3 COLLEGE ALGEBRA - TRIGONOMETRY Business and Finance (MAT115) This course will start with a review of basic algebra (factoring, solving linear equations, and equalities, etc.) and proceed to a study of polynomial, exponential, logarithmic, and trigonometric functions. These functions will be used in applications involving simple mathematical modeling where students will engage in inquiry activities aimed at improving critical thinking skills. title following page Modeling Unemployment Rates Data Denise A. Carter Department of Mathematics, Engineering, and Computer Science Objectives To demonstrate to students that math is a part of their daily lives and that the concepts learned in class are relevant to the real world. This lesson will show that actual data taken from January to September 2008 unemployment rates are approximately linear. Linear functions are encountered from Lesson 2 to Lesson 13 in the syllabus of College Algebra and Trigonometry (MAT115). Through this activity, students are expected to achieve the following: To understand the concept of a function and its role in mathematical modeling of real-life problems To know how to recognize a linear function To be able to graph a linear function To be able to write the equation of a linear function To solve textbook applications of linear functions To solve systems of linear equations in two variables by using the graphical, elimination, and substitution methods To learn the MAPLE commands for this activity To read articles about unemployment rates Math Topics Graphing linear equations, equation of a line, applications of linear functions Purpose Synthesis Comments Based on unemployment rates data and news coverage in the Washington Post and ABC News When to Introduce Week 10 (after introducing measurements and plotting graphs) Duration Week 3 Activity Time Frame From a week to a week-and-a-half Reflection I am using economic current events that are pertinent in students' lives to show the relevance and importance of math. It is anticipated that students will become more aware about their economic predicament and take a more active and aggressive role in their daily lives. It is also expected that they will be able to translate the concepts to similar situations. The students at first were somewhat intimidated by the project because it was the first major project and they did not know what to expect. But they later realized that all the preparation they received from classwork, homework, and MAPLE labs was pertinent to the assignment.From the students' comments concerning the project, they seemed to have gained more respect for math because they were able to see the learned concepts used in real situations. Because this project was meant for ePortfolio, I expected students to demonstrate their abilities to read, analyze, think critically, and communicate effectively. I have learned from my experience that this seemingly daunting task makes many students anxious. In the future, I will introduce reading material earlier, and prepare mini practice projects to make a smoother transition into the first major project. Project Quantum Leap SAMPLER 143 Modeling Unemployment Rates Data Activity Overview This project should be introduced after Lesson 13 and after the MAPLE commands are learned. This will be about three-and-a-half weeks into a twelve-week semester. The students are given a week to a week-and-a-half to complete the project. 1. After Lesson 13, distribute \"Modeling Unemployment Rates Data\" Project Handout. Direct students to (i) read the Washington Post and ABC News articles, and (ii) listen to Dan Arnell's report on the ABC News website. 2. Ask students to follow the instructions in the handout to (i) write a brief review of the Washington Post article and (ii) indicate what the hardest hit industries were, based on the ABC News report. 3. In a computer lab, ask students to use MAPLE to analyze the unemployment rates data for the months January through September 2008 provided in the handout. The instructor should encourage collaborative learning, and offer necessary assistance. 4. As homework, ask students to verify the numerical results, and write a reflection based on their quantitative analysis. 5. Collect the complete project, and ask students to deposit the electronic version on ePortfolio. Materials and Resources Background information on the unemployment situation is given in the following articles: 144 Handout Reading #1: Irwin, N., & Rosenwald, M.S. (2008, October 23). Job losses accelerate, signaling deeper distress. Washington Post. Retrieved January 30, 2009, from http://www.washingtonpost.com/wp-dyn/content/article/2008/10/22/AR2008102203709_pf.html Reading #2: Unemployment at 5 year high as U.S. loses 84,000 jobs. ABCNews.com Retrieved January 29, 2009, from http://abcnews.go.com/Business/Economy/ Story?id=5733848&page=1 (Listen to Dan Arnell's report on 84,000 jobs slashed in August 2008.) Unemployment data percentage rates can be obtained from the U.S. Misery Index (2008, November 9). Retrieved January 29, 2009, from http://www.miseryindex.us. PowerPoint presentations were used when teaching functions in general, linear functions, and solving systems of linear functions. Handouts were also used to reinforce topics. Please contact Dr. Denise A. Carter for these materials. LaGuardia Community College/CUNY Modeling Unemployment Rates Data Handout: Modeling Unemployment Rates Data Carter | Algebra and Trigonometry - Business and Finance (MAT115) Due: _________________ Late papers will lose 10 points. Objective To enhance the understanding of linear functions through the use of MAPLE and current newspaper articles on unemployment in the United States. Directions Read the following articles: (i) Irwin, N., & Rosenwald, M.S. (2008, October 23). Job losses accelerate, signaling deeper distress attached to this handout and (ii) the ABC News Unemployment at 5-year high as U.S. loses 84,000 jobs. Listen to Dan Arnell's report on 84,000 jobs slashed at http:// abcnews.go.com/Business/Economy/Story?id=5733848&page=1. Use MAPLE to solve the following 1. Write a brief review of the Irwin & Rosenwald article. [25 points] 2. What industries does the ABC news article state were hit the hardest this time around? [15 points] 3. The data in the table below gives the unemployment rates for the months January through September 2008 (obtained from the U.S. Misery Index website: http://www.miseryindex.us). [January = 1, February = 2, etc.] Month Unemployment Percentage Rate 1 4.9 2 4.8 3 5.1 4 5.0 5 5.5 6 5.5 7 5.7 8 6.1 9 6.1 Make a scatter plot of the data. [20 points] 4. Find and graph the regression line that models the data above. Interpret the slope with respect to the unemployment percentage rates. [20 points] 5. Assuming that this trend continues, use the model found in Question 3: a. To predict the unemployment rate in July 2009. [8 points] b. To find when the unemployment rate will be 10%. [12 points] Project Quantum Leap SAMPLER 145 Modeling Unemployment Rates Data 6. Based on your analysis, write a reflection essay. Consider the following guiding questions: a. If the unemployment situation continues, what will the unemployment rate be at the time you graduate from college? b. What do you think the government should do to stop the job loss trend? c. How do you remain competitive in this tough job market? 