1.6: Problem Solving Strategies
Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.
Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solve them), you learn strategies and techniques that can be useful. But no single strategy works every time.
George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, How to Solve it . Pólya died at the age 98 in 1985. [1]
George Pólya, circa 1973
- Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 ( http://creativecommons.org/licenses/by/2.0 )], via Wikimedia Commons ↵
In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:
- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look back on your work. How could it be better?
This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.
Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!
We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:
- What if the picture was different?
- What if the numbers were simpler?
- What if I just made up some numbers?
You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.
This brings us to the most important problem solving strategy of all:
A Problem Solving Strategy: Try Something!
If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.
Note that being "good at mathematics" is not about doing things right the first time. It is about figuring things out. Practice being okay with having done something incorrectly. Try to avoid using an eraser and just lightly cross out incorrect work (do not black out the entire thing). This way if it turns out that you did something useful, you still have that work to reference! If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.
Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what was left after paying Brianna. Finally, Alex saw David and gave him 1/2 of the remaining money. Who got the most money from Alex?
After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.
A Problem Solving Strategy: Draw a Picture
Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?
Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?
After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.
A Problem Solving Strategy: Make Up Numbers
Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!
Try this: Assume (that is, pretend) Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person.
Or try working backward: suppose Alex has some specific amount left at the end, say $10. Since he gave David half of what he had before seeing David, that means he had $20 before running into David. Now, work backwards and figure out how much each person got.
Watch the solution only after you tried this strategy for yourself.
If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!
How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)
Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!
After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?
Most people want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. Instead of asking the teacher, “Is this right?”, you should be ready to justify it and say, “Here’s my answer, and here is how I got it.”
A Problem Solving Strategy: Try a Simpler Problem
Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said, “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?
The ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).
A Problem Solving Strategy: Work Systematically
If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.
For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:
| size of board | # of 1 × 1 squares | # of 2 × 2 squares | # of 3 × 3 squares | # of 4 × 4 squares | … |
| 1 by 1 | 1 | 0 | 0 | 0 | |
| 2 by 2 | 4 | 1 | 0 | 0 | |
| 3 by 3 | 9 | 4 | 1 | 0 | |
| … |
A Problem Solving Strategy: Use Manipulatives to Help You Investigate
Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!
For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.
 For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.
If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:
- Describe all of the patterns you see in the table. If possible, actually describe these to a friend.
- Explain and justify any of the patterns you see (if possible, actually do this with a friend). If you don't have a partner to work with, imagine they asked you, "How can you be sure the patterns will continue?"
- Expand this to find what calculation(s) you would perform to find the total number of squares on a 100 × 100 chess board.
(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)
This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)
Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2).
Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?
After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What progress have you made?
A Problem Solving Strategy: Find the Math, Remove the Context
Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.
In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:
- What is the sum of all the numbers on the clock’s face?
- Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
- How do I know when I am done? When should I stop looking?
Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.
A Problem Solving Strategy: Check Your Assumptions
When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?
In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:
Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.


1.8: Teen Idols and Surf
Changes to the Music Industry: Payola Scandal
Payola – The paying for airplay or other publicity by composers/musicians to publishers or radio programmers. It had been a music business practice in some form or another since the 19th century. By the late 1950’s payola meant a payment to radio programmers to “encourage” them to give more airtime to a particular song. Independent record companies would pay for airplay in a town and make profit from the record sale in that town as a result of the airplay. They would then move to another town and repeat the process.
In 1958, the Special Committee on Legislative Oversight formed. ASCAP requested this committee look into the bribing of Djs for airplay (the Payola Scandal). The result was that Payola was deemed illegal and many small, independent record companies that relied on payola for airplay went out of business.
Disc Jockeys
Alan Freed – (1921–1965) Disc jockey who fought against racism in the music industry, promoting music by African American artists. Freed had a TV show on ABC called Dance Party showcasing rhythm and blues. He also had accepted payola, and once the investigations into payola began his career was investigated. When Freed was found guilty of charges of accepting bribery/payola, he lost his television show.
Dick Clark – (1929–2012) Another important disc jockey and television personality, Clark avoided ruin during the payola scandal. Clark was the host of the show American Bandstand . American Bandstand would occasionally showcase a rock and roll or rhythm and blues artist, but was generally more conservative in its choices. More often clean cut teen idols were featured.
The Ed Sullivan Show was also an important television program that provided a variety of entertainment. Sullivan realized that booking rock and roll acts attracted the teenage audience and began regularly featuring musical performances.
Pat Boone and Teen Idol Pop
The image of teen idols and the content of their music was very clean cut, acceptable by the most conservative parent. The themes of teen idol songs were “as youthful and wholesome as the performer’s image”. Musically, teen idol pop represented a departure from many aspects of Rock & Roll, with softer singing styles, a significantly less use of electric guitar and saxophones, and orchestral instrumentals often replacing guitar/saxophone solos. Elements of the Blues were often absent, but not entirely. The performers were often well known for their clean cut and safe image, sometimes more than their music. Popular music seemed to leave Rock behind by the early 1960s, but it would come back in a big way within a few years, only with different artists leading the charge.
Pat Boone – (Born 1934) Singer of popular music, became a “teen idol”. He primarily sang pop ballads but also sang blues and rhythm and blues covers. Boone’s image was very boy-next-door. This was a contrast to the rebellious image of Elvis. In his cover version of Little Richard’s “Tutti Frutti” from 1956 ( Ch. 8 Listening Examples ), he attempted to achieve a similar rhythmic singing style to Little Richard but without the aggressive growling tone quality. Boone, like Elvis, was involved in movies including April Love (1957) and Journey to the Center of the Earth (1959).
On average, teen idol singers were successful more for their clean-cut, popular image than their musical talents. They often lip-synced their lip movements to recordings on TV, in movies, and even live. Rarely did any of them play an instrument and generally didn’t write their own songs. Other “teen idols” include Frankie Avalon, Fabian, Bobbie Rydell, and Ritchie Valens who had a hit with “La Bamba” (1959). Valens was unique in that he was a talented singer, guitarist, and wrote his own music. Chubby Checker (his name is a very obvious variation on Fats Domino) covered a song called “The Twist” by blues singer Hank Ballard, and it turned into a hit record, starting a dance craze in the United States.
Listening Examples 8.1
Listen Boone’s version of Little Richard’s “Tutti Frutti” and notice the contrast in style between this and Richard’s version . Boone’s version still has elements of rock and gospel even, but the aggressive elements from Richards’ version are played down a bit.  Notice in the following verse the subtle lyrical change in Boone’s version from “rock to the east” to “I been to the east”. The word “Rock” often had sexual connotations, so just to be safe the Boone version leaves it out.
Richards’ version:
Got a girl named Sue, she knows just what to do
Got a girl named Sue, she knows just what to do
She rocks to the east, she rocks to the west
She’s the girl that I know best
Boone’s version:
I got a girl, named Sue, she knows just what to do
I got a gal, named Sue, she knows just what to do
I’ve been to the east, I’ve been to the west, but
She’s the gal that I love the best
Here is Pat Boone singing a more typical song in his style, “Moody River” from June 1961 which reached No. 1 on the Billboard Top 100 charts.
The Brill Building
Many pop hits in the late 1950’s and early 1960’s were written in the Brill Building, located on Broadway in New York City. There were numerous music offices and studios in this one building. Professional songwriters were used because in many cases of pop musicians, they didn’t write their own music. Manager Don Kirshner had many of these songwriters under contract and was responsible for giving many of them their first breaks. Some of the songwriters who later found fame in their own performance careers included Neil Diamond, Paul Simon, and Carole King.
Carole King and then-husband Gerry Goffin wrote many hits including the song “Will You Love Me Tomorrow?” for the girl group the Shirelles ( Listening Examples 8.2 ). This song is in AABAA form, a departure from the many early rock songs in 12-bar blues form. One advantage in having professional songwriters was that the songs had more complex arrangements, forms, and instrumentation, adding to the variety of sounds on a given record. Lyrically, the song is from a female perspective, the context being a romantic encounter between lovers where the singer is cautious about the motives of her lover. We get a set up of a seemingly perfect new relationship, but the singer wants to make sure that the encounter is part of a “lasting treasure” and not simply a “moments’ pleasure”, and asks, as a refrain at the end of each A section, “Will you still love me tomorrow?”. At the time of the songs’ release in 1960, this was a powerful message from a female perspective, something of an assertion of female independence at a time when women were generally expected to remain reserved and not overly opinionated.
