Crime Scene Decoders Algebra: Relations, Functions & Patterns

© 21st Century Math Projects

Crime Scene Decoders Algebra: Relations, Functions & Patterns

Standard Focus : Patterns Algebra and Functions, Geometry and Spatial Sense

Time Range : 1-3 Days

Supplies : Pencil and Paper

Topics of Focus :

- Coordinate Planes

- Domain & Range

- Inverses

- Patterns & Arithmetic Sequences

This particular was mapped to the curriculum in Glencoe Algebra 1 Chapter 4 and can be used as an enrichment or review activity.

Functions 8.F 1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1

Functions 8.F 3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

Interpreting Functions

F-IF 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range.


F-IF 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.


F-IF 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.


F-IF 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Building Functions F-BF 1. Write a function that describes a relationship between two quantities.★

Building Functions F-BF 4a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

Building Functions F-BF 4b. (+) Verify by composition that one function is the inverse of another.


CSD: The eviDenCe

name:_______________________________

1.

Clue

___________ 2.

Clue

___________ 3.

Clue

___________


4.

Clue

___________ 5.

Clue

___________ 6.

Clue

___________ CrypTiC TexT meSSage

SuSpeCT

___________


Crime Scene Decoders Algebra: Relations, Functions & Patterns

Detectives,

The evil genius group, the Mathemagicians, have taken a European vacation. Indications are that they are behind a number of robberies across the continent and they are making progress in the construction of a world conquering device. Notes have been found at each site and have been signed “The Diophantus of the Opera”. Through our intelligence we’ve determined this individual is a high ranking member of the group.

Once again we have received a cryptic text message and we’ve been told the result will calculate to the Diophantus’s favorite number. Thus far there are six suspects that police have questioned. It is hoped that someone with a relatively strong number sense and spatial reasoning skills can crack some codes that have puzzled the detectives on the case so far.

Your job is to bring the Diophantus of the Opera to justice and save the planet. Since you are being brought in as a specialist you have to have definitive proof in order for any arrest to hold up in court. You need to be prepared to state your case and demonstrate your understanding of the following skills that the Diophantus is known to use in the notes.

- Coordinate Planes

- Domain & Range

- Patterns & Arithmetic Sequences

Be sure to include:

- Other examples of the concepts

- Definitions

- Any other relevant information.

This is not a time to be sloppy. The slightest miscalculation or illegible footnote could result in a not guilty verdict. Oh, did I mention that use of a calculator might prematurely set off his world conquering device? Good luck to you, gumshoe.

Chief Harris


Who is the Diophantus of the Opera?

Name: Hailey

Occupation: Panini Maker

Favorite Number : 99

Name: Vernon

Occupation: Network Administrator

Favorite Number : -10

Name: Jeff

Occupation: Blogger

Favorite Number: π

Name: Owen

Occupation: Graphic Designer

Favorite Number : 1

Name: Jessica

Occupation: Musician

Favorite Number: φ

Name: Rishi

Occupation: App Developer

Favorite Number: 19


Scene #1 Descartes Home -- Ville de Descartes, France

Yesterday the Diophantus of the Opera broke into the childhood home of esteemed geometer, René Descartes. The owners woke up to find the lucky straightedge of Descartes was missing.

René Descartes worked awfully hard creating his Cartesian Coordinate System, but it strikes me as a little too trivial for our grand premiere. The Diophantus of the Opera is here… inside your mind, but how about later?

Here are the directions to my odyssey. The name of the final destination will be equal to -8

1. From my starting point (the star), I’m going 5o N and 7o E. I am in _______________

2. Then 8o S. I am in ________________________

3. Then I’m gracefully traveling 5o S and 10o E. I am in _______________

4. Finally, 14o N and 24o W to my final destination _____________

______= -8

47oN 5oE


Scene #2 Eiffel Tower –- Paris, France

It is suspected that the Diophantus of the Opera broke into the Eiffel Tower, climbed to the top and proceeded to drop euro coins. A figure was seen taking measurements and investigators later found this note.

Although most of the great French mathematicians were not around to see it, I’m sure they would have loved to have thrown coins from atop the Eiffel Tower. The distance the coin will fall is 4.9 meters the first second, 14.7 meters the next second and so on in an arithmetic sequence. If the coin continues to fall at this rate, how many meters will it fall during the 8th second?

You might like this equation... an = a1 + (n – 1)d e=_______ Scene #3 Remains of the Berlin Wall, Germany

It is believe that after his raid in France, the Diophantus hitchhiked to Germany. Our researchers seems chunks of the mortar from the remains of the Berlin Wall was chiseled away and stolen.

Brava, Brava, Bravissima. I seem to have reached the Point of No Return. Is this yours?

f(x) = 3x - 2 g(x) = x2 – 4 q(x) = PHANTOMx

P = f(2) + 4 H= 2[g(3)] A= f(0) + 5 N= g(-2) + 3

T= 5[f(4)] – 49 O= -3[g(1) + 9] M= f(1) + g(1)

q(x) = 6480? X=_______

t (seconds) 1 2 3 4 5 6 7 8

meters fallen during that second 4.9 14.7 24.5 e


Scene #4 Aristotle University of Thessaloniki -- Greece

A Greek manuscript filled with philosophical writings was found on a table next to a dozen empty cans of Mountain Dew. Investigators believe the Diophantus was attempting to translate the document.

