9.3: Transferring Patients
By the end of this section, you will be able to:
- Recognize safety techniques when transferring, lifting, and ambulating patients
- Identify proper usage for various assistive devices
- Develop key aspects for patient education related to transfers
The nurse’s assessment of the patient’s mobility, strength, balance, and ADL status is critical to assisting the patient with transfer needs. The most common patient transfers completed by nurses are from a bed to a stretcher, from a bed to a wheelchair, and from a wheelchair to a bed (Bergman & De Jesus, 2022). Patient transfers that are poorly managed are associated with higher fall rates, morbidity, and mortality (Bergman & De Jesus, 2022).
While there are many methods to transfer patients, the methods used must be safe for the patient and the nurse. One of the most common injuries reported by the worldwide healthcare workforce is musculoskeletal injuries. Nurses make up 37 percent of those injuries, and healthcare assistants make up 46 percent of those injuries (Albanese et al., 2022). Often these are the result of improper lifting, inadequate assistance or equipment, and lack of preparation for the transfer. The following section will discuss safe techniques to move patients including the use of transfer equipment.
Safety with Transferring, Lifting, and Ambulating Patients
Transferring a patient is moving a patient from one place to another. For example, patients may be moved from bed to chair, bed to stretcher, or stretcher to wheelchair. Safety is a top priority when transferring patients. This can be achieved using best practice standards and tailoring techniques based on the patient’s unique needs. There are several factors to consider including:
- Is the patient physically able to transfer?
- Does the nurse have the proper equipment and have they been trained to use it?
- Has the nurse clearly communicated the steps with the patient?
The U.S. Department of Labor’s Occupational Safety and Health Administration (OSHA) developed Safe Patient Handling Programs to reduce musculoskeletal injuries in healthcare workers during patient transfer.
Falls can have serious consequences, especially for older adults, leading to declines in quality of life and functional performance. The Centers for Medicare and Medicaid Services focus on preventable falls. Hospitals that have a high number of preventable falls are financially penalized or receive incomplete reimbursement from the patient’s insurance. Modifiable risk factors are scrutinized to determine if the fall could have been prevented with the use of assisted devices, changes in medications, or standardized safety measures. When assessing a patient using a standardized tool, modifiable risk factors for falls should be documented with interventions in place, especially before attempting to transfer a patient (Kiyoshi-Teo et al., 2019).
Scenario: Carlton is visiting his brother, Sam, at the hospital and observes the different equipment available for Sam and his roommate, Perry. Sam is in the hospital for a cellulitis infection in his right lower extremity and Perry, a total care patient who has dimentia, was diagnosed with a urinary tract infection but is nearly recovered. Natasha is the registered nurse who is caring for both patients.
Patient’s brother: Excuse me, Natasha. I don’t mean to bother you, but I was wondering if you could answer some of my questions.
Nurse: Sure, Carlton, ask away.
Patient’s brother: My brother will be discharged soon but he still has a lot of weakness in his right leg. How will he manage at home?
Nurse: Physical therapy has been working with him and has given Sam crutches to take home. He’s practiced using them and is partial weight-bearing , meaning he can put some weight on his right leg to help keep himself supported and stabilized.
Patient’s brother: That’s great. I didn’t realize that physical therapy was involved. Can I ask you another question?
Nurse: Of course!
Patient’s brother: I see that Perry cannot get out of bed and needs to use that big contraption where he hangs. How does that work?
Nurse: Yes, that is the Hoyer lift. A pad that is specialized for the Hoyer lift is placed underneath Perry. The nurses and patient care assistants can get the pad underneath him by rolling Perry from side to side. Once the pad is underneath Perry, there are designated hooks and loops that will securely hold him when he is lifted up. The Hoyer can fit under the bed so it’s easy to move the patient from the bed to the wheelchair. It’s safe for Perry and helps us transfer him without compromising our backs.
Patient’s brother: That’s so interesting. It’s great to have equipment in place to keep you and Perry safe.
Weight-Bearing Transfer
A person is weight-bearing if they can put force or body weight on a specific limb. There are several types of weight-bearing grades (Table 9.5). If a patient can bear weight, nurses should encourage their help with transfers. The nurse should stand by for safety.
| Weight-Bearing Grade | Description |
|---|---|
| Full weight-bearing | The leg can tolerate the entire body weight, and can ambulate. |
| Weight-bearing as tolerated | Dependent on the circumstance, the patient can support 50 percent to 100 percent of entire body weight on the leg. |
| Partial weight-bearing | A small amount of weight can be tolerated on the leg. The patient needs an assistive device to ambulate, such as a cane or crutches. |
| Touch-down weight-bearing | The patient can use their toes of the affected leg to touch the ground for balance, but cannot support any weight on the leg. |
| Non-weight-bearing | The leg cannot touch the floor or support any body weight. |
For patients who can partially bear weight, nurses should use the stand-and-pivot technique, preferably using a gait or transfer belt or standing lift assist. A stand and pivot is a technique for patients, who can partially weight-bear, by moving from sitting to standing and then just turning with one leg to sitting again (Figure 9.15). A gait belt is a strap that is a few inches wide made of nylon, canvas, or leather. It is placed around a patient’s waist to help nurses assist with transfers. In addition, the patient should transfer toward their stronger side if they have partial weight-bearing capabilities.
See the competency checklist for Transferring a Patient from the Bed to a Chair. You can find the checklists on the Student resources tab of your book page on openstax.org.
Non-Weight-Bearing Transfer
A non-weight-bearing injury is when the affected limb or limbs cannot support any weight. In injury or recovery, patients may have practitioner orders to maintain non-weight-bearing status with a certain limb to aid in healing. Premature weight-bearing can delay recovery and lead to complications. Transfer devices, such as a full body sling lift or slide board, should be utilized as appropriate.
See the competency checklist for Transferring a Patient from the Bed to a Stretcher. You can find the checklists on the Student resources tab of your book page on openstax.org.
Sling Transfer
A sling transfer is a patient transfer from a bed, wheelchair, toilet, or shower using a lift machine and flat piece of material placed under the patient (Figure 9.16). The use of the sling carries the full weight of the patient, reducing the risk of injury for the patient and caregiver. The slings are fitted to the patient to provide neck, head, and shoulder support.
When transferring with a sling, the equipment used to transfer should be appropriate for the sling and applied correctly. Equipment should never be substituted if the correct item is not available. Most facilities and lift manufacturers require two caregivers to safely operate a patient lift. If the lift has wheels, make sure they are locked. Slide the sling under the patient by either having them roll side to side or lean forward. The patient’s arms should be as close to the sides of their body to help keep the weight centralized. During transfer it is important for the nurse to continuously communicate with the patient as they may be fearful or anxious about falling. Each transfer should be carefully planned, as all have varied requirements.
This video provides step-by step instruction on how to safely operate a Hoyer lift to transfer a patient. A Hoyer lift is a type of sling transfer.
Sliding Board
A sliding board is a piece of equipment that is flat and rigid, used for patients who cannot complete a transfer between two surfaces or if a standing transfer is not safe (Figure 9.17). The board makes a “bridge” between the bed and the other surface, such as a stretcher, and the patient can slide between the two. The nurse should always use a gait/transfer belt for safety and instruct the patient to move slowly and deliberately across the board if they are assisting with the transfer. Skin precautions are essential to prevent sheering and friction.
Various Assistive Devices
Assistive devices can give patients mobility and the ability to transfer from one location to another. Allowing patients to participate in active transfers, if they are able, can help promote functional independence. At home, patients may find ADLs challenging; therefore, the use of assistive devices is vital, especially for older adults, to maintain mobility, remain safe, and enable communication with others. Assistive devices can include self-care devices to accomplish daily tasks, mobility devices that help with moving around and walking, and safety assistive technology that can help with lack of balance , weakened limb function, decreased eyesight, and other sensory issues.
Gait Belt
A gait belt can also be used as a device to assist an unsteady patient with ambulation. The nurse holds the gait belt to steady themselves and the patient as they move around together. Gait belts have been found to reduce the incidence of patient falls as well as physical stress on the nurse (Figure 9.18).
When utilizing a gait belt as an assistive device:
- Ensure the patient is wearing nonslip footwear.
- Place the gait belt snugly over a layer of clothing.
- Ensure two fingers fit between the belt and the patient, ensuring it is not too tight.
- Face the patient with the back straight, knees bent, in a wide stance while preparing to lift the patient from sitting to standing.
- Grab the gait belt on both sides of the patient and straighten your body as the patient rises.
- Stand behind and to the side of the patient when ambulating, and use an underhand grip to grasp the loop on the back of the gait belt.
- Maintain open communication with the patient throughout the procedure, and encourage the patient to promptly report any discomfort or fatigue.
Contraindications in using a gait belt include patients who have abdominal aneurysms, G tubes, hernias, or severe cardiac or respiratory conditions and patients who have had recent chest, back, or abdominal surgery.
Crutches
Devices called crutches are an aid used for standing, walking, and mobility for patients who cannot support the weight of their bodies with their leg, knee, or ankle. The patient’s weight-bearing status will determine how much pressure can be put on the affected limb. There are three types of crutches: axilla, elbow, and gutter.
An axillary (underarm) crutch should be two fingers distant between the axilla and axilla pad of the crutch. The patient’s weight should be focused where their hands grip the crutch and not under their armpits. This is to prevent axillary nerve damage.
This video provides instruction on how to properly use axillary crutches and includes different gait techniques.
Forearm ( lofstrand , elbow, or Canadian) crutches have a forearm cuff and handgrip. These are often used long term in patients with lower extremity disabilities.
Gutter crutches are modified forearm crutches and a modified hand grip for patients who may not have mobility and strength in their hands. This type of crutch can be recommended for patients with rheumatoid arthritis. Crutches should be tailored to the patient and fit according to their height.
Nurses can teach patients how to walk with crutches by having them clasp the hand grips, move both crutches forward at the same, swing their “good” limb through, and land with a shorter stride.
Knee walkers (knee scooters or mobility scooters) are used as well to help with lower limb mobility. This wheeled crutch alternative allows the injured foot and/or leg to be elevated on a padded platform. The patient can balance naturally on two legs and can move by pushing with the good leg and steering with the handlebar.