You give your love so sweetly
Tonight the light of love is in your eyes
But will you love me tomorrow?
Or just a moment’s pleasure
Can I believe the magic in your sighs
Will you love me tomorrow?
You say that I’m the only one
But will my heart be broken
When the night meets the morning sun
Is a love I can be sure of
So tell me now and I won’t ask again
Will you love me tomorrow?
Will you still love me tomorrow
Will you still love me tomorrow
Will you still love me tomorrow
Phil Spector
Phil Spector – (Born 1940) Producer whose distinctive style became known as the “Wall of Sound”. The term is derived from the fact that Spector’s productions were full of instruments, overdubs, and mixed so well that many of his recordings had an orchestral quality and power to them, unique in popular music. One of Spector’s influences was the classical composer Richard Wagner, known for his massive orchestral works featuring huge instrumentation including percussion instruments like timpani, gong, etc. Indeed, Spector called his records “symphonies for teens.” In the late 1960’s, Spector worked with the Beatles, producing Let It Be, and with George Harrison and John Lennon on their solo records after the breakup of the Beatles.
Listening Examples 8.3
Frankie Valli and the Four Seasons:
The Shirelles: (See lyrics above) Here is a live version of “Will You Love Me Tomorrow?” by the Shirelles in 1964. Here is another version. The song uses a relatively new form (for us) called AABA form. Simply put, the A sections always use the same music with different lyrics in each iteration. For contrast, the B section provides different music and often leads back to one last A section in a satisfying way. Sometimes the B section is called a “bridge” section as it bridges two A sections together. In this version, an orchestral break adds an instrumental A section as well. This replaces what would have been a guitar or saxophone solo in a Rock & Roll song.
The Ronnettes: “Be My Baby” This song uses another new form (for us) called Verse-Chorus form. This a two part form (the verse and the chorus being the two parts). The verses and choruses usually have distinct music. Lyrically, the verse features different lyrics each iteration, while the chorus (Sometimes called a refrain) features the same words every iteration. Here is a video outlining the new form along with the song itself: “Be My Baby” Formal Outline Video Again, you’ll notice an orchestral break where a guitar or saxophone solo would have likely been if this were a Rock & Roll song.
Spring Reverb was an important element of the surf guitar sound. In a nutshell, on certain amplifiers from the 1960s and later, the signal from the guitar can be transmitted through a metal spring which vibrates and extends the length of a given sound, giving the effect of the ambience of a hall. A knob on the amplifier allows the guitarist to dial in the amount of reverb, and the length of decay (how long the ambience lingers). In this video below I demonstrate various levels of reverb as well as a “vibrato” effect on a 1965 Reissue amplifier that is modeled after the ones used by many surf guitarists.
Duane Eddy
Duane Eddy – (born 1938) Guitarist who developed a distinctive style the epitomized surf music. The style is described as “twang” where the guitar strings were plucked close to the bridge where they are tightest, resulting in a bright, punchy sound.
Eddy’s use of echo and vibrato effects as well as his repetitive picking style resulted in a wavering sound that may have given rise to the inclusion of surfing topics into the music. His music such as the instrumental “Rebel-Rouser” (1958) used call and response techniques like those used in gospel music and jazz.
Surf Music
Musicians on the west coast began developing their own brand of danceable pop music known as Surf Rock . The music was different in a number of ways. For one, the vocals weren’t based on the crooner vocal tradition the way many pop singers of the day were; in fact, many popular surf hits were instrumental, a rare phenomenon in popular music. Other differences were the fact that many groups wrote their own material, the music was often more simple than the songs written by trained professionals at places like the Brill Building, and the bands played their own instruments, unlike pop stars of the day. The instrumentation of guitars, bass, and drums was very common. The surf instrumentation holds similarity to the older rock traditions, but much of the music was lighter and less aggressive. (That’s why it is considered in the pop tradition). The electric guitar (particular the solid body) became the primary lead instrument, and often replaced the vocals as the primary melodic voice for instrumentals.
The link between Surfing and Music described by Paul Johnson of the Bel-Airs
Paul Johnson, guitarist for The Bel-Airs who had one of the earliest Surf hit songs with “Mr. Moto” released January 1962, describes the link between the instrumental pop known as Surf Music as coincidental. Simply, surfers showed up to their gigs sheerly by proximity/geography. He attributes this primarily to the association between the music and surfing. Later on, the advent of spring reverb at Fender (discussed in a video in Ch. 8.2 Listening Examples ) became a sound inextricably linked to these groups. To hear Johnson discuss the first time he heard the term “Surf Music” and how the genre developed, listen to the podcast linked here from ca. 5:00. Particularly interesting is the reaction of surfers themselves to the Beach Boys at 8:15 into the Podcast. Surfers seemed to associate instrumental music with their sport much more so than vocal music with surf-based lyrics.
Dick Dale
Dick Dale – (Born Richard Monsour 1937-2019) Known as the “King of Surf Guitar” because he originated the style that was to be copied by many of the surf groups of the 1960’s. His group, Dick Dale and the Del-Tones had a string of successful singles and albums in the early 1960s until Dale was sidelined by a cancer diagnosis in the mid 1960s. Dale began using a guitar style similar Duane Eddy, but took it one step further when he incorporated a technique inspired by the tremolo picking of the Middle Eastern and Greek instruments like the bouzouki. Tremolo picking refers to fast repetitions of a note which sustains the note and adds a level of intensity to the sound.
Dale’s music was led by his innovative guitar sound, but he also sang and featured other instrumentalists such as horns or even orchestras ala Phil Spector’s wall of sound. Many of his songs featured a mix of all of the above features, but he is best remembered for his forceful guitar playing and the signature Fender sound featuring a Fender Stratocaster guitar, Fender Spring Reverb, and the Fender Showman Amp, the most advanced and most powerful (ie. Loudest) guitar amplifier available in the early 1960s which Dale helped develop with Leo Fender. His most enduring composition is “Misirlou” which features his unique guitar tone combined with “tremelo” picking and a middle eastern scale, all featured in the main theme of the song. The song was featured in the film Pulp Fiction in 1994 and led to a minor resurgence of Dale’s career.
The Beach Boys
The Beach Boys were formed by the Wilson Brothers Carl, Brian, and Dennis, their cousin, and a neighbor. The early hits were about surfing and teenage issues, though they later developed far beyond their initial image, much like the Beatles.
From the beginning they were not strictly a surf group. Brian Wilson, the primary songwriter, often used the group vocal harmony style of jazz groups. Their guitarist Carl Wilson did use the twang guitar style of Duane Eddy but also used the riff-based style of Chuck Berry. In fact, “Surfin’ USA” (1963) had the same melody, chord progression, and form as Berry’s “Sweet Little Sixteen” (1958).
The groups serene vocal sound was apparent in many of their early songs such as “Surfer Girl” and “In My Room” (both 1963). A testament to their live performance abilities, the group was able to successfully perform these complex vocals in concert.
By 1966, Brian Wilson’s songwriting had matured considerably but he suffered a nervous breakdown and was forced to stop touring. He kept writing and recording though, and that year they released Pet Sounds , their most important and influential album. “Pet Sounds” is notable as an early concept album in pop music where all the songs are thematically related. This predates “Sgt. Pepper’s Lonely Hearts Club Band”, the Beatles’ concept album and magnum opus, which was inspired by the album. The concept was “an expression of hopes, dreams, and anxiety about the present and future felt by a young person growing up”. Pet Sounds features songs written primarily by Brian Wilson as well as his arrangements which featured unusual instruments such as orchestral instruments like strings and percussion, and unusual studio productions such as echoes and reverberation which give the music a haunting yet full and colorful sound. Simple pop-oriented songs such as “Wouldn’t it Be Nice” featured these complex arrangements much like the productions of Phil Spector did earlier in the decade. (Listen to “Wouldn’t it Be Nice” in Ch. 8.3 Listening Examples )
Listening Examples 8.3
Duane Eddy:
Dick Dale: Here is Dick Dale Live on Ed Sullivan. I demonstrate the “Twang” of surf guitar by playing close to the bridge, then I demonstrate the tremolo picking style of Dick Dale.
