1. Grade 7: Data Management &
Probability
>S08E21: The Old Man and the Lisa
>Probabilities expressed in fraction, decimal,
and percent form
Grade 8: Geometry and Spatial Sense
>S05E10: $pringfield (Or, How I Learned to
Stop Worrying and Love Legalized Gambling)
 determine the Pythagorean relationship
Grade 11: Exponential Equations
>S25E10: Married to the Blob
>Noticing and analyzing exponential equations
Grade 12 Advanced Functions:
Exponential & Logarithmic Functions
>S07E06: Treehouse of Horror VI: Homer3
>solve exponential equations in one variable
Grade 12 Calculus & Vectors:
Geometry and Algebra of Vectors
>S07E06: Treehouse of Horror VI: Homer3
>demonstrate an understanding of vectors in
three-space
+ MATH = D’oh!
Not only is The Simpsons the most
watched animated television show
among 18-49 year olds1, but it is also
the most mathematically sophisticated
as well.
Executive Producer Al Jean studied
mathematics ta Harvard at age 16;
Senior research post Jeff Westbrook
left Yale University to be a
scriptwriter on The Simpsons, and the
writer himself, David X. Cohen, who
has a degree in both physics and
computer science2 from Harvard and
UC Berkley respectively.
There are dozens of examples of
mathematical references planted into
episodes of The Simpsons, only some
of which are illustrated in this booklet.
Connecting curriculum requirements to
episodes of The Simpsons will not only
get the attention of your students, but
with effective delivery, will enhance
their understanding of the subject
matter as well.
L e s s o n s I n T h i s
P a c k a g e

2. After Homer’s heart attack, Homer
is convinced that Lisa just gave
away $12,000. Lisa corrects him
“Um, Dad, ten percent of
$120,000,000 isn't $12,000.
It's.…”
What is 10% of $120 000 000?
What is this number expressed as a
fraction? Show all work.
Information Technology
Solutions
SPECIFIC
EXPECTATIONS
Probability - By the end
of Grade 7, students
will:
– research and report
on real-world
applications of
probabilities expressed
in fraction, decimal,
and percent form (e.g.,
lotteries, batting
averages, weather
forecasts, elections)
S08E21: The Old Man and the
Lisa
Grade 7: Data
Management
& Probability
In the United States, your odds of
winning the lottery depend on
where you play. Single state
lotteries have odds of
approximately 18 million to 1, while
multiple state lotteries can have
odds as high as 120 million to 1.
The National Safety Council states
that the odds of getting hit by
lightning is on average 100
people/year. If the US population is
314 million people, then what
percentage of people are struck by
lightning per year?
Do you have a better chance of
winning the lottery or getting struck
by lightning? How much of a better
chance do you have? Explain your
answer.
Solutions for Lottery Odds:
1
18 million
= 0.0000000556 x 100% =
0.00000556%
1
120 million
= 0.00000000833 x 100%
= 0.000000833%
Solutions for Lightning Strikes:
100 people/year
314,000,00 people
= 0.000000318 x
100%
=0.0000318% of people in the US get
struck by lightning each year.
0.000000318
0.00000000833
= 38
Thus, you have a higher chance of
getting struck by lightning than
winning the lottery by 38 times.
FRACTIONS,
RATIOS &
PERCENTS

3. OVERALL
EXPECTATIONS
Develop geometric
relationships involving
lines, triangles, and
polyhedra, and solve
problems involving
lines and triangles.
SPECIFIC
EXPECTATIONS
Determine the
Pythagorean
relationship, through
investigation using a
variety of tools (e.g.,
dynamic geometry
software; paper and
scissors; geoboard) and
strategies.
Solve problems
involving right
triangles geometrically,
using the Pythagorean
relationship.
S05E10:
$pringfield (Or, How I Learned to Stop
Worrying and Love Legalized Gambling)
PYTHAGOREAN THEORM
 Explain Pythagorean
Theorem to be: c2 = a2 + b2
or
 and: 𝒄 𝟐 = 𝒂 𝟐 + 𝒃 𝟐
“The sum of the squares of the
two shortest sides of a right
triangle is equal to the square
of the hypotenuse”.
 Define term hypotenuse
Grade 8:
Geometry &
Spatial Sense
Play this clip for your class, then
pause it at 00:06.
Homer states that “the sum of the
square roots of any two sides of
an isosceles triangle is equal to
the square root of the remaining
side”. Have your students test this
with the isosceles triangles above.
What conclusion do they come to?
3 cm 3 cm
5 cm 2 cm
3 cm3 cm
Now try a right angle triangle.
Have them watch what happens when
you put a square on each side.
 What’s the area of the biggest
square?
 What’s the area of the two smaller
squares put together?
 Have students come up with a rule
to explain this (Pythagorean
Theorem)
3 cm
4 cm
5 cm
PYTHAGOREAN
THEOREM
Replay the clip above fully, and ask
the class what was wrong in Homer’s
statement that the man in the stall
didn’t catch?
Solution: Homer’s statement, ‘the sum of the
square roots’ implies:
𝑎 + 𝑏 = 𝑐; and:
Homer implies that the sum of the square
roots of any 2 sides will give you the square
root of the 3rd side, but it only works for the
2 shortest sides