Walker
A walker is an aide that has four points of contact with the ground. Walkers provide a greater base of support, assisting in stability for balance and mobility. There are many types of walkers including standard, two-wheeled, upright, and four-wheeled walkers. Standard four-point walkers (four-point or Zimmer frame ) are used with the patient taking one step forward and then the patient moving into the walker (Figure 9.19). Two-wheeled walkers are used the same way except the walker can slide forward instead of the patient needing to pick it up because there are two rear wheels (Figure 9.20a). Four-wheeled walkers have handles to control brakes and may also have a seat, tray, or basket (Figure 9.20b). The patient must remember to lock the brakes when stationary to avoid the risk of a fall. The gutter frame walkers are wheeled and adapted to patients who need more support. Forearm supports fix the elbows at 90 degrees.
Cane
A cane is a singular assistive device that assists in balance when walking or helps compensate for an injury or disability. A cane can either have a single tip or four points that touch the ground, called a quad cane . The quad cane provides a broader base of support. The grip on canes can vary by size and material. Some may be made of foam or plastic, and sizes may vary for arthritic hands. The patient’s elbow bend and wrist height should be taken into consideration when fitting a cane for a patient (Figure 9.21). The elbow should bend at a comfortable angle, approximately 15 degrees, and the top of the cane should line up with the crease on the patient’s wrist. Patients should hold the cane in the opposite hand of the weak or painful limb. The cane should move with the affected leg.
Racial and ethnic disparities exist in health care, which include the use of assistive devices among older adults. Assistive devices are meant to extend independence in aging; however, the rate of use of assistive devices among older adults living in Hispanic communities is lower than other populations (Orellano-Colon et al., 2016).
Barriers reported by Hispanic community-living older adults include the following:
Human Barriers
- lack of knowledge about assistive devices including availability, acquisition, or the use of devices for ADL assistance
- safety concerns with many participants fearing that the equipment will break or they may fall
- dislike for assistive devices and a lack of perceived usefulness
- concern of losing functional capacities if assistive devices are used
Assistive Device Barriers
- complexity in the use of assisted devices
- the equipment failed or is unattractive in appearance
Context Barriers
- financial limitations
- lack of access to the physical environment including not having enough space
- institutional systems barriers that prevent access to equipment and services
- social stigma with use of assistive device
There are multifactorial reasons why the use of assistive devices among older people in the Hispanic community is lower than other populations. Social determinants of health, language, and understanding, as well as education, are factors that need to be considered when working with patients of varied cultures.
Refer back to Unfolding Case Study #2: Part 4 to review the patient data.
| Nursing Notes | 1430: Patient educated about importance of maintaining mobility to improve functional status. Patient expressed understanding of teaching and willingness to work with physical therapy to improve strength. Patient reminded to use call light before getting up and fall risk precautions in place. |
| Provider's Orders |
Physical therapy consult
Orthostatic vital signs daily |
Patient Education Related to Transfers
In transfers and assistance with mobility , patient education is important to maintain safety and ensure a successful process. Be sure to include education about
- the importance of remaining engaged to maintain safety;
- maneuvers during transfer, such as pivoting, and ensure the patient understands directions before proceeding;
- any equipment or assistive devices being used, including safety measures;
- signs and symptoms of any distress during the procedure to report to the nurse; and
- if unable to transfer independently, provide education on calling for assistance and use of call light.
The nurse should always ask permission and inform the patient of all the steps of the transfer or equipment use. Explain all steps in simple and clear terms, gather all the necessary equipment for the transfer, and cue the patient to perform as much as possible. Ensure that the patient has proper alignment of the spine and limbs and is wearing the proper clothing (e.g., nonslip footwear, a nonrestrictive hospital gown). Before transfer, the nurse should institute skin precautions, request additional personnel assistance, if needed, use proper body mechanics, and lock and stabilize equipment.


3.4: Your Bodies Response to Stress
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Fight or Flight Response
When we experience excessive stress, either from internal worry or external circumstance, a bodily reaction called the "fight-or-flight" response will be triggered. Harvard physiologist Walter Cannon originally defined it. The response system represents the genetic impulse to protect ourselves from bodily harm, but also can result in negative health effects. According to Cannon’s theory, during stress-response processes, the sympathetic nervous system increases the heart rate and releases chemicals to prepare our body to either fight or flee. When the fight-or-flight response system get activated, it tends to perceive everything in the environment as a potential threat to survival.
In modern life, we do not get the option of "flight" very often. We have to deal with those stressors all the time and find a solution. When you need to take an SAT test, there is no way for you to avoid it; sitting in the test room for five hours is the only choice. Lacking the "flight" option in stress-response process leads to higher stress levels in modern society.


18.6: Exercises
LAB 17 EXERCISES \(\PageIndex{1}\)
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Match the following
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Match the following
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LAB 17 EXERCISES \(\PageIndex{3}\)
Tonsils
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Label tonsils
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Identify the lymphatic structure found in the ileum and colon
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LAB 17 EXERCISES \(\PageIndex{1}\)
Abdominal Region
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Label the following
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Fill in and identify all the boxes
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LAB 17 EXERCISES \(\PageIndex{4}\)
Peyer’s patches (Mucosal associated lymphatic tissue, MALT) and Tonsil
- Obtain a slide of each of the gland listed below from the slide box at your table.
- Follow the checklist above to set up your slide for viewing.
- View the slide on the objective which provides the best view. Find the representative object.
- In the circle below the name, draw a representative sample of the tissue, taking care to correctly and clearly draw their true shape in the slide. Draw your structures proportionately to their size in your microscope’s field of view.
- Fill in the blanks next to your drawing and identify the structures listed on the last page under, “Histology”
Repeat this for each of the tissue types seen below.
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Lymph Node
Total Magnification: _________________
Type of eptihelium: _________________
Source of tissue: ____________________
Function of this tissue: _______________
__________________________________ |
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Spleen
Total Magnification: _________________
Type of eptihelium: _________________
Source of tissue: ____________________
Function of this tissue: _______________
__________________________________ |
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Peyer’s Patches
Total Magnification: _________________
Type of eptihelium: _________________
Source of tissue: ____________________
Function of this tissue: _______________
__________________________________ |
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Tonsil
Total Magnification: _________________
Type of eptihelium: _________________
Source of tissue: ____________________
Function of this tissue: _______________
__________________________________ |
F


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3.2.1: Introduction
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It is one thing to not be able to imagine how one would compute complicated functions, and quite another to actually prove that they are uncomputable.
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3.2.2: Enumerating Turing Machines
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We can show that the set of all Turing machines is countable. This follows from the fact that each Turing machine can be finitely described.
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3.2.3: The Halting Problem
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The halting problem is one example of a larger class of problems of the form “can \(X\) be accomplished using Turing machines.” (The answer is no: the halting problem is unsolvable.)
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3.2.4: The Decision Problem
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We say that first-order logic is decidable iff there is an effective method for determining whether or not a given sentence is valid. As it turns out, there is no such method: the problem of deciding validity of first-order sentences is unsolvable.
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3.2.5: Representing Turing Machines
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In order to represent Turing machines and their behavior by a sentence of first-order logic, we have to define a suitable language. The language consists of two parts: predicate symbols for describing configurations of the machine, and expressions for numbering execution steps (“moments”) and positions on the tape.
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3.2.6: Verifying the Representation
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In order to verify that our representation works, we have to prove two things. First, we have to show that if \(M\) halts on input \(w\), then \(T(M, w) \rightarrow E(M, w)\) is valid. Then, we have to show the converse, i.e., that if \(T(M, w) \rightarrow E(M, w)\) is valid, then \(M\) does in fact eventually halt when run on input \(w\).
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3.2.7: The Decision Problem is Unsolvable
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If the decision problem were solvable, we could use it to solve the halting problem.
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3.2.8: Summary
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19.2.3: Cleavage
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Cleavage refers to the early cell divisions that occur as a fertilized egg begins to develop into an embryo.
Holoblastic Cleavage
In eggs that contain no (mammals) or only moderate amounts (frog) of yolk, cytokinesis divides the cells completely. The figure shows the results of the first two cleavages in the frog embryo.
Meroblastic Cleavage
In eggs that contain a large amount of yolk, cytokinesis does not divide the egg completely.
The hen's egg consists of just a tiny patch of cytoplasm resting on the surface of a large ball of yolk (the "white" of the egg is noncellular accessory protein). When the first cleavages occur in the hen's egg, the cleavage furrows do not continue down through the mass of yolk. Therefore, each of the cells produced in the earliest stages is bound on the top and on the sides by a plasma membrane, but the bottom of the cell is in direct contact with yolk.
This type of meroblastic cleavage is also found in the eggs of fish, reptiles, and 4 species of mammals — the monotremes. This photo, courtesy of H. W. Beames and Richard G. Kessel, shows the zebrafish (Danio) embryo at the 32-cell stage. Note that the cleavage furrows have not continued down through the yolk of the egg.
Insects use a different type of meroblastic cleavage.
The yolk of the eggs of insects is concentrated in the center of the egg. The daughter nuclei produced by mitosis of the zygote nucleus remain suspended within the single egg compartment. After several thousand nuclei have been produced, they migrate to the cytoplasm-rich margin of the egg. Only then does a plasma membrane form around each one.
What does cleavage accomplish in the development of the organism? First, it provides a stockpile of cells out of which the embryo will be constructed. Second, cleavage establishes a normal relationship between the nucleus and the volume of cytoplasm it regulates (and which in turn regulates it). Even small eggs are enormous when compared with other kinds of cells. The volume of the frog egg is about 1.6 million times larger than that of a normal frog cell. But it, too, contains only a single nucleus. During cleavage, thousands of new nuclei are produced by mitosis all of which finally end up in a cell of normal dimensions. The frog blastula, with its thousands of cells is no larger than the original fertilized egg.


2.3: Predicate Logic
Background
In order to efficiently communicate logic statements we use predicates. These are simply functions with a codomain of ‘true’ and ‘false.’ For example \(R(x)=x\) is red. Thus \(R(stop sign)=true\) and \(R(Seawolf logo)=false\). More complicated statements can be expressed as well. The following example converts the statement “\(n\) is a positive, even number” into symbols using predicates.
Define the following statements.\(E(n)\) is the statement n is even.
\(P(n)\) is the statement n>0.The statement: n is a positive, even number can now be written as \(P(n) \wedge E(n)\).
Quantifiers
Terminology
For predicates we often ask the questions, “does any object satisfy the predicate (condition)” and “what is true of all objects that satisfy the predicate (condition).” The first we call existential and use the symbol \(\exists\) for “there exists.” The second we call universal and use the symbol \(\forall\) for “for all.” These two questions add quantifiers to logic. Another symbol used is \(\ni\) for “such that.”
