Understanding the Progressive Tax Model

Transcription

1 MATHEMATICS Lesson 23 Essential Question: How would ending the 2001 and 2003 tax cuts affect the United States economy? Introduction: This lesson provides a real-world application of piecewise functions. In order for students to understand the application, they must learn about how the United States government uses the progressive tax model to calculate income tax. During this introduction to progressive tax models, students will review percentages while utilizing reading comprehension skills. Afterwards, students will be asked to graph two piecewise functions of marginal tax rates and visually analyze the graph for meaning. Mathematical Content: Piecewise functions, step functions, percentages, graphing on the coordinate plane, drawing conclusions from tables and charts Grade Level: 9-11 Algebra I, Algebra II Pacing: Algebra I - 2 days, Algebra II - 1 day (50 minutes each) Materials Needed: Scientific calculators Key Information The following terms and concepts are used in this lesson: progressive tax adjusted gross income (AGI) proportional tax flat tax debt/deficit 369

2 Mathematics Students Will Understand: Mathematical Understandings: Piecewise functions can be used to model real-world data. Analyzing mathematical models of applied data can inform a viewer about social phenomena. Economics Understandings: The progressive tax model charges a higher rate as an individual s adjusted gross income (AGI) increases. The marginal tax rates influence how different tax brackets will be financially affected. Related Curriculum Standards: The National Council of Teachers of Mathematics Algebra Standards Instructional programs from prekindergarten through grade 12 should enable all students to: understand patterns, relations, and functions; represent and analyze mathematical situations and structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; analyze change in various contexts. Students Will Be Able To: Mathematical Skills: Define and graph a step, piecewise function. Make observations of and draw conclusions about graphical representations of piecewise functions. Economics Skills: Calculate taxes in a Progressive Tax Model. Describe how the marginal tax rate can positively or negatively affect individuals with different income levels. List of Lesson Resources: 1. Exploring Progressive Taxes: Piecewise Function Application 2. Exploring Progressive Taxes: Solutions to Problems 3. Assessment Prompt 370 Understanding Fiscal Responsibility

3 How would ending the 2001 and 2003 tax cuts affect the United States economy? Lesson 23 Time Required: 2 class periods Algebra I Algebra II Day 1 Introduction to Piecewise Functions (resources and strategies provided by classroom teacher) Day 2 Exploring Progressive Taxes: Piecewise Function Application Day 1 Review of Piecewise Functions. Exploring Progressive Taxes: Piecewise Function Application Entry: Tell students to silently read the introduction of Resource 1, Exploring Progressive Taxes: Piecewise Function Application, and Example 1. As they read, they should underline any portion of the introduction they do not understand. Tell students they will have about five minutes to complete the reading and share any questions with a partner. Discussion: (7 minutes) Trouble with Taxes Are there any questions on the reading or the example? Students may ask why each marginal tax amount is taxed at a different rate. For example, If a person earns $37,000 adjusted gross income (AGI) in 2007, why isn t the whole thing just taxed 25%? This is a excellent question. Ask students to calculate the amount of tax a single person should be taxed if they earned $8,350. (Answer: $835) Now according to the chart, how much should a single person who earned $8,351 AGI be taxed? (Answer: $1,252.65) What is wrong with this picture? Students should calculate that if a person made $8,350 AGI, after taxes they would have $7,515. If a person made $8,351 AGI, after taxes they would have $7, Encourage students to complete these calculations independently for it will help reinforce the reading. Students will likely say that it would be unfair for a person who made more money before taxes to receive less money after taxes comparatively. What are some of the benefits of a Progressive Tax Model? Student may think the Progressive Model is more fair because it puts less of a tax burden on the lowest income level and the highest tax burden on people who are making the most money. Other students may think it is unfair that different people would pay different percentages of their incomes or that not every person pays the same flat fee for identical benefits. Understanding Fiscal Responsibility 371

4 Mathematics Lesson Strategies and Activities: Assessment: (7 minutes) Ask students to work individually or in groups on Problem 1. This problem is designed to test the students understanding of the Progressive Tax Model and it should be used as a summative assessment for the teacher. Walk around to different groups to check for understanding. Discussion: (3 minutes) Ask students to reflect on their findings. How did Lee and Jo s tax rates differ? How were they the same? Do you feel that they were taxed fairly? Explain your answer. Transition: Let s read the paragraph about Comparing Two Progressive Tax Rate Plans in Resource 1. Depending on the comfort and skill level you anticipate your students to have with piecewise functions, this would be an appropriate time to review basic information about these functions with students. This review should focus on graphing over intervals, and it may be helpful to review how to graph horizontal lines. Working individually or in small groups, students should complete the graphs of the two piecewise functions and Problems 3 and 4. Give them minutes to complete these problems. Assessment: Distribute Resource 3. This asks students to respond to the following prompt in writing: Some individuals advocate for a flat or proportional tax model, where every individual is taxed the same percentage of their AGI regardless of the amount of their AGI. How would the graph of a flat or proportional tax compare to the piecewise graph of the Progressive Tax Model? Which tax model would you prefer? Explain your answer. Further Engagement (Optional): Ask students to bring in an article about the Bush Tax Cuts and write a paragraph about how the mathematics behind the Progressive Tax Model deepens their understanding of the article. What questions or concerns does the article raise? References Cited: Tax Foundation. (2010, June 15). U. S. federal individual income tax rates history, Retrieved September 7, 2010 from Understanding Fiscal Responsibility