The Beach Boys: First, listen to “In My Room” by the Beach Boys performed live (NOT lip-syncing). The Beach Boys recreate their vocal sound nearly-perfectly (with the exception of the screaming fans). In the song “Wouldn’t it Be Nice” , the form features a bridge (middle section) that brings back the intro theme on the piano, but places it in a brand new context including a different key, effectively creating a moment of “Bitonality” (Music in 2 different keys at once) from around 1:05-1:30. The keys are closely related, so the music sounds OK to our ears, bur the concept is quite advanced. The song “Good Vibrations” was a number one hit and also featured Phil Spector-influenced production. As well, the song introduced the electric-theremin into pop music, building on the haunting, colorful sounds from Pet Sounds .


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Nutrition is a key factor in promoting pulmonary wellness and supporting the function of the pulmonary system.
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A systematic approach to assessment, physical exam, diagnostic tests, and nutritional intervention is essential in supporting the nutritional needs of clients with pulmonary disease.
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Dietary choices including high protein, complex carbs, and healthy fats give nutritional support to clients and provide a mechanism for improved function despite chronic respiratory conditions.
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Any nutritional implementation by the client will likely have barriers or challenges. Recognition of the barriers and considering client preferences, religious beliefs, cultural norms, and social determinants of health in the design of interventions provides a holistic approach when planning goals and implementation strategies.
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Education about and implementation of nutritional health require follow-up care and reinforcement to achieve the client’s desired benefits.


8.4: Reading- Repositioning
When It’s Time to Change Direction
After they are initially introduced to the market, products, services, and brands are constantly being repositioned as a result of changes in competitive and market situations. Repositioning involves changing the market’s perceptions of an offering so that it can compete more effectively in its present market or in other target segments.
Generally it is good to consider repositioning when you see the need or opportunity to improve demand for the offering. Perhaps sales have slowed down, your target segment is getting smaller, or you’ve developed a new innovation you’d like to introduce to the product. Specific factors that can trigger the decision to reposition a product, service, or brand include the following:
- Competition: New competitors entering or leaving the market; competitors joining forces; a competitor’s innovation that threatens to make your offering obsolete; competitive pricing strategies
- Market environment: Economic slow-down or recovery; changes in consumer confidence, the political climate, or social forces like the movement around social responsibility and sustainability
- Consumer trends: Changing tastes and preferences; evolving attitudes and behaviors such as how consumers use technology to learn about, acquire, or interact with your offering; new segments emerging as targets for your offering
- Internal environment: Changes in organizational leadership and strategy; acquisition or development of new technology; introduction of innovation that offers new competitive advantages and differentiators
The tax preparation service H&R Block provides a useful example. As technology-savvy millennials (people born between 1980 and 2000) began entering the workforce and caring about taxes, H&R Block saw that this sizable young segment overwhelmingly preferred TurboTax and other technology-based, do-it-yourself tools, rather than hiring tax professionals like H&R Block. Even after introducing its own online tax preparation tools, H&R was not able to capture this market segment. With its competitive advantage undermined by technology and its established customer base getting older, H&R Block knew that if it wanted to survive, it had to figure out how to appeal to younger taxpayers.
In 2012 and 2014, the company invested in repositioning campaigns to alter the company’s image and appeal to millennials. The campaigns combined satirical humor, social media, and social responsibility (in the form of charitable donations) to get millennials’ attention and create buzz around the H&R Block brand. [1] The following H&R Block video ad from the 2012 “Stache Act” campaign makes the case for a fictional Million Mustache March on Washington to alter the tax code to include a $250 deduction for facial hair grooming materials:
You can view the transcript for “Stache Act Campaign Ad – The Oval Office” here (opens in new window) .
The Repositioning Process
The repositioning process is very similar to the original positioning process, but it has a different starting point. The original positioning process focuses on creating a new position or market niche for an offering that wasn’t there previously. The repositioning process, on the other hand, evaluates the established position of a product, service, or brand and focuses on how to alter the positioning–and, with positioning, market perceptions –in order to improve competitiveness.
To change market perceptions, repositioning may involve changes in the tangible product or in its selling price, but this is not always required. Often the new positioning and differentiation are accomplished through changes in the promotional message and approach. It is very common to see companies launch marketing campaigns focused on repositioning a product or service, but few, if any changes, made to the product or service itself. These repositioning efforts often focus on trying to get a current target segment to take another look at a product or service and see it with a new perspective. Repositioning often aims to shift market perceptions in ways that make an offering more appealing to a broader swath of the market.
An important ongoing part of repositioning is to monitor the position of a product, service, or brand over time. This is necessary in order to evaluate what is working or not working about the current position and generate feedback to inform future positioning strategies. A product position, like the score in a ball game, may change readily; keeping track and making necessary adjustments is very important.
Repositioning Risks and Pitfalls
While repositioning is quite common, it carries risks and complexities that marketers must consider. Repositioning happens after initial market perceptions have already been established. Effective repositioning isn’t just creating something new. Instead, it is trying to preserve what is good from the existing market positions and build or shift thinking toward something new. Repositioning offers the opportunity to make something new and better than what you had previously, but it also has the potential to undermine or weaken market perceptions.
Repositioning must always consider carefully what has come before, as well as what’s ahead. In the repositioning process there’s inevitably baggage: residual issues left over from earlier positioning work, which is what led you to the point of needing to reposition. Your product, service, or brand has a history, and people have memories: some people remember what the offering used to stand for, and they will try to figure out how the new positioning fits with their perceptions. Customers, employees, and other stakeholders will have opinions–sometimes very vocal ones–about whether the new positioning is better or worse, effective or ineffective. All of this represents a potential minefield for marketers. Despite these challenges, repositioning can also be very rewarding if you are successful at reshaping perceptions and creating a more powerful, meaningful product, service, or brand.
As you consider repositioning opportunities, try to avoid the following common pitfalls:
- Insufficient research: Marketing research should inform your choices about how to shift positioning in order to improve market perceptions of your product, service, or brand. You should also conduct research to help you understand how your target segment will react to the repositioning, so you’re not caught off guard by adverse reactions.
- A bridge too far: It can be tempting to get wild and crazy with repositioning, especially if you’re trying to freshen things up. While this strategy can work, sometimes marketers go so far in the new direction that customers no longer believe the claims. Their perceptions of the offering can’t accommodate the new message or image, and the offering loses credibility.
- Underestimating “back to basics”: Sometimes repositioning is undertaken because the target segment isn’t sure what a product, service, or brand stands for. Instead of trying to infuse more new ideas and new meaning, marketers are sometimes better served by stripping positioning down to its bare essentials of competitive advantage, benefit, and message. Reinforcing the simple “basics” can be very powerful: this is what customers usually care about most.
- Overpromising: When faced with strong competitive threats, it can be easy for repositioning to overpromise benefits that a product, service, or brand is really ready to deliver. This can be disastrous because it creates customer expectations that the organization cannot live up to. Rarely does this end well.
- Confusing positioning: Repositioning can introduce confusion between the old positioning and the new, especially if they seem to contradict each other. Repositioning needs to offer a clear message for customers; otherwise they are not sure what to believe.
The risks and pitfalls of repositioning are evident in the example of United Airlines and its “Rising” campaign. For decades, United positioned itself as a passenger-center carrier providing great service embodied in the iconic tagline “Fly the Friendly Skies.” Seeking a change in the late 1990s, United introduced a new positioning approach it called “Rising.” Their strategy was to to highlight common frustrations with air travel and make bold promises about how United Airlines provided a different, better level of service. However, the airline was unable to operationalize the changes needed to live up to these promises. The company abandoned the campaign after just two years because the positioning–and the airline–had lost credibility with the customer. [2]
Repositioning Success
Despite the risks, repositioning can be wildly successful when it is handled effectively. A good case in point is the American Red Cross. In 2009, the U.S. had sunk into the Great Recession, and the American Red Cross (ARC) was also feeling the pain. With its budget relying heavily on charitable donations, and with Americans giving less due to the recession, the nonprofit organization faced a budget deficit going into the fourth quarter.
For many nonprofit organizations, the last quarter of the year is prime fundraising season, since people open their wallets for holiday giving. Up until 2009, this was not the case for the Red Cross. Americans gave generously to the organization during disasters, but the ARC wasn’t people’s top choice for holiday giving. Seeing an opportunity in this apparent disconnect, the ARC engaged a creative agency to help repositioning the organization in the minds of potential donors.