Consider the following predicates for examples of the notation.
\[E(n)=n is even\]
\[P(n)=n is prime\]
\[Q(n)=n is a multiple of 4 \]
Using these predicates (symbols) we can express statements such as those in Table \(\PageIndex{1}\)| There is some number that is even and prime. | \(\exists n \ni E(n) \wedge P(n)\) |
| All multiples of four are even. | \(\forall n \: Q(n) \rightarrow E(n)\) |
Checkpoint \(\PageIndex{2}\) Write the following statement using predicates and quantifiers. “All computer science majors must take Introduction to Discrete Mathematics and Data Structures and Algorithms.”
Negation
Negate a statement
One of the logical operations is negation. We can negate whole statements or parts of statements. A useful skill is to simplify the negation of a statement. This typically enables us to understand the meaning of the negation more easily and to optimize conditional statements.
This example demonstrates negating a statement involving conjunctions, but no quantifiers. Suppose the following claim is false.
“Guido owns a sail boat and has either a type rating for the Cessna Citation X or likes to cook.”
First we define predicates for each part of the statement.
S(x)=x owns a sail boat.
C(x)=x has a type rating for the Cessna Citation X.
K(x)=x likes to cook.
Now we write the negation of this statement as
\(\neg ( S(Guido) \wedge (C(Guido) \vee K(Guido) ) \leftrightarrow\) Using DeMorgan's Law we show this is the same as
\(\neg S(Guido) \vee \neg (C(Guido) \vee K(Guido) ) \leftrightarrow\) Then we apply the other DeMorgan's Law to show it is the same as
\(\neg S(Guido) \vee (\neg C(Guido) \wedge \neg K(Guido) )\).
Returning this to English gives us
“Either Guido does not own a sailboat, or he does not have the Cessna Citation X type rating and he does not like to cook.”
Discover
In addition to DeMorgan's Laws (negation of and and or ), the negations of the two quantifiers are needed. The negation of for all and there exists statements can be recognized from examples. Try the ones below.
| Name | Hair | Eye |
| Guido | Brown | Brown |
| Wolfgang | Red | Brown |
| Ludwig | Brown | Hazel |
| Franz | Brown | Brown |
| Maria | Gray | Brown |
Checkpoint \(\PageIndex{5}\)
How would you show that the statement “All of the five friends have brown hair” is false?
Checkpoint \(\PageIndex{6}\)
How would you show that the statement “Some of the five friends have blue eyes” is false?
Checkpoint \(\PageIndex{7}\)
What is the negation of the following statement. “All Apple fanboys own an iPad.”
Checkpoint \(\PageIndex{8}\)
What is the negation of the following statement. “Some chemist likes to cook.”
Checkpoint \(\PageIndex{9}\)
(a) For Checkpoint \(\PageIndex{5}\) how many entries in the table did you have to check?
(b) For Checkpoint \(\PageIndex{6}\) how many entries in the table did you have to check?
(c) What do your answers to these two questions tell you about the efficiency of the operators and and or ?
Examples
We will write the statement: “Some flower is loved by all bees.” First we must define all the terms.
B(x)=x is a bee.
F(x)=x is a flower.
Y(x)=x is yellow.
L(x,y)=x likes y.
Now we can write the statement in symbols. \(\exists y \: F(y) \ni ( \forall x \: B(x) \rightarrow L(x,y) ).\)
Next we will negate the statement. We will use knowledge of negating all/some and DeMorgan's laws.
\begin{align*}
\neg [\exists y \: F(y) \ni ( \forall x \: B(x) \rightarrow L(x,y) )] \leftrightarrow\\
\forall y \: F(y) \rightarrow \neg ( \forall x \: B(x) \rightarrow L(x,y) ) \leftrightarrow\
\forall y \: F(y) \rightarrow ( \exists x \: B(x) \ni \neg L(x,y) ).
\end{align*}
The negation is the statement: “Each flower is distasteful to some bee.”
Next we will write the statement: “All red berries are loved by some bear.”
U(x)=x is a bear.
B(x)=x is a berry.
R(x)=x is red.
L(x,y)=x likes y.
\(\forall y \: (B(y) \wedge R(y)) \rightarrow [\exists x \: U(x) \ni L(x,y)]\)
We can now negate the statement.
\(\neg (\forall y \: (B(y) \wedge R(y)) \rightarrow [\exists x \: U(x) \ni L(x,y)]) \leftrightarrow\)
\(\exists y \: (B(y) \wedge R(y)) \ni \neg [\exists x \: U(x) \ni L(x,y)]) \leftrightarrow\)
\(\exists y \: (B(y) \wedge R(y)) \ni [\forall x \: U(x) \rightarrow \neg L(x,y)]).\)
The negation is the statement: “Some red berry is not loved by any bear.”
If you wonder why DeMorgan's law was not applied in the first step, look at the first example. When the \(\exists\) (first clause) is negated, the exists statement becomes a for all statement and the negation is passed over the condition to the next clause. The negation changes exists/for all but does not change their condition.
Practice
Checkpoint \(\PageIndex{12}\)
Negate the statement: “Either the person is a math major, or the person is a computer science major and has taken MATH A265.”
Checkpoint \(\PageIndex{13}\)
Negate: \(\forall n \: C(n) \rightarrow \left( \exists m T(m) \ni \left[ R(n,m) \vee S(n,m) \right] \right)\). Meanings for the predicates are not required.
Checkpoint \(\PageIndex{14}\)
Negate: \(\forall x \: (A(x) \vee B(x)) \rightarrow \left( \exists y C(y) \ni \left[ D(x,y) \vee E(x,y) \right] \right).\) Meanings for the predicates are not required.
Checkpoint \(\PageIndex{15}\)
We will negate a natural language statement in steps.
(a) Convert this statement to logic notation.
All moose love some mountain ash.
(b) Negate the statement using the logic notation.
(c) Convert your negated logic statement back to natural language.
Checkpoint \(\PageIndex{16}\)
These demonstrate the full spectrum of what can happen and will require careful attention to grammar. Negate the following statements and re-write in English.
- B(x): x is a bee.
- F(x): x is a flower.
- Y(x): x is yellow.
- L(x,y): x likes y.
(a) \(\forall x \: B(x) \rightarrow ( \forall y \: F(y) \rightarrow L(x,y) ).\)
- Answer
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\(\exists x \: B(x) \ni ( \exists y \: F(y) \ni \neg L(x,y) ).\)
Some bee does not love some flower.
(b) \(\forall x \: B(x) \rightarrow ( \exists y \: F(y) \ni L(x,y) ).\)
- Answer
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\(\exists x \: B(x) \ni ( \forall y \: F(y) \rightarrow \neg L(x,y) ).\)
Some bee hates all flowers.
(c) \(\exists x \: B(x) \ni ( \forall y \: F(y) \rightarrow L(x,y) ).\)
- Answer
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\(\forall x \: B(x) \rightarrow ( \exists y \: F(y) \ni \neg L(x,y) ).\)
Every bee dislikes some flower.
(d) \(\exists x \: B(x) \ni ( \forall y \: F(y) \rightarrow \neg L(x,y) ).\)
- Answer
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\(\forall x \: B(x) \rightarrow ( \exists y \: F(y) \ni L(x,y) ).\)
Every bee loves some flower.
(e) \(\exists x \: B(x) \ni ( \forall y \: [F(y) \wedge Y(y)] \rightarrow \neg L(x,y) ).\)
- Answer
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\(\forall x \: B(x) \rightarrow ( \exists y \: [F(y) \wedge Y(y)] \ni L(x,y) ).\)
Every bee loves some yellow flower.
Checkpoint \(\PageIndex{17}\) The following two statements are not logically equivalent. Demonstrate why by finding a case that is true for one statement but false for the other statement.
- Some tree is loved by all moose.
- Every moose loves some tree.
- ►Hint
-
Consider whether each moose has to love the same tree.
Exercises
Use the following predicates below.
- M(x): x is a moose.
- G(x): x is a patch of grass.
- N(x): x is new.
- E(x,y): x eats y.
1. Write the following statements using predicate notation.
All moose eat some patch of new grass.
Some moose eats all patches of new grass.
Some moose eats no patch of new grass.
All moose do not eat some patch of new grass.
2. Negate the following statements using predicate notation.
All moose eat some patch of new grass.
Some moose eats all patches of new grass.
Some moose eats no patch of new grass.
All moose do not eat some patch of new grass.
3. Write the negations of the following statements in English.
All moose eat some patch of new grass.
Some moose eats all patches of new grass.
Some moose eats no patch of new grass.
All moose do not eat some patch of new grass.
4. What must you show to prove the following true?
All moose eat some patch of new grass.
Some moose eats all patches of new grass.
5. What must you show to prove the following false?
a. Some patch of new grass is eaten by no moose.
b. No moose does not eat some patch of grass.


10.2: Intergroup Relations
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- Erika Gutierrez, Janét Hund, Shaheen Johnson, Carlos Ramos, Lisette Rodriguez, & Joy Tsuhako
- Long Beach City College, Cerritos College, & Saddleback College via ASCCC Open Educational Resources Initiative (OERI)
Intergroup Consequences
A variety of intergroup consequences can be used to explain the experiences of Middle Eastern Americans. At the inhumane extreme, genocide (the systematic killing of an entire people) explains the Holocaust. Between 1941 and 1945, across German-occupied Europe, Nazi Germany, and its collaborators systematically murdered some six million Jews, around two-thirds of Europe's Jewish population. The murders were carried out through mass shootings, extermination through work in concentration camps, and gas chambers. This genocide led many to flee as refugees to the United States; however, thousands of Jews fleeing the horrors of the Nazi regime were denied asylum as they were feared to be Nazi spies (Gross, 2015).
Patterns of Intergroup Relations: Middle Eastern Americans
- Extermination/Genocide: The deliberate, systematic killing of an entire people or nation (e.g. Holocaust).
- Expulsion/Population Transfer: The dominant group expels the marginalized group (e.g. Syrian refugees).
- Segregation: The dominant group structures physical, unequal separation of two groups in residence, workplace & social functions (e.g. detainment after 9-11).
- Separatism: The marginalized group desires physical separation of two groups in residence, workplace & social functions (e.g. Quranic schools).
- Fusion/Amalgamation: Race-ethnic groups combine to form a new group (e.g. intermarriage).
- Assimilation: The process by which a marginalized individual or group takes on the characteristics of the dominant group (e.g. Judaization).
- Pluralism/Multiculturalism: Various race-ethnic groups in a society have mutual respect for one another, without prejudice or discrimination (e.g. Muslims elected to Congress).
