Q\) and has an equivalent capacitance \(C_S\). Entering the expressions for \(V_1\), \(V_2\), and \(V_3\), we get
\[\dfrac{Q}{C_S} = \dfrac{Q}{C_1} + \dfrac{Q}{C_2} + \dfrac{Q}{C_3}.\]
Canceling the charge Q , we obtain an expression containing the equivalent capacitance, \(C_S\), of three capacitors connected in series:
\[\dfrac{1}{C_S} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dfrac{1}{C_3}.\]
This expression can be generalized to any number of capacitors in a series network.
For capacitors connected in a series combination , the reciprocal of the equivalent capacitance is the sum of reciprocals of individual capacitances:
\[\dfrac{1}{C_S} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dfrac{1}{C_3} + \dots \label{capseries}\]
Find the total capacitance for three capacitors connected in series, given their individual capacitances are \(1.000 \mu F\), \(5.000 \mu F\), and \(8.000 \mu F\).
Strategy
Because there are only three capacitors in this network, we can find the equivalent capacitance by using Equation \ref{capseries} with three terms.
Solution
We enter the given capacitances into Equation \ref{capseries}:
\[ \begin{align*} \dfrac{1}{C_S} &= \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dfrac{1}{C_3} \\[4pt] &= \dfrac{1}{1.000 \mu F} + \dfrac{1}{5.000 \mu F} + \dfrac{1}{8.000 \mu F} \\[4pt] &= \dfrac{1.325}{\mu F}.\end{align*} \]
Now we invert this result and obtain
\[ \begin{align*} C_S &= \dfrac{\mu F}{1.325} \\[4pt] &= 0.755 \mu F.\end{align*} \nonumber\]
Significance
Note that in a series network of capacitors, the equivalent capacitance is always less than the smallest individual capacitance in the network.
The Parallel Combination of Capacitors
A parallel combination of three capacitors, with one plate of each capacitor connected to one side of the circuit and the other plate connected to the other side, is illustrated in Figure \(\PageIndex{2a}\). Since the capacitors are connected in parallel, they all have the same voltage V across their plates . However, each capacitor in the parallel network may store a different charge. To find the equivalent capacitance \(C_p\) of the parallel network, we note that the total charge Q stored by the network is the sum of all the individual charges:
\[Q = Q_1 + Q_2 + Q_3.\]
On the left-hand side of this equation, we use the relation \(Q = C_pV\), which holds for the entire network. On the right-hand side of the equation, we use the relations \(Q_1 = C_1 V\), \(Q_2 = C_2V\), and \(Q_3 = C_3V\) for the three capacitors in the network. In this way we obtain
\[C_pV = C_1V + C_2V + C_3V.\]
This equation, when simplified, is the expression for the equivalent capacitance of the parallel network of three capacitors:
\[C_p = C_1 + C_2 + C_3.\]
This expression is easily generalized to any number of capacitors connected in parallel in the network.
For capacitors connected in a parallel combination , the equivalent (net) capacitance is the sum of all individual capacitances in the network,
\[C_p = C_1 + C_2 + C_3 + ... \label{capparallel}\]
Find the net capacitance for three capacitors connected in parallel, given their individual capacitances are \(1.0 \mu F\), \(5.0 \mu F\), and \(8.0 \mu F\).
Strategy
Because there are only three capacitors in this network, we can find the equivalent capacitance by using Equation \ref{capparallel} with three terms.
Solution
Entering the given capacitances into Equation \ref{capparallel} yields
\[\begin{align*} C_p &= C_1 + C_2 + C_3 \\[4pt] &= 1.0 \mu F + 5.0 \mu F + 8.0 \mu F \\[4pt] &= 14.0 \mu F. \end{align*}\]
Significance
Note that in a parallel network of capacitors, the equivalent capacitance is always larger than any of the individual capacitances in the network.
Capacitor networks are usually some combination of series and parallel connections, as shown in Figure \(\PageIndex{3}\). To find the net capacitance of such combinations, we identify parts that contain only series or only parallel connections, and find their equivalent capacitances. We repeat this process until we can determine the equivalent capacitance of the entire network. The following example illustrates this process.