146 LaGuardia Community College/CUNY Modeling Unemployment Rates Data Reading #1: Job Losses Accelerate Carter | Algebra and Trigonometry - Business and Finance (MAT115) Source: Irwin, N., & Rosenwald, M.S. (2008, October 23). Job losses accelerate, signaling deeper distress. Retrieved January 30, 2009, from http://www.washingtonpost.com/wp-dyn/content/article/2008/10/22/AR2008102203709_pf.html Employers are moving to aggressively cut jobs and reduce costs in the face of the nation's economic crisis, preparing for what many fear will be a long and painful recession. The labor market has been weak all year, with a slow drip of workers losing their jobs each month. But the deterioration of the job market is now emerging as a driver of economic distress, according to a wide range of data and anecdotal reports from corporate America. In September, there were more mass layoffs - instances in which employers slashed 50 or more jobs at one time - than in any month since September 2001, the Labor Department said yesterday. And nearly half a million Americans have filed new claims for unemployment benefits in each of the past four weeks, the highest rate of such claims since just after the terrorist attacks seven years ago. Anecdotal reports suggest that the hemorrhaging in the job market has only begun. Companies that announced plans this week to cut jobs include Internet company Yahoo (1,500 positions), pharmaceutical company Merck (7,200), National City bank (4,000) and Comcast, the cable company (300). The weakening employment outlook is part of the reason that investors have become more fearful of a deep, prolonged recession - fears that led to yet another miserable day on Wall Street yesterday, with the Dow Jones industrial average down 514 points, or 5.7 percent. \"The customers I've spoken to are all living under a sense of fear,\" said Paul Villella, chief executive of HireStrategy, a Reston company that matches employers and workers. \"They have very limited visibility into the future and have a great degree of uncertainty, so they just want to sit steady and be conservative in hiring.\" Villella and others who work with employers said that for many companies, the pullback in hiring is not a direct result of tightening credit. Rather, firms simply don't know whether their own customers will be affected by the financial crisis; as a result, they want to hold their breath and delay hiring decisions until they have a better sense of the future. The nation has shed jobs every month this year, but at a slower overall pace than in past economic downturns. The slide accelerated in late summer, with declines similar to those in past recessions. Last month, employers shed 159,000 jobs, the most this year and more than the average number of monthly job losses in the terrible labor markets of 2001 and 2002. More obscure indicators monitored by economists at the Federal Reserve and in the private sector also show an inflection point in late summer. For example, employers had 214,000 fewer job openings in August than in July, according to a Labor Department report. Over the past year, the number of openings dropped by a more modest average of 74,000 per month. Indeed, many companies are imposing hiring freezes. Such moves don't often get the kind of headlines that layoffs do, but because they shrink the number of places people can turn to for jobs, they still hurt the economy. Project Quantum Leap SAMPLER 147 Modeling Unemployment Rates Data VMware, a Palo Alto, Calif., software company, is one firm that has curbed hiring. Earlier this week, after reporting third-quarter earnings that beat Wall Street's expectations, VMware told analysts on a conference call that despite a 32 percent jump in revenue, a \"hiring pause\" had been imposed for all jobs except critical ones. \"We are just being conservative,\" VMware spokeswoman Mary Ann Gallo said yesterday. The nation's unemployment rate was 6.1 percent last month, not astronomical by historical standards. But the rate was up from 5 percent in April, and many forecasters now expect it to hit 7 percent or more by the end of this downturn. The construction and manufacturing sectors have been losing jobs for more than a year. But lately, job losses have begun or accelerated in a wide range of other fields. Retailers, stung by less consumer spending, cut 87,000 jobs in the three months ended in September. Employment services shed 100,000 positions in that span, reflecting the fact that companies are slashing temporary jobs. The leisure and hospitality industry cut 51,000 jobs, as people had less money to stay in hotels and eat in restaurants. In the greater Los Angeles area, Manpower, one of the nation's largest temp agencies, has noticed a steady increase in job seekers since early September. Paul Holley, a spokesman for the company, said there are more applicants for fewer openings and better-qualified candidates seeking work. What's particularly noteworthy, Holley said, is what's happening in Phoenix. Job applications have held steady, but since September more applicants have had backgrounds in general labor and warehouse distribution. That's unusual because warehouse and logistics jobs usually hold steady in the fall to support retailing for holiday shopping. Randstad USA, another large temp agency, reports that job applications are up in the Tucson area and that the firm is even getting inquires from people who still have jobs. \"In general, a lot of people seem to be insecure about their current jobs even if they are still employed,\" said Emily Cline, Randstad's area vice president for Tucson. As reports of layoffs continue to pile up around the country, executives at Randstad said they have noticed a shift in psychology among job seekers. \"Employees are much more willing to work extra hours and to take on additional duties to enhance job 148 LaGuardia Community College/CUNY security and improve their employability,\" said Eric Buntin, managing director for marketing and operations at Randstad. \"In a changing market, they know that's a valuable resource.\" They are also willing to make less money, even as the cost of living goes up. Cline said some call center jobs that were paying $9 an hour in the Tucson area last year are now paying $8.50. \"Their option becomes to take the job or not have the job,\" she said. With workers losing their leverage to negotiate raises, there could be greater downward pressure on wages, which in turn could drive down overall economic growth. Workers are already having a hard time getting raises; inflation-adjusted pay for non-managerial workers fell 1.9 percent in the year ended in September, according to the Labor Department. Staff writer Michael A. Fletcher in Cleveland contributed to this report. Modeling Unemployment Rates Data Reading #2: Unemployment at 5-Year High Carter | Algebra and Trigonometry - Business and Finance (MAT115) Source: Irwin, N., & Rosenwald, M.S. (2008, October 23). Job losses accelerate, signaling deeper distress. Retrieved January 30, 2009, from http://www.washingtonpost.com/wp-dyn/content/ article/2008/10/22/AR2008102203709_pf.html Unemployment at 5-Year High as U.S. Loses 84,000 Jobs The Job Losses Were Higher Than Economists Expected Sept. 5, 2008 WASHINGTON (AP) -- The U.S.'s unemployment rate zoomed to a five-year high of 6.1 percent in August as employers slashed 84,000 jobs, dramatic proof of the mounting damage a deeply troubled economy is inflicting on workers and businesses alike. The Labor Department's report, released Friday, showed the increasing toll the housing, credit and financial crises are taking on the economy. How is the Economy Treating You? Tell ABC News The report was sure to rattle Wall Street again. All the major stock indexes tumbled into bear territory Thursday as investors lost hope of a late-year recovery. With the employment situation deteriorating, there's growing worry that consumers will recoil, throwing the economy into a tailspin later this year or early next year. The jobless rate jumped to 6.1 percent in August, from 5.7 percent in July. And, employers cut payrolls for the eighth month in a row. Job losses in June and July turned out to be much deeper. The economy lost a whopping 100,000 jobs in June and another 60,000 in July, according to revised figures. Previously, the government reported job losses at 51,000 in each of those months. The latest snapshot was worse than economists were forecasting. They were predicting payrolls would drop by around 75,000 in August and the jobless rate to tick up a notch, to 5.8 percent. The grim news comes as the race for the White House kicks into high gear. The economy's troubles are Americans' top worry. Wachovia Corp., Ford Motor Co., Tyson Foods Inc. and Alcoa Inc. were among the companies announcing job cuts in August. GMAC Financial Services this week said it would lay off 5,000 workers. Job losses in