Research confirmed that the competitive advantage of the American Red Cross, in consumers’ minds, was providing help in times of disaster. The organization’s then-current positioning of “Change a Life, Starting with Your Own “ shared a powerful emotional message, but it did not reinforce the competitive advantage or create a sense of urgency around giving to the ARC. The repositioning effort developed a new positioning direction expressed in the tag line “Give the gift that saves the day.”
This message reinforced the powerful role that the Red Cross plays in times of disaster and invited Americans to be part of that important work. With words like “give the gift,” it also implanted the idea of the ARC as a great recipient for holiday giving. The following video was created as part of the 2009 integrated marketing campaign that introduced this new positioning.
You can view the transcript for “American Red Cross – Holiday 2009 Campaign” here (opens in new window) .
The repositioning was a resounding success. Income increased more than 5 percent compared to prior years. People who saw ads associated with the repositioning campaign were twice as likely to donate as people who didn’t see them. The fourth quarter of 2009 was one of the strongest since 2000. Brand awareness increased by 6 percentage points. The benefits didn’t stop in 2009, either. Building on their success, the ARC expanded the repositioning campaign in 2010. By the end of the year, income had increased 26 percent over 2009, and the average gift size increased 43 percent. [3]
These impressive results reveal the power of repositioning when it is handled well.
- http://www.theguardian.com/money/us-money-blog/2014/mar/05/hipsters-taxes-brooklyn-ads-hrblock ↵
- http://www.wsj.com/articles/SB94719666565398524 ↵
- russreid.com/2013/02/american-red-cross-integrated-marketing-campaign/ ↵
Contributors and Attributions
- Revision and Adaptation. Authored by : Lumen Learning. License : CC BY: Attribution
- Repositioning Risks and Pitfalls, Repositioning Success. Authored by : Lumen Learning. License : CC BY: Attribution
- Introducing Marketing, Chapter 7. Introducing and managing the product. Authored by : John Burnett. Provided by : Global Text. Located at : http://solr.bccampus.ca:8001/bcc/file/ddbe3343-9796-4801-a0cb-7af7b02e3191/1/Core%20Concepts%20of%20Marketing.pdf . License : CC BY: Attribution
- Compass Study. Authored by : Calsidyrose. Located at : https://www.flickr.com/photos/calsidyrose/4925267732/ . License : CC BY: Attribution
- Suitcases Stacked to the Ceiling. Authored by : Laura Gilmore. Located at : https://www.flickr.com/photos/genbug/3529776320/ . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives
- Stache Act Campaign Ad - The Oval Office. Authored by : 'Stache Act. Located at : https://youtu.be/oZWeYVrdhSc . License : All Rights Reserved . License Terms : Standard YouTube license
- American Red Cross - Holiday 2009 Campaign. Authored by : BlankTV. Located at : https://youtu.be/1QmOw7Snu_M . License : All Rights Reserved . License Terms : Standard YouTube license


8.1A: Structural Classification of Joints
There are three structural classifications of joints: fibrous, cartilaginous, and synovial.
- Describe the three structural categories of joints
Key Points
-
The type
and characteristics of a given joint determine the degree and type of movement. -
Structural classification categorizes joints based on the type of
tissue involved in their formations. -
There are three structural
classifications of joints: fibrous, cartilaginous, and synovial. - Of the three types of fibrous joints, syndesmoses are the most movable.
-
Cartilaginous
joints allow more movement than fibrous joints
but less than synovial joints. - Synovial joints ( diarthroses ) are the most movable joints of the body and contain synovial fluid.
Key Terms
- periosteum : A membrane that covers the outer surface of all bones.
- manubrium : The broad upper part of the sternum.
-
synovial fluid
: A viscous fluid found in the cavities of synovial
joints that reduces friction between the articular cartilage during movement.
A joint, also known as an articulation or articular surface, is a connection that occurs between bones in the skeletal system. Joints provide the means for movement. The type and characteristics of a given joint determines its degree and type of movement. Joints can be classified based on structure and function.
Structural classification of joints categorizes them based on the type of tissue involved in formation. There are three structural classifications of joints: fibrous, cartilaginous, and synovial.
Fibrous Joints
Fibrous joints are connected by dense, tough connective tissue that is rich in collagen fibers. These fixed or immovable joints are typically interlocked with irregular edges. There are three types of fibrous joints.
Sutures are the types of joint found in the cranium (skull). The bones are connected by Sharpey’s fibres. The nature of cranial sutures allows for some movement in the fetus. However, they become mostly immovable as the individual ages, although very slight movement allows some necessary cranial elasticity. These rigid joints are referred to as synarthrodial.
Syndesmoses are found between long bones of the body, such as the radio-ulnar and tibio-fibular joints. These moveable fibrous joints are also termed amphiarthrodial. They have a lesser range of movement than synovial joints.
Gomphosis is a type of joint found at the articulation between teeth and the sockets of the maxilla or mandible (dental-alveolar joint). The fibrous tissue that connects the tooth and socket is called the periodontal ligament.
Fibrous joints : Image demonstrating the three types of fibrous joints. (a) Sutures (b) Syndesmosis (c) Gomphosis.
Cartilaginous Joints
Cartilaginous joints are connected by fibrocartilage or hyaline cartilage. They allow more movement than fibrous joints but less than that of synovial joints. These types of joints are further subdivided into primary (synchondroses) and secondary (symphyses) cartilaginous joints. The epiphyseal (growth) plates are examples of synchondroses. Symphyses are found between the manubrium and sternum (manubriosternal joint), intervertebral discs, and the pubic symphysis.
Cartilaginous Joints : Image demonstrates a synchondrosis joint with epiphyseal plate (temporary hyaline cartilage joint) indicated (a) and a symphysis joint (b).
Synovial Joint : This diagram of a synovial joint delineates the articular cartilage, articular capsule, bone, synovial membrane, and joint cavity containing synovial fluid.
Synovial Joints
This is the most common and movable joint type in the body. These joints (also called diarthroses) have a synovial cavity. Their bones are connected by dense irregular connective tissue that forms an articular capsule surrounding the bones’ articulating surfaces.
A synovial joint connects bones with a fibrous joint capsule that is continuous with the bones’ periosteum. This joint capsule constitutes the outer boundary of a synovial cavity and surrounds the bones’ articulating surfaces.
Synovial cavities are filled with synovial fluid. The knees and elbows are examples of synovial joints.


13.5: Women’s Rights
Women took part in all the antebellum reforms, from transcendentalism to temperance to abolition. In many ways, traditional views of women as nurturers played a role in encouraging their participation. Women who joined the cause of temperance, for example, amplified their accepted role as moral guardians of the home. Some women advocated a much more expansive role for themselves and their peers by educating children and men in solid republican principles. But it was their work in antislavery efforts that served as a springboard for women to take action against gender inequality. Many, especially northern women, came to the conclusion that they, like slaves, were held in shackles in a society dominated by men.
Despite the radical nature of their effort to end slavery and create a biracial society, most abolitionist men clung to traditional notions of proper gender roles. White and black women, as well as free black men, were forbidden from occupying leadership positions in the AASS. Because women were not allowed to join the men in playing leading roles in the organization, they formed separate societies, such as the Boston Female Anti-Slavery Society, the Philadelphia Female Anti-Slavery Society, and similar groups.
THE GRIMKÉ SISTERS
Two leading abolitionist women, Sarah and Angelina Grimké, played major roles in combining the fight to end slavery with the struggle to achieve female equality. The sisters had been born into a prosperous slaveholding family in South Carolina. Both were caught up in the religious fervor of the Second Great Awakening, and they moved to the North and converted to Quakerism.
In the mid-1830s, the sisters joined the abolitionist movement, and in 1837, they embarked on a public lecture tour, speaking about immediate abolition to “promiscuous assemblies,” that is, to audiences of women and men. This public action thoroughly scandalized respectable society, where it was unheard of for women to lecture to men. William Lloyd Garrison endorsed the Grimké sisters’ public lectures, but other abolitionists did not. Their lecture tour served as a turning point; the reaction against them propelled the question of women’s proper sphere in society to the forefront of public debate.
THE DECLARATION OF RIGHTS AND SENTIMENTS
Participation in the abolitionist movement led some women to embrace feminism, the advocacy of women’s rights. Lydia Maria Child, an abolitionist and feminist, observed, “The comparison between women and the colored race is striking . . . both have been kept in subjection by physical force.” Other women, including Elizabeth Cady Stanton, Lucy Stone, and Susan B. Anthony, agreed (Figure 13.5.1).