Find the total capacitance of the combination of capacitors shown in Figure \(\PageIndex{3}\). Assume the capacitances are known to three decimal places (\(C_1 = 1.000 \mu F, C_2 = 5.000 \mu F, C_3 = 8.000 \mu F\)). Round your answer to three decimal places.
Strategy
We first identify which capacitors are in series and which are in parallel. Capacitors \(C_1\) and \(C_2\) are in series. Their combination, labeled \(C_S\) is in parallel with \(C_3\).
Solution
Since \(C_1\) and \(C_2\) are in series, their equivalent capacitance \(C_S\) is obtained with Equation \ref{capseries}:
\[\begin{align*} \dfrac{1}{C_S} &= \dfrac{1}{C_1} + \dfrac{1}{C_2} \\[4pt] &= \dfrac{1}{1.000 \mu F} + \dfrac{1}{5.000 \mu F} \\[4pt] &= \dfrac{1.200}{\mu F} \end{align*}\]
Therefor
\[ C_S = 0.833 \mu F. \nonumber\]
Capacitance \(C_S\) is connected in parallel with the third capacitance \(C_3\), so we use Equation \ref{capparallel} find the equivalent capacitance C of the entire network:
\[\begin{align*} C &= C_S + C_3 \\[4pt] &= 0.833 \mu F + 8.000 \mu F \\[4pt] &= 8.833 \mu F. \end{align*}\]
Determine the net capacitance C of the capacitor combination shown in Figure \(\PageIndex{4}\) when the capacitances are \(C_1 = 12.0 \mu F, C_2 = 2.0 \mu F\), and \(C_3 = 4.0 \mu F\). When a 12.0-V potential difference is maintained across the combination, find the charge and the voltage across each capacitor.
Strategy We first compute the net capacitance \(C_{23}\) of the parallel connection \(C_2\) and \(C_3\). Then C is the net capacitance of the series connection \(C_1\) and \(C_{23}\). We use the relation \(C = Q/V\) to find the charges \(Q_1, Q_2\), and \(Q_3\), and the voltages \(V_1, V_2\), and \(V_3\) across capacitors 1, 2, and 3, respectively.
Solution The equivalent capacitance for \(C_2\) and \(C_3\) is
\[C_{23} = C_2 + C_3 = 2.0 \mu F + 4.0 \mu F = 6.0 \mu F.\]
The entire three-capacitor combination is equivalent to two capacitors in series,
\[\dfrac{1}{C} = \dfrac{1}{12.0 \mu F} + \dfrac{1}{6.0 \mu F} = \dfrac{1}{4.0 \mu F} \Rightarrow C = 4.0 \mu F.\]
Consider the equivalent two-capacitor combination in Figure \(\PageIndex{2b}\). Since the capacitors are in series, they have the same charge, \(Q_1 = Q_{23}\). Also, the capacitors share the 12.0-V potential difference, so
\[12.0 V = V_1 + V_{23} = \dfrac{Q_1}{C_1} + \dfrac{Q_{23}}{C_{23}} = \dfrac{Q_1}{12.0 \mu F} + \dfrac{Q_1}{6.0 \mu F} \Rightarrow Q_1 = 48.0 \mu C.\]
Now the potential difference across capacitor 1 is
\[V_1 = \dfrac{Q_1}{C_1} = \dfrac{48.0 \mu C}{12.0 \mu F} = 4.0 V.\]
Because capacitors 2 and 3 are connected in parallel, they are at the same potential difference:
\[V_2 = V_3 = 12.0 V - 4.0 V = 8.0 V.\]
Hence, the charges on these two capacitors are, respectively,
\[Q_2 = C_2V_2 = (2.0 \mu F)(8.0 V) = 16.0 \mu C,\]
\[Q_3 = C_3V_3 = (4.0 \mu F)(8.0 V) = 32.0 \mu C.\]
Significance As expected, the net charge on the parallel combination of \(C_2\) and \(C_3\) is \(Q_{23} = Q_2 + Q_3 = 48.0 \mu C.\)
Determine the net capacitance C of each network of capacitors shown below. Assume that \(C_1 = 1.0 pF, C_2 = 2.0 pF, C_3 = 4.0 pF\), and \(C_4 = 5.0 pF\). Find the charge on each capacitor, assuming there is a potential difference of 12.0 V across each network.
