Franklin said, "Nothing is certain but death and taxes." A mathematics
class may not be the right place to discuss death, but it is a good
place to discuss taxes. In the process, students can learn some
important lessons about slopes, split functions, averages, rates,
marginal rates, and percents (and taxes).
Taxes are an important political issue. Just prior to the 1990
elections, the federal deficit was very high, and taxes needed to be
raised in spite of George Bush's 1988 campaign pledge, "Read my lips-no
new taxes." Communicating to the people that any changes would be fair
was important in that election. But most people did not
understand the tax system in place, so they were in no position to
understand the fairness of any changes.
The media did not help much. The terminology of tax rates and tax
brackets was (and is) often incorrectly used in the newspapers and on
television. As teachers, we can use terms properly to clarify what is
really meant. Then, our students will be able to see how the tax
brackets really work. This lesson presents the situation as it was
for 1990 taxes due in April 1991. A source of further exercises can be done with any revisions in the tax laws, each of which is
interesting, realistic, important, long, and manageable by algebra
Rates and Taxes: Tables Overhead
Begin class by displaying Table 1 on the Rates and Taxes: Tables Overhead, and ask, "What does a family with an income of $40,000 pay in U.S. income taxes?" On the basis of the information in Table 1, students usually respond, "twenty-eight percent of forty thousand
dollars." We will see that this answer is intuitive but wrong, and
consequently, the public is easily misled by tax rate arguments.
The Real-World Solution
To correct this misconception, generalize your original question to, "How much would a family of four with an
income of x dollars pay in taxes? What is the tax rate?"
To answer these questions, introduce some
technical tax terms (that appear on Form 1040, which serves as a good
prop). Two technical terms for income need explanation. Adjusted gross income (AG income) is similar to what we normally think of as income.
But taxpayers are entitled to some deductions from their AG income before computing their tax. Only taxable income is
taxed. To find taxable income, a family could take from their AG
income a standard deduction of $5,450 plus an additional $2,050
deduction for each person in the family.
For ease of comparison, use
the example of a family of four, hereinafter referred to as "a family."
Their taxable income is their AG income minus deductions:
$5,450 + 4($2,050) = $13,650
Some families are entitled to even more
deductions, but tell students that the complexity of the U.S. tax code prevents you from discussing them here. To make sure that students understand taxable income, ask probing questions, such as:
- Why does a family with an income of $10,000
pay no federal income tax?
[Because they have no taxable income. In other words, the deduction is greater than their income. Tell students that because there is no taxable income, $10,000 is considered to be nontaxable income.]
- What is the taxable income of a
family with an income of $20,000?
[Their taxable income
is $20,000 - $13,650 = $6,350.]
- Up to what amount would be considered nontaxable income?
[$13,650. After your taxable income exceeds this amount, you will begin to have taxable income.]
Now, come back to the original question asked at the beginning of the class:
"What does a family with an income of $40,000 pay in U.S. income taxes?"
Students should now be able to say that a family with $40,000
of regular income will be in the 15 percent bracket, not the 28 percent bracket, since $40,000-$13,650=$26,350.
Tell students that you will denote AG income by x and taxes by
T(x). Let y denote taxable income. You may wish to write this on the board so that students can refer to it during the lesson.
Ask students to create a general equation for the taxable income of a family. [y = x - 13,650.]
With this notation, tell students that they can interpret the first line of Table 1 algebraically as:
T = 0.15y, for 0 < y < 32,450
Ask students to convert this formula to a function of x (which is AG income):
T(x) = 0.15(x - 13,650) =
0.15x - 2,047.50, for 0 < x-13,650 ≤ 32,450 or 13,650 < x ≤ 46,100
In order to ensure that students understand the function, ask probing questions, such as:
Now ask students, "What, then, is the tax rate?" By definition, a rate is
a quantity of something measured for each unit of something else. That
is, a quotient of quantities. Now, ask students to find the tax rate of the family that earned an AG income of $10,000 and $20,000. [0% and , respectively.] Ask students to note that this is certainly not a 15% rate that was shown in Table 1!
