5.5: One-to-One and Onto Transformations
- Determine if a linear transformation is onto or one to one.
Let \(T: \mathbb{R}^n \mapsto \mathbb{R}^m\) be a linear transformation. We define the range or image of \(T\) as the set of vectors of \(\mathbb{R}^{m}\) which are of the form \(T \left(\vec{x}\right)\) (equivalently, \(A\vec{x}\)) for some \(\vec{x}\in \mathbb{R}^{n}\). It is common to write \(T\mathbb{R}^{n}\), \(T\left( \mathbb{R}^{n}\right)\), or \(\mathrm{Im}\left( T\right)\) to denote these vectors.
Let \(A\) be an \(m\times n\) matrix where \(A_{1},\cdots , A_{n}\) denote the columns of \(A.\) Then, for a vector \(\vec{x}=\left [ \begin{array}{c} x_{1} \\ \vdots \\ x_{n} \end{array} \right ]\) in \(\mathbb{R}^n\),
\[A\vec{x}=\sum_{k=1}^{n}x_{k}A_{k}\nonumber \]
Therefore, \(A \left( \mathbb{R}^n \right)\) is the collection of all linear combinations of these products.
- Proof
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This follows from the definition of matrix multiplication.
Below is a video on determining if function mappings are one to one and/or onto.
This section is devoted to studying two important characterizations of linear transformations, called one to one and onto. We define them now.
Suppose \(\vec{x}_1\) and \(\vec{x}_2\) are vectors in \(\mathbb{R}^n\). A linear transformation \(T: \mathbb{R}^n \mapsto \mathbb{R}^m\) is called one to one (often written as \(1-1)\) if whenever \(\vec{x}_1 \neq \vec{x}_2\) it follows that : \[T\left( \vec{x}_1 \right) \neq T \left(\vec{x}_2\right)\nonumber \]
Equivalently, if \(T\left( \vec{x}_1 \right) =T\left( \vec{x}_2\right) ,\) then \(\vec{x}_1 = \vec{x}_2\). Thus, \(T\) is one to one if it never takes two different vectors to the same vector.
Below is a video on one to one transformations.
The second important characterization is called onto.
Let \(T: \mathbb{R}^n \mapsto \mathbb{R}^m\) be a linear transformation. Then \(T\) is called onto if whenever \(\vec{x}_2 \in \mathbb{R}^{m}\) there exists \(\vec{x}_1 \in \mathbb{R}^{n}\) such that \(T\left( \vec{x}_1\right) = \vec{x}_2.\)
We often call a linear transformation which is one-to-one an injection . Similarly, a linear transformation which is onto is often called a surjection .
Below is a video on one to one and onto transformation definitions.
Below is a video on onto transformations.
The following proposition is an important result.
Let \(T:\mathbb{R}^n \mapsto \mathbb{R}^m\) be a linear transformation. Then \(T\) is one to one if and only if \(T(\vec{x}) = \vec{0}\) implies \(\vec{x}=\vec{0}\).
- Proof
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We need to prove two things here. First, we will prove that if \(T\) is one to one, then \(T(\vec{x}) = \vec{0}\) implies that \(\vec{x}=\vec{0}\). Second, we will show that if \(T(\vec{x})=\vec{0}\) implies that \(\vec{x}=\vec{0}\), then it follows that \(T\) is one to one. Recall that a linear transformation has the property that \(T(\vec{0}) = \vec{0}\).
Suppose first that \(T\) is one to one and consider \(T(\vec{0})\). \[T(\vec{0})=T\left( \vec{0}+\vec{0}\right) =T(\vec{0})+T(\vec{0})\nonumber \] and so, adding the additive inverse of \(T(\vec{0})\) to both sides, one sees that \(T(\vec{0})=\vec{0}\). If \(T(\vec{x})=\vec{0}\) it must be the case that \(\vec{x}=\vec{0}\) because it was just shown that \(T(\vec{0})=\vec{0}\) and \(T\) is assumed to be one to one.
Now assume that if \(T(\vec{x})=\vec{0},\) then it follows that \(\vec{x}=\vec{0}.\) If \(T(\vec{v})=T(\vec{u}),\) then \[T(\vec{v})-T(\vec{u})=T\left( \vec{v}-\vec{u}\right) =\vec{0}\nonumber \] which shows that \(\vec{v}-\vec{u}=0\). In other words, \(\vec{v}=\vec{u}\), and \(T\) is one to one.
Note that this proposition says that if \(A=\left [ \begin{array}{ccc} A_{1} & \cdots & A_{n} \end{array} \right ]\) then \(A\) is one to one if and only if whenever \[0 = \sum_{k=1}^{n}c_{k}A_{k}\nonumber \] it follows that each scalar \(c_{k}=0\).
Below is a video on comparing one to one and onto matrix transformations.
Below is a video on determining if transformation mappings are one to one and/or onto.
We will now take a look at an example of a one to one and onto linear transformation.
Suppose \[T\left [ \begin{array}{c} x \\ y \end{array} \right ] =\left [ \begin{array}{rr} 1 & 1 \\ 1 & 2 \end{array} \right ] \left [ \begin{array}{r} x \\ y \end{array} \right ]\nonumber \] Then, \(T:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}\) is a linear transformation. Is \(T\) onto? Is it one to one?
Solution
Recall that because \(T\) can be expressed as matrix multiplication, we know that \(T\) is a linear transformation. We will start by looking at onto. So suppose \(\left [ \begin{array}{c} a \\ b \end{array} \right ] \in \mathbb{R}^{2}.\) Does there exist \(\left [ \begin{array}{c} x \\ y \end{array} \right ] \in \mathbb{R}^2\) such that \(T\left [ \begin{array}{c} x \\ y \end{array} \right ] =\left [ \begin{array}{c} a \\ b \end{array} \right ] ?\) If so, then since \(\left [ \begin{array}{c} a \\ b \end{array} \right ]\) is an arbitrary vector in \(\mathbb{R}^{2},\) it will follow that \(T\) is onto.
This question is familiar to you. It is asking whether there is a solution to the equation \[\left [ \begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array} \right ] \left [ \begin{array}{c} x \\ y \end{array} \right ] =\left [ \begin{array}{c} a \\ b \end{array} \right ]\nonumber \] This is the same thing as asking for a solution to the following system of equations. \[\begin{array}{c} x+y=a \\ x+2y=b \end{array}\nonumber \] Set up the augmented matrix and row reduce. \[\left [ \begin{array}{rr|r} 1 & 1 & a \\ 1 & 2 & b \end{array} \right ] \rightarrow \left [ \begin{array}{rr|r} 1 & 0 & 2a-b \\ 0 & 1 & b-a \end{array} \right ] \label{ontomatrix}\] You can see from this point that the system has a solution. Therefore, we have shown that for any \(a, b\), there is a \(\left [ \begin{array}{c} x \\ y \end{array} \right ]\) such that \(T\left [ \begin{array}{c} x \\ y \end{array} \right ] =\left [ \begin{array}{c} a \\ b \end{array} \right ]\). Thus \(T\) is onto.
Now we want to know if \(T\) is one to one. By Proposition \(\PageIndex{1}\) it is enough to show that \(A\vec{x}=0\) implies \(\vec{x}=0\). Consider the system \(A\vec{x}=0\) given by: \[\left [ \begin{array}{cc} 1 & 1 \\ 1 & 2\\ \end{array} \right ] \left [ \begin{array}{c} x\\ y \end{array} \right ] = \left [ \begin{array}{c} 0 \\ 0 \end{array} \right ]\nonumber \]
This is the same as the system given by
\[\begin{array}{c} x + y = 0 \\ x + 2y = 0 \end{array}\nonumber \]
We need to show that the solution to this system is \(x = 0\) and \(y = 0\). By setting up the augmented matrix and row reducing, we end up with \[\left [ \begin{array}{rr|r} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right ]\nonumber \]
This tells us that \(x = 0\) and \(y = 0\). Returning to the original system, this says that if
\[\left [ \begin{array}{cc} 1 & 1 \\ 1 & 2\\ \end{array} \right ] \left [ \begin{array}{c} x\\ y \end{array} \right ] = \left [ \begin{array}{c} 0 \\ 0 \end{array} \right ]\nonumber \]
then \[\left [ \begin{array}{c} x \\ y \end{array} \right ] = \left [ \begin{array}{c} 0 \\ 0 \end{array} \right ]\nonumber \]
In other words, \(A\vec{x}=0\) implies that \(\vec{x}=0\). By Proposition \(\PageIndex{1}\) , \(A\) is one to one, and so \(T\) is also one to one.
We also could have seen that \(T\) is one to one from our above solution for onto. By looking at the matrix given by \(\eqref{ontomatrix}\), you can see that there is a unique solution given by \(x=2a-b\) and \(y=b-a\). Therefore, there is only one vector, specifically \(\left [ \begin{array}{c} x \\ y \end{array} \right ] = \left [ \begin{array}{c} 2a-b\\ b-a \end{array} \right ]\) such that \(T\left [ \begin{array}{c} x \\ y \end{array} \right ] =\left [ \begin{array}{c} a \\ b \end{array} \right ]\). Hence by Definition \(\PageIndex{1}\) , \(T\) is one to one.
Let \(T: \mathbb{R}^4 \mapsto \mathbb{R}^2\) be a linear transformation defined by \[T \left [ \begin{array}{c} a \\ b \\ c \\ d \end{array} \right ] = \left [ \begin{array}{c} a + d \\ b + c \end{array} \right ] \mbox{ for all } \left [ \begin{array}{c} a \\ b \\ c \\ d \end{array} \right ] \in \mathbb{R}^4\nonumber \] Prove that \(T\) is onto but not one to one.