In 1848, about three hundred male and female feminists, many of them veterans of the abolition campaign, gathered at the Seneca Falls Convention in New York for a conference on women’s rights that was organized by Lucretia Mott and Elizabeth Cady Stanton. It was the first of what became annual meetings that have continued to the present day. Attendees agreed to a “Declaration of Rights and Sentiments” based on the Declaration of Independence; it declared, “We hold these truths to be self-evident: that all men and women are created equal; that they are endowed by their Creator with certain inalienable rights; that among these are life, liberty, and the pursuit of happiness.” “The history of mankind,” the document continued, “is a history of repeated injuries and usurpations on the part of man toward woman, having in direct object the establishment of an absolute tyranny over her.”
Read the entire text of the Declaration of Rights and Sentiments in the Internet Modern History Sourcebook at Fordham University.
REPUBLICAN MOTHERHOOD IN THE ANTEBELLUM YEARS
Some northern female reformers saw new and vital roles for their sex in the realm of education. They believed in traditional gender roles, viewing women as inherently more moral and nurturing than men. Because of these attributes, the feminists argued, women were uniquely qualified to take up the roles of educators of children.
Catharine Beecher, the daughter of Lyman Beecher, pushed for women’s roles as educators. In her 1845 book, The Duty of American Women to Their Country , she argued that the United States had lost its moral compass due to democratic excess. Both “intelligence and virtue” were imperiled in an age of riots and disorder. Women, she argued, could restore the moral center by instilling in children a sense of right and wrong. Beecher represented a northern, middle-class female sensibility. The home, especially the parlor, became the site of northern female authority.
Section Summary
The spirit of religious awakening and reform in the antebellum era impacted women lives by allowing them to think about their lives and their society in new and empowering ways. Of all the various antebellum reforms, however, abolition played a significant role in generating the early feminist movement in the United States. Although this early phase of American feminism did not lead to political rights for women, it began the long process of overcoming gender inequalities in the republic.
Review Questions
Why did William Lloyd Garrison’s endorsement of the Grimké sisters divide the abolitionist movement?
- They advocated equal rights for women.
- They supported colonization.
- They attended the Seneca Falls Convention.
- They lectured to co-ed audiences.
D
Which female reformer focused on women’s roles as the educators of children?
- Lydia Maria Child
- Sarah Grimké
- Catherine Beecher
- Susan B. Anthony
C
How did the abolitionist movement impact the women’s movement?
Women’s involvement in the abolitionist movement, where they were unable to take leadership roles in traditional male organizations, led them to create their own organizations, where they were thrust into the public sphere.
Critical Thinking Questions
In what ways did the Second Great Awakening and transcendentalism reflect and react to the changes in antebellum American thought and culture?
What did the antebellum communal projects have in common? How did the ones most influenced by religion differ from those that had other influences?
In what ways do temperance, health reforms, and phrenology offer reflections on the changes in the United States before the Civil War? What needs did these reforms fill in the lives of antebellum Americans?
Of the various approaches to the problem of slavery, which one do you find to be the most effective and why?
In what ways were antebellum feminists radical? In what ways were they traditional?
Glossary
- Seneca Falls
- the location of the first American conference on women’s rights and the signing of the “Declaration of Rights and Sentiments” in 1848


4.5: Multiply and Divide Mixed Numbers and Complex Fractions (Part 1)
- Multiply and divide mixed numbers
- Translate phrases to expressions with fractions
- Simplify complex fractions
- Simplify expressions written with a fraction bar
Before you get started, take this readiness quiz.
- Divide and reduce, if possible: \((4 + 5) ÷ (10 − 7)\). If you missed this problem, review Example 3.2.8 .
- Multiply and write the answer in simplified form: \(\dfrac{1}{8} \cdot \dfrac{2}{3}\). If you missed this problem, review Example 4.2.7 .
- Convert \(2 \dfrac{3}{5}\) into an improper fraction. If you missed this problem, review Example 4.1.11 .
Multiply and Divide Mixed Numbers
In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction. And you learned how to do that in Visualize Fractions .
Multiply: \(3 \dfrac{1}{3} \cdot \dfrac{5}{8}\)
Solution
| Convert \(3 \dfrac{1}{3}\) to an improper fraction. | \(\dfrac{10}{3} \cdot \dfrac{5}{8}\) |
| Multiply. | \(\dfrac{10 \cdot 5}{3 \cdot 8}\) |
| Look for common factors. | \(\dfrac{\cancel{2} \cdot 5 \cdot 5}{3 \cdot \cancel{2} \cdot 4}\) |
| Remove common factors. | \(\dfrac{5 \cdot 5}{3 \cdot 4}\) |
| Simplify. | \(\dfrac{25}{12}\) |
Notice that we left the answer as an improper fraction, \(\dfrac{25}{12}\), and did not convert it to a mixed number. In algebra, it is preferable to write answers as improper fractions instead of mixed numbers. This avoids any possible confusion between \(2 \dfrac{1}{12}\) and \(2 \cdot \dfrac{1}{12}\).
Multiply, and write your answer in simplified form: \(5 \dfrac{2}{3} \cdot \dfrac{6}{17}\).
- Answer
-
\(2\)
Multiply, and write your answer in simplified form: \(\dfrac{3}{7} \cdot 5 \dfrac{1}{4}\).
- Answer
-
\(\dfrac{9}{4}\)
Step 1. Convert the mixed numbers to improper fractions.
Step 2. Follow the rules for fraction multiplication or division.
Step 3. Simplify if possible.
Multiply, and write your answer in simplified form: \(2 \dfrac{4}{5} \left(− 1 \dfrac{7}{8}\right)\).
Solution
| Convert mixed numbers to improper fractions. | \(\dfrac{14}{5} \left(-1 \dfrac{7}{8}\right)\) |
| Multiply. | \(- \dfrac{14 \cdot 15}{5 \cdot 8}\) |
| Look for common factors. | \(- \dfrac{\cancel{2} \cdot 7 \cdot \cancel{5} \cdot 3}{\cancel{5} \cdot \cancel{2} \cdot 4}\) |
| Remove common factors. | \(- \dfrac{7 \cdot 3}{4}\) |
| Simplify. | \(- \dfrac{21}{4}\) |
Multiply, and write your answer in simplified form. \(5 \dfrac{5}{7} \left(− 2 \dfrac{5}{8}\right)\).
- Answer
-
\(-15\)
Multiply, and write your answer in simplified form. \(-3 \dfrac{2}{5} \cdot 4 \dfrac{1}{6}\).
- Answer
-
\(-\dfrac{85}{6}\)
Divide, and write your answer in simplified form: \(3 \dfrac{4}{7} ÷ 5\).
Solution
| Convert mixed numbers to improper fractions. | \(\dfrac{25}{7} \div \dfrac{5}{1}\) |
| Multiply the first fraction by the reciprocal of the second. | \(\dfrac{25}{7} \cdot \dfrac{1}{5}\) |
| Multiply. | \(\dfrac{25 \cdot 1}{7 \cdot 5}\) |
| Look for common factors. | \(\dfrac{\cancel{5} \cdot 5 \cdot 1}{7 \cdot \cancel{5}}\) |
| Remove common factors. | \(\dfrac{5 \cdot 1}{7}\) |
| Simplify. | \(\dfrac{5}{7}\) |
Divide, and write your answer in simplified form: \(4 \dfrac{3}{8} ÷ 7\).
- Answer
-
\(\dfrac{5}{8}\)
Divide, and write your answer in simplified form: \(2 \dfrac{5}{8} ÷ 3\).
- Answer
-
\(\dfrac{7}{8}\)
Divide: \(2 \dfrac{1}{2} \div 1 \dfrac{1}{4}\).
Solution
| Convert mixed numbers to improper fractions. | \(\dfrac{5}{2} \div \dfrac{5}{4}\) |
| Multiply the first fraction by the reciprocal of the second. | \(\dfrac{5}{2} \cdot \dfrac{4}{5}\) |
| Multiply. | \(\dfrac{5 \cdot 4}{2 \cdot 5}\) |
| Look for common factors. | \(\dfrac{\cancel{5} \cdot \cancel{2} \cdot 2}{\cancel{2} \cdot 1 \cdot \cancel{5}}\) |
| Remove common factors. | \(\dfrac{2}{1}\) |
| Simplify. | \(2\) |
Divide, and write your answer in simplified form: \(2 \dfrac{2}{3} \div 1 \dfrac{1}{3}\).