The mass expulsion (when the dominant group expels the marginalized group) of Jews during WWII was followed many decades later by another Middle Eastern group, Syrians, who fled the Syrian Civil War. In the last five years, the pre-war population of the Syrian Arab Republic was estimated at 22 million; of that number, the United Nations identified 13.5 million as displaced persons, requiring humanitarian assistance. Of these, since the start of the Syrian Civil War in 2011, more than six million were internally displaced, and around five million had crossed into other countries. While not a world leader in accepting Syrian refugees, the U.S. did accept 16,218 Syrian refugees by 2016. In 2017, President Donald Trump signed an executive order suspending any further resettlement of Syrian refugees to the U.S. indefinitely until further notice due to security concerns.
Segregation (physical separation of a marginalized group from the dominant group) is yet another intergroup consequence that can be used to understand the experience of some Middle Easterners, particularly Arab and Muslim American men post 9-11. As a result of the 2001 terrorist attack, as David Cole describes below, the U.S. government rounded up more than 5,000 foreign nationals from Middle Eastern countries, many of whom were deported or detained for months. Similar to the internment of Japanese Americans during WWII, these individuals were rounded up merely because of "guilt by association," as being foreigners "associated with" the terrorist act - but the overwhelming majority actually had no proven association to the attacks.
As quoted in an American Civil Liberties Union (ACLU) press release:
Immigrants weren't the enemy...But, the war on terror quickly became a war on immigrants. The Inspector General's findings confirm our long-held view that civil liberties and the rights of immigrants were trampled in the aftermath of 9/11.
Another intergroup consequence of relevance is separatism , physical separation of race-ethnic groups as desired by a marginalized group, in the case of schooling. Some communities offer a private schooling experience that caters to the needs of Muslim or Jewish families and their children. Quranic or Sunday schools or schools for Black Muslims offer specific religious instruction to those attending mosque schools or as a supplement for children who attend public schools (Schaefer, 2019). Similarly, Hebrew school can be either an educational regimen separate from secular education similar to the Christian Sunday school, education focusing on topics of Jewish history and learning the Hebrew language, or a primary, secondary or college level educational institution where some or all of the classes are taught in Hebrew.
With the increasing practice of intermarriage, Jews marrying non-Jews, fusion has become a norm in the 21st century. In the 1970s, more than 64% of Jews married other Jews (Schaefer, 2019). From 2000 to 2013, that percentage dropped to 42% (ibid). Hence, in contemporary society, intermarriage is common practice. For some, this represents a threat to the faith of Judaism. For others, this represents an opportunity to be raised as bi-cultural - practicing both Hanukkah and Christmas, speaking Hebrew and English. This can also feed into assimilation , conforming to the norms of dominant culture, which lessens, or in some cases, eliminates ties to one's ethnic background. Judaization is the "lessening importance of Judaism as a religion and the substitution of cultural traditions as the ties that bind Jews" (Schaefer, 2019, p. 304).
Finally, pluralism , exemplified by mutual respect and appreciation for diverse cultures, may be understood to relate to Middle Easterners by considering ethnic enclaves and officials elected to office. Consider New York City. Several Middle Eastern ethnic groups have immigrated to New York and formed several neighborhoods with a high concentration of people who are of Arab descent. Between the 1870s and the 1920s, the first wave of Arab immigrants brought mostly Syrian and Lebanese people to New York City, the majority of them being Christian. There are now around 160,000 Arabic people in New York City and more than 480,000 in New York State. According to the Arab American Institute the population of people who identify themselves as Arab, grew by 23% between 2000 and 2008. New York today has the second largest number of Jews in a metropolitan area, behind Tel Aviv (in Israel). Borough Park, Brooklyn is one of the largest Orthodox Jewish communities in the world.
A growing number of Muslims have been elected to political office. A Somali-born woman, Ilhan Omar has served as the U.S. Representatives for Minnesota's 5th congressional district since 2019. She is also one of the first two Muslim women (along with Rashida Tlaib) to serve in Congress. A member of the Congressional Progressive Caucus, Omar has advocated for a living wage, affordable housing, universal healthcare, student loan debt forgiveness, protection of Deferred Action for Childhood Arrivals, abolition of Immigration and Customs Enforcement, and Lesbian Gay Bisexual Transgender rights. However, she is not well received by pro-Israel Jewish groups, as she has frequently denounced Israel's occupied Palestinian territories; she has been accused of anti-semitic remarks, for which she has apologized.
Islamophobia
Let us turn to a stark contrast to pluralism. Relations between Muslim and Arab Americans and the dominant majority group have been marked by mistrust, misinformation, and deeply entrenched beliefs. Helen Samhan of the Arab American Institute suggests that Arab-Israeli conflicts in the 1970s contributed significantly to cultural and political anti-Arab sentiment in the United States (2001). The United States has historically supported the state of Israel, while some Middle Eastern countries deny the existence of the Israeli state. Disputes over these issues have involved Egypt, Syria, Iraq, Jordan, Lebanon, and Palestine. A more detailed discussion of the Israeli-Palestinian conflict is provided later in the chapter.
As is often the case with stereotyping and prejudice, the actions of extremists come to define the entire group, regardless of the fact that most U.S. citizens with ties to the Middle Eastern community condemn terrorist actions, as do most inhabitants of the Middle East. Would it be fair to judge all Catholics by the events of the Inquisition? Of course, the United States was deeply affected by the events of September 11, 2001. This event has left a deep scar on the American psyche, and it has fortified anti-Arab sentiment for a large percentage of Americans. In the first month after 9/11, hundreds of hate crimes were perpetrated against people who looked like they might be of Arab descent.
Arab Americans are still victims of racism and prejudice. Racial profiling has proceeded against Arab Americans as a matter of course since 9/11. Particularly when engaged in air travel, being young and Arab-looking is enough to warrant a special search or detainment. This Islamophobia (irrational fear of or hatred against Muslims) does not show signs of abating. A recent survey of 5,000 respondents showed that many do not consider Muslims to be sufficiently "American," with 67% of Democrats and only 36% of Republicans agreeing with the statement that "Muslim Americans want to fit in as American citizens."
Anti-Semitism
For centuries, the Jewish people have struggled to overcome hatred. Religious observances such as passover, Hanukkah and Purim commemorate some of these struggles. Anti-Semitism (anti Jewish prejudice and discrimination) has existed since before Christianity and continues to exist today. The most horrific example of this was the Holocaust. The Holocaust was the state sponsored, systematic persecution and annihilation of Jews by Nazi Germany. As a result, two-thirds of the Jewish population in Europe was killed.
The Anti-Defamation League (ADL), founded in 1913, publishes an annual report that details incidents of anti-Semitism in the United States. In 2018, the ADL recorded 1,879 anti-Semitic incidents. These incidents included: vandalism, graffiti in the form of swastikas or anti Jewish sentiments, harassment, assault and murder. Some of these incidents were carried out by neo-nazis or skinheads, who are known to perpetuate anti Semitic ideologies. Recent attacks on synagogues in the United States (Pittsburgh, PA in 2018 and Poway, CA in 2019 are just two recent examples) have reminded people all over the world of the dangers of anti-Semitism.
Muslim and Jewish Relations: The Israeli-Palestinian Conflict
Throughout history, few groups have been so closely linked as Muslims and Jews. Tension and conflict among these groups have arisen throughout history as a result of religious differences, political differences, and conflict over land and natural resources. Today, the most contentious example of this conflict is represented in the continuing struggle between Israel and Palestine. Jews and Muslims both claim a religious tie to the land in Israel and Palestine, not only because both religions had major events take place there and are deeply rooted in the area, but because both claim that they were promised the land by God, through Abraham. Abraham had more than one son, however, and descendants from Isaac were predominantly Jewish and descendants from Ishmael became predominantly Muslim.
In the early 20th century, Jews fleeing persecution in Europe wanted to establish a national homeland in what was an Arab and Muslim majority territory. The Arabs resisted, seeing the land as rightfully theirs. Israel and the surrounding Arab nations fought several wars over the territory. The 1967 war left Israel in control of the West Bank and Gaza Strip, two territories that are home to large Palestinian populations. One approach to resolving the conflict would establish Palestine as an independent state in Gaza and most of the West Bank, leaving the rest of the land to Israel. Another approach would give all of the land to either Israel or Palestine. The conflict over who gets what land and how it's controlled is one that remains today.
Although the United States has historically been a strong supporter of Israel, the U.S. government has traditionally supported advancing a solution that would reconcile the claims of the two parties: Israel and Palestine. Multiple administrations have attempted to initiate a process that would result in two distinct states. However, many critics have claimed that the potential for this outcome has diminished as a result of President Trump's policies.
Contributors and Attributions
- Rodriguez, Lisette. (Long Beach City College)
- Ramos, Carlos. (Long Beach City College)
- Hund, Janét. (Long Beach City College)
- Introduction to Sociology (Lumen) ( CC By 4.0 )
- Introduction to Sociology 2e (OpenStax) ( CC BY 4.0 )
- Hebrew School (Wikipedia) ( CC BY-SA 3.0 )
- The Holocaust (Wikipedia) ( CC BY-SA 3.0 )
- Ilhan Omar (Wikipedia) ( CC BY-SA 3.0 )
- New York City Ethnic Enclaves (Wikipedia) ( CC BY-SA 3.0 )
Works Cited
- American Civil Liberties Union. (2003, June). Internal justice department report details 9/11 detainees' plight; Arab, Muslim, South Asian immigrant languished in detention for months. American Civil Liberties Union.
- Gross, D. (2015, November 18). The U.S. government turned away thousands of Jewish refugees, fearing that they were nazi spies. Smithsonian Magazine .
- Samhan, H.H. (2001). Who are Arab Americans? Arab American Institute Foundation.
- Schaefer, R.T. (2019). Racial and Ethnic Groups. 15th ed. New York, NY: Pearson.


24.1: Fractional Lengths
Lesson
Let's solve problems about fractional lengths.
Exercise \(\PageIndex{1}\): Number Talk: Multiplication Strategies
Find the product mentally.
\(19\cdot 14\)
Exercise \(\PageIndex{2}\): Info Gap: How Many Would It Take?
Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner.
If your teacher gives you the problem card :
- Silently read your card and think about what information you need to be able to answer the question.
- Ask your partner for the specific information that you need.
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Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem. - Share the problem card and solve the problem independently.
- Read the data card and discuss your reasoning.
If your teacher gives you the data card :
- Silently read your card.
-
Ask your partner
“What specific information do you need?”
and wait for them to
ask
for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. - Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions.
- Read the problem card and solve the problem independently.
- Share the data card and discuss your reasoning.
Are you ready for more?