August were widespread. Factories cut 61,000 jobs, construction firms eliminated 8,000 jobs, retailers axed 20,000 slots, professional and business services slashed 53,000 positions and leisure and hospitality got rid of 4,000. Those losses swamped employment gains in the government, education and health. Job losses at all private employers -- not including government -- came to 101,000 in August. The government said workers age 25 and older accounted for all the increase in unemployment in August. Workers saw wage gains in August, however. \u0003Average hourly earning rose to $18.14 in August, a 0.4 percent increase from July. Economists were forecasting a 0.3 percent gain. Over the past year, wages have grown 3.6 percent, but paychecks aren't stretching as far because of high food and energy prices. Caught between dueling concerns of slow growth and inflation, the Fed is expected to leave a key interest rate alone at 2 percent when it meets next on Sept. 16 and probably through the rest of this year. Concerned about inflation, the Fed at its last two meetings didn't budge the rate. Before that, though, the Fed had aggressively cut rates to shore up the economy. With the Fed on the sidelines, Democratic presidential nominee Barack Obama has called for a second round of government stimulus, while his Republican rival John McCain has favored free-trade and other business measures to spur the economy. Copyright 2009 ABC News Internet Ventures Project Quantum Leap SAMPLER 149 Car Buyers in Asia Gordon Crandall Department of Mathematics, Engineering, and Computer Science Objectives Students will read an article that describes in \"grassroots\" terms the arrival of new car dealers in rural China and the consequences of vehicle ownership where scant years before, the bicycle provided the sole means of local transportation. Apart from sociological \"color,\" data is given on the explosive growth of vehicle sales in China and in India in the past several years, along with the (slight) decline in vehicle sales in the United States. Students will model this data using exponential, quadratic, and linear models, and they will make predictions on sales volume in future years. Coming later in the semester - at the point of introduction of exponential functions - the assignment reviews linear and quadratic functions, and serves as a capstone activity on the types of growth over time featured in the course. Math Topics Linear, quadratic, exponential modeling Purpose Review, synthesis Comments Based on New York Times article describing bittersweet arrival of car dealerships in rural China When to Introduce Week 10 Activity Time Frame Two weeks Reflection Discussion of the dynamic surge in Chinese and Indian car sales could lead naturally to cautionary tales of increased competition for finite petro-resources and increasing emissions of greenhouse gases. With prior exposure to curve fitting/modeling, students found this activity engaging. Making the graphs by hand or with Excel is visually appealing and students enjoyed creating the final product. These calculations were more natural \"live\" versions of the drier Educo textbook problems. However, quantitative issues are not really the main thrust of this article, although it does offer eyeopening data on the dynamic growth of automobile ownership in Asia. Instructors selecting this activity should not deprive themselves of class discussion on the impact of vehicle ownership on insulated traditional cultures. The article evokes powerfully the warm spring winds of entrepreneurial capitalism, followed - ironically - by the destructive 2008 summer earthquake in Hunan and the crash of world markets in the early fall of 2008. Activity Overview Week 10 (Period 1): Activity handout and \"cultural\" discussion (30 minutes) Week 10 (Period 2): Review data entry and regression modeling features for TI-83 or Excel (40 minutes) Week 11 (Period 1): Hints and straggler management (20 minutes) Week 11 (Period 2): Wrap-up (15 minutes) Project Quantum Leap SAMPLER 151 Car Buyers in Asia Materials and Resources 152 Handout Reading: Bradsher, K. (2008, April 24). With first car, a new life in China. New York Times. Retrieved January 30, 2009, from http://www.nytimes.com/2008/04/24/business/ worldbusiness/24firstcar.html TI-83 graphing calculator (or equivalent modeling tool, such as Excel) LaGuardia Community College/CUNY Car Buyers in Asia Handout: Car Buyers in Asia Crandall | Algebra and Trigonometry - Business and Finance (MAT115) Due: _________________ Late papers will lose 10 points. Car Buyers in Asia and in the U.S. 1. The table below gives data on passenger vehicle sales (cars, minivans, and sport utility vehicles in China. Year Vehicles sold (in millions) 2000 2001 2002 2003 2004 2005 2006 2007 2008 .6 .8 1.2 2.2 2.4 3.1 4.2 5.3 6.1 a. Taking the year 2000 as t=0, make a scatter plot of the data using an Excel spreadsheet or a piece of graph paper. b. Using Excel (or a TI-83 graphing calculator) enter years since 2000 as x-values and vehicle sales in millions as y-values, and fit the data to an exponential equation. c. Using the equation in b), predict vehicle sales volume in China in 2015. What about 2030? d. In what year will sales volume reach 12 million vehicles? 2. The table below gives similar data on vehicle sales in India. Year 2003 2004 2005 2006 2007 2008 0.8 1.0 1.2 1.4 1.6 2.0 Vehicles sold (in millions) a. Taking the year 2003 as t=0, make a scatter plot of the data using an Excel spreadsheet or a piece of graph paper. b. Using Excel (or a TI-83 calculator), enter years since 2003 as x-values and vehicle sales in millions as y-values, and fit the data to a quadratic best-fit model. c. Using your equation in b), predict the sales volume in India in 2015. What about 2030? d. In what year will sales volume in India reach 12 million vehicles? 3. Year Vehicles sold (in millions) 2000 2001 2002 2003 2004 2005 2006 2007 2008 13.8 13.5 13.3 13.0 13.1 13.1 12.8 12.4 12.0 The table below gives data on vehicle sales in the United States. a. Taking the year 2000 as t=0, make a scatter plot of the data using an Excel spreadsheet or a piece of graph paper. b. Using Excel (or a TI-83 graphing calculator), enter years since 2000 as x-values and vehicle sales in millions in the U.S. as y-values; find the equation of the best line through the data. Project Quantum Leap SAMPLER 153 Car Buyers in Asia c. Using the linear equation that you found in b), predict the sales volume in 2015. What about 2030? 4. Assuming that vehicle sales in India continue to grow quadratically, and that vehicle sales in the United States continue to decrease linearly, in what year will the sales volume in these two countries be equal? What is the vehicle sales volume at this time? Do you think that this will ever happen? What assumptions must be questioned? Explain. 5. According to the article, \"With First Car, a New Life in China,\" are Chinese domestic carmakers like Geely and Chery the main beneficiaries of the explosive growth in vehicle sales in China? Explain your answer. 6. Based on the reading, write a brief essay discussing how the status of automobile ownership is beginning to reconfigure traditional folkways in rural China. 7. Discuss the environmental consequences of the explosive growth of car sales in countries like India and China in terms of emissions/greenhouse gases and in terms of competition for (ultimately finite) energy resources. 