Solution
You can prove that \(T\) is in fact linear.
To show that \(T\) is onto, let \(\left [ \begin{array}{c} x \\ y \end{array} \right ]\) be an arbitrary vector in \(\mathbb{R}^2\). Taking the vector \(\left [ \begin{array}{c} x \\ y \\ 0 \\ 0 \end{array} \right ] \in \mathbb{R}^4\) we have \[T \left [ \begin{array}{c} x \\ y \\ 0 \\ 0 \end{array} \right ] = \left [ \begin{array}{c} x + 0 \\ y + 0 \end{array} \right ] = \left [ \begin{array}{c} x \\ y \end{array} \right ]\nonumber \] This shows that \(T\) is onto.
By Proposition \(\PageIndex{1}\) \(T\) is one to one if and only if \(T(\vec{x}) = \vec{0}\) implies that \(\vec{x} = \vec{0}\). Observe that \[T \left [ \begin{array}{r} 1 \\ 0 \\ 0 \\ -1 \end{array} \right ] = \left [ \begin{array}{c} 1 + -1 \\ 0 + 0 \end{array} \right ] = \left [ \begin{array}{c} 0 \\ 0 \end{array} \right ]\nonumber \] There exists a nonzero vector \(\vec{x}\) in \(\mathbb{R}^4\) such that \(T(\vec{x}) = \vec{0}\). It follows that \(T\) is not one to one.
Below is a video on determining if a linear transformation is one to one and/or onto.
Below is another video on determining if a linear transformation is one to one and/or onto.
The above examples demonstrate a method to determine if a linear transformation \(T\) is one to one or onto. It turns out that the matrix \(A\) of \(T\) can provide this information.
Let \(T: \mathbb{R}^n \mapsto \mathbb{R}^m\) be a linear transformation induced by the \(m \times n\) matrix \(A\). Then \(T\) is one to one if and only if the rank of \(A\) is \(n\). \(T\) is onto if and only if the rank of \(A\) is \(m\).
Consider Example \(\PageIndex{2}\) . Above we showed that \(T\) was onto but not one to one. We can now use this theorem to determine this fact about \(T\).
Let \(T: \mathbb{R}^4 \mapsto \mathbb{R}^2\) be a linear transformation defined by \[T \left [ \begin{array}{c} a \\ b \\ c \\ d \end{array} \right ] = \left [ \begin{array}{c} a + d \\ b + c \end{array} \right ] \mbox{ for all } \left [ \begin{array}{c} a \\ b \\ c \\ d \end{array} \right ] \in \mathbb{R}^4\nonumber \] Prove that \(T\) is onto but not one to one.
Solution
Using Theorem \(\PageIndex{1}\) we can show that \(T\) is onto but not one to one from the matrix of \(T\). Recall that to find the matrix \(A\) of \(T\), we apply \(T\) to each of the standard basis vectors \(\vec{e}_i\) of \(\mathbb{R}^4\). The result is the \(2 \times 4\) matrix A given by \[A = \left [ \begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{array} \right ]\nonumber \] Fortunately, this matrix is already in reduced row-echelon form. The rank of \(A\) is \(2\). Therefore by the above theorem \(T\) is onto but not one to one.
Below is a video on classifying matrix multiplication: onto and one to one.
Below is a video on determining if a 4x4 matrix represents a one to one and/or onto transformation.
Recall that if \(S\) and \(T\) are linear transformations, we can discuss their composite denoted \(S \circ T\). The following examines what happens if both \(S\) and \(T\) are onto.
Let \(T: \mathbb{R}^k \mapsto \mathbb{R}^n\) and \(S: \mathbb{R}^n \mapsto \mathbb{R}^m\) be linear transformations. If \(T\) and \(S\) are onto, then \(S \circ T\) is onto.
Solution
Let \(\vec{z}\in \mathbb{R}^m\). Since \(S\) is onto, there exists a vector \(\vec{y}\in \mathbb{R}^n\) such that \(S(\vec{y})=\vec{z}\). Furthermore, since \(T\) is onto, there exists a vector \(\vec{x}\in \mathbb{R}^k\) such that \(T(\vec{x})=\vec{y}\). Thus \[\vec{z} = S(\vec{y}) = S(T(\vec{x})) = (ST)(\vec{x}),\nonumber \] showing that for each \(\vec{z}\in \mathbb{R}^m\) there exists and \(\vec{x}\in \mathbb{R}^k\) such that \((ST)(\vec{x})=\vec{z}\). Therefore, \(S \circ T\) is onto.
The next example shows the same concept with regards to one-to-one transformations.
Let \(T: \mathbb{R}^k \mapsto \mathbb{R}^n\) and \(S: \mathbb{R}^n \mapsto \mathbb{R}^m\) be linear transformations. Prove that if \(T\) and \(S\) are one to one, then \(S \circ T\) is one-to-one.
Solution
To prove that \(S \circ T\) is one to one, we need to show that if \(S(T (\vec{v})) = \vec{0}\) it follows that \(\vec{v} = \vec{0}\). Suppose that \(S(T (\vec{v})) = \vec{0}\). Since \(S\) is one to one, it follows that \(T (\vec{v}) = \vec{0}\). Similarly, since \(T\) is one to one, it follows that \(\vec{v} = \vec{0}\). Hence \(S \circ T\) is one to one.


13.7: Health Disparity
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Despite prevention efforts, some groups of people are affected by HIV/AIDS, viral hepatitis, STI’s, and TB, more than other groups of people. The occurrence of these diseases at greater levels among certain population groups more than among others is often referred to as a health disparity. Differences may occur by gender, race or ethnicity, education, income, disability, geographic location and sexual orientation among others. Social determinants of health like poverty, unequal access to health care, lack of education, stigma, and racism are linked to health disparities.
Health in the United States is a complex and often contradictory issue. One the one hand, as one of the wealthiest nations, the United States fares well in health comparisons with the rest of the world. However, the United States also lags behind almost every industrialized country in terms of providing care to all its citizens. The following sections look at different aspects of health in America.
Health by Race and Ethnicity
When looking at the social epidemiology of the United States, it is hard to miss the disparities among races. The discrepancy between African Americans and white Americans shows the gap clearly; in 2008, the average life expectancy for white males was approximately five years longer than for African American males: 75.9 compared to 70.9. An even stronger disparity was found in 2007: the infant mortality rate for African Americans was nearly twice that of white Americans at 13.2 compared to 5.6 per 1,000 live births (U.S. Census Bureau 2011). According to a report from the Henry J. Kaiser Foundation (2007), African Americans also have higher incidence of several other diseases and causes of mortality, from cancer to heart disease to diabetes. In a similar vein, it is important to note that ethnic minorities, including Mexican Americans and Native Americans, also have higher rates of these diseases and causes of mortality than whites. Lisa Berkman (2009) notes that this gap started to narrow during the Civil Rights movement in the 1960s, but it began widening again in the early 1980s.
What accounts for these perpetual disparities in health among different ethnic groups?
Much of the answer lies in the level of health care that these groups receive. The National Healthcare Disparities Report (2010) shows that even after adjusting for insurance differences, racial and ethnic minority groups receive poorer quality of care and less access to care than dominant groups. The Report identified these racial inequalities in care: Black Americans, American Indians, and Alaskan Natives received inferior care than Caucasian Americans for about 40 percent of measures Asian ethnicities received inferior care for about 20 percent of measures Among whites, Hispanic whites received 60 percent inferior care of measures compared to non-Hispanic whites (Agency for Health Research and Quality 2010). When considering access to care, the figures were comparable.
Health by Socioeconomic Status
Discussions of health by race and ethnicity often overlap with discussions of health by socioeconomic status, since the two concepts are intertwined in the United States. As the Agency for Health Research and Quality (2010) notes, “racial and ethnic minorities are more likely than non-Hispanic whites to be poor or near poor,” so many of the data pertaining to subordinate groups is also likely to be pertinent to low socioeconomic groups. Marilyn Winkleby and her research associates (1992) state that “one of the strongest and most consistent predictors of a person's morbidity and mortality experience is that person's socioeconomic status (SES). This finding persists across all diseases with few exceptions, continues throughout the entire lifespan, and extends across numerous risk factors for disease.”It is important to remember that economics are only part of the SES picture; research suggests that education also plays an important role. Phelan and Link (2003) note that many behavior-influenced diseases like lung cancer (from smoking), coronary artery disease (from poor eating and exercise habits), and AIDS initially were widespread across SES groups. However, once information linking habits to disease was disseminated, these diseases decreased in high SES groups and increased in low SES groups. This illustrates the important role of education initiatives regarding a given disease, as well as possible inequalities in how those initiatives effectively reach different SES groups.
Health by Gender
Women are affected adversely both by unequal access to and institutionalized sexism in the health care industry. According a recent report from the Kaiser Family Foundation, women experienced a decline in their ability to see needed specialists between 2001 and 2008. In 2008, one quarter of females questioned the quality of her health care (Ranji and Salganico 2011). In this report, we also see the explanatory value of intersection theory. Feminist sociologist Patricia Hill Collins developed this theory, which suggests we cannot separate the effects of race, class, gender, sexual orientation, and other attributes. Further examination of the lack of confidence in the health care system by women, as identified in the Kaiser study, found, for example, women categorized as low income were more likely (32 percent compared to 23 percent) to express concerns about health care quality, illustrating the multiple layers of disadvantage caused by race and sex.
