Ahadie, Julia, S02, Math Lens Assignment (no video playback)

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 Th i s P a c k a g e

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

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: 𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 “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 cm 3 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

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

Pe

rce

nta

ge o

f To

tal P

ow

er

Stages of Power Loss

Radioactive Man's Power Decline 2. 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.

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

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

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!)

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)

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.