154 LaGuardia Community College/CUNY Car Buyers in Asia Reading: Car Buyers in Asia Crandall | Algebra and Trigonometry - Business and Finance (MAT115) Source: Bradsher, K. (2008, April 24). With first car, a new life in China. New York Times. Retrieved January 30, 2009, from http://www.nytimes.com/2008/04/24/business/worldbusiness/24firstcar. html. SHUANG MIAO, China - Li Rifu packed a lot of emotional freight into his first car. Mr. Li, a 46-year-old farmer and watch repairman, and his wife secretly hoped a car would better the odds of their sons, then 22 and 24, finding girlfriends, marrying and producing grandchildren. A year and a half later, the plan seems to be working. After Mr. Li purchased his Geely King Kong for the equivalent of $9,000, both sons quickly found girlfriends. His older son has already married, after a short courtship that included a lot of cruising in the family car, where the couple stole their first furtive kisses. \"It's more enclosed, more clandestine,\" said Li Fengyang, Mr. Li's elder son, during a recent family dinner, as his bride blushed deeply. Western attention to China's growing appetite for automobiles usually focuses on its link to mounting dependence on foreign oil, escalating demand on natural resources like iron ore, and increasing emissions of global warming gases. But millions of Chinese families, like millions of American families, do not make those connections. For them, a car is something both simpler and more complicated. J. D. Power and Associates calculates that four-fifths of all new cars sold in China are bought by people who have never bought a car before - not even a used car. That number has remained at that level for each of the last four years. By contrast, less than a tenth of new cars in the United States are purchased by people who have never bought a new car before, and less than 1 percent of all new cars are sold to people who have never bought a new or used care before. China's explosive growth in first-time buyers is the driving force behind the country's record car sales, up more than eight-fold since 2000. It is the reason China just passed Japan to become the world's second-largest car market, behind the United States. One change in Chinese attitudes is already clear and likely to have broad implications worldwide: even firsttime buyers are becoming more sophisticated and want better cars. China's domestic carmakers like Geely and Chery, once feared by Detroit and European automakers as eventual exporters to Western markets, have watched their sales gain modestly, stagnate or drop in the last year - even while the overall Chinese market has continued to grow roughly 20 percent a year. The beneficiaries have been the joint ventures of multinationals that sell cars here that are designed overseas, like the Buick Excelle, Volkswagen Jetta and Toyota Camry. Practically every auto expert had expected the multinationals to lose market share rapidly to low-cost domestic automakers. Instead, Chinese car buyers, including first-time buyers, have become more discriminating about the comfort, styling and reliability of the cars they buy. As a result, instead of planning to conquer overseas markets, local manufacturers are having to redouble their efforts in this market. \"Customers are moving up, they want the bigger, more established brands,\" said Michael Dunne, the managing director for China at J. D. Power. \"They'd rather wait, save and buy higher on the ladder instead of buying a smaller car.\" Back in the fall of 2006, the Li family did not want to wait, especially Mr. Li. When the Li family bought their car, they agreed to extensive interviews with each family member in Shuang Miao, a rural village in east-central China's Zhejiang province. They later agreed to follow-up telephone interviews over the last year and a half and then a long family dinner in Shuang Miao last week to review their experience as first-time car owners. What emerges is a portrait of the rapidly expanding role of cars in the fast-changing ways in which China's people socialize, marry, raise families and, possibly, die. Li Rifu was so excited on the day that he bought his first car in September, 2006, that he woke before dawn. He fixed breakfast for his wife and two grown sons, then climbed on his white motorcycle for a short trip he had been anticipating for many years. Mr. Li had spent most of his life here in his ancestral farm village, nestled at the base of a steep hill. The embodiment of China's version of the American dream, he is largely self-taught. He learned to fix watches, and got a job as a foreman in a coal mine in nearby Anhui province by fixing the mine owner's watch. After saving some money, he came home to start a successful busiProject Quantum Leap SAMPLER 155 Car Buyers in Asia ness that now employs five peasants raising flowers for landscapers. That September morning, Mr. Li rode down the dirt alleys of his village and over a muddy, bamboo-lined stream where local women washed clothing on rocks jutting out into the sluggish current. He reached a four-lane paved road, then a six-lane road, and puttered on to his destination in the nearby city of Taizhou: a car dealership. Over the course of the half-hour journey, Mr. Li was too excited to heed the persistent and unexplained pain at the base of his back. He had really wanted a black car. But his sons preferred white, saying that it was a more popular choice for their generation, and Mr. Li had given in before he ever set out for the dealership. \"Without this car, my two sons wouldn't be able to find wives - the girls would not marry them,\" he said, reminiscing how when he courted his wife in the early 1980s, he only needed a bicycle. He ruined a half-dozen bicycle tires carrying her on the back of the bicycle for their outings together. Mr. Li took a white Geely King Kong compact sedan for a short test drive, then returned to the dealership and climbed three flights of stairs to a cashier's office. He pulled a stack of currency thicker than a brick out of a black shoulder bag and paid the equivalent of $9,000 for the car; he would later pay another $1,000 in fees for a license plate. \"The next few days, everyone will want to drive it,\" he said proudly, a prediction that proved true. Mr. Li talked of his dream of someday driving across China to visit Beijing and Tibet, while acknowledging he would need more driver's education classes before those dayslong journeys would be possible. Car ownership helped Mr. Li bid for bigger contracts for more flowers. \"My customers said, 'Wow, you came to visit me in a car' - it puts the negotiation on a whole different level,\" he said. Several months after he bought the car, Mr. Li's elder son, Fengyang, did indeed find a girlfriend, Jin Ya, a beautiful young saleswoman for China Mobile, a cellphone service. In the space of five months, they had gone to the local marriage registry and been legally wed. Today, both say they want a child someday. At the family dinner this week, Ms. Jin bridled at the idea that young women in China only consider a man to be marriage material if he can take them on dates in a car. \"Not me, not me!\" she said passionately, before reluctantly acknowledging that, \"Other girls do say that you need a car.\" 156 LaGuardia Community College/CUNY But as their Geely King Kong was bringing the Li family new joy - Mr. Li's increased business, Fengyang and Ya's courtship - tragedy struck: Li Rifu and his wife, Chen Yanfe, were each diagnosed with cancer. Ms. Chen's reproductive tract cancer has gone into remission after $7,000 in medical bills. But Mr. Li's fistsized malignant prostate cancer tumor - which turned out to be the cause of the mysterious back pain that was bothering him when he first bought the car - has resisted two surgeries and four rounds of chemotherapy. The cost: more than $40,000. With payments from the local health insurance fund capped at $4,300 a person per year, Mr. Li has had to sell many of his possessions, and still he has had to go into debt. He wore a cap to the family dinner this week, sensitive about the loss of his hair due to the chemotherapy. In two weeks, he will go to a leading hospital in Shanghai for more surgery, a five-hour drive to the north, followed by two more rounds of chemotherapy. But he will not be going in the family car: he sold it for nearly $8,000 last year to help cover his medical expenses, and cannot afford another one right now. It is a common occurrence in this country, nominally communist, but with little or no safety net. While many families are scrambling into the middle class and buying cars, others are falling out of the middle class because of business reversals, medical bills or other problems, and are unable to buy replacements for their first car. Zhu Jinyung, a machinery repairman who lives close to Shuang Miao, said that his family had bought a cheap, domestic car in 1994 after enjoying initial success in the plastic injection molding business. \"The business didn't work out,\" and the car had to be sold, he said. Sadly, the Li family has known new tragedy recently. Their younger son, Fengwei, had also found a girlfriend with the help of the family car, the daughter of a manager at a large factory, an impressive person to Li Rifu. But the girlfriend's father was killed two weeks ago when a construction crane at the factory accidentally dropped its load on him after a crucial steel pin broke. Despite it all, Li Rifu tries to remain optimistic. He now dreams of regaining his health, earning back the money he has spent on medical care and then - like a growing number of his countrymen - buying a bigger, more impressive car than the Geely compact he had to sell. \"If I get another car,\" he said, \"I'll get a better-quality car, with even nicer seats and better steering.