1.2: Functions- Modeling Relationships
- How can we use the mathematical idea of a function to represent the relationship between two changing quantities?
- What are some formal characteristics of an abstract mathematical function? how do we think differently about these characteristics in the context of a physical model?
A mathematical model is an abstract concept through which we use mathematical language and notation to describe a phenomenon in the world around us. One example of a mathematical model is found in Dolbear's Law 1 . In the late 1800s, the physicist Amos Dolbear was listening to crickets chirp and noticed a pattern: how frequently the crickets chirped seemed to be connected to the outside temperature. If we let \(T\) represent the temperature in degrees Fahrenheit and \(N\) the number of chirps per minute, we can summarize Dolbear's observations in the following table.
| \(N\) (chirps per minute) | \(40\) | \(80\) | \(120\) | \(160\) |
| \(T\) (\(^\circ\) Fahrenheit) | \(50^\circ\) | \(60^\circ\) | \(70^\circ\) | \(80^\circ\) |
For a mathematical model, we often seek an algebraic formula that captures observed behavior accurately and can be used to predict behavior not yet observed. For the data in Table \(\PageIndex{1}\) , we observe that each of the ordered pairs in the table make the equation
true. For instance, \(70 = 40 + 0.25(120)\text{.}\) Indeed, scientists who made many additional cricket chirp observations following Dolbear's initial counts found that the formula in Equation (1.2.1) holds with remarkable accuracy for the snowy tree cricket in temperatures ranging from about \(50^\circ\) F to \(85^\circ\) F.
Use Equation (1.2.1) to respond to the questions below.
- If we hear snowy tree crickets chirping at a rate of \(92\) chirps per minute, what does Dolbear's model suggest should be the outside temperature?
- If the outside temperature is \(77^\circ\) F, how many chirps per minute should we expect to hear?
- Is the model valid for determining the number of chirps one should hear when the outside temperature is \(35^\circ\) F? Why or why not?
- Suppose that in the morning an observer hears \(65\) chirps per minute, and several hours later hears \(75\) chirps per minute. How much has the temperature risen between observations?
- Dolbear's Law is known to be accurate for temperatures from \(50^\circ\) to \(85^\circ\text{.}\) What is the fewest number of chirps per minute an observer could expect to hear? the greatest number of chirps per minute?
Functions
The mathematical concept of a function is one of the most central ideas in all of mathematics, in part since functions provide an important tool for representing and explaining patterns. At its core, a function is a repeatable process that takes a collection of input values and generates a corresponding collection of output values with the property that if we use a particular single input, the process always produces exactly the same single output.
For instance, Dolbear's Law in Equation (1.2.1) provides a process that takes a given number of chirps between \(40\) and \(180\) per minute and reliably produces the corresponding temperature that corresponds to the number of chirps, and thus this equation generates a function. We often give functions shorthand names; using “\(D\)” for the “Dolbear” function, we can represent the process of taking inputs (observed chirp rates) to outputs (corresponding temperatures) using arrows:
Alternatively, for the relationship “\(80 \xrightarrow{D} 60\)” we can also use the equivalent notation “\(D(80) = 60\)” to indicate that Dolbear's Law takes an input of \(80\) chirps per minute and produces a corresponding output of \(60\) degrees Fahrenheit. More generally, we write “\(T = D(N) = 40 + 0.25N\)” to indicate that a certain temperature, \(T\text{,}\) is determined by a given number of chirps per minute, \(N\text{,}\) according to the process \(D(N) = 40 + 0.25N\text{.}\)
Tables and graphs are particularly valuable ways to characterize and represent functions. For the current example, we summarize some of the data the Dolbear function generates in Table \(\PageIndex{2}\) and plot that data along with the underlying curve in Figure \(\PageIndex{3}\)
| \(N\) | \(T\) |
| \(40\) | \(50\) |
| \(80\) | \(60\) |
| \(120\) | \(70\) |
| \(160\) | \(80\) |
| \(180\) | \(85\) |
When a point such as \((120,70)\) in Figure \(\PageIndex{3}\) lies on a function's graph, this indicates the correspondence between input and output: when the value \(120\) chirps per minute is entered in the function \(D\text{,}\) the result is \(70\) degrees Fahrenheit. More concisely, \(D(120) = 70\text{.}\) Aloud, we read “\(D\) of \(120\) is \(70\)”.
For most important concepts in mathematics, the mathematical community decides on formal definitions to ensure that we have a shared language of understanding. In this text, we will use the following definition of the term “function”.
A function is a process that may be applied to a collection of input values to produce a corresponding collection of output values in such a way that the process produces one and only one output value for any single input value.
If we name a given function \(F\) and call the collection of possible inputs to \(F\) the set \(A\) and the corresponding collection of potential outputs \(B\text{,}\) we say “\(F\) is a function from \(A\) to \(B\text{,}\)” and sometimes write “\(F : A \to B\text{.}\)” When a particular input value to \(F\text{,}\) say \(t\text{,}\) produces a corresponding output \(z\text{,}\) we write “\(F(t) = z\)” and read this symbolic notation as “\(F\) of \(t\) is \(z\text{.}\)” We often call \(t\) the independent variable and \(z\) the dependent variable , since \(z\) is a function of \(t\text{.}\)
Let \(F\) be a function from \(A\) to \(B\text{.}\) The set \(A\) of possible inputs to \(F\) is called the domain of \(F\text{;}\) the set \(B\) of potential outputs from \(F\) is called the codomain of \(F\text{.}\)
For the Dolbear function \(D(N) = 40 + 0.25N\) in the context of modeling temperature as a function of the number of cricket chirps per minute, the domain of the function is \(A = [40,180]\) 2 and the codomain is “all Fahrenheit temperatures”. The codomain of a function is the collection of possible outputs, which we distinguish from the collection of actual ouputs.
Let \(F\) be a function from \(A\) to \(B\text{.}\) The range of \(F\) is the collection of all actual outputs of the function. That is, the range is the collection of all elements \(y\) in \(B\) for which it is possible to find an element \(x\) in \(A\) such that \(F(x) = y\text{.}\)
In many situations, the range of a function is much more challenging to determine than its codomain. For the Dolbear function, the range is straightforward to find by using the graph shown in Figure \(\PageIndex{3}\) : since the actual outputs of \(D\) fall between \(T = 50\) and \(T = 85\) and include every value in that interval, the range of \(D\) is \([50,80]\text{.}\)
The range of any function is always a subset of the codomain. It is possible for the range to equal the codomain.
Consider a spherical tank of radius \(4\) m that is filling with water. Let \(V\) be the volume of water in the tank (in cubic meters) at a given time, and \(h\) the depth of the water (in meters) at the same time. It can be shown using calculus that \(V\) is a function of \(h\) according to the rule
- What values of \(h\) make sense to consider in the context of this function? What values of \(V\) make sense in the same context?
- What is the domain of the function \(f\) in the context of the spherical tank? Why? What is the corresponding codomain? Why?
- Determine and interpret (with appropriate units) the values \(f(2)\text{,}\) \(f(4)\text{,}\) and \(f(8)\text{.}\) What is important about the value of \(f(8)\text{?}\)
- Consider the claim: “since \(f(9) = \frac{\pi}{3} 9^2(12-9) = 81\pi \approx 254.47\text{,}\) when the water is \(9\) meters deep, there is about \(254.47\) cubic meters of water in the tank”. Is this claim valid? Why or why not? Further, does it make sense to observe that “\(f(13) = -\frac{169\pi}{3}\)”? Why or why not?
- Can you determine a value of \(h\) for which \(f(h) = 300\) cubic meters?
Comparing models and abstract functions
Again, a mathematical model is an abstract concept through which we use mathematical language and notation to describe a phenomenon in the world around us. So far, we have considered two different examples: the Dolbear function, \(T = D(N) = 40 + 0.25N\text{,}\) that models how Fahrenheit temperature is a function of the number of cricket chirps per minute and the function \(V = f(h) = \frac{\pi}{3}h^2(12-h)\) that models how the volume of water in a spherical tank of radius \(4\) m is a function of the depth of the water in the tank. While often we consider a function in the physical setting of some model, there are also many occasions where we consider an abstract function for its own sake in order to study and understand it.