5 Lesson 23 MATHEMATICS Understanding the Progressive Tax Model Resources The following section is formatted for the easy reproduction of resources intended for use by students. They appear in the order in which they are listed in the Introduction and are essential to the lesson. These resources may also be downloaded from the Understanding Fiscal Responsibility website: Understanding Fiscal Responsibility 373

6 Mathematics Resource 1. Exploring Progressive Taxes: Piecewise Function Application Name: Date: Mark Twain once wrote, The only difference between a tax man and a taxidermist is that the taxidermist leaves the skin. While most Americans dread tax season, it would be difficult to go through a typical day without utilizing a service or good provided by the United States government. The federal budget is used to fund national defense, Social Security, Medicare, transportation systems, environmental protection agencies, educational facilities, and much more. These services play an important role in the high quality of life Americans enjoy. Individual income taxes are an important source of revenue for the federal budget. The United States uses a Progressive Tax Model that taxes individuals at greater rates as their Adjusted Gross Income, or AGI, increases. An Adjusted Gross Income is a person s yearly income after tax deductions have been taken. Deductions are provided for certain expenses, including business expenses, health savings account payments or paid alimony. Use the following example to understand how marginal taxes models calculate what a taxpayer should owe. Progressive Tax Rates in 2007 Marginal Tax Rates 2007 Single Married Filing Separately 10% $0-$8,350 $0-$8,350 15% $8,351-$33,950 $8,351-$33,950 25% $33,951-$82,250 $33,951-$68,525 28% $82,251-$171,550 $68,526-$104,425 33% $171,551-$372,950 $104,426-$186,475 35% $372,951+ $186,476+ Data collected from Example 1: In 2007, Lee, who is unmarried, earned $68,900 and earned $565 in interest in a savings account. Lee received a $345 deduction for business expenses throughout the year and a $102 deduction for moving costs. What is Lee s AGI? Using the Single Progressive Tax Rates in 2007, how much money would Lee owe in taxes? How does the amount owed in taxes compare to Lee s AGI? Solution: Lee s AGI equals total deduction costs earnings subtracted from net income. $68,900 + $565 - $345 - $102 = $69,018 Lee s AGI equals $69,018. This is the amount of income that is currently taxable. 374 Understanding Fiscal Responsibility

7 How would ending the 2001 and 2003 tax cuts affect the United States economy? Lesson 23 Notice that Lee s AGI puts Lee in the third tax bracket. Thus, Lee will be taxed 25% of her income between and ( , or 35068), 15% of her income between and 8350 ( , or 25600), and 10% of the first $8350. Calculating, we obtain: 8, , , Total taxes due equals = $13,442. $13,442/$69,010 = 19.5%. Lee s taxes are equivalent to 19.5% of Lee s AGI. Problem 1 A. Jo, who is unmarried, had an estimated AGI of $300,000 in How much would Jo need to pay in taxes? B. Jo s taxes represent what portion of Jo s AGI? C. How are Lee and Jo taxed differently? Do you believe the different tax amounts are fair? Explain your answer. Understanding Fiscal Responsibility 375

8 Mathematics Comparing Two Progressive Tax Rate Plans In an effort to stimulate the economy after 9/11, the government approved tax cuts in 2001 and The size of the growing national debt and the federal deficit is a concern to the strength and autonomy of the U.S. economy. America is no longer paying as it goes. Could the nation afford the tax cuts in 2001 and 2003? If Congress alters the progressive tax rates, how would it affect individuals differently? In order to explore this question, we need a method of comparing different marginal tax rates. While there are many ways to explore this question, we are going to compare different marginal tax rates by utilizing piecewise functions. Piecewise functions are defined by different equations for specific intervals of time. For example, absolute value functions are examples of piecewise function graphs because they are defined by two different linear functions on two separate intervals. Problem 2 Use graph paper to graph the following functions over the specified x interval. If an interval uses a strict inequality, use an open circle to signify the value. If the value is included in the interval, use a filled-in circle. This type of piecewise functions is called a step function, because the graph should look like steps! Graph the Single Marginal Tax Rates from 2000 and 2007 on the same graph and answer the following questions Single Marginal Tax Rates x f(x) $ 0 x $8,350 10% $8,350 x $33,951 15% $33,951 x $82,250 25% $82,250 x $171,550 28% $171,550 x $372,950 33% $372,950 x 35% 2000 Single Marginal Tax Rates x f(x) $0 x $26,250 15% $26,250 x $63,550 28% $63,550 x $132,600 31% $ 132,601 x $228,350 36% $ 228,350 x 39.60% 376 Understanding Fiscal Responsibility