Lin has a work of art that is \(14\) inches by \(20\) inches. She wants to frame it with large paper clips laid end to end.
- If each paper clip is \(1\frac{3}{4}\) inch long, how many paper clips would she need? Show your reasoning and be sure to think about potential gaps and overlaps. Consider making a sketch that shows how the paper clips could be arranged.
- How many paper clips are needed if the paper clips are spaced \(\frac{1}{4}\) inch apart? Describe the arrangement of the paper clips at the corners of the frame.
Exercise \(\PageIndex{3}\): How Many Times as Tall or as Far?
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A second-grade student is 4 feet tall. Her teacher is \(5\frac{2}{3}\) feet tall.
- How many times as tall as the student is the teacher?
- What fraction of the teacher’s height is the student’s height?
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Find each quotient. Show your reasoning and check your answer.
- \(9\div\frac{3}{5}\)
- \(1\frac{7}{8}\div\frac{3}{4}\)
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Write a division equation that can help answer each of these questions. Then find the answer. If you get stuck, consider drawing a diagram.
- A runner ran \(1\frac{4}{5}\) miles on Monday and \(6\frac{3}{10}\) miles on Tuesday. How many times her Monday’s distance was her Tuesday’s distance?
- A cyclist planned to ride \(9\frac{1}{2}\) miles but only managed to travel \(3\frac{7}{8}\) miles. What fraction of his planned trip did he travel?
Exercise \(\PageIndex{4}\): Comparing Paper Rolls
The photo shows a situation that involves fractions.
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Complete the sentences. Be prepared to explain your reasoning.
- The length of the long tube is about ______ times the length of a short tube.
- The length of a short tube is about ______ times the length of the long tube.
- If the length of the long paper roll is \(11\frac{1}{4}\) inches, what is the length of each short paper roll?
Summary
Division can help us solve comparison problems in which we find out how many times as large or as small one number is compared to another. For example, a student is playing two songs for a music recital. The first song is \(1\frac{1}{2}\) minutes long. The second song is \(3\frac{3}{4}\) minutes long.
We can ask two different comparison questions and write different multiplication and division equations to represent each question.
- How many times as long as the first song is the second song?
\(?\cdot 1\frac{1}{2}=3\frac{3}{4}\)
\(3\frac{3}{4}\div 1\frac{1}{2}=?\)
- What fraction of the second song is the first song?
\(?\cdot 3\frac{3}{4}=1\frac{1}{2}\)
\(1\frac{1}{2}\div 3\frac{3}{4}=?\)
We can use the algorithm we learned to calculate the quotients.
\(\begin{aligned} &=\frac{15}{4}\div\frac{3}{2}\\&=\frac{15}{4}\cdot\frac{2}{3}\\&=\frac{30}{12}\\&=\frac{5}{2}\end{aligned}\)
This means the second song is \(2\frac{1}{2}\) times as long as the first song.
\(\begin{aligned} &=\frac{3}{2}\div\frac{15}{4}\\&=\frac{3}{2}\cdot\frac{4}{15}\\&=\frac{12}{30}\\&=\frac{2}{5}\end{aligned}\)
This means the first song is \(\frac{2}{5}\) times as long as the second song.
Practice
Exercise \(\PageIndex{5}\)
One inch is around \(2\frac{11}{20}\) centimeters.
- How many centimeters long is 3 inches? Show your reasoning.
- What fraction of an inch is 1 centimeter? Show your reasoning.
- What question can be answered by finding \(10\div 2\frac{11}{20}\) in this situation?
Exercise \(\PageIndex{6}\)
A zookeeper is \(6\frac{1}{4}\) feet tall. A young giraffe in his care is \(9\frac{3}{8}\) feet tall.
- How many times as tall as the zookeeper is the giraffe?
- What fraction of the giraffe’s height is the zookeeper’s height?
Exercise \(\PageIndex{7}\)
A rectangular bathroom floor is covered with square tiles that are \(1\frac{1}{2}\) feet by \(1\frac{1}{2}\) feet. The length of the bathroom floor is \(10\frac{1}{2}\) feet and the width is \(6\frac{1}{2}\) feet.
- How many tiles does it take to cover the length of the floor?
- How many tiles does it take to cover the width of the floor?
Exercise \(\PageIndex{8}\)
The Food and Drug Administration (FDA) recommends a certain amount of nutrient intake per day called the “daily value.” Food labels usually show percentages of the daily values for several different nutrients—calcium, iron, vitamins, etc.
Consider the problem: In \(\frac{3}{4}\) cup of oatmeal, there is \(\frac{1}{10}\) of the recommended daily value of iron. What fraction of the daily recommended value of iron is in 1 cup of oatmeal?
Write a multiplication equation and a division equation to represent the question. Then find the answer and show your reasoning.
(From Unit 4.3.2)
Exercise \(\PageIndex{9}\)
What fraction of \(\frac{1}{2}\) is \(\frac{1}{3}\)? Draw a tape diagram to represent and answer the question. Use graph paper if needed.
(From Unit 4.2.4)
Exercise \(\PageIndex{10}\)
Noah says, “There are \(2\frac{1}{2}\) groups of \(\frac{4}{5}\) in \(2\).” Do you agree with him? Draw a tape diagram to show your reasoning. Use graph paper, if needed.
(From Unit 4.2.3)


9.3: Women
There are over 11,000 women in prison in California, constituting approximately seven percent of the overall prison population. Although a seemingly small percentage, the female prison population has been growing more rapidly than its male counterpart since the mid-1980s. Since the 1970s, the female prison population in California has increased more than ten-fold. In 1977, California penal institutions housed 671 female prisoners; in 1987, there were 4,152 women incarcerated in CDCR facilities; in 2007, there were 11,416. As with the general prison population, experts have attributed the increase in the numbers of incarcerated females to California's shift away from the indeterminate sentencing regime in the 1970s, various "tough on crime" policies, legislation enacted as part of the "war on drugs," and other developments in law enforcement and parole policy generally, including a policy shift from reform and rehabilitation to punishment and deterrence.
Figure 9.3 Graph of Women Arrested vs. Women in the labor force. Image is in the public domain.
Most incarcerated women have been convicted of nonviolent offenses. In 2007, only 32.1% of women prisoners were convicted of crimes against persons; the remaining 67.9% were convicted for nonviolent property crimes, drug crimes, and other nonviolent crimes. In contrast, 53.5% of the men incarcerated in 2007 were convicted of crimes against persons. Because of the nature of the crimes for which they are convicted, females tend to have sentences that are far shorter than their male counterparts. The median sentence for women prisoners in CDCR institutions in 2007 was nine months. Similar to the male population, formerly incarcerated females have a high rate of recidivism. A recent study by CDCR found that forty-six percent of women released in 2003 returned to prison within three years; another CDCR study indicated that between twenty-seven to thirty-three percent of female parolees recidivate within a year.
Figure 9.4 Graph of Women arrested (%) for Violent crimes vs. Property crimes. Image is in the public domain.
 As with the prison population generally, the majority of women prisoners are women of color; many come from neighborhoods entrenched in poverty; and many have substance abuse issues. However, a disproportionately greater percentage of women prisoners have experienced physical and sexual abuse prior to admission into prison. There are also higher levels of physical and mental health issues among female prisoners. In addition, almost half of incarcerated women are the sole caregivers for their children. Thus, incarceration has an immediate effect on the children who often end up in foster care, as well as a fragmenting effect on their families generally and their communities.
Given the relatively unique characteristics of this population - including the nonviolent nature of the offenses, the shorter sentences, socio-economic factors, and motherhood - advocates, academics, and community members have strongly criticized the placement and treatment of incarcerated women in the existing institutions and programs and have strongly urged for the adoption of gender responsive approaches. These sentiments were echoed by the Little Hoover Commission, an independent state oversight agency in 2004.
Figure 9.5 Pie Chart of numbers and percentages of inmates by gender.
Since the Commission's findings in 2004, California has undertaken various reform measures to address some of these concerns. In 2005, CDCR established the office of Female Offender Programs and Services within the Division of Adult Institutions and created the Gender Responsive Strategies Commission ("GRSC") as an advisory committee. In addition, CDCR developed a phased housing plan to shift Level I and II women prisoners to community based facilities and generally has begun a new gender-responsive classification system for women. In 2007, the legislature enacted California Penal Code section 3430 that required CDCR to develop a "Female Offender Reform Master Plan" and to create policies, operational practices, and programs designed "to ensure a safe and productive institutional environment for female offenders, as well as gender-responsive rehabilitative programs and needs-based case and risk assessment.
Despite these efforts, Women's prisons still remain at over two hundred percent overcrowding levels. Many of the rehabilitative programs have yet to be implemented, and the construction of many of the community-based facilities have been delayed.


8.9: Blood Buffers
- To define buffer and describe how it reacts with an acid or a base.
- Understand the pH of blood is kept in range by the carbonic acid and bicarbonate ion buffer system.
Buffers
Weak acids are relatively common, in the foods we eat. But we occasionally come across a strong acid, such as stomach acid, that has a strongly acidic pH of 1 to 2. By definition, strong acids and bases can produce a relatively large amount of hydronium or hydroxide ions and, as a consequence, have a marked chemical activity. In addition, very small amounts of strong acids and bases can change the pH of a solution very quickly. If 1 mL of stomach acid [which we will approximate as 0.05 M HCl(aq)] is added to the bloodstream, and if no correcting mechanism is present, the pH of the blood would go from about 7.4 to about 4.9. Such a drop in pH that is not conducive to continued living. Fortunately, the body has a mechanism for minimizing such dramatic pH changes.
The mechanism involves a buffer , a solution that resists dramatic changes in pH. A buffer (or buffered ) solution is one that resists a large change in its pH when small amounts of H 3 O + or OH – ions are added or removed owing to a neutralizing reaction taking place in the same solution.
Composition of Buffers
Buffers contain either a weak acid plus its conjugate base or a weak base plus its conjugate acid . For example, a buffer can be composed of dissolved acetic acid (HC 2 H 3 O 2 , a weak acid) and sodium acetate ( NaC 2 H 3 O 2 ) . Sodium acetate is a salt that dissociates into sodium ions and acetate ions in solution. For as long as acetic acid and acetate ions are present in significant amounts a solution, this can resist dramatic pH changes. Another example of a buffer is a solution containing ammonia (NH 3 , a weak base) and ammonium chloride ( NH 4 Cl). Ammonium chloride is also a salt that dissociates into ammonium ions and chloride ions in solution. The presence of ammonium ions with ammonia molecules satisfies the requisite condition for a buffer solution.