- Answer
-
\(2\)
Divide, and write your answer in simplified form: \(3 \dfrac{3}{4} \div 1 \dfrac{1}{2}\).
- Answer
-
\(\dfrac{5}{2}\)
Translate Phrases to Expressions with Fractions
The words quotient and ratio are often used to describe fractions. In Subtract Whole Numbers , we defined quotient as the result of division. The quotient of \(a\) and \(b\) is the result you get from dividing \(a\) by \(b\), or \(\dfrac{a}{b}\). Let’s practice translating some phrases into algebraic expressions using these terms.
Translate the phrase into an algebraic expression: “the quotient of \(3x\) and \(8\).”
Solution
The keyword is quotient ; it tells us that the operation is division. Look for the words of and and to find the numbers to divide.
The quotient of \(3x\) and \(8\).
This tells us that we need to divide \(3x\) by \(8\). \(\dfrac{3x}{8}\)
Translate the phrase into an algebraic expression: the quotient of \(9s\) and \(14\).
- Answer
-
\(\dfrac{9s}{14}\)
Translate the phrase into an algebraic expression: the quotient of \(5y\) and \(6\).
- Answer
-
\(\dfrac{5y}{6}\)
Translate the phrase into an algebraic expression: the quotient of the difference of \(m\) and \(n\), and \(p\).
Solution
We are looking for the quotient of the difference of \(m\) and \(n\), and \(p\). This means we want to divide the difference of \(m\) and \(n\) by \(p\).
\[\dfrac{m − n}{p} \nonumber \]
Translate the phrase into an algebraic expression: the quotient of the difference of \(a\) and \(b\), and \(cd\).
- Answer
-
\(\dfrac{a-b}{cd}\)
Translate the phrase into an algebraic expression: the quotient of the sum of \(p\) and \(q\), and \(r\).
- Answer
-
\(\dfrac{p+q}{r}\)
Simplify Complex Fractions
Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction , which is a fraction in which the numerator or the denominator contains a fraction. Some examples of complex fractions are:
\[\dfrac{\dfrac{6}{7}}{3} \quad \dfrac{\dfrac{3}{4}}{\dfrac{5}{8}} \quad \dfrac{\dfrac{x}{2}}{\dfrac{5}{6}} \nonumber \]
To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction \(\dfrac{\dfrac{3}{4}}{\dfrac{5}{8}}\) can be written as \(\dfrac{3}{4} \div \dfrac{5}{8}\).
Simplify: \(\dfrac{\dfrac{3}{4}}{\dfrac{5}{8}}\).
Solution
| Rewrite as division. | \(\dfrac{3}{4} \div \dfrac{5}{8}\) |
| Multiply the first fraction by the reciprocal of the second. | \(\dfrac{3}{4} \cdot \dfrac{8}{5}\) |
| Multiply. | \(\dfrac{3 \cdot 8}{4 \cdot 5}\) |
| Look for common factors. | \(\dfrac{3 \cdot \cancel{4} \cdot 2}{\cancel{4} \cdot 5}\) |
| Remove common factors and simplify. | \(\dfrac{6}{5}\) |
Simplify: \(\dfrac{\dfrac{2}{3}}{\dfrac{5}{6}}\).
- Answer
-
\(\dfrac{4}{5}\)
Simplify: \(\dfrac{\dfrac{3}{7}}{\dfrac{6}{11}}\).
- Answer
-
\(\dfrac{11}{14}\)
Step 1. Rewrite the complex fraction as a division problem.
Step 2. Follow the rules for dividing fractions.
Step 3. Simplify if possible.
Simplify: \(\dfrac{− \dfrac{6}{7}}{3}\).
Solution
| Rewrite as division. | \(- \dfrac{6}{7} \div 3\) |
| Multiply the first fraction by the reciprocal of the second. | \(- \dfrac{6}{7} \cdot \dfrac{1}{3}\) |
| Multiply; the product will be negative. | \(- \dfrac{6 \cdot 1}{7 \cdot 3}\) |
| Look for common factors. | \(- \dfrac{\cancel{3} \cdot 2 \cdot 1}{7 \cdot \cancel{3}}\) |
| Remove common factors and simplify. | \(- \dfrac{2}{7}\) |
Simplify: \(\dfrac{− \dfrac{8}{7}}{4}\).
- Answer
-
\(-\dfrac{2}{7}\)
Simplify: \(− \dfrac{3}{\dfrac{9}{10}}\).
- Answer
-
\(-\dfrac{10}{3}\)
Simplify: \(\dfrac{\dfrac{x}{2}}{\dfrac{xy}{6}}\).
Solution
| Rewrite as division. | \(\dfrac{x}{2} \div \dfrac{xy}{6}\) |
| Multiply the first fraction by the reciprocal of the second. | \(\dfrac{x}{2} \cdot \dfrac{6}{xy}\) |
| Multiply. | \(\dfrac{x \cdot 6}{2 \cdot xy}\) |
| Look for common factors. | \(\dfrac{\cancel{x} \cdot 3 \cdot \cancel{2}}{\cancel{2} \cdot \cancel{x} \cdot y}\) |
| Remove common factors and simplify. | \(\dfrac{3}{y}\) |
Simplify: \(\dfrac{\dfrac{a}{8}}{\dfrac{ab}{6}}\).
- Answer
-
\(\dfrac{3}{4b}\)
Simplify: \(\dfrac{\dfrac{p}{2}}{\dfrac{pq}{8}}\).
- Answer
-
\(\dfrac{4}{q}\)
Simplify: \(\dfrac{2 \dfrac{3}{4}}{\dfrac{1}{8}}\).
Solution
| Rewrite as division. | \(2 \dfrac{3}{4} \div \dfrac{1}{8}\) |
| Change the mixed number to an improper fraction. | \(\dfrac{11}{4} \div \dfrac{1}{8}\) |
| Multiply the first fraction by the reciprocal of the second. | \(\dfrac{11}{4} \cdot \dfrac{8}{1}\) |
| Multiply. | \(\dfrac{11 \cdot 8}{4 \cdot 1}\) |
| Look for common factors. | \(\dfrac{11 \cdot \cancel{4} \cdot 2}{\cancel{4} \cdot 1}\) |
| Remove common factors and simplify. | \(22\) |
Simplify: \(\dfrac{\dfrac{5}{7}}{1 \dfrac{2}{5}}\).
- Answer
-
\(\dfrac{25}{49}\)
Simplify: \(\dfrac{\dfrac{8}{5}}{3 \dfrac{1}{5}}\).
- Answer
-
\(\dfrac{1}{2}\)
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at<EMAIL_ADDRESS>."


4.8: Applications
View Nondimensionalization on YouTube
RLC Circuit
View \(RLC\) circuit on YouTube
Consider a resister \(R\), an inductor \(L\), and a capacitor \(C\) connected in series as shown in Fig. \(\PageIndex{1}\). An AC generator provides a time-varying electromotive force (emf), \(\mathcal{E}(t)\), to the circuit. Here, we determine the differential equation satisfied by the charge on the capacitor.
The constitutive equations for the voltage drops across a capacitor, a resister, and an inductor are given by \[\label{eq:1}V_C=q/C,\quad V_R=iR,\quad V_L=\frac{di}{dt}L,\] where \(C\) is the capacitance, \(R\) is the resistance, and \(L\) is the inductance. The charge \(q\) and the current \(i\) are related by \[\label{eq:2}i=\frac{dq}{dt}.\]
Kirchhoff’s voltage law states that the emf \(\mathcal{E}\) applied to any closed loop is equal to the sum of the voltage drops in that loop. Applying Kirchhoff’s voltage law to the \(RLC\) circuit results in \[\label{eq:3}V_L+V_R+V_C=\mathcal{E}(t);\] or using \(\eqref{eq:1}\) and \(\eqref{eq:2}\), \[L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac{1}{C}q=\mathcal{E}(t).\nonumber\]
The equation for the \(RLC\) circuit is a second-order linear inhomogeneous differential equation with constant coefficients.