Calculus shows that for a tennis ball tossed vertically from a window \(48\) feet above the ground at an initial vertical velocity of \(32\) feet per second, the ball's height above the ground at time \(t\) (where \(t = 0\) is the instant the ball is tossed) can be modeled by the function \(h = g(t) = -16t^2 + 32t + 48\text{.}\) Discuss the differences between the model \(g\) and the abstract function \(f\) determined by \(y = f(x) = -16x^2 + 32x + 48\text{.}\)
- Solution
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We start with the abstract function \(y = f(x) = -16x^2 + 32x + 48\text{.}\) Absent a physical context, we can investigate the behavior of this function by computing function values, plotting points, and thinking about its overall behavior. We recognize the function \(f\) as quadratic 3 , noting that it opens down because of the leading coefficient of \(-16\text{,}\) with vertex located at \(x = \frac{-32}{2(-16)} = 1\text{,}\) \(y\)-intercept at \((0,48)\text{,}\) and with \(x\)-intercepts at \((-1,0)\) and \((3,0)\) because
\[ -16x^2 + 32x + 48 = -16(x^2 - 2x - 3) = -16(x-3)(x+1)\text{.} \nonumber \]Computing some additional points to gain more information, we see both the data in Table \(\PageIndex{8}\) and the corresponding graph in Figure \(\PageIndex{9}\)
Table \(\PageIndex{8}\) Data for the function \(y = f(x) = -16x^2 + 32x + 48\text{.}\) \(x\) \(f(x)\)
\(-80\)
\(0\)
\(48\)
\(64\)
\(48\)
\(0\)
\(-80\)\(-2\) \(-1\) \(0\) \(1\) \(2\) \(3\) \(4\) For this abstract function, its domain is “all real numbers” since we may input any real number \(x\) we wish into the formula \(f(x) = -16x^2 + 32x + 48\) and have the result be defined. Moreover, taking a real number \(x\) and processing it in the formula \(f(x) = -16x^2 + 32x + 48\) will produce another real number. This tells us that the codomain of the abstract function \(f\) is also “all real numbers.” Finally, from the graph and the data, we observe that the largest possible output of the function \(f\) is \(y = 64\text{.}\) It is apparent that we can generate any \(y\)-value less than or equal to \(64\text{,}\) and thus the range of the abstract function \(f\) is all real numbers less than or equal to \(64\text{.}\) We denote this collection of real numbers using the shorthand interval notation \((-\infty, 64]\text{.}\) 4
Next, we turn our attention to the model \(h = g(t) = -16t^2 + 32t + 48\) that represents the height of the ball, \(h\text{,}\) in feet \(t\) seconds after the ball in initially launched. Here, the big difference is the domain, codomain, and range associated with the model. Since the model takes effect once the ball is tossed, it only makes sense to consider the model for input values \(t \ge 0\text{.}\) Moreover, because the model ceases to apply once the ball lands, it is only valid for \(t \le 3\text{.}\) Thus, the domain of \(g\) is \([0,3]\text{.}\) For the codomain, it only makes sense to consider values of \(h\) that are nonnegative. That is, as we think of potential outputs for the model, then can only be in the interval \([0, \infty)\text{.}\) Finally, we can consider the graph of the model on the given domain in Figure \(\PageIndex{11}\) and see that the range of the model is \([0,64]\text{,}\) the collection of all heights between its lowest (ground level) and its largest (at the vertex).
Table \(\PageIndex{10}\) Data for the model \(h = g(t) = -16t^2 + 32t + 48\text{.}\) \(t\) \(g(t)\) 0 \(48\) 1 \(64\) 2 \(48\) 3 \(0\) -
Consider a spherical tank of radius \(4\) m that is completely full of water. Suppose that the tank is being drained by regulating an exit valve in such a way that the height of the water in the tank is always decreasing at a rate of \(0.5\) meters per minute. Let \(V\) be the volume of water in the tank (in cubic meters) at a given time \(t\) (in minutes), and \(h\) the depth of the water (in meters) at the same time. It can be shown using calculus that \(V\) is a function of \(t\) according to the model
In addition, let \(h = q(t)\) be the function whose output is the depth of the water in the tank at time \(t\text{.}\)
- What is the height of the water when \(t = 0\text{?}\) When \(t = 1\text{?}\) When \(t = 2\text{?}\) How long will it take the tank to completely drain? Why?
- What is the domain of the model \(h = q(t)\text{?}\) What is the domain of the model \(V = p(t)\text{?}\)
- How much water is in the tank when the tank is full? What is the range of the model \(h = q(t)\text{?}\) What is the range of the model \(V = p(t)\text{?}\)
- We will frequently use a graphing utility to help us understand function behavior, and strongly recommend Desmos because it is intuitive, online, and free. 5
- How does the model \(V = p(t) = \frac{256\pi}{3} - \frac{\pi}{24} t^2(24-t)\) differ from the abstract function \(y = r(x) = \frac{256\pi}{3} - \frac{\pi}{24} x^2(24-x)\text{?}\) In particular, how do the domain and range of the model differ from those of the abstract function, if at all?
- How should the graph of the height function \(h = q(t)\) appear? Can you determine a formula for \(q\text{?}\) Explain your thinking.
Determining whether a relationship is a function or not
To this point in our discussion of functions, we have mostly focused on what the function process may model and what the domain, codomain, and range of a model or abstract function are. It is also important to take note of another part of Definition 1.2.4: “\(\ldots\) the process produces one and only one output value for any single input value”. Said differently, if a relationship or process ever associates a single input with two or more different outputs, the process cannot be a function.
Is the relationship between people and phone numbers a function?
- Solution
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No, this relationship is not a function. A given individual person can be associated with more than one phone number, such as their cell phone and their work telephone. This means that we can't view phone numbers as a function of people: one input (a person) can lead to two different outputs (phone numbers). We also can't view people as a function of phone numbers, since more than one person can be associated with a phone number, such as when a family shares a single phone at home.
The relationship between \(x\) and \(y\) that is given in the following table where we attempt to view \(y\) as depending on \(x\text{.}\)
| \(x\) | 1 | 2 | 3 | 4 | 5 |
| \(y\) | \(13\) | \(11\) | \(10\) | \(11\) | \(13\) |
- Solution
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The relationship between \(y\) and \(x\) in Table \(\PageIndex{14}\) allows us to think of \(y\) as a function of \(x\) since each particular input is associated with one and only one output. If we name the function \(f\text{,}\) we can say for instance that \(f(4) = 11\text{.}\) Moreover, the domain of \(f\) is the set of inputs \(\{1,2,3,4,5\}\text{,}\) and the codomain (which is also the range) is the set of outputs \(\{10,11,13\}\text{.}\)
Each of the following prompts describes a relationship between two quantities. For each, your task is to decide whether or not the relationship can be thought of as a function. If not, explain why. If so, state the domain and codomain of the function and write at least one sentence to explain the process that leads from the collection of inputs to the collection of outputs.
1. The relationship between \(x\) and \(y\) in each of the graphs below (address each graph separately as a potential situation where \(y\) is a function of \(x\)). In Figure 1.2.15, any point on the circle relates \(x\) and \(y\text{.}\) For instance, the \(y\)-value \(\sqrt{7}\) is related to the \(x\)-value \(-3\text{.}\) In Figure 1.2.16, any point on the blue curve relates \(x\) and \(y\text{.}\) For instance, when \(x = -1\text{,}\) the corresponding \(y\)-value is \(y = 3\text{.}\) An unfilled circle indicates that there is not a point on the graph at that specific location.
2. The relationship between the day of the year and the value of the S&P500 stock index (at the close of trading on a given day), where we attempt to consider the index's value (at the close of trading) as a function of the day of the year.
3. The relationship between a car's velocity and its odometer, where we attempt to view the car's odometer reading as a function of its velocity.
4. The relationship between \(x\) and \(y\) that is given in the following table where we attempt to view \(y\) as depending on \(x\text{.}\)
| \(x\) | 1 | 2 | 3 | 2 | 1 |
| \(y\) | \(11\) | \(12\) | \(13\) | \(14\) | \(15\) |
For a relationship or process to be a function, each individual input must be associated with one and only one output. Thus, the usual way that we demonstrate a relationship or process is not a function is to find a particular input that is associated with two or more outputs. When the relationship is given graphically, such as in Figure \(\PageIndex{15}\) , we can use the vertical line test to determine whether or not the graph represents a function.
A graph in the plane represents a function if and only if every vertical line intersects the graph at most once. When the graph passes this test, the vertical coordinate of each point on the graph can be viewed as a function of the horizontal coordinate of the point.
Since the vertical line \(x = -3\) passes through the circle in Figure \(\PageIndex{15}\) at both \(y = -\sqrt{7}\) and \(y = \sqrt{7}\text{,}\) the circle does not represent a relationship where \(y\) is a function of \(x\text{.}\) However, since any vertical line we draw in Figure 1.2.16 intersects the blue curve at most one time, the graph indeed represents a function.
We conclude with a formal definition of the graph of a function.
Let \(F : A \to B\text{,}\) where \(A\) and \(B\) are each collections of real numbers. The graph of \(F\) is the collection of all ordered pairs \((x,y)\) that satisfy \(y = F(x)\text{.}\)
When we use a computing device such as Desmos to graph a function \(g\text{,}\) the program is generating a large collection of ordered pairs \((x,g(x))\text{,}\) plotting them in the \(x\)-\(y\) plane, and connecting the points with short line segments.
Summary
- A function is a process that generates a relationship between two collections of quantities. The function associates each member of a collection of input values with one and only one member of the collection of output values. A function can be described or defined by words, by a table of values, by a graph, or by a formula.