- Answer a
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\(C = 0.86 pF, Q_1 = 10 pC, Q_2 = 3.4 pC, Q_3 = 6.8 pC\)
- Answer b
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\(C = 2.3 pF, Q_1 = 12 pC, Q_2 = Q_3 = 16 pC\)
- Answer c
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\(C = 2.3 pF, Q_1 = 9.0 pC, Q_2 = 18 pC, Q_3 = 12 pC, Q_4 = 15 pC\)
Circuits often contain both capacitors and resistors. Table \(\PageIndex{1}\) summarizes the equations used for the equivalent resistance and equivalent capacitance for series and parallel connections.
| Series combination | Parallel combination | |
|---|---|---|
| Equivalent capacitance | \[\frac{1}{C_{S} }= \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + . . . \nonumber\] | \[C_{P} = C_1 + C_2 + C_3 + . . . \nonumber\] |
| Equivalent resistance | \[R_{S} = R_1 + R_2 + R_3 + . . . = \sum_{i=1}^N R_i \nonumber\] | \[\frac{1}{R_{P}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + . . . \nonumber\] |


26.4: Gibbs Energy of a Reaction vs. Extent of Reaction is a Minimum at Equilibrium
As discussed in Section 26.4 , we defined the extent of the reaction (\(\xi\)) as a quantitative measure of how far along the reaction has evolved. For simple reactions like \(\ce{A <=> B}\), the extent of reaction is easy to define and is simply the number of moles of \(\ce{A}\) that has been converted to \(\ce{B}\). Since \(d\xi\) has the same value as \(dn\), we can write that
\[\begin{align*} dG &= \mu_A dn_A + \mu_B dn_B \\[4pt] &= -\mu_A d\xi + \mu_B d\xi. \end{align*} \]
The minus sign comes from the fact that when the reaction goes in the left to right direction, the amount of \(\ce{A}\) is decreasing, while the amount of \(\ce{B}\) is increasing. Looking at these equations, it is reasonable to suggest that:
\[\left(\frac{\partial G}{\partial \xi}\right)_{P T}=\mu_{B}-\mu_{A} \nonumber \]
This is the slope of the free energy with respect to the extent of the reaction. This relationship will have a region where the sign is negative, one point where the value is zero and a region where the value is positive (Figure \(\PageIndex{1}\)). If we look at a plot of \(G\) as a function of \(\xi\), we can see that the point where is the minimum of the curve:
\[\left(\frac{\partial G}{\partial \xi}\right)_{PT}=0 \nonumber \]
This is the point where \(\mu_A\) and \(\mu_B\) are the same. On one side of the minimum, the slope is negative and on the other side, the slope is positive. In both cases, the reaction is spontaneous (i.e, \(dG < 0\)) as long as the reaction is evolving towards the minimum, which will be called the equilibrium position . Hence, all reactions spontaneously move towards equilibrium; the immediate question is what value of \(\xi\) corresponds to the equilibrium position of a reaction?
- A reaction for which \(Δ_rG < 0\) is called exergonic .
- A reaction for which \(Δ_r G > 0\) is called endergonic .
- \(Δ_r G < 0\), the forward reaction is spontaneous.
- \(Δ_r G > 0\), the reverse reaction is spontaneous.
- \(Δ_r G = 0\), the reaction is at equilibrium.
Ideal Gas Equilibrium
To understand how we can find the minimum and what the Gibbs free energy of a reaction depends on, let's first start with a reaction that converts one ideal gas into another.
\[\ce{A<=>B} \nonumber \]
Let's assume the reaction enthalpy is zero and hence the only thing that determines what the ratio of product to reactant should be is the entropy term (the mixing term), which is the most favorable when the mixture is half and half.
\[\Delta G_{\text {mix }}=n R T\left(\chi_{A} \ln \chi_{A}+ \chi_{B} \ln \chi_{B}\right) \label{Smixing} \]
Remember that the Gibbs free energy of mixing is not a molar quantity and depends on \(n\) (unlike the reaction Gibbs free energy). Also, the Gibbs free energy of mixing is defined relative to pure \(\ce{A}\) and \(\ce{B}\). The free energy of mixing in Equation \ref{Smixing} is at a minimum when the amounts of \(\ce{A}\) equal \(\ce{B}\).