How Buffers Work
The essential component of a buffer system is a conjugate acid-base pair whose concentration is fairly high in relation to the concentrations of added H 3 O + or OH – . Let us use an acetic acid–sodium acetate buffer to demonstrate how buffers work. If a strong base, a source of OH − (aq) ions is added to the buffer solution, those hydroxide ions will react with the acetic acid in an acid-base reaction:
\[HC_2H_3O_{2(aq)} + OH^−_{(aq)} \rightarrow H_2O_{(ℓ)} + C_2H_3O^−_{2(aq)} \label{Eq1} \]
Rather than changing the pH dramatically by making the solution basic, the added hydroxide ions react to make water, and the pH does not change much.
Many people are aware of the concept of buffers from buffered aspirin , which is aspirin that also has magnesium carbonate, calcium carbonate, magnesium oxide, or some other salt. The salt acts like a base, while aspirin is itself a weak acid.
If a strong acid, a source of H 3 O + ions is added to the buffer solution, the H 3 O + ions will react with the anion from the salt. Because HC 2 H 3 O 2 is a weak acid, it is not ionized much. This means that if hydronium ions and acetate ions (from sodium acetate) are present in the same solution, they will come together to make acetic acid.
H 3 O + (aq) + C 2 H 3 O 2 - (aq) → HC 2 H 3 O 2 (aq) + H 2 O(l)
Rather than changing the pH dramatically and making the solution acidic, the added hydrogen ions react to make molecules of a weak acid. Figure \(\PageIndex{1}\) illustrates both actions of a buffer.
Figure 8.9.1 shows acetic acid HAc and its conjugate base, the acetate ion Ac – . The idea is that this conjugate pair "pool" will be available to gobble up any small (≤ 10 –3 M ) addition of H 3 O + or OH – . For simplicity in the diagram, H 3 O + is addressed as H + , acetic acid as HAc, and acetate ion as Ac - .
Buffers work well only for limited amounts of added strong acid or base. Once either solute is all reacted and depleted, the solution is no longer a buffer, and rapid changes in pH may occur. We say that a buffer has a certain capacity. Buffers that have more solute dissolved in them to start with have larger capacities, as might be expected.
Buffers made from weak bases and salts of weak bases act similarly. For example, in a buffer containing NH 3 and NH 4 Cl, ammonia molecules can react with any excess hydrogen ions introduced by strong acids:
\[NH_{3(aq)} + H^+_{(aq)} \rightarrow NH^+_{4(aq)} \label{Eq3} \]
while the ammonium ion [NH 4 + (aq)] can react with any hydroxide ions introduced by strong bases:
\[NH^+_{4(aq)} + OH^−_{(aq)} \rightarrow NH_{3(aq)} + H_2O_{(ℓ)} \label{Eq4} \]
Buffer solutions are essential components of all living organisms.
- Our blood is buffered to maintain a pH of 7.35 to 7.45 that must remain unchanged as metabolically-generated CO 2 (carbon dioxide) is added and then removed by our lungs.
- Buffers in the oceans, in natural waters such as lakes and streams, and within soils help maintain their environmental stability against acid rain and increases in atmospheric CO 2 .
- Many industrial processes, such as brewing, require buffer control, as do research studies in biochemistry and physiology that involve enzymes, are active only within certain pH ranges.
The pH in living systems is maintained by buffer systems.
| Compartment | pH |
|---|---|
| Gastric Acid | 1 |
| Lysosomes | 4.5 |
| Granules of Chromaffin Cells | 5.5 |
| Human Skin | 5.5 |
| Urine | 6 |
| Neutral H 2 O at 37 °C | 6.81 |
| Cytosol | 7.2 |
| Cerebrospinal Fluid | 7.3 |
| Blood | 7.35-7.45 |
| Mitochondrial Matrix | 7.5 |
| Pancreas Secretions | 8.1 |
Human blood has a buffering system to minimize extreme changes in pH. One buffer in blood is based on the presence of HCO 3 − and H 2 CO 3 [H 2 CO 3 is another way to write CO 2 dissolved in water]. With this buffer present, even if some stomach acid were to find its way directly into the bloodstream, the change in the pH of blood would be minimal. Inside many of the body’s cells, there is a buffering system based on H 2 PO 4 - and HPO 4 2 - .
The pH of the blood is maintained between 7.35 and 7.45 by buffers in the body. If blood pH goes above or below that range, it can destroy cells in the blood. The pH of the blood is kept constant by several buffer solutions. One important buffer solution is made of carbonic acid, H 2 CO 3 (weak acid), and bicarbonate anion, HCO 3 - (conjugate base). The equilibrium equation is shown below:
H 2 CO 3 (aq) + H 2 O (l) ⇌ HCO 3 - (aq) + H 3 O + (aq)
When acid (H 3 O + ) enters the blood, it reacts with the HCO 3 - anion and forms water and carbonic acid as follows:
H 3 O + + HCO 3 - → H 2 O + H 2 CO 3
When base (OH - ) is added, it reacts with the carbonic acid in the buffer and produces bicarbonate ion and water as follows:
OH - + H 2 CO 3 → HCO 3 - + H 2 O
Diabetes mellitus is a disorder of glucose metabolism in which insulin production by the pancreas is impaired. Since insulin helps glucose enter the cells, a decrease of this hormone means that glucose cannot be used in its normal fashion. When this happens, the body begins to break down fats, producing an excess of acetyl-CoA which is then converted to ketone bodies. This results in the decrease in blood pH. Buffer systems in the body can balance this pH shift for a while, but excessive acid production can create serious problems if not corrected by administering insulin to restore normal glucose use.
Lactic acid is produced in our muscles when we exercise. As the lactic acid enters the bloodstream, it is neutralized by the \(\ce{HCO3-}\) ion, producing H 2 CO 3 . An enzyme then accelerates the breakdown of the excess carbonic acid to carbon dioxide and water, which can be eliminated by breathing. In fact, in addition to the regulating effects of the carbonate buffering system on the pH of blood, the body uses breathing to regulate blood pH. If the pH of the blood decreases too far, an increase in breathing removes CO 2 from the blood through the lungs driving the equilibrium reaction such that [H 3 O + ] is lowered. If the blood is too alkaline, a lower breath rate increases CO 2 concentration in the blood, driving the equilibrium reaction the other way, increasing [H 3 O + ] and restoring an appropriate pH.
At this point in this text, you should have the idea that the chemistry of blood is fairly complex. Because of this, people who work with blood must be specially trained to work with it properly.
A blood bank technology specialist is trained to perform routine and special tests on blood samples from blood banks or transfusion centers. This specialist measures the pH of blood, types it (according to the blood’s ABO+/− type, Rh factors, and other typing schemes), tests it for the presence or absence of various diseases, and uses the blood to determine if a patient has any of several medical problems, such as anemia. A blood bank technology specialist may also interview and prepare donors to give blood and may actually collect the blood donation.
Blood bank technology specialists are well trained. Typically, they require a college degree with at least a year of special training in blood biology and chemistry. In the United States, training must conform to standards established by the American Association of Blood Banks.
Although medicines are not exactly "food and drink," we do ingest them, so let's take a look at an acid that is probably the most common medicine: acetylsalicylic acid, also known as aspirin. Aspirin is well known as a pain reliever, antipyretic (fever reducer) and anti-inflammatory.
The structure of aspirin is shown in the accompanying figure. The acid part is circled; it is the H atom in that part that can be donated as aspirin acts as a Brønsted-Lowry acid. Acetylsalicylic acid is a weak acid. Some people consume relatively large amounts of aspirin daily, its acidic nature can cause problems in the stomach lining, despite the stomach's defenses against its own stomach acid.
Because the acid properties of aspirin may be problematic, many aspirin brands offer a "buffered aspirin" form of the medicine. In these cases, the aspirin also contains a buffering agent-usually MgO-that regulates the acidity of the aspirin to minimize its acidic side effects.
As useful and common as aspirin is, it was formally marketed as a drug starting in 1899. The US Food and Drug Administration (FDA), the governmental agency charged with overseeing and approving drugs in the United States, wasn't formed until 1906. Some have argued that if the FDA had been formed before aspirin was introduced, aspirin may never have gotten approval due to its potential for side effects-gastrointestinal bleeding, ringing in the ears, Reye's syndrome (a liver problem), and some allergic reactions. However, recently aspirin has been touted for its effects in lessening heart attacks and strokes, so it is likely that aspirin is here to stay.
Key Takeaway
- A buffer is a solution that resists sudden changes in pH.
Concept Review Exercise
Explain how a buffer prevents large changes in pH.
Answer
A buffer has components that react with both strong acids and strong bases to resist sudden changes in pH.
Exercises
- Describe a buffer. What two related chemical components are required to make a buffer?
- Can a buffer be made by combining a strong acid with a strong base? Why or why not?
-
Which solute combinations can make a buffer? Assume all are aqueous solutions.
- HCl and NaCl
- HNO 2 and NaNO 2
- NH 4 NO 3 and HNO 3
- NH 4 NO 3 and NH 3
-
Which solute combinations can make a buffer? Assume all are aqueous solutions.
- H 3 PO 4 and Na 3 PO 4
- NaHCO 3 and Na 2 CO 3
- NaNO 3 and Ca(NO 3 ) 2
- HN 3 and NH 3
- For each combination in Exercise 3 that is a buffer, write the chemical equations for the reactions of the buffer components when a strong acid and a strong base is added.
- For each combination in Exercise 4 that is a buffer, write the chemical equations for the reaction of the buffer components when a strong acid and a strong base is added.
- The complete phosphate buffer system is based on four substances: H 3 PO 4 , H 2 PO 4 − , HPO 4 2 − , and PO 4 3 − . What different buffer solutions can be made from these substances?
- Explain why NaBr cannot be a component in either an acidic or a basic buffer.
- Explain why Mg(NO 3 ) 2 cannot be a component in either an acidic or a basic buffer.
Answers
- A buffer resists sudden changes in pH. It has a weak acid or base and a salt of that weak acid or base.
- No. Combining a strong acid and a strong base will produce salt and water. Excess strong acid or strong base will not act as a buffer.
-
- not a buffer
- buffer
- not a buffer
- buffer
4. 1. not a buffer
2. buffer
3. not a buffer
4. not buffer
- 3b: strong acid: H + + NO 2 − → HNO 2 ; strong base: OH − + HNO 2 → H 2 O + NO 2 − ; 3d: strong acid: H + + NH 3 → NH 4 + ; strong base: OH − + NH 4 + → H 2 O + NH 3
- 4b: strong acid: H + + CO 3 2 − → HCO3 − ; strong base: OH − + HCO 3 − → H 2 O + CO 3 2 − ;
- Buffers can be made by combining H 3 PO 4 and H 2 PO 4 − , H 2 PO 4 − and HPO 4 2 − , and HPO 4 2 − and PO 4 3 − .