- Functions may be viewed as mathematical objects worthy of study for their own sake and also as models that represent physical phenomena in the world around us. Every function or model has a domain (the set of possible or allowable input values), a codomain (the set of possible output values), and a range (the set of all actual output values). Both the codomain and range depend on the domain. For an abstract function, the domain is usually viewed as the largest reasonable collection of input values; for a function that models a physical phenomenon, the domain is usually determined by the context of possibilities for the input in the phenomenon being considered.
Exercises
Consider an inverted conical tank (point down) whose top has a radius of \(3\) feet and that is \(2\) feet deep. The tank is initially empty and then is filled at a constant rate of \(0.75\) cubic feet per minute. Let \(V=f(t)\) denote the volume of water (in cubic feet) at time \(t\) in minutes, and let \(h= g(t)\) denote the depth of the water (in feet) at time \(t\text{.}\)
- Recall that the volume of a conical tank of radius \(r\) and depth \(h\) is given by the formula \(V = \frac{1}{3} \pi r^2 h\text{.}\) How long will it take for the tank to be completely full and how much water will be in the tank at that time?
- On the provided axes, sketch possible graphs of both \(V = f(t)\) and \(h = g(t)\text{,}\) making them as accurate as you can. Label the scale on your axes and points whose coordinates you know for sure; write at least one sentence for each graph to discuss the shape of your graph and why it makes sense in the context of the model.
3. What is the domain of the model \(h = g(t)\text{?}\) its range? why?
4. It's possible to show that the formula for the function \(g\) is \(g(t) = \left( \frac{t}{\pi} \right)^{1/3}\text{.}\) Use a computational device to generate two plots: on the axes at left, the graph of the model \(h = g(t) = \left( \frac{t}{\pi} \right)^{1/3}\) on the domain that you decided in (c); on the axes at right, the graph of the abstract function \(y = p(x) = \left( \frac{t}{\pi} \right)^{1/3}\) on a wider domain than that of \(g\text{.}\) What are the domain and range of \(p\) and how do these differ from those of the physical model \(g\text{?}\)
 Their velocity, \(v\) (in feet per second), which is a function of time \(t\) (in seconds), is given by the graph in Figure \(\PageIndex{19}\)
- What is the person's velocity when \(t = 2\text{?}\) when \(t = 7\text{?}\)
- Are there any times when the person's velocity is exactly \(v = 3\) feet per second? If yes, identify all such times; if not, explain why.
- Describe the person's behavior on the time interval \(4 \le t \le 5\text{.}\)
- On which time interval does the person travel a farther distance: \([1,3]\) or \([6,8]\text{?}\) Why?
A driver of a new car periodically keeps track of the number of gallons of gas remaining in their car's tank, while simultaneously tracking the trip odometer mileage. Their data is recorded in the following table. Note that at mileages where they add fuel to the tank, they record the mileage twice: once before fuel is added, and once afterward.
| \(D\) (miles) | \(0\) | \(50\) | \(100\) | \(100\) | \(150\) | \(200\) | \(250\) | \(300\) | \(300\) | \(350\) |
| \(G\) (gallons) | \(4.5\) | \(3.0\) | \(1.5\) | \(10.0\) | \(8.5\) | \(7.0\) | \(5.5\) | \(4.0\) | \(11.0\) | \(9.5\) |
Use the table to respond to the questions below.
- Can the amoung of fuel in the gas tank, \(G\text{,}\) be viewed as a function of distance traveled, \(D\text{?}\) Why or why not?
- Does the car's fuel economy appear to be constant or does it appear to vary? Why?
- At what odometer reading did the driver put the most gas in the tank?
<1>You can read more in the Wikipedia entry for Dolbear's Law, which has proven to be remarkably accurate for the behavior of snowy tree crickets. For even more of the story, including a reference to this phenomenon on the popular show The Big Bang Theory , see this article.
<2>The notation “\([40,180]\)” means “the collection of all real numbers \(x\) that satisfy \(40 \le x \le 80\)” and is sometimes called “interval notation”.
<3>We will engage in a brief review of quadratic functions in Section 1.5
<4>The notation \((−infty,64]\) stands for all the real numbers that lie to the left of an including 64.64. The “\(-\infty\)” indicates that there is no left-hand bound on the interval.
<5>To learn more about Desmos , see their outstanding online tutorials .


1.2.3: Special Properties of Relations
Some kinds of relations turn out to be so common that they have been given special names. For instance, \(\le\) and \(\subseteq\) both relate their respective domains (say, \(\Nat\) in the case of \(\le\) and \(\Pow{A}\) in the case of \(\subseteq\) ) in similar ways. To get at exactly how these relations are similar, and how they differ, we categorize them according to some special properties that relations can have. It turns out that (combinations of) some of these special properties are especially important: orders and equivalence relations.
A relation \(R \subseteq A^2\) is reflexive iff, for every \(x \in A\) , \(Rxx\) .
A relation \(R \subseteq A^2\) is transitive iff, whenever \(Rxy\) and \(Ryz\) , then also \(Rxz\) .
A relation \(R \subseteq A^2\) is symmetric iff, whenever \(Rxy\) , then also \(Ryx\) .
A relation \(R \subseteq A^2\) is anti-symmetric iff, whenever both \(Rxy\) and \(Ryx\) , then \(x=y\) (or, in other words: if \(x\neq y\) then either \(\lnot Rxy\) or \(\lnot Ryx\) ).
In a symmetric relation, \(Rxy\) and \(Ryx\) always hold together, or neither holds. In an anti-symmetric relation, the only way for \(Rxy\) and \(Ryx\) to hold together is if \(x = y\) . Note that this does not require that \(Rxy\) and \(Ryx\) holds when \(x = y\) , only that it isn’t ruled out. So an anti-symmetric relation can be reflexive, but it is not the case that every anti-symmetric relation is reflexive. Also note that being anti-symmetric and merely not being symmetric are different conditions. In fact, a relation can be both symmetric and anti-symmetric at the same time (e.g., the identity relation is).
A relation \(R \subseteq A^2\) is connected if for all \(x,y\in A\) , if \(x \neq y\) , then either \(Rxy\) or \(Ryx\) .
Give examples of relations that are (a) reflexive and symmetric but not transitive, (b) reflexive and anti-symmetric, (c) anti-symmetric, transitive, but not reflexive, and (d) reflexive, symmetric, and transitive. Do not use relations on numbers or sets.
A relation \(R \subseteq A^2\) is called irreflexive if, for all \(x \in A\) , not \(Rxx\) .
A relation \(R \subseteq A^2\) is called asymmetric if for no pair \(x,y\in A\) we have both \(Rxy\) and \(Ryx\) .
Note that if \(A \neq \emptyset\) , then no irreflexive relation on \(A\) is reflexive and every asymmetric relation on \(A\) is also anti-symmetric. However, there are \(R \subseteq A^2\) that are not reflexive and also not irreflexive, and there are anti-symmetric relations that are not asymmetric.


6.3: Price Elasticity of Supply
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Definition of Price Elasticity of Supply
The price elasticity of supply is the measure of the responsiveness in quantity supplied to a change in price for a specific good.
learning objectives
- Differentiate between the price elasticity of demand for elastic and inelastic goods
In economics, elasticity is a summary measure of how the supply or demand of a particular good is influenced by changes in price. Elasticity is defined as a proportionate change in one variable over the proportionate change in another variable:
\[\text{Elasticity}=\dfrac{\% \text{Change in quantity}}{\% \text{Change in price}}\]
The price elasticity of supply (PES) is the measure of the responsiveness in quantity supplied (QS) to a change in price for a specific good (% Change QS / % Change in Price). There are numerous factors that directly impact the elasticity of supply for a good including stock, time period, availability of substitutes, and spare capacity. The state of these factors for a particular good will determine if the price elasticity of supply is elastic or inelastic in regards to a change in price.
The price elasticity of supply has a range of values:
- PES > 1: Supply is elastic.
- PES < 1: Supply is inelastic.
- PES = 0: The supply curve is vertical; there is no response of demand to prices. Supply is “perfectly inelastic.”
- PES = ∞∞ (i.e., infinity): The supply curve is horizontal; there is extreme change in demand in response to very small change in prices. Supply is “perfectly elastic.”
Inelastic goods are often described as necessities. A shift in price does not drastically impact consumer demand or the overall supply of the good because it is not something people are able or willing to go without. Examples of inelastic goods would be water, gasoline, housing, and food.
Elastic goods are usually viewed as luxury items. An increase in price for an elastic good has a noticeable impact on consumption. The good is viewed as something that individuals are willing to sacrifice in order to save money. An example of an elastic good is movie tickets, which are viewed as entertainment and not a necessity.
The price elasticity of supply is determined by:
- Number of producers: ease of entry into the market.
- Spare capacity: it is easy to increase production if there is a shift in demand.
- Ease of switching: if production of goods can be varied, supply is more elastic.