Rates and Taxes: Graphs Overhead
Now display Figure 1 in the Rates and Taxes: Graphs Overhead. Before continuing, define marginal tax rate to students as the percentage of tax applied to your income based off of your earnings. Ask students to discuss, in a think-pair-share manner, what the graph is depicting. You may also wish to have students write down their thoughts as a form of assessment. In general, here is what the graph is portraying:
Too many people think that the rates in
the tax rate schedules refer to the actual tax rate (total
taxes divided by total income). This notion is incorrect. The
tax-bracket percents are actually marginal rates, not average rates. In
the 15 percent bracket, "15 percent" refers to the slope of the tax
The average rate is the actual rate the taxpayer pays, that is, total
tax divided by total income. That is the slope of the line segment
(s1) through the origin and (x,T(x)), which is less than 15 percent (see the dashed line in Figure 1). The slope of the dashed line is the geometric representation of the average rate at (x, T(x)).
Slopes alone do not determine heights. The intercept is important.
With a marginal rate of 15 percent, each additional $100 of taxable
income raises taxes by $15. Note that not every $100 of income is taxed
$15. Figure 1 makes clear why the marginal rate is higher than the actual rate.
After students understand this concept, derive a formula to determine the actual tax rate (for the 15 percent bracket) as a class. Let R(x) denote the actual tax rate.
for 13,650 < x ≤ 46,100. Ask students to find how much the actual tax rate varies. [By substitution 13,650 and 46,100, students should be able to determine that the rate varies from 0% to ≈10.56% for the 15 percent bracket]. It is never close to the nominal (marginal) rate of 15 percent!
Now move up a bracket and interpret the second line of Table 1 with your students. Ask students how much a family with $50,000 of taxable income will be taxed. Some students may hold the misconception that all $50,000 will be taxed in the 28 percent tax bracket. But this is not true. A family with an income of
$32,450–$78,400 pays 15 percent tax on a taxable income of
$32,450 (the top of the lower bracket) and additional tax at a 28
percent marginal rate on taxable income above that level.
The taxes on
$32,450 are $4,867.50. Thus,
T = 4,867.50 + 0.28 (y-32,450), for 32,450 < y ≤ 78,400
Changing to x, or AG income, we get:
T(x) = 4,867.50 + 0.28((x-13,650)-32,450), for 32,450 < x-13,650 ≤ 78,400, or
T(x)= 0.28x - 8,040.50, for 46,100 < x ≤ 92,050
A graph (extending Figure 1)
of the tax split function over this wider interval is easy to create.
In this bracket, the tax carried over from the previous bracket is
$4,867.50 and the maximum tax is $17,733.50.
, for 46,100 < x ≤ 92,050
The marginal rate is 28 percent, but the actual rate varies from 10.56% through 19.27% (see Figure 2). Higher-income families not only pay higher taxes, they pay at
a higher rate. And no family in this bracket pays close to 28 percent of income in taxes.
Now is a good time to discuss the so-called bubble. Most people do
not understand the bubble, that is, the 33 percent bracket. This
misunderstanding caused major political problems. Taxpayers are
concerned that the wealthiest people are not paying their fair share,
and the bubble appeared to be a clear example. People thought that
those families with the highest incomes (y > $162,770) paid
taxes at a lower rate (28%) than those with lower incomes in the bubble
bracket (33%). This discrepancy seemed to them unfair and wrong. Have the class do the mathematics and then decide whether the tax
discrepancy is unfair.
Most students understand Table 1 by this point
in the lesson. Again, the taxes of families in any bracket are
calculated by computing the taxes owed by a family at the top end of
the next-lower bracket plus a fixed percent of the income exceeding the
bottom end of the bracket.
Here are the calculations from the bubble bracket.