- NaBr splits up into two ions in solution, Na + and Br − . Na + will not react with any added base knowing that NaOH is a strong base. Br- will not react with any added acid knowing that HBr is a strong acid. Because NaBr will not react with any added base or acid, it does not resist change in pH and is not a buffer.
- Mg(NO 3 ) 2 includes two types of ions, Mg 2 + and NO 3 − . Mg(OH) 2 is strong base and completely dissociates (100% falls apart), so Mg 2 + will not react with any added base (0% combines with OH − ). HNO 3 is strong acid and completely dissociates (100% falls apart), so NO 3 − will not react with any added acid (0% combines with H + ). Because Mg(NO 3 ) 2 will not react with any added base or acid, it does not resist change in pH and is not a buffer.


8: Estimating Unknown Quantities from a Sample
At the start of the last chapter I highlighted the critical distinction between descriptive statistics and inferential statistics . As discussed in Chapter 5, the role of descriptive statistics is to concisely summarise what we do know. In contrast, the purpose of inferential statistics is to “learn what we do not know from what we do”. Now that we have a foundation in probability theory, we are in a good position to think about the problem of statistical inference. What kinds of things would we like to learn about? And how do we learn them? These are the questions that lie at the heart of inferential statistics, and they are traditionally divided into two “big ideas”: estimation and hypothesis testing. The goal in this chapter is to introduce the first of these big ideas, estimation theory, but I’m going to witter on about sampling theory first because estimation theory doesn’t make sense until you understand sampling. As a consequence, this chapter divides naturally into two parts Sections 10.1 through 10.3 are focused on sampling theory, and Sections 10.4 and 10.5 make use of sampling theory to discuss how statisticians think about estimation.


12.1.7: Media Mix - Convergence
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Learning Objectives
- Define convergence and discuss examples of it in contemporary life.
- Name the five types of convergence identified by Henry Jenkins.
- Examine how convergence is affecting culture and society.
Each cultural era is marked by changes in technology. What happens to the “old” technology? When radio was invented, people predicted the end of newspapers. When television was invented, people predicted the end of radio and film. It’s important to keep in mind that the implementation of new technologies does not mean that the old ones simply vanish into dusty museums. Today’s media consumers still read newspapers, listen to radio, watch television, and get immersed in movies. The difference is that it’s now possible to do all those things and do all those things through one device—be it a personal computer or a smartphone—and through the medium of the Internet. Such actions are enabled by media convergence, the process by which previously distinct technologies come to share content, tasks and resources. A cell phone that also takes pictures and video is an example of the convergence of digital photography, digital video, and cellular telephone technologies. A news story that originally appeared in a newspaper and now is published on a website or pushed on a mobile phone is another example of convergence.
Kinds of Convergence
Convergence isn’t just limited to technology. Media theorist Henry Jenkins has devoted a lot of time to thinking about convergence. He argues that convergence isn’t an end result but instead a process that changes how media is both consumed and produced. Jenkins breaks convergence down into five categories:
- Economic convergence is the horizontal and vertical integration of the entertainment industry, in which a single company has interests across and within many kinds of media. For example, Rupert Murdoch’s News Corporation owns numerous newspapers ( The New York Post , The Wall Street Journal ) but is also involved in book publishing (HarperCollins), sports (the Colorado Rockies), broadcast television (Fox), cable television (FX, National Geographic Channel), film (20th Century Fox), Internet (Facebook), and many others.
- Organic convergence is what happens when someone is watching television while chatting online and also listening to music—such multitasking seems like a natural outcome in a diverse media world.
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Cultural convergence
has several different aspects. One important component is stories flowing across several kinds of media platforms—for example, novels that become television series (
Dexter
or
Friday Night Lights
); radio dramas that become comic strips (
The Shadow
); even amusement park rides that become film franchises (
Pirates of the Caribbean
). The character Harry Potter exists in books, films, toys, amusement park rides, and candy bars. Another aspect of cultural convergence is participatory culture—that is, the way media consumers are able to annotate, comment on, remix, and otherwise talk back to culture in unprecedented ways.
Figure 1.5. Nigeria’s Nollywood produces more films annually than any other country besides India.
- Global convergence is the process of geographically distant cultures influencing one another despite the oceans and mountains that may physically separate them. Nigeria’s “Nollywood” cinema takes its cues from India’s Bollywood, which of course stemmed from Hollywood; old Tom and Jerry cartoons and newer Oprah shows are popular on Arab satellite television channels; successful American horror movies like The Ring and The Grudge are remakes of Japanese hits; the hit television show, “American Idol” was a remake of a British show. The advantage of global convergence is worldwide access to a wealth of cultural influence. Its downside, some critics posit, is the threat of cultural imperialism, defined by Herbert Schiller as the way that developing countries are “attracted, pressured, forced, and sometimes bribed into shaping social institutions to correspond to, or even promote, the values and structures of the dominating centre of the system.” That is, less powerful nations lose their cultural traditions as more powerful nations spread their culture through their media and other forms. Cultural imperialism can be a formal policy or can happen more subtly, as with the spread of outside influence through television, movies, and other cultural projects.
- Technological convergence is the merging of technologies. When more and more different kinds of media are transformed into digital content, as Jenkins notes, “we expand the potential relationships between them and enable them to flow across platforms.”Henry Jenkins, “Convergence? I Diverge,” Technology Review , June 2001, p. 93.
Effects of Convergence
Jenkins’ concept of “organic convergence”—particularly, multitasking—is perhaps most evident in your own lives. To many who grew up in a world dominated by so-called old media, there is nothing organic about today’s mediated world. As a New York Times editorial sniffed, “Few objects on the planet are farther removed from nature—less, say, like a rock or an insect—than a glass and stainless steel smartphone.”Editorial, “The Half-Life of Phones,” The New York Times , Week in Review section, June 18, 2010, http://www.nytimes.com/2010/06/20/opinion/20sun4.html?_r=1 (accessed July 15, 2010). But modern American culture is plugged in as never before, and many students today have never known a world where the Internet didn’t exist. Such a cultural sea change causes a significant generation gap between those who grew up with new media and those who didn’t.
A 2010 study by the Kaiser Family Foundation found that Americans aged 8 to 18 spend more than 7.5 hours with electronic devices each day—and, thanks to multitasking, they’re able to pack an average of 11 hours of media content into that 7.5 hours.Tamar Lewin, “If Your Kids Are Awake, They’re Probably Online,” The New York Times , Education section, January 20, 2010, http://www.nytimes.com/2010/01/20/education/20wired.html (accessed July 15, 2010). These statistics highlight some of the aspects of the new digital model of media consumption: participation and multitasking.  Instead, they are sending text messages to friends, linking news articles on Facebook, commenting on YouTube videos, writing reviews of television episodes to post online, and generally engaging with the culture they consume. Convergence has also made multitasking much easier, as many devices allow users to surf the Internet, listen to music, watch videos, play games, and reply to emails and texts on the same machine.
However, this multitasking is still quite new and we do not know how media convergence and immersion are shaping culture, people, and individual brains. In his 2005 book Everything Bad Is Good for You , Steven Johnson argues that today’s television and video games are mentally stimulating, in that they pose a cognitive challenge and invite active engagement and problem-solving. Poking fun at alarmists who see every new technology as making children more stupid, Johnson jokingly cautions readers against the dangers of book reading: it “chronically understimulates the senses” and is “tragically isolating.” Even worse, books “follow a fixed linear path. You can’t control their narratives in any fashion—you simply sit back and have the story dictated to you.…This risks instilling a general passivity in our children, making them feel as though they’re powerless to change their circumstances. Reading is not an active, participatory process; it’s a submissive one.”Steven Johnson, Everything Bad Is Good for You (Riverhead, NY: Riverhead Books, 2005).
A 2010 book by Nicholas Carr, The Shallows: What the Internet Is Doing to Our Brains is more pessimistic. Carr worries that the vast array of interlinked information available through the Internet is eroding attention spans and making contemporary minds distracted and less capable of deep, thoughtful engagement with complex ideas and arguments. He mourns the change in his own reading habits. “Once I was a scuba diver in a sea of words,” Carr reflects ruefully. “Now I zip along the surface like a guy on a Jet Ski.” Carr cites neuroscience studies showing that when people try to do two things at once, they give less attention to each and perform the tasks less carefully. In other words, multitasking makes us do a greater number of things poorly. Whatever the ultimate cognitive, social, or technological results, though, convergence is changing the way we relate to media today.
Video Killed the Radio Star: Convergence Kills Off Obsolete Technology—Or Does It?
When was the last time you used a rotary phone? How about a payphone on a street? Or a library’s card catalog? When you need brief, factual information, when was the last time you reached for a handy volume of Encyclopedia Britannica ? Odds are, it’s been a while. Maybe never. All of these habits, formerly common parts of daily life, have been rendered essentially obsolete through the progression of convergence.
But convergence hasn’t erased old technologies; instead, it may have just altered the way we use them. Take cassette tapes and Polaroid film, for example. The underground music tastemaker Thurston Moore of Sonic Youth recently claimed that he only listens to music on cassette. Polaroid Corporation, creators of the once-popular instant film cameras, was driven out of business by digital photography in 2008, only to be revived two years later—with pop star Lady Gaga as the brand’s creative director. Several iPhone apps promise to apply effects to photos to make them look more like Polaroids.
Cassettes, Polaroids, and other seemingly obsolete technologies have been able to thrive—albeit in niche markets—both despite and because of Internet culture. Instead of being slick and digitized, cassette tapes and Polaroid photos are physical objects that are made more accessible and more human, according to enthusiasts, because of their flaws. “I think there’s a group of people—fans and artists alike—out there to whom music is more than just a file on your computer, more than just a folder of mp3s,” says Brad Rose, founder of a Tulsa-based cassette label. The distinctive Polaroid look—caused by uneven color saturation, under- or over-development, or just daily atmospheric effects on the developing photograph—is emphatically analog. In an age of high resolution, portable printers, and camera phones, the Polaroid’s appeal has something to do with ideas of nostalgia and authenticity. Convergence has transformed who uses these media and for what purposes, but it hasn’t run them out of town yet.
Key Takeaways
- Twenty-first century media culture is increasingly marked by convergence, or the coming together of previously distinct technologies, as in a cell phone that also allows users to take video and check email.