- Ease of storage: when goods can be stored easily, the elastic response increases demand.
- Length of production period: quick production responds to a price increase easier.
- Time period of training: when a firm invests in capital the supply is more elastic in its response to price increases.
- Factor mobility: when moving resources into the industry is easier, the supply curve in more elastic.
- Reaction of costs: if costs rise slowly it will stimulate an increase in quantity supplied. If cost rise rapidly the stimulus to production will be choked off quickly.
The result of calculating the elasticity of the supply and demand of a product according to price changes illustrates consumer preferences and needs. The elasticity of a good will be labelled as perfectly elastic, relatively elastic, unit elastic, relatively inelastic, or perfectly inelastic.
Price elasticity over time : This graph illustrates how the supply and demand of a product are measured over time to show the price elasticity.
Perfectly Inelastic Supply : A graphical representation of perfectly inelastic supply.
Measuring the Price Elasticity of Supply
The price elasticity of supply is the measure of the responsiveness of the quantity supplied of a particular good to a change in price.
learning objectives
- Calculate elasticities and describe their meaning
The price elasticity of supply (PES) is the measure of the responsiveness of the quantity supplied of a particular good to a change in price (PES = % Change in QS / % Change in Price). The intent of determining the price elasticity of supply is to show how a change in price impacts the amount of a good that is supplied to consumers. The price elasticity of supply is directly related to consumer demand.
Elasticity
The elasticity of a good provides a measure of how sensitive one variable is to changes in another variable. In this case, the price elasticity of supply determines how sensitive the quantity supplied is to the price of the good.
Calculating the PES
When calculating the price elasticity of supply, economists determine whether the quantity supplied of a good is elastic or inelastic. The percentage of change in supply is divided by the percentage of change in price. The results are analyzed using the following range of values:
- PES > 1: Supply is elastic.
- PES < 1: Supply is inelastic.
- PES = 0: Supply is perfectly inelastic. There is no change in quantity if prices change.
- PES = infinity: Supply is perfectly elastic. An decrease in prices will lead to zero units produced.
Factors that Influence the PES
There are numerous factors that impact the price elasticity of supply including the number of producers, spare capacity, ease of switching, ease of storage, length of production period, time period of training, factor mobility, and how costs react.
The price elasticity of supply is calculated and can be graphed on a demand curve to illustrate the relationship between the supply and price of the good.
Applications of Elasticities
In economics, elasticity refers to how the supply and demand of a product changes in relation to a change in the price.
learning objectives
- Give examples of inelastic and elastic supply in the real world
In economics, elasticity refers to the responsiveness of the demand or supply of a product when the price changes.
The technical definition of elasticity is the proportionate change in one variable over the proportionate change in another variable. For example, to determine how a change in the supply or demand of a product is impacted by a change in the price, the following equation is used: Elasticity = % change in supply or demand / % change in price.
The price is a variable that can directly impact the supply and demand of a product. If a change in the price of a product significantly influences the supply and demand, it is considered “elastic.” Likewise, if a change in product price does not significantly change the supply and demand, it is considered “inelastic.”
For elastic demand, when the price of a product increases the demand goes down. When the price decreases the demand goes up. Elastic products are usually luxury items that individuals feel they can do without. An example would be forms of entertainment such as going to the movies or attending a sports event. A change in prices can have a significant impact on consumer trends as well as economic profits. For companies and businesses, an increase in demand will increase profit and revenue, while a decrease in demand will result in lower profit and revenue.
For inelastic demand, the overall supply and demand of a product is not substantially impacted by an increase in price. Products that are usually inelastic consist of necessities like food, water, housing, and gasoline. Whether or not a product is elastic or inelastic is directly related to consumer needs and preferences. If demand is perfectly inelastic, then the same amount of the product will be purchased regardless of the price.
Economists study elasticity and use demand curves in order to diagram and study consumer trends and preferences. An elastic demand curve shows that an increase in the supply or demand of a product is significantly impacted by a change in the price. An inelastic demand curve shows that an increase in the price of a product does not substantially change the supply or demand of the product.
Inelastic Demand : For inelastic demand, when there is an outward shift in supply and prices fall, there is no substantial change in the quantity demanded.
Elastic Demand : For elastic demand, when there is an outward shift in supply, prices fall which causes a large increase in quantity demanded.
Key Points
- Elasticity is defined as a proportionate change in one variable over the proportionate change in another variable: \(\text{Elasticity}=\dfrac{\% \text{Change in quantity}}{\% \text{Change in price}}\)
- The impact that a price change has on the elasticity of supply also directly impacts the elasticity of demand.
- Inelastic goods are often described as necessities, while elastic goods are considered luxury items.
- The elasticity of a good will be labelled as perfectly elastic, relatively elastic, unit elastic, relatively inelastic, or perfectly inelastic.
- The price elasticity of supply = % change in quantity supplied / % change in price.
- When calculating the price elasticity of supply, economists determine whether the quantity supplied of a good is elastic or inelastic.
- PES > 1: Supply is elastic. PES < 1: Supply is inelastic. PES = 0: if the supply curve is vertical, and there is no response to prices. PES = infinity: if the supply curve is horizontal.
- To determine the elasticity of a product, the proportionate change of one variable is placed over the proportionate change of another variable (Elasticity = % change of supply or demand / % change in price ).
- For elastic demand, a change in price significantly impacts the supply and demand of the product.
- For inelastic demand, a change in the price does not substantially impact the supply and demand of the product.
- Economists use demand curves in order to document and study elasticity.
Key Terms
- luxury : Something very pleasant but not really needed in life.
- supply : The amount of some product that producers are willing and able to sell at a given price, all other factors being held constant.
- demand : The desire to purchase goods and services.
- mobility : The ability for economic factors to move between actors or conditions.
- capacity : The maximum that can be produced on a machine or in a facility or group.
- elastic : Sensitive to changes in price.
- demand : The desire to purchase goods and services.
- inelastic : Not sensitive to changes in price.
- supply : The amount of some product that producers are willing and able to sell at a given price, all other factors being held constant.
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- IB Economics/Microeconomics/Elasticities. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/IB_Econ...s/Elasticities . License : CC BY-SA: Attribution-ShareAlike
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- IB Economics/Microeconomics/Elasticities. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/IB_Econ...s/Elasticities . License : CC BY-SA: Attribution-ShareAlike
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8: Social and Emotional Development in Middle Childhood (Ages 7 to 12)
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8.0: What Does Psychology Say?
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The page discusses the experiences of Andre and Omar, two brothers who moved from Brazil to Boston. It highlights the challenges they face in adjusting to a new school and cultural environment during middle childhood, a critical period for identity development. Andre struggles with English and making friends, while Omar adapts more easily. Their parents are concerned about their cultural identity and social development.
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8.1: Identity, Self-Concept, and Self-Esteem in Middle Childhood
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This section focuses on development during middle childhood, highlighting key aspects such as Erikson's industry versus inferiority challenge, self-efficacy, self-esteem, and identity development. Children in this stage begin to compare their abilities with peers, influencing their self-concept. Gender and racial identities also develop as children navigate societal expectations and stereotypes, progressively attaining more complex self-concepts.
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8.2: Emotional Development and Socioemotional Learning in Middle Childhood
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This section discusses the social and emotional development in middle childhood, focusing on the growth in social skills, emotional regulation, and social cognition. It describes how children develop the ability to manage their emotions and cope with stress, highlighting techniques such as problem-solving and support seeking. Social cognition skills, including empathy and perspective-taking, assist children in maintaining healthy relationships.
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8.3: Social Contexts- Peers, Family, and Media in Middle Childhood
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This page discusses key aspects of middle childhood development, including friendships, family influences, aggression, bullying, and media use. It highlights how friendships evolve beyond proximity and demographic similarities, stressing shared interests and social skills. The role of family remains crucial, providing emotional support and influencing peer perceptions. Aggression in childhood can manifest physically or relationally, potentially leading to bullying.
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8.4: Context- School and Extracurricular Activities in Middle Childhood
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This section explores the influence of school climate and teacher-student relationships on child development. It emphasizes the importance of positive classroom environments in fostering student engagement and highlights how different teaching styles can impact student outcomes. The page also discusses the significance of cultural sensitivity, extracurricular activities, and parental expectations on academic performance.
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8.5: Atypical Development and Interventions in Middle Childhood
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The section discusses understanding and supporting children's socioemotional well-being and mental health during middle childhood.


10.3: Preventing Management Problems by Focusing Students on Learning
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The easiest management problems to solve are ones that do not happen in the first place! Even before the school year begins, you can minimize behavior problems by arranging classroom furniture and materials in ways that encourage a focus on learning as much as possible. Later, once school begins, you can establish procedures and rules that support a focus on learning even more.