4. Grade 11:
Exponential
Functions
OVERALL
EXPECTATIONS
2. make connections
between the numeric,
graphical, and algebraic
representations of
exponential
functions;
3. identify and represent
exponential functions, and
solve problems involving
exponential functions,
including problems arising
from real-world
applications.
SPECIFIC
EXPECTATIONS
1.4 determine, through
investigation, and
describe key properties
relating to domain
and range, intercepts,
increasing/decreasing
intervals, and asymptotes
(e.g., the domain is the set
of real numbers; the range
is the set of positive real
numbers; the function
either increases or
decreases throughout its
domain) for exponential
functions represented
in a variety of ways [e.g.,
tables of values,
mapping diagrams, graphs,
equations of the
form
f(x) =ax (a>0, a≠1),
function
machines]
S25E10: Married to the Blob
At 01:39, Radioactive Man appears defeated and starts losing his life
force. The radioactive symbol on his chest shows how his power
declines. Answer the following questions:
1. Does Radioactive Man’s power decline linearly? If not, how does his
power decline?
2. Graph your results and find what equation models his power decline.
3. Hypothetically, if the radioactive symbol had 6 bars instead of 3, in
what stages would his power decline? Would this have helped him
survive?
Solutions:
1. His power declines exponentially in powers of 2. 1 to ½ to ¼ to 1/8.
y = 2e-0.693x
0
0.2
0.4
0.6
0.8
1
1 2 3 4
PercentageofTotalPower
Stages of Power Loss
Radioactive Man's Power Decline2. 3. If the radioactive
symbol had 6 bars, the
power would decline as:
1 to ½ to ¼ to 1/8 to
1/16 to 1/32 to 1/64.
This wouldn’t have been
much more helpful to
Radioactive Man.

5. Information Technology
Solutions
S07E06: Treehouse of Horror VI:
Homer3
Grade 12
Advanced
Functions:
Exponential &
Logarithmic
Functions
OVERALL
EXPECTATIONS
3. Solve exponential
and simple logarithmic
equations in one
variable algebraically,
including those in
problems arising from
real-world applications.
SPECIFIC
EXPECTATIONS
3.2 Solve exponential
equations in one
variable by determining
a common base (e.g.,
solve 4x = 8x+3 by
expressing each side as
a power of 2) and by
using logarithms (e.g.,
solve 4 = 8 by taking
the logarithm base 2 of
both sides), recognizing
that logarithms base 10
are commonly used
(e.g., solving 3 = 7 by
taking the logarithm
base 10 of both sides).
Recall Exponent Laws from Gr. 11
Multiplication Law:
xa *xb = xa+b
Division Law: xa/xb = xa-b
Power of a Power: (xa)b = xab
Power of a Product:
(xy)a = xaya
Power of a Quotient:
(x/y)a = xa/ya
Negative Exponents:
x-a = 1/xa
Zero Exponents: x0 = 1, x≠0
Try some Examples
1) 6x5*3x-2 = ?
2) 5x-4*2x-3 = ?
3) 6x5/3x-2 = ?
Solving Exponential Equations
4) 32-x = 3
Solution: Since the bases are the
same (3), set their exponents
equal: 2-x = 1, x = 1.
5) 42x+1 = (0.5)3x+5
Solution: First write both sides with
the same base (2). (22)2x+1=(2-2)3x+5,
then use power of a power rule:
24x+2 = 2-6x-10, set exponents equal:
4x+2=6x-10; 12=2x; x=6.
6) 2x-1 + 2x + 2x+1 = 7
Solution: Use graphing technology,
and graph y = 2x-1 + 2x + 2x+1. Find
the x-value where y=7 (x=2)
Fermat’s Last Theorem
Fermat looked at solutions to the
Diophantine Equation,
ax + bx = cx, for x>2 and a,b,c ≠ 0,
and discovered something quite
peculiar – this equation has no
solutions for x being an integer.
Notice that x=2 returns the
Pythagorean Theorem.
In this episode of Homer3, pause
the video at approximately 03:20,
and have students reflect on the
equation in the background. Has
Homer stumbled into a 3D universe
where Fermat’s Last Theorem is
incorrect? Why or why not? How
can you tell by just looking at the
equation(see solutions at end of
booklet)?
EXPONENTIAL
EQUATIONS