\" Price Per Square Foot Ladji Doumbia Department of Mathematics, Engineering, and Computer Science Objectives Students will read articles about real estate pricing from Real Estate Libraries and Wikipedia in order to understand the concept of price per square foot of floor area. Then they will analyze data from the National Association of Home Builders (NAHB) about the average price per square foot of floor area of a one-family house in the USA from 1992 to 2006 using an Excel spreadsheet. Math skills relevant to this activity include (1) locating and graphing points, (2) finding the slope of a line, (3) graphing and solving linear equations, and (4) finding the equation of the line of best fit through a scatter plot. Furthermore, students will use that \"regression line\" to predict price trends in the future. Math Topics Linear functions Purpose Review, synthesis Comments Students need to know Excel When to Introduce Week 3 Activity Time Frame Three weeks Reflection Many students plan to buy a house some day. To compare prices in different markets, locations, and months, the real estate industry uses either the median home price, the mean (average) home price, or the price per square foot. The price per square foot of a house is obtained by dividing the median or average price of that house by its area. According to the real estate industry, the best tool to compare homes with different prices as well as homes of different sizes is the price per square foot of floor. Therefore, this activity focuses on the trends in price per square foot over time. After reading the background article, students are asked to analyze and interpret trends in the value of a one-family home based on price per square foot of floor. Students first study the broad shape of the data, then translate specific terms (slope, linear equation) in the handout questions (where units involved were price per square foot and time in years) into mathematical language, and relate these mathematical objects to concepts that they have learned in class. In the first iteration of this activity, most students could relate questions to concepts via the language of math. Oddly, they did not use the TI83 graphing calculator to do the project, and, when asked why, they said that they could not get a printout of the graph on the calculator screen. Instead, they used Excel to conduct different parts of the project. Students were also asked to write a short essay about whether the price per square foot was always the best way of determining property values. Most of them submitted sound arguments to support their views. After discussion with many colleagues, I improved the activity by asking students to give the price or rent per square foot of their own house or apartment and compare those to the average price per square foot in the northeast U.S. Project Quantum Leap SAMPLER 157 Price Per Square Foot Activity Overview This activity should be introduced in the third week, at the end of the chapter on linear functions. The time breakdown should be: Week 3 1.5 hour class time: Distribute the project handout and reading: \"Price per Square Foot??\" 30 minutes class time: Discussion and examples related to buying a house: Median price of a house Mean (average) price of a house Price per square foot of a house Rent per square foot of a house 1 hour class time: Teach students how to use Excel to: make a scatter plot, find coefficients of a linear regression, get determination or correlation coefficients, and plot the regression line. Week 4 (30 minutes class time): Continue activities undertaken in the previous week. Review key concepts about lines/linear equations. Week 5 (30 minutes class time): Wind up and help students who are still struggling with the project. Materials and Resources 158 Handout Reading #1: Blank, S. Price per square foot. Real Estate Libraries. Retrieved January 30, 2009, from http://www.relibraries.com/pricePerSqft.aspx Reading #2: Wikipedia. Real estate pricing. (2009, January 23). Retrieved January 30, 2009, from http://en.wikipedia.org/wiki/Real_estate_pricing. National Association of Home Builders. (2008, August 7). Price per square foot by location. Retrieved February 10, 2009, from http://www.nahb.org/search_simple.aspx?txtkeyword=pri ce+per+square+foot&x=0&y=0 LaGuardia Community College/CUNY Price Per Square Foot Handout: Price Per Square Foot Doumbia | Algebra and Trigonometry - Business and Finance (MAT115) Average price per square foot of floor area in new one-family houses sold. Price per square foot of floor area excludes value of improved lot Year Price per square foot 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 $55.18 $57.41 $60.21 $62.45 $64.38 $66.81 $67.83 $67.70 $70.43 $71.93 $75.68 $79.21 $85.13 $90.63 $91.99 Source: The National Association of Home Builders 1. To simplify years, take t = 0 for year 1992; take t = 1 for year 1993; take t = 2 for year 1994; take t = 3 for year 1995; take t = 14 for year 2006. a. What is the relationship between t and year? b. Treating time t as input and price P as output, enter the data in an Excel spreadsheet to make a scatter plot. 2. a. What is the slope m1 of the line passing through the points (0, 55.18) and (5,66.81)? b. Interpret the slope m1 c. What is the P-intercept? d. What is the equation of the line passing through the points (0, 55.18) and (5, 66.81)? 3. a. What is the slope m2 of the line passing through the points (10, 75.68) and (13,90.63)? b. Interpret the slope m2. c. What is the equation of the line passing through the points (10, 75.68) and (13,90.63)? 4. Instead of working on portions of the scatter plots, we want to approximate the data by a line we can use to predict prices in the future. The line that will best fit the given data is called the linear regression. The approximation will be acceptable whether the determination coefficient R (r**2) is close to 1 or the correlation coefficient r is close to -1 or +1. a. Use your entered data to find the equation y = ax + b of the best line through the data points and give the values of coefficients a and b, and of the determination coefficient R or correlation coefficient r. b. Use Excel to plot the scatter plot and the regression line on the same graph. c. Is the linear regression a visually good approximation of the data? Sustain your argument by a mathematical proof. d. Use the linear regression to predict the price per square foot of floor area in 2009. e. In what year will the price per square foot be greater than $100? Project Quantum Leap SAMPLER 159 Price Per Square Foot 5. If your family owns a house or an apartment: a. Calculate, to the nearest foot, the floor area of your home. b. Knowing the buying price of your home, find the price per square foot of floor area of your house. c. Comparing the price per square foot of your home to those of your classmates, what borough offers the cheapest price per square foot and why? d. Should you buy a home in the New York area, or in another part of the northeast? Explain your reasoning. 