We can see an example of institutionalized sexism in the way that women are more likely than men to be diagnosed with certain kinds of mental disorders. Psychologist Dana Becker notes that 75 percent of all diagnoses of Borderline Personality Disorder (BPD) are for women according to the Diagnostic Statistical Manual of Mental Disorders. This diagnosis is characterized by instability of identity, of mood, and of behavior, and Becker argues that it has been used as a catch-all diagnosis for too many women. She further decries the pejorative connotation of the diagnosis, saying that it predisposes many people, both within and outside of the profession of psychotherapy, against women who have been so diagnosed (Becker).
Many critics also point to the medicalization of women’s issues as an example of institutionalized sexism. Medicalization refers to the process by which previously normal aspects of life are redefined as deviant and needing medical attention to remedy. Historically and contemporaneously, many aspects of women’s lives have been medicalized, including menstruation, pre-menstrual syndrome, pregnancy, childbirth, and menopause.
The medicalization of pregnancy and childbirth has been particularly contentious in recent decades, with many women opting against the medical process and choosing a more natural childbirth. Fox and Worts (1999) find that all women experience pain and anxiety during the birth process, but that social support relieves both as effectively as medical support. In other words, medical interventions are no more effective than social ones at helping with the difficulties of pain and childbirth. Fox and Worts further found that women with supportive partners ended up with less medical intervention and fewer cases of postpartum depression. Of course, access to quality birth care outside of the standard medical models may not be readily available to women of all social classes.
Summary
Despite generally good health in the U.S. compared with less-developed countries, America is still facing challenging issues such as a prevalence of obesity and diabetes. Moreover, Americans of historically disadvantaged racial groups, ethnicities, socioeconomic status, and gender experience lower levels of health care. Mental health and disability are health issues that are significantly impacted by social norms.


5: In-depth Object Studies Last updated Save as PDF Page ID 8258 Ann Smart Martin University of Wisconsin-Madison via University of Wisconsin Pressbooks 5.10: The Rosemaling of Patricia Edmundson: Expressing Norwegian-American Folk Art in Mount Horeb, Wisconsin 5.11: Martin Cliff’s Hardanger Fiddle: An Epitomic Norwegian Cultural Icon 5.12: Explore the Material Culture Meanings of A Frame at Mt. Horeb Historical Society 5.13: The Paradox of the Rocking Kubbestol 5.14: Olaf ‘Ole’ Colberson’s Wall Murals: Finding a Voice Against Stigma 5.15: The Mystery of the Mount Horeb Mangle Board 5.1: Stugard Family Plant Stand: A Rustic Handicraft Tradition 5.2: Cunneen's Mini Dress 5.3: Goli Music Cabinet 5.4: Aslak Lie Cupboard 5.5: Unraveling the Past: The Resonance of Ellen Donald Sweet’s Mid-Nineteenth Century Quilt 5.6: The Sweet Family Dulcimer - Object Study 5.7: Emigrant Trunks 5.8: The Little Kubbestol: A View of Norwegian-American Ethnic Identity through an Autobiography of an Object Biography 5.9: Community, Kindness, and Humor: Looking at the Trolls of Mount Horeb, WI


21.3: The H-R and the Study of Stellar Evolution
By the end of this section, you will be able to:
- Determine the age of a protostar using an H–R diagram and the protostar’s luminosity and temperature
- Explain the interplay between gravity and pressure, and how the contracting protostar changes its position in the H–R diagram as a result
One of the best ways to summarize all of these details about how a star or protostar changes with time is to use a Hertzsprung-Russell (H–R) diagram. Recall from The Stars: A Celestial Census that, when looking at an H–R diagram, the temperature (the horizontal axis) is plotted increasing toward the left. As a star goes through the stages of its life, its luminosity and temperature change. Thus, its position on the H–R diagram, in which luminosity is plotted against temperature, also changes. As a star ages, we must replot it in different places on the diagram. Therefore, astronomers often speak of a star moving on the H–R diagram, or of its evolution tracing out a path on the diagram. In this context, “tracing out a path” has nothing to do with the star’s motion through space; this is just a shorthand way of saying that its temperature and luminosity change as it evolves.
Watch an animation of the stars in the Omega Centauri cluster as they rearrange according to luminosity and temperature, forming a Hertzsprung-Russell (H–R) diagram.
To estimate just how much the luminosity and temperature of a star change as it ages, we must resort to calculations. Theorists compute a series of models for a star, with each successive model representing a later point in time. Stars may change for a variety of reasons. Protostars, for example, change in size because they are contracting, and their temperature and luminosity change as they do so. After nuclear fusion begins in the star’s core (see Stars from Adolescence to Old Age), main-sequence stars change because they are using up their nuclear fuel.
Given a model that represents a star at one stage of its evolution, we can calculate what it will be like at a slightly later time. At each step, the model predicts the luminosity and size of the star, and from these values, we can figure out its surface temperature. A series of points on an H–R diagram, calculated in this way, allows us to follow the life changes of a star and hence is called its evolutionary track .
Evolutionary Tracks
Let’s now use these ideas to follow the evolution of protostars that are on their way to becoming main-sequence stars. The evolutionary tracks of newly forming stars with a range of stellar masses are shown in Figure 21.12. These young stellar objects are not yet producing energy by nuclear reactions, but they derive energy from gravitational contraction—through the sort of process proposed for the Sun by Helmhotz and Kelvin in this last century (see the chapter on The Sun: A Nuclear Powerhouse).
Initially, a protostar remains fairly cool with a very large radius and a very low density. It is transparent to infrared radiation, and the heat generated by gravitational contraction can be radiated away freely into space. Because heat builds up slowly inside the protostar, the gas pressure remains low, and the outer layers fall almost unhindered toward the center. Thus, the protostar undergoes very rapid collapse, a stage that corresponds to the roughly vertical lines at the right of Figure 21.12. As the star shrinks, its surface area gets smaller, and so its total luminosity decreases. The rapid contraction stops only when the protostar becomes dense and opaque enough to trap the heat released by gravitational contraction.
When the star begins to retain its heat, the contraction becomes much slower, and changes inside the contracting star keep the luminosity of stars like our Sun roughly constant. The surface temperatures start to build up, and the star “moves” to the left in the H–R diagram. Stars first become visible only after the stellar wind described earlier clears away the surrounding dust and gas. This can happen during the rapid-contraction phase for low-mass stars, but high-mass stars remain shrouded in dust until they end their early phase of gravitational contraction (see the dashed line in Figure 21.12).
To help you keep track of the various stages that stars go through in their lives, it can be useful to compare the development of a star to that of a human being. (Clearly, you will not find an exact correspondence, but thinking through the stages in human terms may help you remember some of the ideas we are trying to emphasize.) Protostars might be compared to human embryos—as yet unable to sustain themselves but drawing resources from their environment as they grow. Just as the birth of a child is the moment it is called upon to produce its own energy (through eating and breathing), so astronomers say that a star is born when it is able to sustain itself through nuclear reactions (by making its own energy.)
When the star’s central temperature becomes high enough (about 12 million K) to fuse hydrogen into helium, we say that the star has reached the main sequence (a concept introduced in The Stars: A Celestial Census). It is now a full-fledged star, more or less in equilibrium, and its rate of change slows dramatically. Only the gradual depletion of hydrogen as it is transformed into helium in the core slowly changes the star’s properties.
The mass of a star determines exactly where it falls on the main sequence. As Figure 21.12 shows, massive stars on the main sequence have high temperatures and high luminosities. Low-mass stars have low temperatures and low luminosities.
Objects of extremely low mass never achieve high-enough central temperatures to ignite nuclear reactions. The lower end of the main sequence stops where stars have a mass just barely great enough to sustain nuclear reactions at a sufficient rate to stop gravitational contraction. This critical mass is calculated to be about 0.075 times the mass of the Sun. As we discussed in the chapter on Analyzing Starlight, objects below this critical mass are called either brown dwarfs or planets. At the other extreme, the upper end of the main sequence terminates at the point where the energy radiated by the newly forming massive star becomes so great that it halts the accretion of additional matter. The upper limit of stellar mass is between 100 and 200 solar masses.
Evolutionary Timescales
How long it takes a star to form depends on its mass. The numbers that label the points on each track in Figure 21.12 are the times, in years, required for the embryo stars to reach the stages we have been discussing. Stars of masses much higher than the Sun’s reach the main sequence in a few thousand to a million years. The Sun required millions of years before it was born. Tens of millions of years are required for stars of lower mass to evolve to the lower main sequence. (We will see that this turns out to be a general principle: massive stars go through all stages of evolution faster than low-mass stars do.)
We will take up the subsequent stages in the life of a star in Stars from Adolescence to Old Age, examining what happens after stars arrive in the main sequence and begin a “prolonged adolescence” and “adulthood” of fusing hydrogen to form helium. But now we want to examine the connection between the formation of stars and planets.


5.7: How to Read Residential Electric and Natural Gas Meters
You can read your own meters to help monitor your electric or gas energy use. During the heating season, your energy use should be compared to the number of heating degree days for the same time period; during the cooling season, compare your energy use to the number of cooling degree days.
Heating and cooling degree days are a simple measure of the effect of weather on your energy needs: using the average temperature for each day, each degree Fahrenheit below 65°F is counted as one heating degree day, and each degree Fahrenheit above 65°F is counted as one cooling degree day. Your heating and cooling use should be proportional to the number of heating and cooling degree days for the time period in question.
You may also wish to contact your local utility companies for more information about reading your meter. If monthly information is good enough, your utility bills could have all the information you need. Just be sure the bills are based on actual, not estimated, meter readings, and be aware of when the meter was read, because the time period between readings can vary. Contact your local utility if you are uncertain about this.