The AC voltage can be written as \(\mathcal{E}(t) = \mathcal{E}_0 \cos \omega t\), and the governing equation for \(q = q(t)\) can be written as \[\label{eq:4}\frac{d^2q}{dt^2}+\frac{R}{L}\frac{dq}{dt}+\frac{1}{LC}q=\frac{\mathcal{E}_0}{L}\cos\omega t.\]
Nondimensionalization of this equation will be shown to reduce the number of free parameters.
To construct dimensionless variables, we first define the natural frequency of oscillation of a system to be the frequency of oscillation in the absence of any driving or damping forces. The iconic example is the simple harmonic oscillator, with equation given by \[\overset{..}{x}+\omega_0^2x=0,\nonumber\] and general solution given by \(x(t) = A \cos \omega_0 t + B \sin\omega_0 t\). Here, the natural frequency of oscillation is \(\omega_0\), and the period of oscillation is \(T = 2π/\omega_0\). For the \(RLC\) circuit, the natural frequency of oscillation is found from the coefficient of the \(q\) term, and is given by \[\omega_0=\frac{1}{\sqrt{LC}}.\nonumber\]
Making use of \(\omega_0\), with units of one over time, we can define a dimensionless time \(\tau\) and a dimensionless charge \(Q\) by \[\tau =\omega_0t,\quad Q=\frac{\omega_0^2L}{\mathcal{E}_0}q.\nonumber\]
The resulting dimensionless equation for the \(RLC\) circuit is then given by \[\label{eq:5}\frac{d^2Q}{d\tau ^2}+\alpha\frac{dQ}{d\tau}+Q=\cos\beta\tau ,\] where \(\alpha\) and \(\beta\) are dimensionless parameters given by \[\alpha=\frac{R}{L\omega_0},\quad\beta=\frac{\omega}{\omega_0}.\nonumber\]
Notice that the original five parameters in \(\eqref{eq:4}\), namely \(R,\: L,\: C,\: \mathcal{E}_0\) and \(\omega\), have been reduced to the two dimensionless parameters \(\alpha\) and \(\beta\). We will return later to solve \(\eqref{eq:5}\) after visiting two more applications that will be shown to be governed by the same dimensionless equation.
Mass on a Spring
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Consider a mass connected by a spring to a fixed wall, with top view shown in Fig. \(\PageIndex{2}\). The spring force is modeled by Hooke’s law, \(F_s = −kx\), and sliding friction is modeled as \(F_f = −cdx/dt\). An external force is applied and is assumed to be sinusoidal, with \(F_e = F_0 \cos\omega t\). Newton’s equation, \(F = ma\), results in \[m\frac{d^2x}{dt^2}=-kx-c\frac{dx}{dt}+F_0\cos\omega t.\nonumber\]
Rearranging terms, we obtain \[\frac{d^2x}{dt^2}+\frac{c}{m}\frac{dx}{dt}+\frac{k}{m}x=\frac{F_0}{m}\cos\omega t.\nonumber\]
Here, the natural frequency of oscillation is given by \[\omega_0=\sqrt{\frac{k}{m}},\nonumber\] and we define a dimensionless time \(\tau\) and a dimensionless position \(X\) by \[\tau =\omega_0t,\quad X=\frac{m\omega_0^2}{F_0}x.\nonumber\]
The resulting dimensionless equation is given by \[\label{eq:6}\frac{d^2X}{d\tau ^2}+\alpha\frac{dX}{d\tau }+X=\cos\beta\tau, \] where here, \(\alpha\) and \(\beta\) are dimensionless parameters given by \[\alpha=\frac{c}{m\omega_0},\quad\beta =\frac{\omega}{\omega_0}.\nonumber\]
Notice that even though the physical problem is different, and the dimensionless variables are defined differently, the resulting dimensionless equation \(\eqref{eq:6}\) for the mass-spring system is the same as that for the \(RLC\) circuit \(\eqref{eq:5}\).
Pendulum
Here, we consider a mass that is attached to a massless rigid rod and is constrained to move along an arc of a circle centered at the pivot point (see Fig. \(\PageIndex{3}\)). Suppose \(l\) is the fixed length of the connecting rod, and \(θ\) is the angle it makes with the vertical.
We can apply Newton’s equation, \(F = ma\), to the mass with origin at the bottom and axis along the arc with positive direction to the right. The position \(s\) of the mass along the arc is given by \(s = lθ\). The relevant gravitational force on the pendulum is the component along the arc, and from Fig. \(\PageIndex{3}\) is observed to be \[F_g=-mg\sin\theta.\nonumber\]
We model friction to be proportional to the velocity of the pendulum along the arc, that is \[F-f=-c\overset{.}{s}=-cl\overset{.}{\theta}.\nonumber\]
With a sinusoidal external force, \(F_e = F_0 \cos\omega t\), Newton’s equation \(m\overset{..}{s}= F_g + F_f + F_e\) results in \[ml\overset{..}{\theta}=-mg\sin\theta -cl\overset{.}{\theta}+F_0\cos\omega t.\nonumber\]
Rewriting, we have \[\overset{..}{\theta}+\frac{c}{m}\overset{.}{\theta}+\frac{g}{l}\sin\theta=\frac{F_0}{ml}\cos\omega t.\nonumber\]
At small amplitudes of oscillation, we can approximate \(\sin θ\approx θ\), and the natural frequency of oscillation \(\omega_0\) of the mass is given by \[\omega_0=\sqrt{\frac{g}{l}}.\nonumber\]
Nondimensionalizing time as \(\tau = \omega_0t\), the dimensionless pendulum equation becomes \[\frac{d^2\theta}{d\tau ^2}+\alpha\frac{d\theta}{d\tau}+\sin\theta=\gamma\cos\beta\tau ,\nonumber\] where \(\alpha\), \(\beta\), and \(\gamma\) are dimensionless parameters given by \[\alpha=\frac{c}{m\omega_0},\quad\beta=\frac{\omega}{\omega_0},\quad\gamma=\frac{F_0}{ml\omega_0^2}.\nonumber\]
The nonlinearity of the pendulum equation, with the term \(\sin \theta\), results in the additional dimensionless parameter \(\gamma\). For small amplitude of oscillation, however, we can scale \(\theta\) by \(\theta =\gamma\Theta\), and the small amplitude dimensionless equation becomes \[\label{eq:7}\frac{d^2\Theta}{d\tau ^2}+\alpha\frac{d\Theta}{d\tau}+\Theta=\cos\beta\tau ,\] the same equation as \(\eqref{eq:5}\) and \(\eqref{eq:6}\).


3.4: Paired T-test Assumptions, Interpretation, and Write Up
At the end of this chapter you should be able to answer the following questions:
- How does the Paired t-test differ from the Independent t-test?
- What assumptions that need to be met before performing a Paired-samples t-test ?
- What elements or individual statistics should be reported when writing up a Paired T-test?
The Paired t-test is also known as the Paired Samples t-test or the Dependent t-test.
Paired T-test Assumptions
Just like in the Independent t-test from our previous chapter, there are a number of assumptions that need to be met before performing a Paired-samples t-test:
- The dependent variable (the variable of interest) needs a continuous scale (i.e., the data needs to be at either an interval or ratio measurement). An example of this variable could be weight or level of anxiety.
- The independent variable must have two groups that are related or have “matched pairs”. Matched pairs, as mentioned above, mean that the same participants are present in both groups.
- There should be no spurious outliers across either of the groups being used in the test.
- There should be an approximate normal distribution of differences across the two related groups.
Paired T-test Interpretation
A Paired t-tests examines the within-group differences of a single group. So the same subjects are the respondents for both pairs of measurements, and this indicates that you have related groups of scores.
This means that the “related groups” or “matched pairs” are the same participants who have been measured in each of the two groups. Specifically this means the two groups both contain the same people, who are producing the same scores or measurements in both groups. Unlike the independent samples t-test, researchers can measure the same participants in each level of the independent variable because the measurement or scoring for each participant will be taken at two distinct time points.
In the worked example for this test, we will be looking at the levels of perceived social support a group of Australians reported before engaging with a social skills building program and after completing the program.
Red: Test statistics
Green: Differences in means for effect size
Blue: Means and standard deviation
This example shows perceived social support as an outcome variable. This outcome is expected to change after the intervention of a social skills building program.
The output shows levels of perceived social support across two groups of scores: Scores from a group of Australians reported before engaging with a social skills building program, and scores the same people reported after completing the program.