Thus, if we have a reaction \(\ce{A<=>B}\) and there is no enthalpy term (and no change in the inherent entropy of A vs. B), we would expect the system to have the minimum Gibbs free energy when the mole fraction of \(\ce{A}\) and \(\ce{B}\) are each 0.5.
If \(\ce{A}\) can form \(\ce{B}\) and \(\ce{B}\) can form \(\ce{A}\) and there is no other forces involved (no heat/enthalpy) that would favor one product over the other, probability would just state that eventually you will have statistically the same amount of both A and B present -- this is the lowest free energy state. (This, of course, will not be true if the enthalpy for the reaction is not zero or if \(\ce{A}\) and \(\ce{B}\) have different inherent entropies). Usually there are additional terms for the reaction.
Lets work through this for an ideal gas reaction:
\[\begin{align*}
\Delta_{r} G &=\mu_{B}-\mu_{A}
\\[4pt] &=\mu_{B}^{o}+R T \ln \dfrac{P_{B}}{P^{\theta}}-\mu_{A}^{o}-R T \ln \dfrac{P_{A}}{P^{\theta}} \\[4pt]
&=\mu_{B}^{o}-\mu_{A}^{o}+R T \ln \dfrac{P_{B}}{P_{A}}
\end{align*} \]
What we normally do at this point is to give the first two terms a special name. Since it is the difference between the chemical potentials at standard conditions, we refer to it as the Gibbs free energy of reaction at standard conditions or the standard Gibbs free energy of reaction .
\[\begin{align} \Delta_{r} G^{o} &= \mu_B^{o}-\mu_A^{o} \nonumber \\[4pt] \Delta_r G &= \Delta_{r} G^{o} + R T \ln \dfrac{P_{B}}{P_{A}} \label{eq30} \end{align} \]
There are two parts to Equation \ref{eq30}. The first term (\(\Delta_{r} G^{o}\)) is the Gibbs free energy for converting one mole of \(\ce{A}\) to \(\ce{B}\) under standard conditions (1 bar of both \(\ce{A}\) and \(\ce{B}\)). The second term is the mixing term that is minimized when the amounts of A and B are equal to one another (Figure \(\PageIndex{2}\)). The first term you look up in a reference book and is specific for a specific reaction. The second term is calculated knowing the partial pressures of \(\ce{A}\) and \(\ce{B}\) in the gas.
Now the minimum absolute Gibbs free energy will occur at the bottom of the curve where the slope is zero. Thus, the lowest free energy will occur when the reaction free energy (i.e., the slope of that curve in Figure \(\PageIndex{1}\)) is equal to zero. The chemical potentials of \(\ce{A}\) and \(\ce{B}\) are equal.
\[\mu_A = \mu_B \nonumber \]
At this point, the reaction will neither go forwards or backwards and we call this equilibrium. Hence, at equilibrium:
\[ \begin{align*} \Delta_{r} G &=\Delta_{r} G^{o}+R T \ln \dfrac{P_{B}}{P_{A}} \\[4pt] &=0 \end{align*} \]
and the specific ratio of \(P_A\) and \(P_B\) necessary to ensure \(\Delta_{r} G=0\) is characteristic of the reaction and is called the equilibrium constant for that reaction.
\[K_{e q}=\dfrac{P_{B}}{P_{A}} \nonumber \]
We can now relate thermodynamic quantities to concentrations of molecules at equilibrium. We can also see that at equilibrium:
\[\begin{aligned}
&0=\Delta_{r}G^{o}+R T \ln K_{e q} \\[4pt]
&\Delta_{r} G^{o}=-R T \ln K_{e q} \\[4pt]
&K_{eq}=e^{-\dfrac{\Delta_{r} G^{o}}{R T}}
\end{aligned} \nonumber \]


5.5: Sample Documentation
Sample Documentation of Expected Findings
The patient’s skin tone is appropriate for ethnicity and they are not diaphoretic. Facial movements are symmetrical. Nares are patent and no drainage is present. Uvula and tongue are midline. No dental caries and gums are pink, moist, without bleeding. Patient is able to swallow without difficulty. Trachea is midline. There is no enlargement of the lymph nodes.