Arranging classroom space
Viewed broadly, classrooms may seem to be arranged in similar ways, but there are actually important alternative arrangements to consider. Variations exist because of grade level, the subjects taught, the teacher's philosophy of education, and of course the size of the room and the furniture available. Whatever the arrangement that you choose, it should help students to focus on learning tasks as much as possible and minimize the chances of distractions. Beyond these basic principles, however, the "best" arrangement depends on what your students need and on the kind of teaching that you prefer and feel able to provide (Boyner, 2003; Nations & Boyett, 2002). The next sections describe some of the options. In considering them (and before moving too much furniture around your room!), you might want to try experimenting with spatial arrangements "virtually" by using one of the computer programs available on the Internet (see: https://www.pinterest.com/pin/178666310189912023/ for example).
Displays and wall space
All classrooms have walls, of course, and how you fill them can affect the mood or feeling of a classroom. Ample displays make a room interesting and can be used to reinforce curriculum goals and display (and hence publicly recognize) students' work. But too many displays can also make a room seem "busy" or distracting as well as physically smaller. They can also be more work to maintain. If you are starting a new school year, then, a good strategy is to decorate some of the wall or bulletin board space, but not to fill it all immediately. Leaving some space open leaves flexibility to respond to ideas and curriculum needs that emerge after the year is underway. The same advice applies especially for displays that are high maintenance, such as aquariums, pets, and plants. These can serve wonderfully as learning aids, but do not have to be in place on the first day of school. Not only the students, but also you yourself, may already have enough to cope with at that time.
Computers in the classroom
If you are like the majority of teachers, you will have only one computer in your room, or at most just a few, and their placement may be pre-determined by the location of power and cable outlets. If so, you need to think about computer placement early in the process of setting up a room. Once the location of computers is set, locations for desks, high-usage shelves, and other moveable items can be chosen more sensibly— in general, as already mentioned, so as to minimize distractions to students and to avoid unnecessary traffic congestion.
Visibility of and interactions with students
Learning is facilitated if the furniture and space allow you to see all students and to interact with them from a comfortable distance. Usually this means that the main, central part of the room— where desks and tables are usually located— needs to be as open and as spacious as possible. While this idea may seem obvious, enacting it can be challenging in practice if the room itself is small or shaped unusually. In classrooms with young students (kindergarten), furthermore, open spaces tend to allow, if not invite, physical movement of children— a feature that you may consider either constructive or annoying, depending on your educational goals and the actual level of activity that occurs.
Spatial arrangements unique to grade levels or subjects
The best room arrangement sometimes depends on the grade level or subject area of the class. If you teach in elementary school, for example, you may need to think especially about where students can keep their daily belongings, such as coats and lunches. In some schools, these can be kept outside the classroom— but not necessarily. Some subjects and grade levels, furthermore, lend themselves especially well to small group interaction, in which case you might prefer not to seat students in rows, but instead around small-group tables or work areas. The latter arrangement is sometimes preferred by elementary teachers, but is also useful in high schools wherever students need lots of counter space, as in some shops or art courses, or where they need to interact, as in English as a Second Language courses (McCafferty, Jacobs, & Iddings, 2006). The key issue in deciding between tables and rows, however, is not grade level or subject as such, but the amount of small group interaction you want to encourage, compared to the amount of whole-group instruction. As a rule, tables make working with peers easier, and rows make listening to the teacher more likely and group work slightly more awkward physically.
Ironically, some teachers also experience challenges about room arrangement because they do not actually have a classroom of their own, because they must move each day among other teachers' rooms. "Floating" is especially likely for specialized teachers (e.g. music teachers in elementary schools, who move from class to class) and in schools have an overall shortage of classrooms. Floating can sometimes be annoying to the teacher, though it actually also has advantages, such as not having to take responsibility for how other teachers' rooms are arranged. If you find yourself floating, it helps to consider a few key strategies, such as:
• consider using a permanent cart to move crucial supplies from room to room
• make sure that every one of your rooms has an overhead projector (do not count on using chalkboards or
computers in other teachers' rooms)
• talk to the other teachers about having at least one shelf or corner in each room designated for your
exclusive use
Establishing daily procedures and routines
Procedures or routines are specific ways of doing common, repeated classroom tasks or activities. Examples include checking daily attendance, dealing with students who arrive late, or granting permission to leave the classroom for an errand. Academically related procedures include ways of turning in daily homework (e.g. putting it on a designated shelf at a particular time), of gaining the teacher's attention during quiet seat work (e.g. raising your hand and waiting), and of starting a "free choice" activity after completing a classroom assignment.
Procedures serve the largely practical purpose of making activities and tasks flow smoothly— a valuable and necessary purpose in classrooms, where the actions of many people have to be coordinated within limited time and space. As such, procedures are more like social conventions than like moral expectations. They are only indirectly about what is ethically right or ethically desirable to do (Turiel, 2006). Most procedures or routines can be accomplished in more than one way, with only minor differences in outcomes. There is more than one way, for example, for the procedure of taking attendance: the teacher could call the role, delegate a student to call the role,or note students' presence on a seating chart. Each variation accomplishes essentially the same task, and the choice may be less important than the fact that the class coordinates its actions somehow , by committing to some sort of choice.
For teachers, of course, an initial management task is to establish procedures and routines as promptly as possible. Because of the conventional quality of procedures, some teachers find that it works well simply to announce and explain key procedures without inviting much discussion from students ("Here is how we will choose partners for the group work"). Other teachers prefer to invite input from students when creating procedures (asking the class, "What do you feel is the best way for students to get my attention during a quiet reading time?"). Both approaches have advantages as well as disadvantages. Simply announcing key procedures saves time and insures consistency in case you teach more than one class (as you would in high school). But it puts more responsibility on the teacher to choose procedures that are truly reasonable and practical. Inviting students' input, on the other hand, can help students to become aware of and committed to procedures, but at the cost of requiring more time to settle on them. It also risks creating confusion if you teach multiple classes, each of which adopts different procedures. Whatever approach you choose, of course, they have to take into account any procedures or rules imposed by the school or school district as a whole. A school may have a uniform policy about how to record daily attendance, for example, and that policy may determine, either partly or completely, how you take attendance with your particular students.
Establishing classroom rules
Unlike procedures or routines, rules express standards of behavior for which individual students need to take responsibility. Although they are like procedures in that they sometimes help in insuring the efficiency of classroom tasks, they are really about encouraging students to be responsible for learning and showing respect for each other.
Exhibit \(\PageIndex{1}\): Sample set of classroom rules
- Treat others with courtesy and politeness.
- Make sure to bring required materials to class and to activities.
- Be on time for class and other activities.
- Listen to the teacher and to others when they are speaking.
- Follow all school rules
Note three things about the examples in Exhibit \(\PageIndex{1}\) . One is that the rules are not numerous; the table lists only five. Most educational experts recommend keeping the number of rules to a minimum in order to make them easier to remember (Thorson, 2003; Brophy, 2004). A second feature is that they are stated in positive terms ("Do X...") rather than negative terms ("Do not do Y..."), a strategy that emphasizes and clarifies what students should do rather than what they should avoid. A third feature is that each rule actually covers a collection of more specific behaviors. The rule "Bring all materials to class", for example, covers bringing pencils, paper, textbooks, homework papers, and permission slips— depending on the situation. As a result of their generality, rules often have a degree of ambiguity that sometimes requires interpretation. Infractions may occur that are marginal or "in a grey area", rather than clear cut. A student may bring a pen, for example, but the pen may not work properly. You may therefore wonder whether this incident is really a failure to follow the rule, or just an unfortunate (and in this case minor) fault of the pen manufacturer.
As with classroom procedures, rules can be planned either by the teacher alone, or by the teacher with advice from students. The arguments for each approach are similar to the arguments for procedures: rules "laid on" by the teacher may be more efficient and consistent, and in this sense more fair, but rules influenced by the students may be supported more fully by the students. Because rules focus strongly on personal responsibility, however, there is a stronger case for involving students in making them than in making classroom procedures (Brookfield, 2006; Kohn, 2006). In any case the question of who plans classroom rules is not necessarily an either/or choice. It is possible in principle to impose certain rules on students (for example, "Always be polite to each other") but let the students determine the consequences for violations of certain rules (for example, "If a student is discourteous to a classmate, he/she must apologize to the student in writing"). Some mixture of influences is probably inevitable, in fact, if only because the class needs to take into account your own moral commitments as the teacher as well as any imposed by the school (like "No smoking in the school" or "Always walk in the hallways").
Pacing and structuring lessons and activities
One of the best ways to prevent management problems is by pacing and structuring lessons or activities as smoothly and continuously as possible. This goal depends on three major strategies:
• selecting tasks or activities at an appropriate level of difficulty for your students
• providing a moderate level of structure or clarity to students about what they are supposed to do, especially
during transitions between activities
• keeping alert to the flow and interplay of behaviors for the class as a whole and for individuals within it.
Each strategy presents special challenges to teachers, but also opportunities for helping students to learn.