Electric Meters
The basic unit of measure of electric power is the watt. One thousand watts are called a kilowatt. If you use one thousand watts of power in one hour you have used a kilowatt-hour (kWh).
The standard electric power meter is a clock-like device driven by the electricity moving through it. As the home draws current from the power lines, a set of small gears inside the meter move. The number of revolutions is recorded by the dials that you can see on the face of the meter. The speed of the revolutions depends on the amount of current drawn -- the more power consumed at any one instant, the faster the gears will rotate.
When reading an electric meter, read and write down the numbers as shown on the dials from right to left . When the pointer is directly on a number, look at the dial to the right. If it has passed zero, use the next higher number. If it has not passed zero, use the lower number. Record the numbers shown by writing down the value of the dial to your extreme right first and the rest as you come to them. Should the hand of a dial fall between two numbers, use the smaller of the two numbers.
Natural Gas Meters
Natural gas is commonly measured by the cubic foot, and you are billed by the thousands of cubic feet (MCF) or hundreds of cubic feet (CCF). You may also be billed by the therm, which is about the same as a CCF or 100 cubic feet. To measure the amount of electricity or gas that you use, the utility installs a meter between the incoming electric power or gas lines and the point of distribution at the house.
A gas meter is driven by the force of the moving gas in the pipe, and also turns faster as the flow increases. Each time the dial with the lower value makes one complete revolution, the pointer on the next higher value dial moves ahead one digit.
When reading a gas meter, read and write down the numbers as shown on the dials from left to right (opposite of an electric meter). It is important to note that on both types of meters, the hands of adjacent dials turn in opposite directions to each other.
A natural gas meter on a house. | Photo courtesy of ©iStockphoto/fstockfoto


3.1: Matter
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Atoms, Molecules, & Compounds
At its most fundamental level, life is made of matter . Matter is something that occupies space and has mass. All matter is composed of elements , substances that cannot be broken down or transformed chemically into other substances. Each element is made of atoms, each with a constant number of protons and unique properties. A total of 118 elements have been defined; however, only 92 occur naturally and fewer than 30 are found in living cells. The remaining 26 elements are unstable and therefore do not exist for very long or are theoretical and have yet to be detected. Each element is designated by its chemical symbol (such as H, N, O, C, and Na), and possesses unique properties. These unique properties allow elements to combine and to bond with each other in specific ways.
An atom is the smallest component of an element that retains all of the chemical properties of that element. For example, one hydrogen atom has all of the properties of the element hydrogen, such as it exists as a gas at room temperature and it bonds with oxygen to create a water molecule. Hydrogen atoms cannot be broken down into anything smaller while still retaining the properties of hydrogen. If a hydrogen atom were broken down into subatomic particles, it would no longer have the properties of hydrogen. At the most basic level, all organisms are made of a combination of elements. They contain atoms that combine together to form molecules. In multicellular organisms, such as animals, molecules can interact to form cells that combine to form tissues, which make up organs. These combinations continue until entire multicellular organisms are formed.
All matter, whether it be a rock or an organism, is made of atoms. Often, these atoms combine to form molecules . Molecule are chemicals made from two or more atoms bonded together. Some molecules are very simple, like O2, which is comprised of just two oxygen atoms. Some molecules used by organisms, such as DNA, are made of many millions of atoms. All atoms contain protons, electrons, and neutrons (Figure \(\PageIndex{1}\) below). The only exception is hydrogen (H), which is made of one proton and one electron. A proton is a positively charged particle that resides in the nucleus (the core of the atom) of an atom and has a mass of 1 and a charge of +1. An electron is a negatively charged particle that travels in the space around the nucleus. In other words, it resides outside of the nucleus. It has a negligible mass and has a charge of –1. Neutrons , like protons, reside in the nucleus of an atom. They have a mass of 1 and no charge. The positive (protons) and negative (electrons) charges balance each other in a neutral atom, which has a net zero charge.
Each element contains a different number of protons and neutrons, giving it its own atomic number and mass number. The atomic number of an element is equal to the number of protons that element contains. The mass number is the number of protons plus the number of neutrons of that element. Therefore, it is possible to determine the number of neutrons by subtracting the atomic number from the mass number.
Isotopes are different forms of the same element that have the same number of protons, but a different number of neutrons. Some elements, such as carbon, potassium, and uranium, have naturally occurring isotopes. Carbon 12, the most common isotope of carbon, contains six protons and six neutrons. Therefore, it has a mass number of 12 (six protons and six neutrons) and an atomic number of 6 (which makes it carbon). Carbon 14 contains six protons and eight neutrons. Therefore, it has a mass number of 14 (six protons and eight neutrons) and an atomic number of 6, meaning it is still the element carbon. These two alternate forms of carbon are isotopes. Some isotopes are unstable and will lose protons, other subatomic particles, or energy to form more stable elements. These are called radioactive isotopes or radioisotopes.
EVOLUTION IN ACTION: Carbon dating
Carbon-14 (14C) is a naturally occurring radioisotope that is created in the atmosphere by cosmic rays. This is a continuous process, so more 14C is always being created. As a living organism develops, the relative level of 14C in its body is equal to the concentration of 14C in the atmosphere. When an organism dies, it is no longer ingesting 14C, so the ratio will decline. 14C decays to 14N by a process called beta decay; it gives off energy in this slow process. After approximately 5,730 years, only one-half of the starting concentration of 14C will have been converted to 14N. The time it takes for half of the original concentration of an isotope to decay to its more stable form is called its half-life.
Because the half-life of 14C is long, it is used to age formerly living objects, such as fossils. Using the ratio of the 14C concentration found in an object to the amount of 14C detected in the atmosphere, the amount of the isotope that has not yet decayed can be determined. Based on this amount, the age of the fossil can be calculated to about 50,000 years (Figure \(\PageIndex{2}\) below). Isotopes with longer half-lives, such as potassium-40, are used to calculate the ages of older fossils. Through the use of carbon dating, scientists can reconstruct the ecology and biogeography of organisms living within the past 50,000 years.
Chemical Bonds
How elements interact with one another depends on the number of electrons and how they are arranged. When an atom does not contain equal numbers of protons and electrons it is called an ion . Because the number of electrons does not equal the number of protons, each ion has a net charge . For example, if sodium loses an electron, it now has 11 protons and only 10 electrons, leaving it with an overall charge of +1. Positive ions are formed by losing electrons and are called cations . Negative ions are formed by gaining electrons and are called anions . Elemental anionic names are changed to end in -ide. As an example, when chlorine becomes an ion it is referred to as chloride.
Ionic and covalent bonds are strong bonds formed between two atoms. These bonds hold atoms together in a relatively stable state. Ionic bonds are formed between two oppositely charged ions (an anion and a cation). Because positive and negative charges attract, these ions are held together much like two oppositely charged magnets would stick together. Covalent bonds form when electrons are shared between two atoms. Each atom shares one of their electrons, which then orbits the nuclei of both atoms, holding the two atoms together. Covalent bonds are the strongest and most common form of chemical bond in organisms. Unlike most ionic bonds, covalent bonds do not dissociate in water.
Covalent bonds come in two varieties: polar and non-polar. A non-polar covalent bond occurs when electrons are shared equally between the two atoms. Polar covalent bonds form when the electrons are shared unequally. Why does this occur? Each element has a known electronegativity : a measure of their affinity for electrons. Some elements, such as oxygen, are very electronegative because they strongly attract electrons from other atoms. Hydrogen, meanwhile, has low electronegativity and thus weakly attracts electrons, in comparison. Polar covalent bonds form when the two atoms involved have significantly different electronegativities. In biological systems, this occurs when oxygen bonds with hydrogen and when nitrogen (also quite electronegative) bonds with hydrogen.
When oxygen and hydrogen bond, for example, the shared electrons are pulled more strongly toward oxygen and thus farther away from hydrogen’s nucleus. Because the electrons move farther away from hydrogen, it becomes slightly positively charged (δ+). The oxygen becomes slightly negatively charged as the electrons become closer to it (δ–). If two molecules with polar covalent bonds approach one another, they can interact due to the attraction of opposite electrical charges. For example, the slight positive charge of hydrogen in a water molecule can be attracted to the slight negative charge of oxygen in a different water molecule (Figure \(\PageIndex{3}\)). This interaction between two polar molecules is called a hydrogen bond . This type of bond is very common in organisms. Notably, hydrogen bonds give water the unique properties that sustain life. If it were not for hydrogen bonding, water would be a gas rather than a liquid at room temperature.
WATER IS CRUCIAL TO MAINTAINING LIFE
Do you ever wonder why scientists spend time looking for water on other planets? It is because water is essential to life; even minute traces of it on another planet can indicate that life could or did exist on that planet. Water is one of the more abundant molecules in living cells and the one most critical to life as we know it.  Without it, life simply would not exist.
- Water is polar. The hydrogen and oxygen atoms within water molecules form polar covalent bonds. The shared electrons spend more time associated with the oxygen atom than they do with hydrogen atoms. There is no overall charge to a water molecule, but there is a slight positive charge on each hydrogen atom and a slight negative charge on the oxygen atom. Because of these charges, the slightly positive hydrogen atoms repel each other and form the unique shape. Each water molecule attracts other water molecules because of the positive and negative charges in the different parts of the molecule. Water also attracts other polar molecules (such as sugars) that can dissolve in water and are referred to as hydrophilic (“water-loving”).