9 How would ending the 2001 and 2003 tax cuts affect the United States economy? Lesson 23 Using your graph, answer the following questions. Problem 3 What year was the highest marginal tax rate for individuals with an AGI of $300,000? $140,000? Were the marginal tax rates in 2000 ever greater than the marginal tax rates in 2007? Problem 4 What income level benefited the most from the 2001 and 2003 tax cuts? How is that illustrated visually on your graph? Pick one specific AGI amount in that interval and calculate the amount of taxes owed in 2000 and From this, explain the possible impact of not extending the 2001 and 2003 tax cuts in the future. Understanding Fiscal Responsibility 377

10 Mathematics 378 Understanding Fiscal Responsibility

11 How would ending the 2001 and 2003 tax cuts affect the United States economy? Lesson 23 Resource 2. Exploring Progressive Taxes: Solutions to Problems Problem 1 A. Jo, who is unmarried, had an estimated AGI of $300,000 in How much would Jo need to pay in taxes? For the solution, it is necessary to find out how much Jo will pay at each marginal rate and then sum the answers together. 8, , ,840 48, ,075 89, , , , Total: $84, B. Jo s taxes represent what portion of Jo s AGI? $84,142.50/$300,000 = 28% C. How are Lee and Jo taxed differently? Do you believe the different tax amounts are fair? Explain your answer. Jo pays 28% of his AGI in taxes, while Lee pays 19.5%. That means Jo pays 8.5% more than Lee. Looking at the flat amount paid, Jo pays more than $70,000 more in taxes than Lee. This amount is more than Lee s total income! But while Jo s AGI was 4.3 times as much as Lee s (300,000/69018), Jo s percentage of AGI paid in taxes was 1.4 times Lee s percentage of AGI paid in taxes (28%/19.5%). Student answers will vary about fairness based on their own views. Some students may think it is fair that Jo pays more in taxes because Jo earns so much more than Lee. Other students may feel that it is unfair that they pay different rates, or different amounts, for the same services provided by the U.S. government. Understanding Fiscal Responsibility 379

12 Mathematics Problem 2 Use graph paper to graph the following functions over the specified x interval. If an interval uses a strict inequality, use an open circle to signify the value. If the value is included in the interval, use a filled-in circle. This type of piecewise functions is called a step function, because the graph should look like steps! Graph the Single Marginal Tax Rates from 2000 and 2007 on the same graph and answer the following questions. [see attached sample student graph] Problem 3 What year was the highest marginal tax rate for individuals with an AGI of $300,000? $140,000? Were the marginal tax rates in 2000 ever greater than the marginal tax rates in 2007? The tax rates in 2000 are always equal to or greater than the tax rates in For people with AGIs between $8,350 and $26,250 the tax rate stayed the same at 15%. Problem 4 What income level benefited the most from the 2001 and 2003 tax cuts? How is that illustrated visually on your graph? Pick one specific AGI amount in that interval and calculate the amount of taxes owed in 2000 and From this, explain the possible impact of not extending the 2001 and 2003 tax cuts in the future. Student answers will vary. Students may reply that the highest tax cuts came for individuals who made an AGI between $228,350 and $372,950. This is illustrated in the graph because this interval shows the largest gap between the different marginal levels. Some students might say that individuals with AGIs greater than $372,950 experienced the most tax cuts because they benefited from all of the other marginal rate deductions and saved the most money overall. It is also possible to argue that individuals who made an AGI between $0 and $8,350 benefited most from the tax cut because they experienced a 5% tax cut. For people with a very low income, this amount could be significant for improving their quality of life. The impact of not extending the 2001 and 2003 tax cuts will depend on the students viewpoints. Some students may believe that raising taxes will help decrease the federal debt and deficit and make the U.S. economy sounder. Other students may believe higher taxes would hurt the economy because people will have less money to spend in the free market. 380 Understanding Fiscal Responsibility

13 How would ending the 2001 and 2003 tax cuts affect the United States economy? Lesson 23 Understanding Fiscal Responsibility 381

14 Mathematics Resource 3. Assessment Prompt Respond to the following prompt in writing: Some individuals advocate for a flat or proportional tax model, where every individual is taxed the same percentage of their AGI regardless of the amount of their AGI. How would the graph of a flat or proportional tax compare to the piecewise graph of the Progressive Tax Model? Which tax model would you prefer? Explain your answer. 382 Understanding Fiscal Responsibility