Sample Documentation of Unexpected Findings
The patient has a history of asthma and uses a fluticasone inhaler daily. White patches are present on the tongue and inner buccal membranes. Dr. Smith was notified at 1530 and a new order for nystatin medication was received and administered. Patient was instructed to rinse their mouth after every use of fluticasone to prevent thrush from recurring.


21.9: Paragraphs and Transitions
Paragraphs help readers make their way through prose writing by presenting it in manageable chunks. Transitions link sentences and paragraphs so that readers can clearly understand how the points you are making relate to one another. (See Editing Focus: Paragraph and Transitions for a related discussion of paragraphs and transitions. See Evaluation: Transitions for a related discussion of transitions in multimodal compositions.)
Effective Paragraphs
Paragraphs help readers make their way through prose writing by presenting it in manageable chunks. Transitions link sentences and paragraphs so that readers can clearly understand how the points you are making relate to one another. (See Editing Focus: Paragraph and Transitions for a related discussion of paragraphs and transitions. See Evaluation: Transitions for a related discussion of transitions in multimodal compositions.) evidence and examples to back up your claims. When a paragraph is too long, readers can lose the point you are making.
Developing a Main Point
A paragraph is easier to write and easier to read when it centers on a main point. The main point of the paragraph is usually expressed in a topic sentence . The topic sentence frequently comes at the start of the paragraph, but not always. No matter the position, however, the other sentences in the paragraph support the main point.
Supporting Evidence and Analysis
All the sentences that develop the paragraph should support or expand on the main point given in the topic sentence. Depending on the type of writing you are doing, support may include evidence from sources—such as facts, statistics, and expert opinions—as well as examples from your own experience. Paragraphs also may include an analysis of your evidence written in your own words. The analysis explains the significance of the evidence to the reader and reinforces the main point of the paragraph.
In the following example, the topic sentence is underlined. The supporting evidence discussed through cause-and-effect reasoning comes in the next three sentences. The paragraph concludes with two sentences of analysis in the writer’s own words.
Millions of retired Americans rely on Social Security benefits to make ends meet after they turn 65 . According to the Social Security Administration, about 46 million retired workers receive benefits, a number that reflects about 90 percent of retired people. Although experts disagree on the exact numbers, somewhere between 12 percent and 40 percent of retirees count on social security for all of their income, making these benefits especially important (Konish). These benefits become more important as people age. According to Eisenberg, people who reach the age of 85 become more financially vulnerable because their health care and long-term care costs increase at the same time their savings have been drawn down. It should therefore come as no surprise that people worry about changes to the program. Social Security keeps millions of retired Americans out of poverty.
Opening Paragraphs
Readers pay attention to the opening of a piece of writing, so make it work for you. After starting with a descriptive title, write an opening paragraph that grabs readers’ attention and alerts them to what’s coming. A strong opening paragraph provides the first clues about your subject and your stance. In academic writing, whether argumentative, interpretative, or informative, the introduction often ends with a clear thesis statement , a declarative sentence that states the topic, the angle you are taking, and the aspects of the topic the rest of the paper will support.
Depending on the type of writing you’re doing, you can open in a variety of ways.
- Open with a conflict or an action . If you’re writing about conflict, a good opening may be to spell out what the conflict is. This way of opening captures attention by creating a kind of suspense: Will the conflict be resolved? How will it be resolved?
- Open with a specific detail, statistic, or quotation . Specific information shows that you know a lot about your subject and piques readers’ curiosity. The more dramatic your information, the more it will draw in readers, as long as what you provide is credible.
- Open with an anecdote . Readers enjoy stories. Particularly for reflective or personal narrative writing, beginning with a story sets the scene and draws in readers. You may also begin the anecdote with dialogue or reflection.
The following introduction opens with an anecdote and ends with the thesis statement, which is underlined.