- Water stabilizes temperature. The hydrogen bonds in water allow it to absorb and release heat energy more slowly than many other substances. Temperature is a measure of the motion (kinetic energy) of molecules. As the motion increases, energy is higher and thus temperature is higher. Water absorbs a great deal of energy before its temperature rises. Increased energy disrupts the hydrogen bonds between water molecules. Because these bonds can be created and disrupted rapidly, water absorbs an increase in energy and temperature changes only minimally. This means that water moderates temperature changes within organisms and in their environments.
- Water is an excellent solvent. Because water is polar, with slight positive and negative charges, ionic compounds and polar molecules can readily dissolve in it. Water is, therefore, what is referred to as a solvent—a substance capable of dissolving another substance. The charged particles will form hydrogen bonds with a surrounding layer of water molecules.
- Water is cohesive. Have you ever filled up a glass of water to the very top and then slowly added a few more drops? Before it overflows, the water actually forms a dome-like shape above the rim of the glass. This water can stay above the glass because of the property of cohesion. In cohesion, water molecules are attracted to each other (because of hydrogen bonding), keeping the molecules together at the liquid-air (gas) interface, although there is no more room in the glass. Cohesion gives rise to surface tension, the capacity of a substance to withstand rupture when placed under tension or stress. When you drop a small scrap of paper onto a droplet of water, the paper floats on top of the water droplet, although the object is denser (heavier) than the water. This occurs because of the surface tension that is created by the water molecules. Cohesion and surface tension keep the water molecules intact and the item floating on the top. It is even possible to “float” a steel needle on top of a glass of water if you place it gently, without breaking the surface tension. These cohesive forces are also related to the water’s property of adhesion, or the attraction between water molecules and other molecules. This is observed when water “climbs” up a straw placed in a glass of water. You will notice that the water appears to be higher on the sides of the straw than in the middle. This is because the water molecules are attracted to the straw and therefore adhere to it. Cohesive and adhesive forces are important for sustaining life. For example, because of these forces, water can flow up from the roots to the tops of plants to feed the plant.
Buffers, pH, Acids, and Bases
The pH of a solution is a measure of its acidity or alkalinity . The pH scale ranges from 0 to 14. A change of one unit on the pH scale represents a change in the concentration of hydrogen ions by a factor of 10, a change in two units represents a change in the concentration of hydrogen ions by a factor of 100. Thus, small changes in pH represent large changes in the concentrations of hydrogen ions. Pure water is neutral. It is neither acidic nor basic and has a pH of 7.0. Anything below 7.0 (ranging from 0.0 to 6.9) is acidic, and anything above 7.0 (from 7.1 to 14.0) is alkaline. The blood in your veins is slightly alkaline (pH = 7.4). The environment in your stomach is highly acidic (pH = 1 to 2). Orange juice is mildly acidic (pH = approximately 3.5), whereas baking soda is basic (pH = 9.0).
Acids are substances that provide hydrogen ions (H+) and lower pH, whereas bases provide hydroxide ions (OH–) and raise pH. The stronger the acid, the more readily it donates H+. For example, hydrochloric acid and lemon juice are very acidic and readily give up H+ when added to water. Conversely, bases are those substances that readily donate OH–. The OH– ions combine with H+ to produce water, which raises a substance’s pH. Sodium hydroxide and many household cleaners are very alkaline and give up OH– rapidly when placed in water, thereby raising the pH.
How is it that we can ingest or inhale acidic or basic substances and not die? Buffers are the key. Buffers readily absorb excess H+ or OH–, keeping the pH of the body carefully maintained in the aforementioned narrow range. Carbon dioxide is part of a prominent buffer system in the human body; it keeps the pH within the proper range. This buffer system involves carbonic acid (H 2 CO 3 ) and bicarbonate (HCO 3 –) anion. If too much H+ enters the body, bicarbonate will combine with the H+ to create carbonic acid and limit the decrease in pH. Likewise, if too much OH– is introduced into the system, carbonic acid will combine with it to create bicarbonate and limit the increase in pH. While carbonic acid is an important product in this reaction, its presence is fleeting because the carbonic acid is released from the body as carbon dioxide gas each time we breathe. Without this buffer system, the pH in our bodies would fluctuate too much and we would fail to survive.
Biological Molecules
Besides water, the molecules necessary for life are organic. Organic molecules are those that contain carbon covalently bonded to hydrogen. In addition, they may contain oxygen, nitrogen, phosphorus, sulfur, and additional elements.There are four major classes of organic molecules: carbohydrates , lipids , proteins , and nucleic acids. Each is an important component of the cell and performs a wide array of functions.
It is often said that life is “carbon-based.” This means that carbon atoms, bonded to other carbon atoms or other elements, form the fundamental components of many of the molecules found uniquely in living things. Other elements play important roles in biological molecules, but carbon certainly qualifies as the “foundation” element for molecules in living things. It is the bonding properties of carbon atoms that are responsible for its important role.
Carbon can form four covalent bonds with other atoms or molecules. The simplest organic carbon molecule is methane (CH4), in which four hydrogen atoms bind to a carbon atom (Figure \(\PageIndex{5}\)). Carbohydrates include what are commonly referred to as simple sugars, like glucose, and complex carbohydrates such as starch. While many types of carbohydrates are used for energy, some are used for structure by most organisms, including plants and animals. For example, cellulose is a complex carbohydrate that adds rigidity and strength to the cell walls of plants. The suffix “-ose” denotes a carbohydrate, but note that not all carbohydrates were given that suffix when names (e.g., starch).
Lipids include a diverse group of compounds that are united by a common feature. Lipids are hydrophobic (“water-fearing”), or insoluble in water, because they are non-polar molecules (molecules that contain non-polar covalent bonds) . Lipids perform many different functions in a cell. Cells store energy for long-term use in the form of lipids called fats. Lipids also provide insulation from the environment for plants and animals. For example, they help keep aquatic birds and mammals dry because of their water-repelling nature. Lipids are also the building blocks of many hormones and are an important constituent of cellular membranes. Lipids include fats, oils, waxes, phospholipids, and steroids.
Proteins are one of the most abundant organic molecules in living systems and have the most diverse range of functions of all macromolecules. They are all polymers of amino acids. The functions of proteins are very diverse because there are 20 different chemically distinct amino acids that form long chains, and the amino acids can be in any order. Proteins can function as enzymes, hormones, contractile fibers, cytoskeleton rods, and much more. Enzymes are vital to life because they act as catalyst in biochemical reactions (like digestion). Each enzyme is specific for the substrate (a reactant that binds to an enzyme) upon which it acts. Enzymes can function to break molecular bonds, to rearrange bonds, or to form new bonds.
Nucleic acids are very large molecules that are important to the continuity of life. They carry the genetic blueprint of a cell and thus the instructions for its functionality. The two main types of nucleic acids are deoxyribonucleic acid ( DNA ) and ribonucleic acid (RNA). DNA is the genetic material found in all organisms, ranging from single-celled bacteria to multicellular mammals. The other type of nucleic acid, RNA , is mostly involved in protein synthesis. The DNA molecules never leave the nucleus, but instead use an RNA intermediary to communicate with the rest of the cell. Other types of RNA are also involved in protein synthesis and its regulation. DNA and RNA are made up of small building blocks known as nucleotides . The nucleotides combine with each other to form a polynucleotide: DNA or RNA.  DNA has a beautiful double-helical structure (Figure \(\PageIndex{6}\)).
Additional Resources:
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- Concepts of Biology by OpenStax is licensed under CC BY 4.0. Modified from original by Matthew R. Fisher.


10.0: Chapter Scenario
WorldCorp’s Fairfax, Virginia, location is a manufacturer of consumer electronics, and it regularly fulfills orders for its Portland, Maine, location, where their products are shipped worldwide through the port. WorldCorp’s sales are increasing, but the Fairfax plant is struggling to keep up with the orders. Their Fairfax location is more than ten years old and no longer has the capacity they need. The corporate headquarters has approved an expansion of their facility. The regional manager wants to keep the original facility but increase the amount of square footage through adjacent additions. Executing this expansion will involve project planning, acquiring the needed materials, determining the needed funding, working with outside suppliers, and so forth. The data generated during the expansion will need to be evaluated and regularly tracked to ensure the project stays on time and within the budgeted resources.
Microsoft Excel can facilitate this type of analysis to aid the decision makers as the projects progress. In this chapter, you will use the “what-if” scenarios of each location in the walk-throughs and exercises to rename cell ranges, use the AutoFill feature in tables, and practice simultaneous calculations of formulas. Excel also supports the associated statistical and financial analyses required for WorldCorp.


3.9: Paraphrasing and Summarizing
Paraphrasing
The essay assignments for most first-year composition classes call on you to use information from various texts. However, you cannot just copy the exact words out of the text you are writing about. If you do, you are plagiarizing and may put your academic career in jeopardy. If you are not citing direct quotes and citing them, you must put the words from the readings fully into your own words and cite them.
When paraphrasing, there are two basic kinds of changes you can make:
- Change the words so you are using synonyms wherever possible. For example, imagine you are paraphrasing the sentence “After work, Judy rushed back to her house.” Instead of “her house” you can say “her home” or “her place.” Instead of “rushed” you could say “hurried.”
- Change the sentence structure so that it is different. One method is to switch the order of dependent and independent clauses. For example, instead of “After work, Judy rushed back to her house” you might write, “Judy was in a hurry to get back home at the end of the day.”
However, this is really just a bit simplistic. It still may not give you a full paraphrase. A full paraphrase, which should convey another's idea fully and truly, should be in your own words (except for, perhaps, key words and data measures). One of the best ways to paraphrase another's idea is to put aside the text (don't look at it) and write down the gist (the general idea) of what the author is saying.
