The Function DogPutting the “Fun” in Function Since 2002

Corey Cogswell

The Function DogPutting the “Funk” in Function Since 2002

Wrong Funk

Right Funk

Gross-out math is intended for grossing out students, not teachers. If at any point during this presentation, you are offended, then it’s totally on you and your sensibilities.

This presentation is meant for purely instructional purposes and not for personal use. What that use might be, no one knows… but no judgments…

Finally, this is not intended to change your life, and if it does so, your life probably needed changing any way…

Disclaimers

I graduated with a degree in mathematics from Rice University in 2002.

I have taught 8 years in Pearland ISD. I have taught Algebra 1, Geometry, Algebra 2, SAT

Prep, AMDM, PAP Precalculus, and AP Caclulus AB, and I will be adding AP Stats to that list this year.

I am currently writing a textbook for a fourth year math class, Pre-College Math, to satisfy the four by four rule.

I just began my master’s program through Lamar Academic Partnerships in Educational Technology Leadership.

Brief Biography

I understand the irony in a 2-page “brief” biography.

I am the math / speech / interview coach for Pearland’s Two-Time State Champion Academic Decathlon team.

I am a playwright in my spare time. I dislike guacamole. I have a Welsh corgi named Gozer the

Destructor.

Brief Biography (cont.)

The Evolution of the Function Dog

2002 - 20052005 - 20062006 - Present

Order of Operations Functions Conceptually Function Notation Properties Composite Functions Function Transformation (New for 2010) Function Inverses (Updated for 2010) Trigonometry Chain Rule for Differentiation

When to use the Function Dog

Pre-Algebra

Please Excuse My Dear Aunt Sally

Order of Operations

Parentheses Exponents Multiplication Division Addition Subtraction

Order of Operations

GroupingsExponentsMultiplicationAddition

New Order of Operations

Grandmothers

ExpoundMultipleAnecdotes

GirlsExpectMemorizedAnniversaries

GorillasEatMimesAddictively

The simplest operation one can do with two numbers is add or subtract them.

Order of Operations

35 8

+

The simplest operation one can do with two numbers is add or subtract them.

Order of Operations

35 2-

The next simplest operation one can do with two numbers is multiply or divide them.

Order of Operations

35 15

×

The next simplest operation one can do with two numbers is multiply or divide them.

Order of Operations

35 53

÷

The most complex operation one can do with numbers (in Pre-Algebra) is applying an exponent or radical to them.

Order of Operations

5 25^2

The most complex operation one can do with numbers (in Pre-Algebra) is applying an exponent or radical to them.

Order of Operations

25 5

√

The most complex dogs eat first.

Order of Operations

^2 √

The next most complex dogs eat second.

Order of Operations

× ÷

The least complex dogs eat last.

Order of Operations

+ -

Groups of dogs, or packs, can be found inside parentheses, numerators, denominators, or radicals. Packs eat before everyone else, maintaining internal pecking order.

Dogs of equal complexity eat in the order they line up.

Order of Operations

Example 1

There are four symbols in this expression and thus four dogs have lined up for dinner.

Based on our rules, × will eat first followed by ÷, then +, and finally -.

5

210

×

Example 1

10

52

÷

Example 1

2

4

6+

Example 1

6

3

3

-

Example 2

18

2

36××

Example 2

36 6×√

Example 2

2

6

12

×÷

Example 2

2

-1

4

5

×-

Example 2

2

-1

1×+

Algebra 1

Relations

Functions

Example

x4x+5

3 17

y

If all functions are named y, then we run into a slight problems when there are multiple functions…

Function Notation

yy

y

y

y

yy

y

y y

y

Which y were you looking for?

Function Notation

x

f

f(x)

g

g(x)

Example

x

f

f(x)4 f(4)

=3(4)2+5=

Input Output

Coordinate: (4,53)

53

Geometry

Commutative Property

Properties

3 +85

Associative Property

Properties

× ×

Distributive Property

Properties

+3

5

2

8

× 16

2×

6

10

Trigonometry

sin

sin(θ)θ

ANGLERATIO

cos

cos(θ)

tan

tan(θ)

Algebra 2

Composite Functions

xf(g(h(x)))g(h(x))h(x)

ghf

Example and

2g

6f

26

x g f

External TransformationsFunction Transformations

xf

f(x)

Internal TransformationsFunction Transformations

x f(x)x+c

b(x+c)f(b(x+c))

×b+cf

Example 1

f(2(x+5))x fx

Function Inverses

x f(x)

f-1f

Example 1

5 10f-1

2 4f

Example 2

5 25f-1

-5f

Example 3

f3 27

f-1

Logarithmic Aside

3x = 27

log

Logarithmic Aside

Logarithmic Aside

Example 3

f3 27

f-1

Reciprocal Functions

x 1xf

Precalculus

Inverse Trig Functions

θ sin(θ)sin-1sin

Reciprocal Trig Functions

θ 1 sin θ

csc

Comparison

x sin(x)sin-1

x 1 sin x

csc

This x is an angle.

This x is a ratio.

Calculus

Differentiation

ff’x

f(x)f’(x)

Chain Rule

ff(g(h(x)))

xh’(x)

g’h’ f’f’(g(h(x)))g’(h(x))

gh(x) g(h(x))h

Example

sin usin(e3x2)

x6x

eu6u cos u

cos(e3x2)e3x2eu

3x2 e3x23u2

Closing Thoughts

Everyone likes puppies… even when they have accidents…

Visual representations of abstract concepts help students of all grades and abilities learn, regardless of maturity level.

Vertically aligning the presentation of concepts ties the different mathematical subjects together.

For many high school students, the range of a dog, and in fact most biological entities, is funny.

Benefits of Using the Function Dog

Contact Information

Pearland High Schoolcogswellc@pearlandisd.org

This presentation and other materials can be found at:

http://www.pearlandisd.org/webpages/ccogswell/

Corey Cogswell

GOZER THE DESTRUCTOR COGSWELL

THIS SESSION WAS SPONSERED BY