Betty stood outside the salon, wondering how to get in. It was June of 2020, and the door was locked. A sign posted on the door provided a phone number for her to call to be let in, but at 81, Betty had lived her life without a cell phone. Betty’s day-to-day life had been hard during the pandemic, but she had planned for this haircut and was looking forward to it: she had a mask on and hand sanitizer in her car. Now she couldn’t get in the door, and she was discouraged. In that moment, Betty realized how much Americans’ dependence on cell phones had grown in the months she and millions of others had been forced to stay at home. Betty and thousands of other senior citizens who could not afford cell phones or did not have the technological skills and support they needed were being left behind in a society that was increasingly reliant on technology .
Closing Paragraphs
The conclusion is your final chance to make the point of your writing stick in readers’ minds by reinforcing what they have read. Depending on the purpose for your writing and your audience, you can summarize your main points and restate your thesis, draw a logical conclusion, speculate about the issues you have raised, or recommend a course of action, as shown in the following conclusion:
Although many senior citizens purchased and learned new technologies during the COVID-19 pandemic, a significant number of older people like Betty were unable to buy and/or learn the technology they needed to keep them connected to the people and services they needed. As society becomes increasingly dependent on technology, social service agencies, religious institutions, medical providers, senior centers, and other organizations that serve the elderly need to be equipped to help them access and become proficient in the technologies essential to their daily lives.
Transitions
Transitional words and phrases show the connections or relationships between sentences and paragraphs and help your writing flow smoothly from one idea to the next.
Flow
A paragraph flows when ideas are organized logically and sentences move smoothly from one to the next. Transitional words and phrases help your writing flow by signaling to readers what’s coming in the next sentence. In the paragraph below, the topic sentence and transitional words and phrases are underlined.
Some companies court the public by mentioning environmental problems and pointing out that they do not contribute to these problems . For example , the natural gas industry often presents natural gas as a good alternative to coal. However , according to the Union of Concerned Scientists, the drilling and extraction of natural gas from wells and transporting it through pipelines leaks methane, a major cause of global warming (“Environmental Impacts”). Yet leaks are rarely mentioned by the industry. By taking credit for problems they don’t cause and being silent on the ones they do, companies present a favorable environmental image that often obscures the truth.
Transitional Words and Phrases
Following are some transitional words and phrases and their functions in paragraphs. Use this list when drafting or revising to help guide readers through your writing. (See Editing Focus: Paragraphs and Transitions for another discussion on transitions.)
| Type of Transition | Examples |
| to compare or show similarity | likewise, similarly, in like manner |
| to contrast or change direction | but, yet, however, nevertheless, still, at the same time, on the other hand, conversely |
| to add to | also, and, furthermore, next, then, in addition |
| to give examples | for example, for instance, to illustrate, specifically, thus |
| to agree or concede | certainly, of course, to be sure, granted |
| to summarize or conclude | finally, in conclusion, in short, in other words, thus, in summary |
| to show time | first, second, third, next, then, soon, meanwhile, later, currently, concurrently, at the same time, eventually, at last, finally |
| to show a spatial relationship | here, there, in the background, in the foreground, in the distance, to the left, to the right, near, above, below |


10.3: How can we teach to meet all students' needs?
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By Honorina Maristela
Learning Objectives
- Define differentiated instruction and understand the use of this approach in the classroom.
- Identify the teacher’s role in differentiated instruction.
Introduction
In a traditional classroom setting, the teachers talk while the students listen. Nowadays, traditional settings have completely changed in the field of education. The way the classroom are set-up, the teaching and learning style, the accommodation of individual students and application of small group instruction made the school and teaching challenging both for teachers and students.In addition to these changes, differentiated instruction has been emphasized.
The district and state and federal governments have established our standards and have handed our curriculum down to us. These standards make up goals established for all students. How we reach the goals may require different paths. The core of differentiated instruction is flexibility in content, process, and product based on students' strengths, needs and learning styles. (Levy, 2008 cited in The Clearing House, 2008)
What is Differentiated Instruction?
At the most basic level, differentiating instruction means “shaking up” what goes on in the classroom. Students have multiple options for taking in information, making sense of ideas and expressing what they learn. (Tomlinson, 2001) In other words, differentiated instructing is catering to student’s needs and interests so they can acquire learning efficiently..