-
Media theorist Henry Jenkins identifies the five kinds of convergence as the following:
- Economic convergence, when a single company has interests across many kinds of media.
- Organic convergence is multimedia multitasking, or the “natural” outcome of a diverse media world.
- Cultural convergence, when stories flow across several kinds of media platforms, and when readers or viewers can comment on, alter, or otherwise talk back to culture.
- Global convergence, when geographically distant cultures are able to influence one another.
- Technological convergence, in which different kinds of technology merge. The most extreme example of technological convergence would be the as-yet hypothetical “black box,” one machine that controlled every media function.
- The jury is still out on how these different types of convergence will affect people on an individual and cultural level. Some theorists believe that convergence and new media technologies make people smarter by requiring them to make decisions and interact with the media they’re consuming; others fear that the digital age is giving us access to more information but leaving us shallower.
Exercise \(\PageIndex{1}\)
Which argument do you find more compelling, Johnson’s or Carr’s? Make a list of points, examples, and facts that back up the theory that you think best explains the effects of convergence. Alternatively, come up with your own theory of how convergence is changing individual and society as a whole. Stage a mock debate with a member of the class who holds a view different from your own.


4.2.1: Lin-Manuel Miranda - Hamilton
Hamilton is considered by many to be the greatest American musical theater production of the current era. The musical tells the story of Alexander Hamilton, an immigrant who travelled from the West Indies to the American colonies before the time of the American Revolution, and who came to be one of the founding fathers of the United States and the first Secretary of the Treasury under George Washington. The musical is based upon the biography by Ron Chernow (b. 1949), Alexander Hamilton, published in 2004.
Miranda and Hamilton
Hamilton is the singular creation of the actor, writer, and musician Lin-Manuel Miranda (b. 1980), who, in addition to premiering the title role, wrote the music and lyrics. Miranda is one of the biggest musical theater celebrities of his generation, having won multiple Emmys, Grammys, and Tony Awards for his music, lyrics, and performances. Much of Miranda’s work is influenced by his origins. As the child of parents of Puerto Rican descent, he grew up in a largely Latino neighborhood in Inwood, New York City, regularly visiting family in Puerto Rico. His musical preferences—Miranda grew up listening to salsa and American musicals (such as West Side Story)—reflected both an early interest in his musical heritage and a fascination with the stage. Miranda was gifted musically and academically, attending the highly competitive primary and secondary schools at Hunter College before enrolling at Wesleyan University.
In his theatrical work, Miranda foregrounds his multi-cultural upbringing and Latino identity. As a director, he has always made a conscious decision to cast actors from various cultural and racial backgrounds. This feature of Hamilton has created some controversy, given the historical fact that the men and women represented in the musical were primarily white. Miranda, however, has defended this approach by drawing parallels between his casting and musical decisions. Just as he did not attempt to write music in the style of the 1770s, he did not attempt to create a visual image representative of the 1770s. Miranda’s Hamilton was created to represent America in the current age. He makes the point that the United States was founded by immigrants. The founding fathers themselves—despite their skin color—were either immigrants or the children or grandchildren of immigrants. Hamilton himself was born in the West Indies, coming to America at a young age to make a name for himself. What is more American than that?
The musical numbers in Hamilton incorporate hip-hop beats and rap. The use of popular styles in a musical is not unique to
Hamilton : Hair (1967) and Jesus Christ Superstar (1970) were the first modern- era musicals to incorporate rock styles, and Lin-Manuel Miranda had already used rap and hip-hop in his first musical, In the Heights (on Broadway from 2008 to 2011). In writing Hamilton, Miranda found that rap was not only musically exciting but also very useful, for he was able to fit more words into a shorter span of time. Hamilton the man was known for being poetically loquacious. In addition, rap allowed Miranda to incorporate a large amount of information from Chernow’s biography, in addition to Hamilton’s own words. The entire musical is 150 minutes long, but includes over 20,000 words.
Miranda also uses music to communicate with the Hamilton audience in ways that both complement and transcend the script and stage action. One of his favorite techniques for doing so is allegory , or the use of musical sounds to signify hidden meaning. We will address several ways in which Miranda employs instruments or musical styles in an allegorical manner.
Allegory
|
Time |
Form |
What to listen for |
|---|---|---|
|
0’00” |
Introduction |
King George III of England addresses America |
|
“You say…” |
to a homophonic accompaniment of sustained piano chords in a four-chord descending pattern |
|
|
0’33” |
Verse 1 “You’ll be back…” |
The accompaniment switches to short harpsichord accents; the overall tone of the text becomes more sinister, which offers an unusual juxtaposition with happy music |
|
1’15” |
Hook “Da-da-da-da-da…” |
This catchy passage is sung to nonsense words and is reminiscent of the outro of The Beatles’ “Hey Jude,” which is similar in terms of key and melodic arc |
|
1’29” |
Bridge “You say our love is draining…” |
Stop-time accompaniment; double entendre on the word subject at 1’46” |
|
2’12” |
Verse 2 |
Same musical material as Verse 1; harpsichord “You’ll be returns to the front of the mix, one octave back, like higher than before; tempo slows, emphasizing before…” text, until the word “Love” at 2’36” |
|
2’46” |
Hook/Outro |
The performer is joined in this catchy tune by “Da-da-da-da-da. ” the rest of the cast members on stage |
There are several additional musical allegories in Hamilton . Let us consider, for example, the number “You’ll Be Back,” which is sung from the perspective of King George III. The music for this number is in the style of 1960s British Invasion pop. (Incidentally, The Beatles—the most famous and influential of the British Invasion bands—prominently featured harpsichord in their 1967 album Sgt. Pepper’s Lonely Hearts Club Band .) Again, Miranda makes a connection between a British musical style (an invasive one at that) and an imperialistic British character. But that’s not all: Miranda further uses the pop genre to heighten the impact of his character’s words. King George sings this number to a personified America. The music is relentlessly upbeat, as if expressing the emotional state of a delirious ex-boyfriend who cannot grasp that the relationship is over. The text, which is quite disturbing at times, is hilariously juxtaposed with the some of the happiest and tuneful music of the show.
Let us also consider Miranda’s use of the harpsichord. Miranda writes for two keyboard instruments in his musical: the harpsichord and the piano. The two instruments really did not coexist in history. During the course of the 1700s, the harpsichord was gradually replaced by the fortepiano, which was itself an early predecessor to the piano. In Hamilton , however, we do not hear a fortepiano but rather a modern, 20th-century piano—an instrument historically separated from the harpsichord by over one hundred years. The harpsichord, therefore, represents an old way of life—and, more particularly, an old way of conducting politics. Specifically, Miranda uses the harpsichord to represent British rule before the revolution.
The sound of the harpsichord was, historically speaking, the sound of the 18th-century British monarchy. In order to draw a historical comparison, we might consider the music of George Frederic Handel. Handel served as music director (German: Kapellmeister ) to the German Prince George of Hanover, who later became King George I of England. While living in London, Handel composed many great works for the King, most notably his Water Music and Music for the Royal Fireworks 1 . Handel was also one of the best-known and highly celebrated harpsichordists of his time. It is therefore fitting that the harpsichord should represent King George III, England, political oppression, and a deranged narcissistic ex-suitor.
We encounter this allegory in several musical examples. In “Farmer Refuted,”2 Bishop Samuel Seabury, accompanied by harpsichord, defends his decision to remain loyal to England and begs Hamilton to do the same. “You’ll Be Back” begins with King George III singing to a subdued piano accompaniment. When his lyrics turn sinister, however, we hear the harpsichord enter. In both of these cases, therefore, the sound of the harpsichord is a symbol for the old-fashioned values of the British monarchy.
G.F. Handel’s Music for the Royal Fireworks incorporates the harpsichord in a way that is typical of 18th-century European music.
In “Farmer Refuted,” Miranda uses the harpsichord to symbolize loyalty to England.
Cyclicism
King George later returns to perform “I Know Him . ” This song is a musical reprise of the first number he sang (“You’ll Be Back”). At this point in the musical, the Revolution is long over and John Adams has been elected president. The harpsichord can still be heard, but it is less prominent than it was in “You’ll Be Back.” Instead, it is lurks behind a texture that consists primarily of strings played pizzicato. This is significant: England (harpsichord) has lost the war and King George (harpsichord) no longer rules over the Americans.
|
Time |
Form |
What to listen for |
|---|---|---|
|
0’00” |
Introduction “They say…” |
Same introductory chords as the original introduction, despite plot changes |
|
0’24” |
“I’m perplexed...” |
Unusual harmonies |
|
0’42” |
“John Adams…” is spoken in surprise. |
|
|
0’45” |
Verse 1 “I know him…” |
Begins with the same tune and feel of Verse 1 from “You’ll Be Back,” but there is no harpsichord; instead, the texture consists of plucked strings, which recall the sound of the harpsichord but offer a warmer timbre; harpsichord returns to the mix in the background at 0’54” on the words, “Years ago” |
|
1’18” |
Hook |
Only one iteration of this, unlike before, and there is no “everybody” joining in |
“I Know Him” is not the only example of a musical reprise. Indeed, in crafting the score to Hamilton , Miranda sought to unify the various parts of the work using cyclical techniques . Hamilton contains many musical ideas that harken back to earlier points in the work. This approach to creating a narrative can be compared to Wagner’s use of leitmotifs.  Miranda treats this passage from Aaron Burr’s opening number, which is entitled “Alexander Hamilton,”3 as a leitmotif , which later returns in “A Winter’s Ball,” “Guns and Ships,” “What’d I Miss,” “The Adams Administration,” and “Your Obedient Servant.” Despite the recurrence of musical material, however, each of these numbers has a distinct character. In “What’d I Miss,” the borrowing takes on a swinging feel as the once-straight sixteenths of the opening become dotted. In “Your Obedient Servant,” on the other hand, the tone is darkened by tremolo strings and the omission of the snaps that were previously heard on offbeats: Burr, now downright angry, is resentful of Alexander Hamilton.
The opening phrases of “Alexander Hamilton” become a leitmotif and return throughout the work.
Success
Upon its Broadway premiere in 2015, Hamilton immediately sold out. As of 2019, it continues to be nearly impossible to secure a ticket, the aftermarket prices of which have soared to record-breaking levels. The musical is currently on tour to cities across the globe, including Puerto Rico and London, England. It has won numerous awards, including ten Lortel Awards, three Outer Critic Circle Awards, eight Drama Desk Awards, the New York Drama Critics Circle Award for Best New Musical, and an Obie for Best New American Play.