Choosing tasks at an appropriate level of difficulty
As experienced teachers know and as research has confirmed, students are most likely to engage with learning when tasks are of moderate difficulty, neither too easy nor too hard and therefore neither boring nor frustrating (Britt, 2005). Finding the right level of difficulty, however, can be a challenge if you have little experience teaching a particular grade level or curriculum, or even if students are simply new to you and their abilities unknown. Whether familiar or not, members of any class are likely to have diverse skills and readiness-a fact that makes it challenging to determine what level of difficulty is appropriate. A common strategy for dealing with these challenges is to begin units, lessons, or projects with tasks that are relatively easy and familiar. Then, introduce more difficult material or tasks gradually until students seem challenged, but not overwhelmed. Following this strategy gives the teacher a chance to observe and diagnose students' learning needs before adjusting content, and it gives students a chance to orient themselves to the teacher's expectations, teaching style, and topic of study without becoming frustrated prematurely. Later in a unit, lesson, or project, students seem better able to deal with more difficult tasks or content (Van Merrionboer, 2003). The principle seems to help as well with "authentic" learning tasks— ones that resemble real-world activities, such as learning to drive an automobile or to cook a meal, and that present a variety of complex tasks simultaneously. Even in those cases it helps to isolate and focus on the simplest subtasks first (such as "put the key in the ignition") and move to harder tasks only later (such as parallel parking).
Sequencing instruction is only a partial solution to finding the best "level" of difficulty, however, because it does not deal with enduring individual differences among students. The fundamental challenge to teachers is to individualize or differentiate instruction fully: to tailor it not only to the class as a group, but to the lasting differences among members of the class. One way to approach this sort of diversity, obviously, is to plan different content or activities for different students or groups of students. While one group works on Task A, another group works on Task B; one group works on relatively easy math problems, for example, while another works on harder ones. Differentiating instruction in this way complicates a teacher's job, but it can be done, and has in fact been done by many teachers (it also makes teaching more interesting!). In the next chapter, we describe some classroom management strategies that help with such multi-tasking.
Providing moderate amounts of structure and detail
Chances are that at some point in your educational career you have wished that a teacher would clarify or explain an assignment more fully, and perhaps give it a clearer structure or organization. Students' desire for clarity is especially common with assignments that are by nature open-ended, such as long essays, large projects, or creative works. Simply being told to "write an essay critiquing the novel", for example, leaves more room for uncertainty (and worry) than being given guidelines about what questions the essay should address, what topics or parts it should have, and what its length or style should be (Chesebro, 2003). As you might suspect, some students desire clarity more than others, and improve their performance especially much when provided with plenty of structure and clarity. Students with certain kinds of learning difficulties, in particular, often learn effectively and stay on task only if provided with somewhat explicit, detailed instructions about the tasks expected of them (Marks, et al., 2003).
As a teacher, the challenge is to accommodate students' need for clarity without making guidance so specific or detailed that students do little thinking for themselves. As a (ridiculously extreme) example, consider a teacher gives "clear" instructions for an essay by announcing not only exactly which articles to read and cite in the essay and which topics or issues to cover, but even requires specific wording of sentences in their essays. This much specificity may reduce students' uncertainties and make the teacher's task of evaluating the essays relatively straightforward and easy. But it also reduces or even eliminates the educational value of the assignment— assuming, of course, that its purpose is to get students to think for themselves.
Ideally, then, structure should be moderate rather than extreme. There should be just enough to give students some sense of direction and to stimulate more accomplishment than if they worked with less structure or guidance. This ideal is an application of Vygotsky's idea of the zone of proximal development that we discussed in Chapter 2: a place (figuratively speaking) where students get more done with help than without it. The ideal amount of guidance —the "location" of the zone of proximal development— varies with the assignment and the student, and it (hopefully) decreases over time for all students. One student may need more guidance to do his or her best in math, but less guidance in order to write her or his best essay. Another student may need the reverse. But if all goes well, both students may need less at the end of the year than at the beginning.
Managing transitions
Transitions between activities is often full of distractions and "lost" time, and is a time when inappropriate behaviors are especially likely to occur. Part of the problem is intrinsic to transitions: students may have to wait before a new activity actually begins, and therefore get bored at the very moment when the teacher is preoccupied with arranging materials for the new activity. From the point of view of the students, transitions may seem essentially like unsupervised group time, when seemingly any behavior is tolerated.
Minimizing such problems requires two strategies, one of which is easier to implement than the other. The easier strategy is for you, as teacher, to organize materials as well as possible ahead of time, so that you minimize the time needed to begin a new activity. The advice sounds simple, and mostly is, but it sometimes takes a bit of practice to implement smoothly. When one of us (Kelvin) first began teaching university, for example, particular papers or overhead transparencies sometimes got lost in the wrong folder in spite of Kelvin's efforts to keep them where they were easy to find. The resulting delays about finding them slowed the pace of class and caused frustrations.
A second, more complex strategy is to teach students ways to manage their own behavior during transitions (Marzano & Marzano, 2004). If students talk too loudly at these times, for example, then discuss with them what constitutes appropriate levels or amounts of talk, and discuss the need for them to monitor their own sound level.Or if students stop work early in anticipation of ending an activity, then talk about— or even practice— waiting for a signal from yourself to indicate the true ending point for an activity. If certain students continue working beyond the end of an activity. On the other hand, try giving them warning of the impending end in advance, and remind them about to take responsibility for actually finishing work once they hear the advance warning, and so on. The point of these tactics is to encourage responsibility for behavior during transitions, and thereby reduce your own need to monitor students at that crucial time.
None of these ideas, of course, mean that you, as teacher, should give up monitoring students' behavior entirely. Chances are that you still will need to notice if and when someone talks too loudly, finishes too early, or continues too long, and you will still need to give some students appropriate reminders. But the amount of reminding will be less to the extent that students can remind and monitor themselves— a welcome trend at any time, but especially during transitions.
Maintaining the flow of activities
A lot of classroom management is really about keeping activities flowing smoothly, both during individual lessons and across the school day. The trouble is that there is never just "one" event happening at a time, even if only one activity has been formally planned and is supposed to be occurring. Imagine, for example, that everyone is supposed to be attending a single whole-class discussion on a topic; yet individual students will be having different experiences at any one moment. Several students may be listening and contributing comments, for example, but a few others may be planning what they want to say next and ignoring the current speakers, still others may be ruminating about what a previous speaker said, and still others may be thinking about unrelated matters—the restroom, food, or sex. Things get even more complicated if the teacher deliberately plans multiple activities: in that case some students may interact with the teacher, for example, while others do work in an unsupervised group or work independently in a different part of the room. How is a teacher to keep activities flowing smoothly in the face of such variety?
A common mistake of beginning teachers in multi-faceted settings like these is to pay too much attention to any one activity, student, or small group, at the expense of noticing and responding to all the others. If you are helping a student on one side of the room when someone on the other side disturbs classmates with off-task conversation, it can be less effective either to finish with the student you are helping before attending to the disruption, or to interrupt yourself to solve the disruption on the other side of the room. Although one of these responses may be necessary, either one involves disruption somewhere . There is a risk that either the student's chatting may spread to others, or the interrupted student may become bored with waiting for the teacher's attention and wander off-task herself.
A better solution, though one that at first may seem challenging, is to attend to both events at once— a strategy that was named withitness in a series of now-classic research studies several decades ago (Kounin, 1970). Withitness does not mean that you focus on all simultaneous activities with equal care, but only that you remain aware of multiple activities, behaviors, and events to some degree. At a particular moment, for example, you may be focusing on helping a student, but in some corner of your mind you also notice when chatting begins on the other side of the room.  Research has found that experienced teachers are much more likely to show withitness than inexperienced teachers, and that these qualities are associated with managing classrooms successfully (Emmer & Stough, 2001).
Simultaneous awareness— withitness— makes possible responses to the multiple events that are immediate and nearly simultaneous— what educators sometimes called overlapping . The teacher's responses to each event or behavior need not take equal time, nor even be equally noticeable to all students. If you are helping one student with seat work at the precise moment when another student begins chatting off-task, for example, a quick glance to the second student may be enough to bring the second one back to the work at hand, and may scarcely interrupt your conversation with the first student, or be noticed by others who are not even involved. The result is a smoother flow to activities overall.
As a new teacher, you may find that withitness and overlapping develop more easily in some situations than in others. It may be easier to keep an eye (or ear) on multiple activities during familiar routines, such as taking attendance, but harder to do the same during activities that are unfamiliar or complex, such as introducing a new topic or unit that you have never taught before. But skill at broadening your attention does increase with time and practice. It helps to keep trying. Merely demonstrating to students that you are "withit", in fact, even without making deliberate overlapping responses, can sometimes deter students from off-task behavior. Someone who is tempted to pass notes in class, for example, might not do so because she believes that you will probably notice her doing it anyway, whether or not you are able to notice in fact.