One important thing to remember is that you choose which information is important. For a simple example,
Joe drove after hours for Lyft for six months to save up enough money to buy himself a laptop.
You could decide that some of the detail is not necessary for your paper. You could leave the unimportant detail out of your paraphrase, like this:
In order to afford a laptop, Joe worked for a long time.
The author of this sentence decided that the specific job Joe held, the number of years, and the fact that it was his dream stereo wasn’t important. That was his choice.
Here is a video that explains the above with additional examples.
How to Paraphrase in Five Easy St eps. Author ed by: Scribbr. License: All Rights Reserved. License Terms: Standard YouTube license.
Remember: When paraphrasing another's idea, be sure to still provide an in-text citation.
How to avoid plagiarism when paraphrasing a text:
Stop, Thief! Avoid Plagiarism by Paraphrasing . Authored by: Emily Nimsakont. All Rights Reserved. Standard YouTube License.
Exercise 1\(\PageIndex{1}\)
Using the methods discussed above (especially the "don't look at the original method") paraphrase the following passage on your own piece of paper.
Infectious diseases are transmitted from person to person by direct or indirect contact. Certain types of viruses, bacteria, parasites, and fungi can all cause infectious disease. Malaria, measles, and respiratory illnesses are examples of infectious diseases. (Source: Healthline.com. No author).
Exercise 2\(\PageIndex{1}\)
Using the methods discussed above (especially the "don't look at the original method") paraphrase the following passage on your own piece of paper.
As the continued spread of COVID-19 puts much of the U.S. economy at a standstill, many are wondering how the economic slowdown will affect the real estate market. While construction is considered an essential business in the majority of states and real estate agents have transitioned to online showings, housing markets nationwide are likely to struggle, and some appear to be far more vulnerable than others. (Source: Evan Comen, May 26, 2020, USA Today)
Summarizing
How to Write a Summary
Proficient students understand that summarizing , identifying what is most important and restating the text (or other media) in your own words, is an important tool for college success.
After all, if you really know a subject, you will be able to summarize it. If you cannot summarize a subject, even if you have memorized all the facts about it, you can be absolutely sure that you have not learned it. And, if you truly learn the subject, you will still be able to summarize it months or years from now.
Proficient students may monitor their understanding of a text by summarizing as they read. They understand that if they can write a one- or two-sentence summary of each paragraph after reading it, then that is a good sign that they have correctly understood it. If they can not summarize the main idea of the paragraph, they know that comprehension has broken down and they need to use fix-up strategies to repair understanding.
How to Write a Summary (A Review)
A summary is a brief restatement of someone else’s points in your own words . It needs to be:
- Brief – much shorter than the original, so not the details but just the main points
- Complete – including all the key information
- Accurate – the correct information
- Objective – not your opinion, but what the writer actually communicates
Step A. Annotating the Text for Main Ideas
1. Annotate for engagement and understanding: be sure you stayed awake and actually understood what you read!
2. Find the main ideas of each of the sections and, when appropriate, each of the paragraphs in your section.
3. You may underline the main ideas, or write them in your own words in the margin.
Remember, a main idea is a complete idea or sentence, not just a couple of words.
- Don’t write for your main idea: “youth unemployment”
- Do say: “Youth unemployment has increased.”
Step B. Planning the Summary
Write the main ideas of each paragraph and each section in your own words – and make sure each is a complete sentence. Do not include any judgment about what is in the article.
Step C. Write a Draft Summary
1. Introduce the name of the article and the author in the first sentence along with the main idea of the article.
2. Type up the paraphrased main ideas in complete sentences.
3. Analyze how the ideas connect and relate to each other. (For example, does one idea cause the next? Are some ideas examples of a larger point?) Organize the sentences into a logical order.
4. Include transition words that help the reader understand these connections.
5. Edit out ideas that repeat, and change the order of sentences as necessary.
6. Write the name of the article, the authors, and the controlling idea (main idea of the whole reading) in the first or second sentence.
7. Make sure that important details are included, but not details that are too specific.
8. Do not give your own opinions or say whether the authors do a good or bad job.
9. If the article discusses a study, the summary should briefly describe the purpose of the study and include the results of the study on a high level.
Summary Writing Format
- When writing a summary, remember that it should be in the form of a paragraph.
- A summary begins with an introductory sentence that states the text’s title, author and main point of the text as you see it.
- A summary is written in your own words.
- A summary contains only the ideas of the original text. Do not insert any of your own opinions, interpretations, deductions or comments into a summary.
Example \(\PageIndex{1}\)
Summary Writing Format Example
In the essay Santa Ana , author Joan Didion’s main point is ( state main point ). According to Didion “… passage 1 …” (para.3). Didion also writes “… passage 2 …” (para.8). Finally, she states “… passage 3 …” (para. 12) Write a last sentence that “wraps” up your summary; often a simple rephrasing of the main point.
Summary: Using it Wisely
Knowing how to summarize something you have read, seen, or heard is a valuable skill, one you have probably used in many writing assignments. It is important, though, to recognize when you must go beyond describing, explaining, and restating texts and offer a more complex analysis. This page will help you distinguish between summary and analysis and avoid inappropriate summary in your academic writing.
It is important that your keep your assignment and your audience in mind as you write. If your assignment requires an argument with a thesis statement and supporting evidence—as many academic writing assignments do—then you should limit the amount of summary in your paper. You might use summary to provide background, set the stage, or illustrate supporting evidence, but keep it very brief: a few sentences should do the trick. Most of your paper should focus on your argument.
Writing a summary of what you know about your topic before you start drafting your actual paper can sometimes be helpful. If you are unfamiliar with the material you’re analyzing, you may need to summarize what you’ve read in order to understand your reading and get your thoughts in order. Once you figure out what you know about a subject, it’s easier to decide what you want to argue.
Why is it so tempting to stick with summary and skip analysis?
Many writers rely too heavily on summary because it is what they can most easily write. If you’re stalled by a difficult writing prompt, summarizing may be more appealing than staring at the computer for three hours and wondering what argument to make about the content.
To write a more analytical paper, you may need to carefully consider your writing assignment before reading, viewing, or listening to the material about which you’ll be writing so that your encounter with the material will be more purposeful.
How do I know if I’m writing an argument instead of just summarizing?
If you answer yes to the questions below, it is a sign that your paper may have analysis rather than just summary (which is usually a good thing):
- Am I making an original argument about the text?
- Have I arranged my evidence around my own points, rather than just stating what the author has said?
- Am I explaining why or how an aspect of the text is significant?
Strategies for Writing a Summary
You can use T.I.P.S. to help you write a summary.
T=topic
I=main idea
P=major supporting points
S=summarize
Read through the selection you are summarizing thoroughly, jotting down the topic, main idea, major supporting points. Then, in a couple of lines, summarize the information you have jotted down in your own words. This is your summary.
How to Write a Summary by Paraphrasing Source Material
When you paraphrase material from a source, you restate the information from an entire sentence or passage in your own words, using your own original sentence structure. A paraphrased source differs from a summarized source in that you focus on restating the ideas, not condensing them.
It is important to check your paraphrase against the source material to make sure it is both accurate and original. Inexperienced writers sometimes use the thesaurus method of paraphrasing—that is, they simply rewrite the source material, replacing most of the words with synonyms. This constitutes a misuse of sources. A true paraphrase restates the gist of the ideas using the writer’s own language and style
Example \(\PageIndex{1}\)
Example Paraphrase Writing Format
In the essay Santa Ana , author Joan Didion’s main point is ( state main point ). According to Didion … PARAPHRASE 1 …”. Didion also CLAIMS… PARAPHRASE 2 …”. Finally, she CLAIMS “… PARAPHRASE 3 …” Write a last sentence that “wraps” up your summary; often a simple rephrasing of the main point.
Exercise \(\PageIndex{1}\)
Summarizing Circles: A small group/ class activity
Practice summarizing by working with teams.
1. Divide the text you are summarizing into small sections.
2. In groups of three or so, ask students to summarize each section. They must use words such as
“In other words,….” “Here the author is saying that….” To summarize,
3. Students should try to write their section down in a single line using these starters. They should not use the words used in the text, but instead come up with their own words to explain the ideas in the text. They should get at the main idea in the section rather than just repeat the details.
4. As a class, determine the main idea of the whole passage. Write up a sample summary on the board based on the summaries from the sections.
Contributors and Attributions
- Adapted from English Composition I: Rhetorical Methods-Based , Provided by Lumen Learning, License: CC BY: Attribution
This page most recently revised on June 6, 2020.


<IP_ADDRESS>: Understanding Your Health Coverage
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Health coverage pays for provider services, medications, hospital care, and special equipment when you’re sick. It’s also important when you’re not sick. Marketplace coverage includes preventive health services like immunizations for children and adults, annual doctor visits for women and seniors, screening and counseling for people of all ages, and more. The best news – it’s free. Here are 4 things to know about your health coverage:
- All Marketplace health plans must provide you with a Summary of Benefits and Coverage , which includes coverage examples showing how the plan might help pay for services.
- The actual costs and care will vary by your health care needs and your coverage.
- You can use our glossary to understand key terms like copayment , deductible , network , and premium .
- If you have questions about your coverage, you can contact your health plan, state Medicaid program , or Children’s Health Insurance Program (CHIP) to get more information.
See why it’s so important to have health coverage and share what you learn with your family and friends.