6. If you are renting a house or an apartment: a. Calculate, to the nearest foot, the floor area of your home. b. Knowing the monthly rent of your house, find the rent per square foot of floor area of your home. c. Comparing your rent per square foot to those of your classmates, what borough offers the cheapest rent per square foot and why? 7. Based on your reading of the article, \"Price Per Square Foot??\" and the Wikipedia article on real estate pricing, write a short essay on whether price per square foot is always the best way to determine property value. Give cogent reasons to support your argument. References 160 National Association of Home Builders. (2008, August 7). Price per square foot by location. Retrieved February 10, 2009, from http://www.nahb.org/search_simple.aspx?txtkeyword=pri ce+per+square+foot&x=0&y=0 LaGuardia Community College/CUNY Price Per Square Foot Reading #1: Price Per Square Foot Doumbia | Algebra and Trigonometry - Business and Finance (MAT115) Source: Blank, S. Price per square foot. Real Estate Libraries. Retrieved January 30, 2009, from http://www.relibraries.com/pricePerSqft.aspx. It's probably time someone addresses a widely used term that is loosely defined and almost never qualified: PRICE PER SQUARE FOOT! These four words have developed into a term almost everyone loves to toss into a conversation when talking about real estate as it relates to a city neighborhood. I rarely hear it used when describing a suburban neighborhood where the price per square foot factor would actually make some sense. Say you have a two-story home (let's call it the Belair model) that contains 2,000 square feet. Let's assume they were all built on a similar sized lot, have two car garages and full, open basements. Let's also say that the neighborhood is about ten years old. Now, if that Belair, or similar two-story home, sold for $200,000, it easily calculates out to $100 per square foot. You can make simple dollar adjustments, higher or lower, for things like central air, basement finishing, decor, etc. But, the bottom line is that the Belair model is selling for about $100 per square foot compared to other two-story homes in that neighborhood. Even ranch or tri-level styled homes will typically sell at a different price level in the same neighborhood. Now, let's talk about PRICE PER SQUARE FOOT and how that term means virtually nothing in most of our city neighborhoods. You can choose almost any older neighborhood to illustrate the extreme, and yet wonderful, blend of value differences in neighborhoods such as Hilltop, Washington Park, Country Club, Cherry Creek, Park Hill, Lodo, etc. The obvious considerations when identifying different factors should include but, not be limited to, location, architectural styles, traffic patterns, newer construction vs. 50-100 years old, quality of construction, lot sizes, landscaping, updates and remodels, garages (1, 2 or 3/attached or detached), basements (finished/not finished) floor plans and room sizes, curb appeal and maintenance, and did I mention location (even in the same neighborhood or down the street). These factors can make tremendous differences in the value. The concept of price per square foot should only be used as a minor gauge in determining the value of a property. Let's pretend you're looking at two bungalows in Washington Park. They were both built in the 1920's and let's presume they're on the same block and are both 1,200 sq. ft. However, one is listed for $195,000 ($162.50 per sq. ft.) and the other is $285,000 (or $237.50 per sq. ft.). Initially, you would probably jump all over the $195,000 property. But, the better the value for you could very possibly be the more expensive home. No, I'm not nuts! Washington Park has plenty of examples of like-kind homes selling for $225 to $250 a foot. So the higher priced home is not a freak of nature. It likely has been completely renovated (new and upgraded electrical, plumbing, roof, new kitchen and baths, decor finishes, etc.) Maybe it includes a wonderfully finished basement, is on a larger, more well landscaped lot, has a better garage arrangement, is surrounded by larger more expensive homes, and various other factors. Unless you are willing, able and have the time and patients to make the improvements to the $195,000 home, the more expensive one could be the better value. Many people might feel that a condo or a loft would be easier to talk price per square foot when comparing similar sized homes in the same or nearby building. Well, it is a little easier, but don't get carried away there, either. Many of the same factors will definitely apply; however, there are other important elements to consider also. Remember location! Now add the location of the floor and the views. Does the building have a storage facility? What is the availability of parking, what kind of amenities are on the premises, and how big is the maintenance fee relative to the benefits? The long and the not-so-short of all this is; when you are shopping for you new home, work with a knowledgeable realtor and most importantly, choose the home that fits your needs the best and have it be the one that you can envision enjoying the most. And then, see how it stacks up to its current competition and possibly other like-kind sold properties. Just compare apples to apples! Project Quantum Leap SAMPLER 161 Price Per Square Foot Reading #2: Real Estate Pricing Doumbia | Algebra and Trigonometry - Business and Finance (MAT115) Wikipedia. Real estate pricing. (2009, January 23). Retrieved January 30, 2009, from http:// en.wikipedia.org/wiki/Real_estate_pricing. Real estate pricing deals with the valuation (finance) and there are three main methods: appraisals with comparable properties, capitalization rate comparisons with similar income producing properties, and discounted present value of expected future cash flows. After realty prices are estimated, recorded or otherwise reported, there remain two major ways in which aggregate home prices are reported: median and mean (average). Prices are also calculated by square foot, using both the mean and median price. Real estate prices have had a profound impact on urban, as well as the suburban and rural landscape. The most important government measurement of home prices in the United States is the house price index. Median house prices are reported for metro areas and regions of the country by the private National Association of Realtors. Median home price The median home price is the threshold which divides the real estate market into two equal halves, in reference to pricing. One half of all homes in the market were sold at a price above the median home price, while the other half were sold below that price. For example, the median home price in the United States was $213,900 in the fourth quarter of 2005, meaning that half of all homes sold in the US were priced above $213,900, and half were priced below $213,900. In California, the median home price was $548,000. The median home price is one of the most common measurements used to compare real estate prices in different markets, areas, and periods. It is said to be less biased than the average since it is not as heavily influenced by the top 2% of homes sold. For example, the average home sale price in the US was $264,000 in October 2005, compared with a median home price of $213,900 for the same time period. Mean (average) home price The mean home price, or average home price is the sum of prices of all homes sold in a certain area in a certain period, divided by the number of properties sold in the same area in that period. For example, say in a hypothetical townhouse complex there were five townhouses sold in March 2006. The properties were priced as follows: one for $450,000, one for $459,000, two for $465,000, 162 LaGuardia Community College/CUNY and one for $499,000. The sum of all of these properties is $2,338,000. This number is then divided by five, which equals $467,600. Thus, the mean home price for a townhouse in this complex was $467,600 in March 2006. Per square foot pricing In per square foot pricing, the total price or rent of a unit is divided by the area of the unit. Sales price per square foot Sometimes real estate prices are measured