The State of Virginia encourages the school districts to apply differentiated instruction in the classrooms. In 2004, the Superintendent of Public Instruction conducted a Professional Development to all School Heads, member of Virginia Association of Elementary School Principals, Assistant Principals and Lead teachers to participate in the conference entitled “ Differentiated Instruction: Meeting the Needs of All Children” the highlight for this conference included the special needs students and at-risk students.
Why differentiate?
Individual students are different in many ways. The use of differentiated instruction will gives the students the opportunity to meet their needs in an intensive step that will expedite the learning skill of the individual student. We as teachers give the support and the careful planning of the instruction to make sure that the activities will be in the students’ instructional level and interest.
Differentitate Why: addresses the teachers’ reason for modifying the leaning experience. Teachers believe modification is important for many reasons. Three key reasons include access to learning, motivation to learn and efficiency of learning. Any or all of these three reasons for differentiating instruction can be tied to students readiness, interest, and learning profile. (Tomlinson, 1999)
In differentiated classroom, teachers begin at the level of the students, not the front of a curriculum guide. They accept and build upon the premise that learners differ in important ways. Thus, they also accept and act on the premise that teachers must be ready to engage students in instruction though different learning modalities, by appealing to differing interest and by using varied rates of instruction along with varied degrees of complexity. In differentiated classrooms, teachers ensure that a student competes agains himself or herself as he or she grows and develops more than he or she competes against other students. (Tomlinson, 1999)
Though differentiated instruction and activities, students take a greater responsibility and ownership of their own learning, activities that are primarily focused on students’ multiple intelligences, higher order thinking, and learning styles. With this in mind, differentiated instruction is an effective tool to implement in the classroom in order to meet students’ learning style and strengthen multiple intelligences and encourage them to use high order thinking. (Bailey and Williams-Black, p. 134)
How to Differentiate?
It has been a big question for teachers how to differentiate instruction in the classroom. It is indeed a challenging task to put into use this approach. I myself can relate to this for it took me five years to attain the effectiveness of this approach and I am still learning until now. The preparation and designed activities should target the individual needs of the students.
Being in an inclusion is a tough job, having four groups of students with individual needs ranging from high, average, low level and special needs students with IEP (Individualized Education Program) and 504 plans. It is a team work of the teacher, special education teacher and the students in carrying out the activities in the classroom.
Use formal and informal assessments in making decision with small group instruction. Most schools are required certain formal assessments such as TPRI(Texas Primary Reading Assessment, DIBLE(Dynamic Indicators of Basic Early Literacy Skill, DRA( Developmental Reading Assessment) and/or QRI(Qualitative Reading Inventory. (Diller, 2007)
These assessments are useful tools for teachers for figuring out how to start with their small groups. Once the teacher determined the level of the individual students she may begin applying the differentiated instruction in the classroom. The teacher can design her own activities that are according to the instructional level of the students. One great example is in reading. Using leveled books that are color coded are useful in targeting the instructional level of the students and work towards it to improve to the fullest. The idea of using the flexible grouping of Debbie Diller is useful for teachers to differentiate. As Diller stated, to create flexible group folders for thinking about small groups as well as in whole group. Once the student show progress he/she can be moved into different group. (Diller, 2007,p. 24)
Flexible groupings were defined by Radencich and McKay (1995) as” grouping that is not static, where members of the group change frequently. This emerged as a practice to address concerns. It acknowledges that all grouping patterns- large groups, small groups, teams and individuals have value because they all offer the reader slightly different experiences with different outcomes. Students may work with a partner, in a small cooperative or teacher-led group, or with the whole class. The basis for this may be students’ interest or needs. ( Ford, 2005 cited in Radencich and McKay, 1995)
In planning flexible grouping teachers should consider the strengths and weaknesses of each grouping approach and then put them together to allow the teacher to best meet the needs of the classroom. The groups are formed and dissolved as needs changes to allow for maximum flexibility, avoiding the static nature of the grouping patterns of the past. ( Ford, 2005 cited in Radencich and McKay, 1995)
Note
Key Principles of a Differentiated Instruction
- The teacher is clear about what matters in subject matter.
- The teacher understands, appreciates, and builds upon student differences.
- Assessment and instruction are inseparable.
- The teacher adjusts content, process and product in response to student readiness, interest, and learning profile.
- All students participate in respectful work.
- Students and teachers are collaborators in learning.