Watch the video: “Understand the Plan” – from Coverage to Care
Watch the video: “Words to Know” – from Coverage to Care
Print your own booklet: Step 2 – Understand Your Health Coverage
Contributors and Attributions
- Authored by : HealthCare.gov Blog. Located at : https://www.healthcare.gov/blog/understanding-your-health-coverage/ . License : Public Domain: No Known Copyright


12.6: Verbs and Verb Tense
Suppose you must give an oral presentation about what you did last summer. How do you make it clear that you are talking about the past and not about the present or the future? Using the correct verb tense can help you do this. Verbs indicate actions or states of being in the past, present, or future using tenses.
It is important to use the proper verb tense. Otherwise, your listener might not understand the time frame you are talking or writing about. In addition, mistakes in tense often leave a listener or reader with a negative impression, especially in a formal or work context.
One way to categorize verbs is whether they follow certain rules or not: regular verbs (verbs that follow tense rules consistently with regard to spelling) and irregular verbs (verbs that don't follow those rules). Unfortunately, in English, some of the most common verbs are also irregular.
Regular Verbs
Simple Tenses
Regular verbs follow the same patterns when shifting from the present to past tense or future tense. For example, to form a past-tense or past-participle verb form, add -ed or -d to the end of a verb. You can avoid mistakes by understanding this basic pattern.
Verb tense identifies the time of action described in a sentence. Verbs take different forms to indicate different tenses. Verb tenses indicate:
- an action or state of being in the present,
- an action or state of being in the past,
- an action or state of being in the future.
Helping verbs, such as be and have, also work to create verb tenses, such as the future tense.
Present Tense:
Tim
walks
to the store. (Singular subject)
Present Tense:
Sue
and
Kimmy
walk
to the store. (Plural subject)
Past Tense:
Yesterday,
they
walked
to the store to buy some bread. (Singular subject)
Future Tense: I will walk to the store tomorrow. (Singular subject)
When he , she , or it doing the present tense action, remember to add – s , or – es to the end of the verb or to change the y to – ies .
When he , she , or it is doing the action in the past tense, remember to add – d or – ed to the end of regular verbs.
Simple future verbs are used when the action has not yet taken place:
- I will work late tomorrow.
- I will kiss my boyfriend when I see him.
- I will erase the board after class.
For the most part, we focus on the simple past, present, and future tenses here. However, a list of all the tense can be found in the chart below. Notice that the examples all use regular verbs, and -- depending on the tense -- helping verbs as well.
Exercises
Exercise 1
Complete the following sentences by selecting the correct form of the verb in simple present, simple past, or simple future tenses.
- The Dust Bowl (is, was, will be) a name given to a period of very destructive dust storms that occurred in the United States during the 1930s.
- Historians today (consider, considered, will consider) The Dust Bowl to be one of the worst weather of events in American history.
- The Dust Bowl mostly (affects, affected, will affect) the states of Kansas, Colorado, Oklahoma, Texas, and New Mexico.
- Dust storms (continue, continued, will continue) to occur in these dry regions, but not to the devastating degree of the 1930s.
- The dust storms during The Dust Bowl (cause, caused, will cause) irreparable damage to farms and the environment for a period of several years.
- When early settlers (move, moved, will move) into this area, they (remove, removed, will remove) the natural prairie grasses in order to plant crops and graze their cattle.
- They did not (realize, realized, will realize) that the grasses kept the soil in place.
- There (is, was, will be) also a severe drought that (affects, affected, will affect) the region.
- The worst dust storm (happens, happened, will happen) on April 14, 1935, a day called Black Sunday.
- The Dust Bowl era finally came to end in 1939 when the rains (arrive, arrived, will arrive).
- Dust storms (continue, continued, will continue) to affect the region, but hopefully they will not be as destructive as the storms of the 1930s.
Exercise 2
The infinitive (tenseless) form of the verb is included before each sentence. Rewrite each sentence with the correct tense of the verb that makes sense within the context of the sentence.
1. (to travel) Tomorrow, I _________________________ to visit my family.
2. (to work) I ____________________ hard every day.
3. (to run) I __________________ all the way to the bus this morning so I could get to class on time.
4. (to request) Your presence is ___________________________ at the party on May 3, 2020.
5. (to carry) They ___________________ their luggage up the hill to the vacation house they rented.
6. (to marry) Tom and John ____________________ on June 6, 2020.
Tip
When describing what happens in something you've read (on the computer or paper) or watched (in a video), use the simple present tense to describe that action. This is an academic convention. For example, you might write: Chewbacca acts as Han Solo's assistant and character commentator in Star Wars .
Progressive Tenses
Present Progressive Tense
Progressive verb tenses describe a continuing or unfinished action, such as I am going, I was going , or I will be going .
The present progressive tense describes an action or state of being that takes place in the present and that continues to take place.
To make verbs in the present progressive tense, combine these two parts:
|
Present tense form of to be |
+ |
-ing (present participle) |
|
am/is/are |
help |
helping |
You should use the present progressive tense to describe a planned activity, to describe an activity that is recurring right now, and to describe an activity that is in progress although not actually occurring at the time of speaking:
- Preeti is starting school on Tuesday.
This sentence describes a planned activity.
- Janetta is getting her teeth cleaned right now.
This sentence describes an activity that is occurring right now.
- I am studying ballet at school.
This sentence describes an activity that is in progress but not actually occurring at the time of speaking.
Past Progressive Tense
The past progressive tense describes an action or state of being that took place in the past and that continues to take place.
To make verbs in the past progressive tense, combine these two parts:
|
Past tense form of to be |
+ |
-ing (present participle) |
|
was/were |
helping |
You should use the past progressive tense to describe a continuous action in the past, to describe a past activity in progress while another activity occurred, or to describe two past activities in progress at the same time:
- Ella and I were planning a vacation.
This sentence describes a continuous action in the past.
- I was helping a customer when I smelled delicious fried chicken.
This sentence describes a past activity in progress while another activity occurred.
- While I was finishing my homework, my wife was talking on the phone.
This sentence describes two past activities in progress at the same time.
Future Progressive Tense
The future progressive tense describes an action or state of being that will take place in the future and that will continue to take place. The action will have started at that future moment, but it will not have finished at that moment.
To make verbs in the future progressive tense, combine these parts:
|
Future tense form of to be |
+ |
-ing (present participle) |
|
will be |
helping |
Use the future progressive tense to describe an activity that will be in progress in the future:
- Samantha and I will be dancing in the school play next week.
- Tomorrow Agnes will be reading two of her poems.
Exercise 3
On a separate sheet of paper, revise the following sentences, written in simple tenses, using the progressive tenses indicated in parentheses.
1. He prepared the food while I watched. (past progressive tense)
2. Jonathan will speak at the conference. (future progressive)
3. Josie traveled to Egypt last July. (past progressive tense)
4. My foot aches, so I know it will rain. (present progressive tense)
5. Micah will talk a lot when I see him. (future progressive)
6. I yawn a lot because I feel tired. (present progressive tense)
Tip
Many people casually speak in the present progressive tense when they really should use the simple present tense. Watch out for this in your writing!
Perfect Tenses
The three perfect tenses indicate that an action is complete. They are present perfect, past perfect, and future perfect. These are the three basic tenses of English. A past participle is used in the perfect tenses. Past participles are often called the – ed form of a verb because they are formed by adding – d or – ed to the base form of regular verbs. Past participles can also end in -t or -en (irregular verbs, see below). Keep in mind, however, the past participle is also formed in various other ways for irregular verbs. The past participle can be used to form the present perfect tense.
Present Perfect Tense
Review the following basic formula for the present perfect tense:
|
Subject |
+ |
has or have |
+ |
past participle |
|
I |
have |
helped |
Use the present perfect tense to describe a continuing situation and to describe an action that has just happened.
- I have worked as a caretaker since June.
This sentence tells us that the subject has worked as a caretaker in the past and is still working as a caretaker in the present.
• Dmitri has just received an award from the Dean of Students. This sentence tells us that Dmitri has very recently received the award.
The word just emphasizes that the action happened very recently.
Past Perfect Tense
Review the following basic formula for the past perfect tense:
| Subject | + | have or had | + | past participle |
| I | had | listened |
Use the past perfect tense to describe a continuing situation that was completed by a certain time in the past.
- The bus had left by the time Theo arrived at the station.
Notice that both actions occurred entirely in the past, but one action occurred before the other. At some time in the past, Theo arrived (simple past tense) at the station, but at some time before that, the bus had left (past perfect).
Future Perfect Tense
See the following basic formula for the future perfect tense:
| Subject | + | will have | + | past participle |
| I | will have | graduated |
The future perfect tense describes an action from the past in the future, as if the past event has already occurred. Use the future perfect tense when you anticipate completing an event in the future, but you have not completed it yet.
- You will have forgotten me after you move to London.
Notice that both actions occur in the future, but one action will occur before the other. At some time in the future, the subject ( you ) will move (future tense) to London, and at some time after that, the subject will have forgotten (future perfect tense) the speaker, me .
Exercise 4
On a separate sheet of paper, complete the following sentences by using the correct perfect verb tense for the verb in parentheses.
1. I plan to start a compost bin because I ________ (to want) one for a long time now.
2. My brother told me he ________ (to argue) with his friend about politics.
3. By the time we reach the mountain top the sun ________ (to set).
4. Denise ________ (to walk) several miles in the past three hours.
5. His mother ________ (to offer) to pay him to work in her office.
Perfect Progressive Tenses
Present Perfect Progressive
Similar to the present perfect tense, the present perfect progressive tense is used to indicate an action that was begun in the past and continues into the present. However, the present perfect progressive is used when you want to stress that the action is ongoing.