by the price of each square foot. This allows for a better comparison between differently priced homes as well as homes of different sizes. In this pricing measurement method, the median or mean price of a home is divided by its area. For example, a 1,243 sq ft (115.5 m2). home was for sale for $465,000. To find the per square foot price, the price of $465,000 is divided by the area of 1,243 sq ft (115.5 m2). The result, $374.09, is the price per square foot for this particular home. To effectively compare neighborhoods, the mean or median home price of a neighborhood is divided by the mean or median area. To refer to our example of Villawood Townhomes in Salinas, CA, the sold units in Willowood range from 1,243 sq ft (115.5 m2). to 1,621 sq ft (150.6 m2). The first four units are each 1,243 sq ft.; the fifth unit is a little larger, measuring 1,621 sq ft (150.6 m2). The area of all units combined is 6,593 sq ft (612.5 m2). This number is divided by five, the number of homes for sale, and the result, 1,318 sq ft., is the mean area. The mean price for the sold townhouses in Villawood of $467,600 is then divided by the mean area of 1,318 sq ft.; the result is a mean price of $354.78 per square foot. This also includes the lot the home is built on (which varies in size). In cities where condos, co-ops and to some extent townhomes and row houses are more prevalent, the prices do not include the land, since that is not owned by the indiviual but the entire association (though older town homes and row houses sometimes are not part of associations). The size of city lots are generally much smaller than those in the suburbs, measuring sometimes a mere 20 feet (6.1 m) wide for a 3 or 4 flat apartment building, with little to no land left for vegetation. Price Per Square Foot Rent per square foot External effects Rent is also calculated by square foot. The rent per square foot is often used as an effective tool in comparing units and different markets, when units are of different sizes. For example, in March 2006, a given 1,600 sq ft (150 m2). single-family home in Aptos, California charged a monthly rent of $1,800. In order to find the rent per square foot, the rent of $1,800 is divided by the area of 1,600 sq ft., which gives a rent per square foot of $1.12. Another given 1,140 sq ft (106 m2) single-family home in Aptos, CA had a rent of $1,595 per month. Again, the rent of $1,595 is divided by the area of 1,140 sq ft., which equals $1.40 per square foot. So, even though this home has a lower overall rent, its rent per square foot is actually higher than that of the first home. Real estate prices can have a profound impact on an area. They can cause urban decay, over crowding as well as an urban renaissance. Factors Influencing Real Estate Prices The variables that drive residential real estate prices can be grouped into macro forces and micro forces. Macro forces include mortgage interest rates, economic strength (the business cycle), demographics, and federal taxes. Micro forces include local economic strength, state and municipal zoning, neighborhood features (such as quality of schools), and the condition of the property itself. The biggest factor influencing home prices are income levels. Home prices are limited in how far they can rise by the incomes of potential buyers. Since eighty percent of all homes purchased are purchased with a mortgage, the ability to make payments, borrow money, and the cost of borrowing money are major influences limiting how far prices can rise before hitting resistance due to prices hitting levels where potential are unable to qualify. In general the ratio in the US are home values at 2-4 times income levels. Responsible lenders use what they refer to as \"frontend\" ratio which consists of income/mortgage payments to determine how much of a mortgage a person or family can qualify for and safely handle. This front-end ratio is generally a maximum of 28% of a borrower's monthly gross income that can be allocated toward housing expenses including the mortgage payment, insurance and property taxes. For example, a lender would allow a person with a $5000 monthly gross income to allocate $1400 per month for mortgage payments, fire insurance and property taxes. ($5000 28% = $1400) So with $1200 available for monthly mortgage payments, after subtracting $200 for insurance and taxes, $1200 would support a $200,000 mortgage. Decline In Harlem and many other inner city areas of large American cities, real estate prices dropped due to lower demand and the resulting surplus in supply. This trend was largely caused by suburban migration. Lower real estate prices had a negative effect on landlords, who in these declining neighborhoods became known as \"slumlords.\" Slumlords are notorious for not caring for the properties and letting them slip into a state of severe disrepair. Often, though, landlords in blighted urban areas did not have funds to care for their worthless and unprofitable properties. In many American slums, the situation became so dire that landlords were known to set their properties on fire in order to commit insurance fraud and collect the insurance policy payouts. Urban Renaissance In some areas, rising home prices have caused an urban renaissance. Rising home prices can transform unprofitable and decaying properties into potential gold mines. In many areas, landlords renovated their now-profitable properties, while other properties were bought by developers, who renovated them, seeing the neighborhoods' potential. Many downtown areas of large American cities have enjoyed a significant reawakening of interest in inner-city properties, which has rejuvenated these once blighted areas. Thanks in part to rising real estate prices, many inner cities have experienced a true urban renaissance. This can lead to gentrification. Project Quantum Leap SAMPLER 163 title following page The Life-Cycle Hypothesis Alioune Khoule Department of Mathematics, Engineering, and Computer Science Objectives The purpose of this project is to search for patterns, observe the parabolic shape of the plotted data, and use quadratic functions to model financial data. Students will review how to find the vertex of a quadratic function, and learn how to use a calculator, Excel or MAPLE to graph a function to fit a data set. Finally, students will be able to better understand the value of the quadratic function and how it enables us to predict the future, if such a model still holds. Reflection Math Topics Quadratic functions and scatter plots Purpose Review, synthesis Comments The abstract nature of math has created a lack of interest for some students, which is one of the reasons they don't strive to understand it. I have come to realize that a project based on quadratic functions is more engaging for the students simply because of the concrete resources available. While I have taught a similar unit before, the development of the PQL project was new to me. It looks so simple to find the vertex of a quadratic function, and graph it. But the biggest question that most students When to Introduce ask is how a quadratic equation can be applied to their Week 6 lives. To capture the students' attention on the day I Activity Time Frame introduced the project, I initiated a discussion with Two weeks them about what goals they have. All students came to the conclusion that education is the way to get a better job, so they can live a better life. The next question I asked them was how they could make such a lifestyle sustainable. During the time that we spent discussing the project in class, I came to realize that students were more engaged in learning about the quadratic function as it models income over a lifetime than when I was teaching the quadratic topic traditionally. The project is a self-contained unit that I will be able to use again and develop further with each use. Activity Overview As a class, students read the assigned questions and discuss what is necessary in order to complete the project (30 minutes ). Review quadratic functions/parabolic graphs (2 hours). Learn how to use Excel (or a graphing calculator or MAPLE) to fit