Communicating the importance of learning and of positive behavior
Altogether, the factors we have discussed— arranging space, procedures, and rules, and developing withitness — help communicate an important message: that in the classroom learning and positive social behavior are priorities. In addition, teachers can convey this message by offering timely feedback to students about performance, by keeping accurate records of the performance, and by deliberately communicating with parents or caregivers about their children and about class activities.
Communicating effectively is so important for all aspects of teaching, in fact, that we discuss it more fully later in this book (see Chapter 8, "The nature of classroom communication"). Here we focus on only one of its important aspects: how communication contributes to a smoothly functioning classroom and in this way helps prevent behavior problems.
Giving timely feedback
The term feedback , when used by educators, refers to responses to students about their behavior or performance. Feedback is essential if students are to learn and if they are to develop classroom behavior that is socially skilled and "mature". But feedback can only be fully effective if offered as soon as possible, when it is still relevant to the task or activity at hand (Reynolds, 1992). A score on a test is more informative immediately after a test than after a six-month delay, when students may have forgotten much of the content of the test. A teacher's comment to a student about an inappropriate, off-task behavior may not be especially welcome at the moment the behavior occurs, but it can be more influential and informative then; later, both teacher and student will have trouble remembering the details of the off-task behavior, and in this sense may literally "not know what they are talking about". The same is true for comments about a positive behavior by a student: hearing a compliment right away makes it easier to the comment with the behavior, and allows the compliment to influence the student more strongly. There are of course practical limits to how fast feedback can be given, but the general principle is clear: feedback tends to work better when it is timely.
The principle of timely feedback is consistent, incidentally, with a central principle of operant conditioning discussed in Chapter 2: reinforcement works best when it follows a to-be-learned operant behavior closely (Skinner, 1957)- I n this case a teacher's feedback serves as a form of reinforcement. The analogy is easiest to understand when the feedback takes the form of praise; in operant conditioning terms, the reinforcing praise then functions like a "reward". When feedback is negative, it functions as an "aversive stimulus" (in operant terms), shutting down the behavior criticized. At other times, though, criticism can also function as an unintended reinforcement. This happens, for example, if a student experiences criticism as a reduction in isolation and therefore as in increase in his importance in the class— a relatively desirable change. So the inappropriate behavior continues, or even increases, contrary to the teacher's intentions. Exhibit 9 diagrams this sequence of events.
Exhibit \(\PageIndex{2}\): Attracting attention as negative reinforcement
Exhibit \(\PageIndex{2}\): Attracting attention as negative reinforcement
Student is isolated socially → Student publicly misbehaves → Student gains others’ attention
Reinforcement can happen in class if an undesirable behavior, leads to a less aversive state for a student. Social isolation can be reduced by public misbehavior, which stimulates attention that is reinforcing. Ironically, the effort to end misbehavior ends up stimulating the misbehavior.
Maintaining accurate records
Although timeliness in responding to students can sometimes happen naturally during class, there are also situations where promptness depends on having organized key information ahead of time. Obvious examples are the scores, marks, and grades returned to students for their work. A short quiz (such as a weekly spelling test) may be possible to return quite soon after the quiz— sometimes you or even the students themselves can mark it during class. More often, though, assignments and tests require longer processing times: you have to read, score, or add comments to each paper individually. Excessive time to evaluate students' work can reduce the usefulness of a teacher's evaluations to students when she finally does return the work (Black, et al., 2004). During the days or weeks waiting for a test or assignment to be returned, students are left without information about the quality or nature of their performance; at the extreme they may even have to complete another test or do another assignment before getting information about an earlier one. (Perhaps you yourself have experienced this particular problem!)
Delays in providing feedback about academic performance can never be eliminated entirely, but they can be reduced by keeping accurate, well-organized records of students' work. A number of computer programs are available to help with this challenge; if your school does not already have one in use, then there are several downloadable either free or at low cost from the Internet . Describing these is beyond the scope of this book. For now we simply emphasize that grading systems benefit students' learning the most when they provide feedback as quickly and frequently as possible (McMillan, 2001), precisely the reason why accurate, well-organized record-keeping is important to keep.
Accurate records are helpful not only for scores on tests, quizzes, or assignments, but also for developing descriptive summaries of the nature of students' academic skills or progress. A common way to develop a description is the student portfolio, which is a compilation of the student's work and on-going assessments of it created by the teacher or in some cases by the student (Moritz & Christie, 2005; White, 2005). To know how a student's science project evolved from its beginning, for example, a teacher and student can keep a portfolio of lab notes, logs, preliminary data, and the like. To know how a student's writing skills developed, they could keep a portfolio of early drafts on various writing assignments. As the work accumulates, the student can discuss it with the teacher, and write brief reflections on its strengths thus far or on the steps needed to improve the work further. By providing a way to respond to work as it evolves, and by including students in making the assessments, portfolios provide relatively prompt feedback, and in any case provide it sooner than by waiting for the teacher to review work that is complete or final.
Communicating with parents and caregivers
Since parents and caregivers in a sense "donate" their children to schools (at least figuratively speaking), teachers are responsible for keeping them informed and involved to whatever extent is practical. Virtually all parents understand and assume that schools are generally intended for learning. Detailed communication can enrich parents' understanding, of how learning is addressed with their particular child's classroom, and show them more precisely what their particular child is doing. The better such understanding in turn encourages parents and caregivers to support their child's learning more confidently and "intelligently". In this sense it contributes indirectly to a positive learning environment in their child's class.
There are various ways to communicate with parents, each with advantages and limitations. Here are three
common examples:
• A regular classroom newsletter: A newsletter establishes a link with parents or caregivers with comparatively little effort on the part of the teacher. At the beginning of the year, for example, a newsletter can tell about special materials that students will need, important dates to remember (like professional development days when there is no school), or about curriculum plans for the next few weeks. But newsletters also have limitations. They can seem impersonal, and they may get lost on the way home and never reach parents or caregivers. They can also be impractical for teachers with multiple classes, as in high school or in specialist subjects (like music or physical education), where each class follows a different program or curriculum.
• Telephone calls: The main advantage of phoning is its immediacy and individuality. Teacher and parent or caregiver can talk about a particular student, behavior, or concern, and do it now. By the same token, however, phone calls are not an efficient way for informing parents about events or activities that affect everyone in common. The individuality of phoning may explain why teachers often use this method when a student has a problem that is urgent or unusual— as when he has failed a test, missed classes, or misbehaved seriously. Rightly or wrongly, a student's successes tend not to prompt phone calls to the student's home (though in fairness students may be more likely to tell parents about their successes themselves, making it less essential for the teacher to do so).
• Parent-teacher conferences: Most schools schedule periodic times— often a day or evening per term— when teachers meet briefly with parents or caregivers who wish to meet. Under good conditions, the conferences have the individuality of phone calls, but also the richness of communication possible only in face-to-face meetings. Since conferences are available to all parents, they need not focus on behavior or academic problems, but often simply help to build rapport and understanding between parents or caregivers and the teacher. Sometimes too, particularly at younger grade levels, teachers involve students in leading their own conferences; the students display and explain their own work using a portfolio or other archive of accumulated materials (Benson & Barnett, 2005; Stiggins & Chappuis, 2005). In spite of all of these advantages, though, parent-teacher conferences have limitations. Some parents cannot get to conferences because of work schedules, child care, or transportation problems. Others may feel intimated by any school- sponsored event because they speak limited English or because they remember painful experiences from their own school days.
Even if you make several efforts to communicate, some parents may remain out of contact. In these cases it is important to remember that the parents may not be indifferent to their child or to the value of education. Other possibilities exist, as some of our comments above imply: parents may have difficulties with child care, for example, have inconvenient work schedules, or feel self-conscious about their own communication skills (Stevens & Tollafield, 2003). Even so, there are ways to encourage parents who may be shy, hesitant, or busy. One is to think about how they can assist the school even from home— for example, by making materials to be used in class or (if they are comfortable using English) phoning other parents about class events. A second way is to have a specific task for the parents in mind— one with clear structure, such as photocopying materials to be used by students later. A third is to remember to encourage, support, and respect the parents' presence and contributions when they do show up at school functions. Keep in mind that parents are experts about their own particular children, and without them, you would have no students to teach!


1.11: Hidden Money- Scholarships
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When the world says, ”Give up,” Hope whispers, ”Try it one more time.”
~ Author Unknown
A scholarship is a financial investment in someone’s potential to succeed. It’s based on past experiences, the possibility of making a difference, and the embodiment of the core values of the organization or person sponsoring the scholarship. As a student applying for scholarships, think about what can you do to demonstrate that you are a worthy investment. What story can you tell that will make someone want to invest in you?
Last year I volunteered with the Art Support Services. Art support is a part of Burning Man’s infrastructure that facilitates everything involving massive art installations. Being my second year I went from being a simple volunteer to a volunteer trusted with the duties and responsibilities of a radio. My job entailed communicating with the artists, figure out where they were in the building process then decide what heavy machinery they needed. Next I would radio Heavy Equipment to inform them of the artist’s needs. The most important skill I gained was appropriate radio communication. I learned how to change channels and proper radio etiquette. On the radio I used terms like “ten-nine”, “copy that”, or “affirmative” a personal favorite. Having a radio was a huge feeling of accomplishment, because I got to play a part in coordinating all the pieces that made the art come together
.