As you can see, the t-test was significant as circled in red, showing a change from the scores before completing the social skills program, and after. By looking at the means as circled in blue you can tell that the mean level of perceived social support was higher following the completion of the program. To calculate the effect size, you divide the mean difference by the standard deviation of the difference as circled in green.
Paired T-test Write Up
When you are writing your report, you will need to include the Means and SD for each group, along with the t test statistic ( t ), its p value, and its effect size d .
A Write-Up for a Paired T-test should look like this:
A paired samples t-test showed that the participant’s level of perceived social support increased from pre-program (M = 32.83, SD = 7.91) to post-program (M = 38.07, SD = 7.23; t = -14.07, p < .001, d = -.73).


27.7.1: Arrays and Functions
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In modular programming, specific task functions are often created and used or reused for array processing. Array processing functions are usually passed the array and any data necessary to process the array for the given task.
It should be noted that arrays are passed by reference in most current programming languages. Array processing functions must take care not to alter the array unless intended.
Arrays are an important complex data type used in almost all programming. We continue to concentrate on simple one dimension arrays also called a list. Most programmers develop a series of user-defined specific task functions that can be used with an array for normal processing. These functions are usually passed the array along with the number of elements within the array. Some functions also pass another piece of data needed for that particular function’s task.
This module covers the displaying the array members on the monitor via calling an array function dedicated to that task.
Function Main Declare Integer Array ages[5] Assign ages = [49, 48, 26, 19, 16] Call DisplayArray(ages) End Function DisplayArray (Integer Array array) Declare Integer index For index = 0 to Size(array) - 1 Output array[index] End End
49 48 26 19 16


2.1: Medical Language Introduction
Learning Objectives
- Use the anatomic reference system to identify the anatomic position of the body
- Use the anatomic reference system to identify the body planes
- Use the anatomic reference system to identify the body cavities
- Use the anatomic reference system to identify the directional terms
- Use the anatomic reference system to identify the divisions of the body
- Describe the structural organization of the body
In Chapter 1, the components of medical terms were introduced. As you continue to learn medical language, it is important to contextualize the meaning of these terms based on human anatomy. This chapter will review anatomical terminology related to organ systems, body cavities, body regions, and tissue membranes. It will also discuss the standardized anatomical position, directional terms, and planes of the body.


1.1: Prelude to Middle East, Near East, and Greece
The texts chosen for this chapter were influential in their own times and beyond. Gilgamesh was an ancient Sumerian king whose story was valued and retold by other cultures who invaded the area. The Bible remains one of the most widely read books in history. Homer’s epics form a cornerstone of western literature, and the two plays selected from ancient Greek drama influenced countless writers after them. Only the plays were originally written works; the other texts were part of an oral tradition before they were written down. Even then, the subject matter of the plays is not original to the authors: The audience knew the stories of Oedipus and Medea already. Homer was not the first (or the last) to compose poems on the Trojan War and its aftermath. Originality was not particularly prized in an oral culture, where only the best works were worth memorizing. Homer’s fame comes from how well he tells his version of events.
When reading the selected texts, remember that the contemporary definition of a hero or leader is often not compatible with the ancient world’s definition of a hero or leader. Each society, and sometimes each time period in each society, can have a different definition, based on what the expectations were. There is also a difference between the modern idea of an action hero and the ancient world’s definition of an epic hero. To be the hero of an epic, the character needs to meet at least some of the following requirements: He receives divine intervention (or is chosen by the gods to win), has superhuman strength or abilities, is of national or international importance, has the ability to overcome and learn from a personal flaw, and goes on a significant journey. The ultimate goal of epic heroes is to be remembered: achieving immortality through their deeds, which will live on in stories. Unlike a modern film hero who might be expected to act in the best interests of others, epic heroes may or may not act with other people’s interests in mind. Some of the epic heroes in this chapter fight to protect others, but many fight for personal glory, regardless of the collateral damage. In other words, an epic hero is an ideal warrior in his society, but not always an ideal human being. In the Iliad , Achilles is the greatest warrior among the Greeks, and his main concern is making a name for himself that will last forever. When he is insulted by Agamemnon, therefore, he asks that Zeus punish Achilles’ own side, slaughtering the Greeks until they beg him for forgiveness. Achilles fights for his own glory, not the glory of others.
In Gilgamesh , the title character begins the story as an impressive epic hero, but a poor leader (as the gods themselves indicate in the story when they respond to the prayers of the citizens of Uruk, who are begging the gods to protect them from their own king). Gilgamesh’s lack of morality stems in part from his demigod status; as the ancient Sumerians recognized, their pantheon of gods was not particularly moral. Since epic heroes need the help of the gods to win, the focus is not on individual strength, but on gaining the favor of the gods. Yes, Gilgamesh is strong, but to fight the supernatural creature Humbaba, Gilgamesh needs help: his mother’s prayers to the gods, his friend Enkidu’s support, supernatural weapons from the god Shamash (namely the winds), and his tears as offerings to Shamash in exchange for his help. The expectations for a good king are clear in the text, but they conflict on some level with the expectations for an epic hero in this case.
The hero who receives divine intervention is the one who wins every time, so being humble to the gods is vital for success. When Brad Pitt plays Achilles in the movie Troy , there are no toddler tantrums; in the Iliad , Achilles cries every time he wants the help of his mother, the goddess Thetis. The modern film expectations for the character of Achilles would be foreign (and strange, and irreligious) to the original audience, just as a modern American film audience would not be impressed by an action hero who sobbed to his mother for help. The original audience, how- ever, would be familiar with example after example of how pointless it is to try to win without the help of the gods: No matter who would have won based on his own strength, the gods determine the final result. Human strength means little in such a universe.
Equally pointless is the attempt to change fate, which is the one force in the Greek stories that is stronger than the gods. Zeus cannot change the outcome of various events in the Iliad , and Oedipus realizes the futility of attempting to change his fate. The fatalistic approach of the Greek texts stems from the belief that the ages of man are in a decline, from the golden age down to the iron age of Homer. This belief in the general decline of humanity is echoed later in Dante’s Inferno , where the Old Man of Crete is composed of the same metals, but this time with a clay foot.
As you read, consider the following questions:
- Using the list of traits above, which traits apply to each epic hero in the texts?
- What is similar and/or different about heores such as Gilgamesh, Achilles, Hector, and Odysseus?
- How do the characters view the gods, and how do the gods treat humans?
- What do we learn about what each society considers proper or improper behavior, again based on the text itself?
- Is family love or romantic love more important in the text, and why?


2.1: p Values
At the end of this section you should be able to answer the following questions:
- What is a p value?
- How can you interpret a p value?
- What question can p value answer?
An important area of statistics is probability, and it is the basis for all of the tests we will be reviewing in this textbook. One important kind of probability is a conditional probability. For example, given the weather forecast for today, what is the likelihood that it will rain?
The p value itself is a figure or numeral – typically represented by a number between 0 and 1.00 – that provides the probability of a result (for a particular test statistic) being due to a true effect rather than chance. The p value is a conditional probability and relies on a number of assumptions about the test statistics used.
Here is an example from psychology that provides an illustration of the p value:
Psychological scientists at your university are evaluating a clinical therapy that is believed to reduce anxiety in young adults. In a field study, these scientists use two groups to test the therapy – one group receives the clinical therapy and a second group that does not receive the clinical therapy – which are respectively known as the experimental and control groups. Anxiety in participants is then measured in both groups after the therapy takes place (or not). Using a T-test statistic – which examines the different means for anxiety between two groups – the result of the test statistic is t(18)= 2.7, p = .01.
The p value is indicated by the statement of p = .01 that appears after the 2.7, which is the value of the T-test statistic. You interpret the significance of a p value based on a critical value for p values, which is often designated as .05. You also note that p values less than .05 are considered significant in most research. In our example p = .01 which is below .05. This means that the test statistic of t(18)= 2.7 provides evidence for a difference between the control and experimental groups.
When concluding there is a difference between the control and experimental groups, a researcher is really referring back to the populations from which the two groups are assumed to be drawn. Hence, there is an inference from the samples back to the populations.
It is critical to remember that a p value does NOT answer “What is the probability that the difference is due to chance?” A p value does answer: ‘Assuming that there is no real difference in the populations (that correspond to the two groups), what is the probability that the difference between the means of randomly selected subjects will be as large as or larger than actually observed?’ This distinction might sound academic, but it is very important.