a curve to data (1 hour). Research the life-cycle hypothesis beyond reading the article (1 week period). As a class, discuss the project output for about 30 minutes, including how to write a reflective essay on the project. Hold a 5 minute one-on-one meeting with each student the week before the due date to gauge their progress and make sure that they are on the right track. Project Quantum Leap SAMPLER 165 The Life-Cycle Hypothesis Materials and Resources Handout Reading - - 166 Schenk, R. The life-cycle hypothesis. Cyber Economics. Retrieved January 30, 2009, from http://ingrimayne.com/econ/FiscalDead/LifeCycle.html Jappelli, T. & Modigliani, F. (1998, November 1). The age-saving profile and the life-cycle hypothesis. Centre for Studies in Economics and Finance (CSEF) Working Papers. Retrieved January 30, 2009, from EconPapers: http://econpapers.repec.org/paper/sefcsefwp/09. htm U.S. Census Bureau. (2009, February 3). American community survey. Retrieved February 11, 2009, from http://www.census.gov/acs/www/index.html LaGuardia Community College/CUNY The Life-Cycle Hypothesis Handout: Life-Cycle Hypothesis Khoule | Algebra and Trigonometry - Business and Finance (MAT115) The following table shows the \"Median Income I\" of individuals of different age groups within the United States for 2001. For each age group, let the class midpoint represent the independent variable, x. For the age group \"65 years and older,\" we will assume that the class midpoint is 69.5. 2001 Age Class Midpoint, x Median Income, I 15-24 $9,301 25-34 $30,510 35-44 $38,340 45-54 $41,104 55-64 $35,637 65 and older 69.5 $19,688 Source: U.S. Census Bureau. (2009, February 3). American community survey. Retrieved February 11, 2009, from http://www.census.gov/acs/www/index.html 1. Find the class midpoint for each class age. 2. Draw a scatter diagram of the data with input as class midpoint and output as median income. 3. Is the relation between the two variables a function? If yes, what type of function is it and what is its domain? 4. Use a graphing utility or Excel to find the function of best fit. 5. Use this function to determine the age at which an individual can expect to earn the most income. 6. Use the function to predict the peak income earned. 7. Graph the quadratic function of best fit and compare it to the scatter diagram in 2. 8. Based on the reading attached to this project, write a reflective essay of at least 600 words. Project Quantum Leap SAMPLER 167 The Life-Cycle Hypothesis Reading: Life-Cycle Hypothesis Khoule | Algebra and Trigonometry - Business and Finance (MAT115) Sources: Schenk, R. The life-cycle hypothesis. Cyber Economics. Retrieved January 30, 2009, from http:// ingrimayne.com/econ/FiscalDead/LifeCycle.html Jappelli, T. & Modigliani, F. (1998, November 1). The age-saving profile and the life-cycle hypothesis. Centre for Studies in Economics and Finance (CSEF) Working Papers. Retrieved January 30, 2009, from EconPapers: http://econpapers.repec.org/paper/sefcsefwp/09.htm In examining why people spend the amount they do, a logical starting point is to ask what goals they have. Two goals seem reasonable for a great many people. First, they prefer a higher standard of living to a lower standard of living. In other words, people want the highest level of consumption spending they can get. Second, most people prefer to have a roughly constant standard of living through time. They do not like to live on a roller coaster, with one year of feast followed by a year of famine. Together, these two goals suggest that people try to maintain the highest, smoothest consumption path that they can get. Presented in this way, discussion of consumption behavior becomes a problem that the tools of microeconomics are designed to examine (with budget lines and indifference curves). Consider, for example, a person who will earn $10,000 this year and $5,000 next year. Ignoring all future years, how should he spend? The idea that he wants a smooth consumption path suggests that he should save part of this year's income and spend it next year. Alternatively, if the person earned the $5,000 this year and the $10,000 next year, the goal of a high smooth consumption path would require that he borrow in the first year and pay back in the second. An important function of financial markets from the point of view of consumers is that these markets help one maintain a constant standard of living despite fluctuations in income. The idea that people have fluctuations in income that they want to smooth is the basis of the life-cycle hypothesis of consumption. In a series of articles in the 1950s and 1960s, Franco Modigliani, Richard Brumberg, and Albert Ando explored why people save. They found that people generally live longer than they earn income - that is, people usually retire. If people are to keep spending after they no longer earn income, they must accumulate assets while they are earning so that they can dissave. (Few are willing to lend to those who have no prospect of future income.) 168 LaGuardia Community College/CUNY Suppose a 20 year-old person expects to live 50 years more, but only to work for 40 of those years. He expects to earn $20,000 each year. Ignoring interest, this person will have earnings of $800,000 to spread over 50 years. If he spends $16,000 a year, he will die with zero assets left. To get this spending pattern, he saves $4,000 each year while he works, and at retirement will have assets of $160,000. Now suppose that the person in the above example begins with assets of $200,000. He will then have a lifetime amount of $1,000,000 that can be spent. He will be able to spend $20,000 each year and die with zero assets. Thus the life-cycle hypothesis introduces wealth as a factor into the consumption function. Consumption can be financed either through income or through the sale of assets, and an increase in either should increase consumption. By looking at what a \"typical\" individual should do, the life-cycle hypothesis builds microeconomic footings for the consumption function. Behavior is goal-directed in the life-cycle hypothesis, while it is not in the original Keynesian consumption function. This latter consumption function is mechanical without a reason. Since economists prefer behavior that can be explained in terms of people pursuing goals, it is no surprise that the life-cycle hypothesis has become popular. The life-cycle hypothesis can be expanded to take into account uncertainty about when death will occur, the existence of social security, the interest rate, savings for bequests for heirs, and various patterns of lifetime earnings. It does not deal well with what should happen if incomes fluctuate erratically over time, but for this situation another theory, the permanent-income hypothesis, provides an answer. The permanent-income and life-cycle hypotheses are not contradictory theories, but theories that nicely complement each other (Schenk). The Life-Cycle Hypothesis The life-cycle hypothesis posits that saving is positive for young households and negative for the retired, so that wealth should be hump-shaped. Yet, if one looks at the microeconomic evidence on saving by age, dissaving by the elderly is limited or absent. But the saving measures usually computed on cross-sections or panel data are based on a concept of income that does not take into account the presence of pension arrangements. In fact, disposable income treats pension contributions as taxes, and pension benefits as transfers. But since contributions entitle the payer to receive a pension after retirement, contributions should be regarded as life-cycle saving and hence included back to income. Similarly, pension benefits accruing to the retired do not represent income produced, but a drawing from the pension wealth accumulated up to retirement (Jappelli and Modigliani). Project Quantum Leap SAMPLER 169 title following page Rising Gas Prices Crisis: How Fast and Why? Rudy Meangru Department of Mathematics, Engineering, and Computer Science Objectives Students will use their mathematical knowledge of graphs and rate of change of a function to gain a better under