To make verbs in the present perfect progressive tense, combine the following parts:
| Present tense form of to have | + | been | + | --ing (present participle) |
| has or have | been | helping |
- He has been talking for the past hour.
This sentence indicates that he started talking in the past and is continuing to talk in the present.
• I have been feeling tired lately.
This sentence indicates that I started feeling tired in the past, and I continue to feel tired in the present. Instead of indicating time, as in the first sentence, the second sentence uses the adverb lately. You can also use the adverb recently when using the present perfect progressive tense.
Past Perfect Progressive
Similar to the past perfect tense, the past perfect progressive tense is used to indicate an action that was begun in the past and continued until another time in the past. The past perfect progressive does not continue into the present but stops at a designated moment in the past.
To make verbs in the past perfect progressive tense, combine the following parts:
| Past tense form of to have | + | been | + | --ing (present participle) |
| had | been | helping |
- The employees had been talking until their boss arrived.
This sentence indicates that the employees were talking in the past and they stopped talking when their boss arrived, which also happened in the past.
- I had been working all day.
This sentence implies that I was working in the past. The action does not continue into the future, and the sentence implies that the subject stopped working for unstated reasons.
Future Perfect Progressive
The future perfect progressive tense is rarely used. It is used to indicate an action that will begin in the future and will continue until another time in the future.
To make verbs in the future perfect progressive tense, combine the following parts:
| Future tense form of to have | + | been | + | --ing (present participle) |
| will have | been | helping |
- By the end of the meeting, I will have been hearing about mortgages and taxes for eight hours.
This sentence indicates that in the future I will hear about mortgages and taxes for eight hours, but it has not happened yet. It also indicates the action of hearing will continue until the end of the meeting, something that is also in the future.
Irregular Verbs
Irregular verbs do not use the same rules to create past tense.Here are some examples.
Present Tense:
Lauren
keeps
all her letters.
Past Tense:
Lauren
kept
all her letters.
Future Tense:
Lauren
will keep
all her letters.
The videos below will give you some general ideas of how to identify and memorize irregular verbs.
Some of the most common irregular verbs are also the most common verbs in English: "to be," "to do," and "to have."
To Be, To Do, and To Have
The following table shows the past, present and future forms of these common irregular verbs.
| Base Form | Past Tense Form | Present Tense Form | Future Tense Form |
| be | was/were | am/is/are | will be |
| do | did | do/does | will do |
| have | had | have/has | will have |
Memorize the present tense forms of to be, to do, and to have. A song or rhythmic pattern will make them easier to memorize.
Review these examples of to be, to do, and to have used in sentences.
| Past | Present | Future |
| To Be | ||
| Yesterday I was angry. | Today I am angry. | Tomorrow I will be angry. |
| To Do | ||
| I did my best yesterday. | I do my best today. | I will do my best tomorrow. |
| To Have | ||
| Yesterday I had ten dollars. | Today I have ten dollars. | Tomorrow I will have ten dollars. |
Remember the following uses of to be, to have and to do:
To Be
• I → am/was/will be
• you/we/they → are/were/will be
• he/she/it → is/was/will be
To Have
• I/you/we/they → have/had/will have
• he/she/it → has/had/will have
To Do
• I/you/we/they → do/did/will do
• he/she/it → does/did/will do
Exercise 5
On a separate sheet of paper, complete the following sentences by circling the correct form of the verbs to be, to have, and to do in the three simple tenses.
1. Stefan always (do, does, will do) his taxes the day before they are due.
2. We (are, is, was) planning a surprise birthday party for my mother.
3. Turtles (have, had, has) the most beautiful patterns on their shells.
4. I always (do, did, will do) my homework before dinner, so I can eat in peace.
5.
Study the table below, “Irregular Verbs”, which lists the most common irregular verbs.
Table 12.6: Irregular Verbs
| Irregular Verbs | |||
|---|---|---|---|
| Simple Present | Past | Simple Present | Past |
| be | was, were | lose | lost |
| become | became | make | made |
| begin | began | mean | meant |
| blow | blew | meet | met |
| break | broke | pay | paid |
| bring | brought | put | put |
| build | built | quit | quit |
| burst | burst | read | read |
| buy | bought | ride | rode |
| catch | caught | ring | rang |
| choose | chose | rise | rose |
| come | came | run | ran |
| cut | cut | say | said |
| dive | dove (dived) | see | saw |
| do | did | seek | sought |
| draw | drew | sell | sold |
| drink | drank | send | sent |
| drive | drove | set | set |
| eat | ate | shake | shook |
| fall | fell | shine | shone (shined) |
| feed | fed | shrink | shrank (shrunk) |
| feel | felt | sing | sang |
| fight | fought | sit | sat |
| find | found | sleep | slept |
| fly | flew | speak | spoke |
| forget | forgot | spend | spent |
| forgive | forgave | spring | sprang |
| freeze | froze | stand | stood |
| get | got | steal | stole |
| give | gave | strike | struck |
| go | went | swim | swam |
| grow | grew | swing | swung |
| have | had | take | took |
| hear | heard | teach | taught |
| hide | hid | tear | tore |
| hold | held | tell | told |
| hurt | hurt | think | thought |
| keep | kept | throw | threw |
| know | knew | understand | understood |
| lay | laid | wake | woke |
| lead | led | wear | wore |
| leave | left | win | won |
| let | let | wind | wound |
Tip
The best way to learn irregular verbs is to memorize them. With the help of a classmate, create flashcards of irregular verbs and test yourselves until you master them.
Exercise 6
Complete the following sentences by selecting the correct form of the irregular verb in simple present, simple past, or simple future tense. Copy the corrected sentence onto your own sheet of paper.
- Marina finally (forgived, forgave, will forgive) her sister for snooping around her room.
- The house (shook, shaked, shakes) as the airplane rumbled overhead.
- I (buyed, bought, buy) several items of clothing at the thrift store on Wednesday.
- She (put, putted, puts) the lotion in her shopping basket and proceeded to the checkout line.
- The prized goose (layed, laid, lay) several golden eggs last night.
- Mr. Batista (teached, taught, taughted) the class how to use correct punctuation.
- I (drink, drank, will drink) several glasses of sparkling cider instead of champagne on New Year’s Eve next year.
- Although Hector (growed, grew, grows) three inches in one year, we still called him “Little Hector.”
- Yesterday our tour guide (lead, led, will lead) us through the maze of people in Times Square.
- The rock band (burst, bursted, bursts) onto the music scene with their catchy songs.
Exercise 7
On your own sheet of paper, write a sentence using the correct form of the verb tense shown below.
- Throw (past)
- Paint (simple present)
- Smile (future)
- Tell (past)
- Share (simple present)Maintaining Consistent Verb Tense
Verb Tense Consistency
Consistent verb tense means the same verb tense is used throughout a sentence or a paragraph. As you write and revise, it is important to use the same verb tense consistently and to avoid shifting from one tense to another unless there is a good reason for the tense shift. In the following box, see whether you notice the difference between a sentence with consistent tense and one with inconsistent tense.
Inconsistent tense:
The crowd
starts
cheering as Melina
approached
the finish line.
Consistent tense:
The crowd
started
cheering as Melina
approached
the finish line.
Consistent tense:
The crowd
starts
cheering as Melina
approaches
the finish line.
Tip:
In some cases, clear communication will call for different tenses. Look at the following example:
When I was a teenager, I wanted to be a fire fighter, but now I am studying computer science.
If the time frame for each action or state of being is different, a tense shift is appropriate.
Exercise 8
Edit the following paragraph by correcting the inconsistent verb tense. Copy the corrected paragraph onto your own sheet of paper.
In the Middle Ages, most people lived in villages and work as agricultural laborers, or peasants. Every village has a “lord,” and the peasants worked on his land. Much of what they produce go to the lord and his family. What little food was leftover goes to support the peasants’ families. In return for their labor, the lord offers them protection. A peasant’s day usually began before sunrise and involves long hours of backbreaking work, which includes plowing the land, planting seeds, and cutting crops for harvesting. The working life of a peasant in the Middle Ages is usually demanding and exhausting.
Writing at Work
Read the following excerpt from a work e-mail:
I would like to highlight an important concern that comes up after our meeting last week. During the meeting, we agree to conduct a series of interviews over the next several months in which we hired new customer service representatives. Before we do that, however, I would like to review your experieces with the Customer Relationship Managment Program. Please suggest a convenient time next week for us to meet so that we can discuss this important matter.
The inconsistent tense in the e-mail will very likely distract the reader from its overall point. Most likely, your coworkers will not correct your verb tenses or call attention to grammatical errors, but it is important to keep in mind that errors such as these do have a subtle negative impact in the workplace.
Contributors and Attributions
CC LICENSED CONTENT, SHARED PREVIOUSLY:
- Adapted from College ESL Writers: Applied Grammar and Composing Strategies for Success . Authored by: Barbara Hall and Elizabeth Wallace. Provided by: GALILEO Open Learning Materials. License: CC BY-NC-SA (3.0): Attribution.
- Adapted from Writing for Success . Provided by: The Saylor Foundation. License: CC-NC-SA 3.0 .
ALL RIGHTS RESERVED CONTENT:
- How Many Verb Tense Are There in English? Authored by: Anna Anunichuk. Provided by: TedEd. License: All Rights Reserved. License Terms: Standard YouTube License.
- Nessy Writing Strategy: Irregular Verbs. Authored by: Nessy. License: All Rights Reserved. License Terms: Standard YouTube License.
- Irregular Verbs: All Irregular Verbs in One Song . Authored by: Lingportal Online School of English. License: All Rights Reserved. License Terms: Standard YouTube License.