~ Student #1
Coming back to school after twenty-five years is an accomplishment I’m very proud of. I didn’t graduate from high school when I was supposed to, so the first strength I used was faith that I could come back and do it now. I definitely needed a little courage, going from a forty year old server in a restaurant to a first time college student was a big change. In order to figure out what I wanted to study I researched online, bounced ideas off of friends and even talked to strangers on the street. I tried to keep an open mind and think creatively about my options, and then used my experience and perspective to narrow down the long list of potential interests. Once I decided on a career path and school, I quit my job and moved to a new city, relying heavily on discipline, humor and hope. I can’t tell you how many times I thought about staying in Portland and working at my old job forever, but now that I’m in school, I’m really grateful that I didn’t.
~Student #2
A few years ago I gathered together a group of children from my neighborhood and together we wrote an adaptation of “The Frog Prince.” We then built a stage in the back yard and spent weeks painting sets and creating costumes. When we had finished all the preparations, we pulled couches and chairs into the backyard and invited our whole neighborhood over to watch our play. It was amazing to help guide and motivate the children as they performed their creation; it took an enormous amount of organization and delegation skills to make our production go smoothly. It was incredible to be able to help our community come together and watch neighbors that had lived next to each other for years finally forging connections and becoming friends. It was wonderful to be able to see the children growing in their confidence and sharing their creation with our community.
~Student #3
Thinking about applying for scholarships can seem like an overwhelming prospect, and students have many excuses for not applying. There are so many scholarships available for college that knowing where to start is the first obstacle to the process. Remember, scholarships are the gift of money for college. A gift does not have to be paid back like a loan does.
Scholarships are offered to students who meet a specific requirement established by the sponsor, who may be an individual or an organization. Scholarships can be offered through local, state, or national sponsors. Each scholarship will have its own requirements based on the purpose of the scholarship. Scholarships are a good way to help pay for college without increasing student debt. Students may apply for multiple scholarships. Receiving a scholarship will affect the student’s overall financial aid award because all the student aid added together cannot be more than the cost of attending college. However, it is important to realize that scholarships are gifts and do not have to be repaid, so trying to include a scholarship your overall financial aid package is a good idea.
Finding scholarships requires research and effort on the part of the student, but the effort can have a financially rewarding outcome. Searching for scholarships today is much easier than in the past. Students used to have to comb through books in counselors’ offices and photocopy applications to be put in the mail, snail mail!
The Internet has changed the search process. In today’s scholarship search process, a student can use several websites to help find the treasure. Never pay for help to search for scholarships. Websites that charge fees to find scholarships may be scams. The Scholarship Fraud Prevention Act of 2000 was passed to help increase the penalties for people convicted of scholarship fraud. Before this Act was passed, the Federal Trade Commission was limited to closing operations defrauding consumers. Now the government has the power to incarcerate or fine perpetrators of scholarship fraud.
Free help can be found through the college you have selected to attend as well as through several great websites. Check with student support services at your college to see what services are offered. Scholarship Junkies , Unigo, Fastweb , and Fin Aid are examples of online resources for finding scholarships to apply for. Unigo even has a section for scholarships that don’t require an essay.
A YouTube element has been excluded from this version of the text. You can view it online here: https://openoregon.pressbooks.pub/collegetransition/?p=161
Video link: https://youtu.be/-VZKL5bcFvI
In the State of Oregon, the Oregon Office of Student Access and Completion (OSAC) is a place to start the scholarship application process. OSAC was established in 1959 by the Oregon Legislature to help fund post-secondary education. OSAC requires students to fill out the FAFSA before applying to OSAC. Each year OSAC awards over $118 million to Oregon residents seeking a college education. According to their website:
As a national leader in public/private partnerships of scholarship and innovate outreach programs, OSAC administers the following programs: Scholarships , the Oregon Opportunity Grant , the Chafee Grant , the Childcare Grant , ASPIRE (Access to Student assistance Programs In Reach of Everyone) as well as smaller funding programs.
Scholarship committees want to give their money away to deserving students. It’s your job to properly sell yourself so they know why you are the right choice. Build a profile that can’t be ignored, one that showcases your originality, your character and your drive to be successful. Avoid these common mistakes students make.
One of the major reasons student fail to earn scholarships is due to missing the application deadline. Deadlines matter and once they pass, the opportunity for that scholarship has ended for that year. It is important to pay attention to the time zone the deadline occurs in. The scholarship website may be located in a different time zone than you are. If the deadline says 11:30 pm EST (Eastern Standard Time) that is 8:30 pm PST (Pacific Standard Time). If the scholarship says it’s due by March 15th, it means it. On March 16th you will not be able to submit the application. This is a harsh reality for some students who put off working on their scholarship applications.
The directions on a scholarship application are not suggestions. These are the basic requirements that you need to fulfill in order to be considered for a scholarship. If you do something careless like emailing your application when you are supposed to mail it or not bothering to format your application correctly, you may not get the scholarship.
Scholarship committees request specific information because they need it. If the scholarship committee does not receive all of that information from you, the scholarship committee will likely look at your application, see that it is incomplete, and move it to the disqualified pile. If your application is submitted online and the information is incomplete, the application will not make it past the computer screening.
Read the requirements of the scholarship carefully. If there is a specific aspect of the scholarship that you do not meet, find a different scholarship to apply for.
Use the Internet to find out as much as possible about the sponsor of the scholarship. If it is a company or organization, find out what their mission is and what they care about. If the sponsor is a person or in memory of a person, what was the person’s passion?
Always have someone proofread your application before you send it in. This will help reduce any spelling or grammar errors or other mistakes that may be in your application before you send it. If you want to earn some money, you’ll want your application to be as polished as possible!
Word count is probably the most common scholarship essay mistake. If the application asks for a word range, hit the range. If it asks for a specific word count, hit the word count as closely as possible. This shows you’re capable of paying attention and satisfying specific requirements. Another common mistake is falling off topic. You want your essay to stand out from all the others. It needs to be unique, but it needs to address the topic given.
While a cute or risqué email address can seem clever among your group of friends, it can send the wrong message to a scholarship committee, or the professors at your college. When applying for scholarships, avoid email addresses that use nicknames, profanity, that are offensive, or that have sexual connotations. Instead, create a professional email address to use for scholarship applications and professional correspondence. Keep it simple and straight forward by using variations of your first, middle, and last name.
Many scholarship applications request a personal statement or essay to gain a perspective on the student in a more personal way. This is an opportunity for a student to build a unique picture of him or herself. OSAC uses 4 topics in their application. At Lane Community College, the Foundation (the source of scholarships specific to Lane) uses the same 4 topics in their applications. A student may use the same answers for both OSAC and Lane Community College’s applications. Check the college you plan to attend and see if you can find their essay questions. Chances are good, if it’s an Oregon school, it will use the same questions as OSAC.
OSAC’s application limits the number of characters you can use in a response. This is different from a word limit. Be sure to find out if the application you are completing uses words or characters in the directions for space limitations of answers.
Sometimes students worry that they don’t have a good answer to the questions posed by the scholarship application. Your answer doesn’t need to be a world-saving event. It needs to show your personality and qualities that will be worth investing in. The examples at the beginning of the chapter demonstrate real-life events that answer the question being posed. Finding small stories to tell will make better statements when you only have limited characters to use.
A YouTube element has been excluded from this version of the text. You can view it online here: https://openoregon.pressbooks.pub/collegetransition/?p=161
Video link: https://youtu.be/VMIpxqeoI1c
The final message in Zach King’s video is that everyone has ideas that matter and thoughts and ideas that inspire people. The personal statements and essays in a scholarship application are the place a student can set him or herself apart from the other applicants. Sitting down and writing an inspiring essay in 1000 characters or 150 words can seem like an impossible task.
One way to get started is to write something less structured. Try writing a random autobiography about yourself. This can be a fun way to start thinking about yourself and your experiences in order to find topics to use for personal statements and essays. It’s your story – you can’t get it wrong. Instructions and examples for writing a random autobiography are in the next chapter.
Student work on this page used with permission.


16.12: Introduction to Tradeoffs in Global Business
What you’ll learn to do: explain the risk tradeoffs in typical forms of global business
If you wanted to expand your business into the global marketplace, how would you go about it? Would you try to buy a competing company and take it over? Would you start a new business from scratch? Would you try to partner with a competing firm on equal terms? What is the most economical way to penetrate a new market that will provide the greatest profits? These are the very questions that business people all over the world today ask themselves (and their managers). And these are the questions that will be answered in the next section.
Contributors and Attributions
CC licensed content, Original
- Introduction to Tradeoffs in Global Business . Authored by : John/Lynn Bruton and Lumen Learning. License : CC BY: Attribution