T = 17,733.50 + 0.33(y - 78, 400), for 78,400 < y ≤ 162,770.
T(x) = 17,733.50 + 0.33(x - 92,050) = 0.33x - 12, 643, for 92,050 < x ≤ 176,420.
, for 92,050 < x ≤ 176,420.
Tax Deductions Decrease Taxable Income
The tax carried over from the previous bracket to the
bubble bracket is $17,733.50, and the maximum tax is $45,575.60. The
actual rate varies from 19.27% to the maximum rate of 25.83%. The actual rate is never near 33 percent. More to the point,
it is never near the 28 percent marginal rate in the next higher
Taxpayers in the top bracket pay 25.83% on the
first $176,420 of AG income and 28 percent on the rest. (Actually,
personal deductions are phased out as income increases in this bracket,
so the truth is slightly more complex. See Schedule Y1 of Form 1040).
If their income is extremely high, their actual rate is close to 28
percent. In fact, in terms of y, the taxable income, the rate is a constant 28
percent; that is,
T = 45,575.60 + 0.28(y-162,770)
= 45,575.60 + 0.28y-45,575.60
for y≤162,770. In the highest bracket, and only
there, the actual rate really is the nominal (marginal) rate. Without
some sort of higher bubble-bracket rate, obtaining a constant rate for
all families in the top bracket is not possible. Note that the people
in the bubble did not pay taxes at a higher actual rate than those in
the highest bracket; they paid only a higher marginal rate.
Table 2 summarizes the tax brackets, taxes, and actual tax rates.
Students can learn a great deal about rates and taxes
if they do the mathematics of possible tax revisions for themselves.
For example, the primary change passed into law in 1990 as stated in
the press was that the bubble bracket was eliminated and that the rates
in that bracket and the highest bracket were both changed to 31
percent. This statement brings up some interesting questions.
Disregarding all the other changes, how does this change in marginal
rates affect taxes? Who will pay more in taxes than before? Who will
pay less? Answer: If x < $92,050, taxpayers pay the same. If $92,050
< x < $232,666.67, they pay less. Only if x > $232,666.67 do they
pay more.) Did the voting public understand this change?
Other problems concern the effect of changing
deductions. For example, the tax brackets were the same in 1989, but
the standard deduction was lower ($5,200) and personal deductions were
lower ($2,000 per person). Was the change especially good for
low-income families? Who benefits, and who benefits most, when the
standard deduction is increased? (Answer: All taxpayers pay less, but
the decrease is larger for families in higher brackets.) Did the voting
public understand the ramifications? By the way, these results
illustrate one reason that some people advocate tax credits rather than
deductions. Tax credits decrease taxes, whereas tax deductions decrease
Many interesting extensions can be developed. Refer to the Extensions section for recommendations.
Semantic confusion has clouded understanding of
important tax issues. The mathematical concepts of actual rates and
marginal rates help explain the reality behind the political arguments.
Of course, in this context marginal rates are simply slopes of lines
and actual rates are slopes of lines through the origin. Slope is a
fundamental concept of both algebra and calculus. Therefore, these
vivid lessons about actual rates and marginal rates are appropriate for
algebra and calculus students alike.
This tax lesson is a real-world application of
mathematics that really catches my students' attention. Few or none of
them appear to know how tax brackets work when I start. At the end of
the lesson, I get many comments about how pleased they are with their
new knowledge. Most often I hear, "I never knew that's how taxes
worked." I am asked how I think taxes ought to be changed, but I stay
out of politics and never tell. My students don't seem to have any
trouble with the mathematics after I explain Table 1. After all, most of the mathematics is just straight lines.
We tell our students that mathematics applies to
real-world problems, but they do not often see really good
applications. This significant problem can be done step by step using
middle-level algebra skills. Not only does it explain taxes, thus
serving also as a good civics lesson, but it also illuminates important
concepts of mathematics.