6. OVERALL
EXPECTATIONS
1. Demonstrate an
understanding of
vectors in two-space
and three-space by
representing them
algebraically and
geometrically and by
recognizing their
applications.
SPECIFIC
EXPECTATIONS
1.1 Recognize a vector
as a quantity with both
magnitude and
direction, and identify,
gather, and interpret
information about real-
world applications of
vectors (e.g.,
displacement, forces
involved in structural
design, simple
animation of computer
graphics, velocity
determined using GPS)
1.4 Recognize that
points and vectors in
three-space can both be
represented using
Cartesian coordinates,
and determine the
distance between two
points and the
magnitude of a vector
using their Cartesian
representations
S07E06: Treehouse of Horror VI: Homer3
INTRODUCTION
TO 3-SPACE, R3
Recall Vectors in 2D Space:
|c| = |𝑎|2 + |𝑏|2
 Any point P(x,y) in R2 can
be thought of as a vector c,
whose magnitude is
|c| = |𝒙| 𝟐 + |𝒚| 𝟐
 The direction of P(a,b) can
be found using the tangent
of the component vectors:
c is θo above the horizontal,
where tanθ = b/a, θ = tan-1(b/a)
a
b
c
Vectors in 3D Space:
z x
x
y
y
z
1) Explain why when falling
through the wormhole, it is not
sufficient to describe Homer’s
position in R2?
Solution: Because there is depth
to his location now as well. You
would measure z from the same
reference level as x and y.
2) Based on how we found the
magnitude for a vector in R2,
predict the formula for the
magnitude of a vector in R3.
Solution: |c| = |𝒙| 𝟐 + |𝒚| 𝟐 + |𝒛| 𝟐
3) How does the position of Homer’s
feet in R3 compare to that of his
head?
y
z
x
Feel free to assign your own coordinates.
4) Predict approximately how you
would determine his exact position
in R3.
Grade 12
Calculus &
Vectors:
Geometry and
Algebra of
Vectors

7. EXTENDED
SOLUTIONS
EXTENDED
SOLUTIONS
 Grade 12
Advanced
Functions:
Exponential &
Logarithmic
Functions,
Exponential
Equations
.
 Grade 12
Calculus &
Vectors:
Geometry and
Algebra of
Vectors,
Introduction to 3-
Space, R3
Grade 12, MHF 4U, Exponential Equations
TRYING SOME EXAMPLES
1) 6x5*3x-2 = ?
Left Side: Multiplying exponentials, therefore add exponents of x and multiply coefficients 6
and 3. LS: 18x3
2) 5x-4*2x-3 = ?
Left Side: Multiply coefficients 5 and 2 together to get 10, and add exponents of x, (-4)+(-3)
= -7. LS: 10x-7
3) 6x5/3x-2 = ?
Left Side: Divide the coefficients, 6 by 3 = 2, and subtract the exponent of the denominator
from the exponent in the numerator, (5)-(-2) = 5 + 2 = 7, thus you end up with: 2x7
FERMAT’S LAST THEOREM – DISPROVEN?
At 03:20 in the clip, there is an equation in the background:
178212 + 184112 = 192212, which seems to be in opposition to Fermat’s Last Theorem,
which states that an equation in the form ax + bx = cx, for x>2 and a,b,c ≠ 0, cannot exist
when x is an integer.
In this case, x=12 is an integer – plugging it into our calculator gives us:
LS: 2.541210259 x 1039 and RS: 2.541210259 x 1039
At first glance, it seems like the laws of this 3D universe have altered and disproven
Fermat’s Last Theorem. However, upon closer inspection, one sees that it still holds true.
Notice that you’re calculator only goes up to 9 decimal places. If we plug the same
numbers into a device/software that generates more decimal places, we would have found
that:
LS: 2,541,210,258,614,589,176,288,669,958,142,428,526,657
RS: 2,541,210,259,314,801,410,819,278,649,643,651,567,616
Now, you might be inclined to say that this is a trick question, because who’s calculator has
that large a display? Well, it turns out you could have answered the question by simply
looking at the equation:
178212 + 184112 = 192212
LS: (even number)(even number) + (odd number)(even number) = even number + odd number
= odd number
RS: (even number)(even number) = even number
Discrepancy: even number ≠ odd number, therefore this is a false solution (but still
quite close!)

8. Grade 12, MCV 4U, Introduction to R3
EXTENDED
SOLUTIONS
CONT’D
VECTORS IN 3-SPACE, R3
For the following problems, the red lines on the axis of the screenshot represent one point
each in the positive direction.
3) The position of Homer’s feet are on the xy-plane, hence z=0. Therefore the location of
his feet are at approximately (x,y,z) = (6.5, 10.5, 0)
The z-coordinate of the position of Homer’s head (top) can be approximated by drawing a
line parallel to the xy-plane from his head to the z-axis. This gives you approximately z=1,
thus giving you coordinates (x,y,z) = (6.5, 10.5, 1).
4) To determine Homer’s exact position in R3 would require finding his centre of mass, but
since we are asked to determine this approximately, simply find the midpoint of the two
extreme points (his head versus his feet). Since the (x,y) coordinates will remain the same,
you only need to take the midpoint of the z points: 0 and 1, which gives you z = 0.5.
Midpoint = (6.5, 10.5, 0.5)

9. DISCLAIMER:
I do not own, nor represent, nor am affiliated
with The Simpsons creators, publishers, media,
images, videos, etc.
These lesson plans are only intended as an
educational tool for the math teacher.