- Goals of a differentiated classroom are maximum growth and individual success.
- Flexibility is the hallmark of a differentiated classroom.( Tomlinson, 1999)
Teachers Role
Mixed-ability classrooms that offer differentiated instruction make good sense for teachers, as well a student. For many teachers, offering differentiated instruction first requires a paradigm shift. ( Tomlinson, 2001)
Teachers who become comfortable with differentiated classroom would probably say their role differs in some significant ways from that of a traditional teacher. When teacher differentiate instruction, they move away from being themselves as keepers and dispensers of knowledge and move toward seeing themselves as organizers of learning opportunities. (Tomlnison, 2001)
Teachers who differentiate instruction focus on their role as coach or mentor(Tomlinson, 2001). We serve as a guide and facilitator for the students in doing their responsibilities in completing their task. We organize the activities for them to discover, explore and share acquired knowledge. ..
Note
A teacher is a compass that activates the magnets of curiosity, knowledge, and wisdom in the pupils. ~Ever Garrison
Conclusion
When applying differentiated instruction time is required in order for it to work. It is not an overnight activity to see the effectiveness of this approach. It takes time and dedication to help the learners be successful.
For most teachers, the classroom is the place where we spend our career. The classroom is the place where we give the better part of our lifetime trying to make a difference. It is a curiosity of teaching that no two days are the alike. We must remember that we have every opportunity to transform ourselves and our practice, just as we have every opportunity to stagnate, remain much the same teacher we were when we began. ( Tomlinson, 1999)
Exercise \(\PageIndex{1}\)
1. It is the “shaking up” what goes in the classroom so that students have multiple options for taking in information, making sense of ideas and expressing what they learn. This is called?
A. Whole group instruction
B. Differentiated instruction
C. Cooperative learning
D. Flexible group
2. The grouping that is not static where members of the group change frequently is called?
A. Flexible group
B. Paired activity
C. Leveled group
D. Mixed group
3. Ms. Parker is planning to differentiate instruction in her class. Which is the best tool for her to use in figuring out her small groups?
A. Use student’s gender
B. Use students formal and informal assessments
C. Use students race
D. Use students’ favorite color
4. Mr. Castro is a third grade teacher applying differentiated instruction in his class. He walks around in each group providing assistance in the group while the students work . Mr. Castro can be describes as?
A. Guest in the classroom.
B. Coach or mentor in the classroom
C. Student in the classroom
D. Principal in the classroom
- Answer
-
1. B
2. A
3. B
4. B
References
Bailey, J.P. and Williams-Black,T.H.(2008).Differentiated instruction three teacher's perspective.College Reading Association Yearbook(2008).pp. 29,134. Retrieved January 30, 2009 from http://search.ebscohost.com.proxy.lib.odu.edu/login.aspx?direct=true&db=ehh&AN=33435085&site=ehost-live
Diller, D. (2007).Making the most of small groups differentiation for all.Markham,Ontario Canada:Pembroke Publishes,pp. 21–24.
Ford, M.(2005).Differentiation through flexible grouping successfully reading all readers.Learning Point Associates(2005),p. 3.Retrieved February 7, 2009 from www.cpsgifted.org/pdf/flexibleGrouping1.pdf
Levy, H.(2008).Meeting the needs of all students through differentiated instruction:Helping every child reach and exceed standtards.Clearing House;Mar/Apr2008.Vol.81(4),2.Retrieved from February 6, 2009 http://web.ebscohost.com.proxy.lib.odu.edu/ehost/dilivery?vid=7&hid=113&sid=1cb60ea3 ...
Tomlinson,C.A. (1999).The differentiated classroom responding to the needs of all learners.Alexandria VA:Association for Supervision and Curriculum Development Publications,pp. 2,48-49,119.
Tomlinson, C.A. (2001).How to differentiate instruction in mixed ability classrooms(2nd ed.)Alexandria VA:Association for Supervision and Curriculum Development Publications,p. 16.
Common Wealth of Virginia.Department of Education.Retrieved February 5, 2009 from www.doe.virginia.gov/VDOE/suptsmemos/2004/inf025.html
Special Needs- Differentiation in Action.Retrieved February 8, 2009 from http://www.teachers.tv/video/21992