Lesson
23
MATHEMATICS
Understanding the Progressive Tax Model
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.
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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:
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progressive tax
adjusted gross income (AGI)
proportional tax
369
flat tax
debt/deficit
Mathematics
Understanding the Progressive Tax Model
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:
„
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.
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.
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
How would ending the 2001 and 2003 tax
cuts affect the United States economy?
Lesson
23
Time Required:
2 class periods
Algebra I
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x
Day 1—Introduction to Piecewise Functions (resources and strategies provided by
classroom teacher)
Day 2—Exploring Progressive Taxes: Piecewise Function Application
Algebra II
x
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
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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 $7,098.35. 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.
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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
Mathematics
Understanding the Progressive Tax Model
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.
x
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 15-20 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, 1913-2010. Retrieved
September 7, 2010 from http://www.taxfoundation.org/taxdata/show/151.html
372 Understanding Fiscal Responsibility
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:
http://understandingfiscalresponsibility.org/
Understanding Fiscal Responsibility 373
Mathematics
Understanding the Progressive Tax Model
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 www.irs.gov
Understanding the Progressive Tax Model
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
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 69018 and 33950 (69018-33950, or 35068), 15% of
her income between 33950 and 8350 (33950-8350, or 25600), and 10% of the
first $8350.
Calculating, we obtain:
8,350 u .1 835
25,600 u .15 3840
35,068 u .25 8767
Total taxes due equals 835 + 3840 + 8767 = $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 2007. 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
Mathematics
Understanding the Progressive Tax Model
Comparing Two Progressive Tax Rate Plans
In an effort to stimulate the economy after 9/11, the government approved tax cuts in 2001 and
2003. 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.
2007 Single Marginal Tax Rates
x
f(x)
$0 d x $8,350
10%
$8,350 d x $33,951
$33,951 d x $82,250
$82,250 d x $171,550
$171,550 d x $372,950
$372,950 d x
15%
25%
28%
33%
35%
2000 Single Marginal Tax Rates
x
f(x)
$0 d x $26,250
$26,250 d x $63,550
$63,550 d x $132,600
31%
$132,601 d x $228,350
36%
$228,350 d x
39.60%
376 Understanding Fiscal Responsibility
15%
28%
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 2007. From this, explain the possible impact of not extending the 2001 and
2003 tax cuts in the future.
Understanding Fiscal Responsibility 377
Mathematics
Understanding the Progressive Tax Model
378 Understanding Fiscal Responsibility
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 2007. 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.
8,350 u .1 835
25,600 u .15 3,840
48,300 u .25 12,075
89,300 u .28 25,004
128,450 u .33 42,388.50
Total: $84,142.50
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
Mathematics
Understanding the Progressive Tax Model
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 2007. 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 2007. 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
How would ending the 2001 and 2003 tax
cuts affect the United States economy?
Lesson
23
Understanding Fiscal Responsibility 381
Mathematics
Understanding the Progressive Tax Model
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