My sweet Christine, tis best if we can improve our relations. Remember that relations may take many different forms and look different ways -- masks of a sort. Unfortunately only one of our famous Greek friends could translate their relation correctly every time. Who’s right?

R = {(1, 4), (-2, 5), (5, 5), (2, 3), (-1, -1)}

Greek Mathematician Domain & Range Table, Mapping

Relation, or Graph Is it a

Function? Inverse

Aristotle

x = 10 D: {-2, -1, 1, 2, 5} R: {-1, 3, 4, 5}

Yes {(4,1), (2, -5), (5, 5), (3, 2), (-1, -1)}

Euclid

e = -5 D: {-2, -1, 1, 2, 5} R: {-1, 3, 4, 5}

Yes {(4,1), (5, -2), (5, 5), (3, 2), (-1, -1)}

Plato

u = -4 D: {-2, -1, 1, 3, 5} R: {-1, 2, 4, 5}

Yes {(4,1), (5, -2), (5, 5), (3, 2), (-1, -1)}

Ptolemy

ω = 0 D: {-2, -1, 1, 2, 5} R: {-1, 3, 4, 5}

Yes {(4,1), (5, -2), (5, 5), (3, 2), (-1, -1)}

Pythagoras

k = 20 D: {-2, -1, 1, 2, 5} R: {-1, 3, 4, 5}

No {(4,1), (5, -2), (5, 5), (3, 2), (-1, -1)}

Who’s right?

____=___


Scene #5 Stonehenge -- England

The curators of the mysterious ruins found that a mythic satchel was missing. They believe the contents are jewels from a departed alien life form. This could potentially be the fuel for a world conquering device. Atop one of the mammoth stones, investigators found nearly a hundred half eaten apple cores and this note written with sidewalk chalk.

The final act is near. I thought it’d be best to honor the locals by paying respects to their most significant mathematical figure – Isaac Newton. I toyed with some

different apple tossing strategies to inspire the genius I’ll soon need to complete this world conquering device.

I leave you with my last puzzle. A bit of calculus. Here are the models for my two different apple tosses.

Complete the tables. Graph them.

f(t) = 2t g(t) = t2

If you shade the region between the intersection points of the two functions, which shape will be drawn? This is the final piece of the puzzle.

y= 9 z= 9 e= 9 x= 9


Scene #6 Gondola Dock -- Venice, Italy

A pair of gondoliers came upon a surprise when thet found the following carved in the floor of their boats. Soon after, they discovered some famous Fibonacci mystical artwork was missing. Later, investigators were sent a cryptic text message.

Blaise Pascal and Fibonnacci join together for a duet aria for the ages. Who will steal the stage? The Triangle or the Golden Spiral?

A + B + C +D = Z

Z = ______

CRYPTIC PUZZLE SOLVER TEXT MESSAGE

It’s golden! Good luck cracking this code. ~The Diophantus of the Opera

D

))(( xUKyUKze

−

−+ω


Who is the Diophantus of the Opera?

Name: Hailey

Occupation: Panini Maker

Favorite Number : 99

Name: Vernon

Occupation: Network Administrator

Favorite Number : -10

Name: Jeff

Occupation: Blogger

Favorite Number: π

Name: Owen

Occupation: Graphic Designer

Favorite Number : 1

Name: Jessica

Occupation: Musician

Favorite Number: φ

Name: Rishi

Occupation: App Developer

Favorite Number: 19


Scene #1 Descartes Home -- Ville de Descartes, France

Yesterday the Diophantus of the Opera broke into the childhood home of esteem geometer, René Descartes. The owners woke up to find the lucky straightedge of Descartes was missing.

René Descartes worked awfully hard creating his Cartesian Coordinate System, but it strikes me as a little too trivial for our grand premiere. The Diophantus of the Opera is here… inside your mind, but how about later?

Here are the directions to my odyssey. The name of the final destination will be equal to -8

1. From my starting point, I’m going 5o N and 7o E. I am in GERMANY

2. Then 8o S. I am in ITALY

3. Then I’m gracefully traveling 5o S and 10o E. I am in GREECE

4. Finally, 14o N and 24o W to my final destination _____________

UK= -8

47oN 5oE


Scene #2 Eiffel Tower –- Paris, France

It is suspected that the Diophantus of the Opera broke into the Eiffel Tower, climbed to the top and proceeded to drop euro coins. A figure was seen taking measurements and investigators later found this note.

Although most of the great French mathematicians were not around to see it, I’m sure they would have loved to have thrown coins from atop the Eiffel Tower. The distance the coin will fall is 4.9 meters the first second, 14.7 meters the next second and so on in an arithmetic sequence. If the coin continues to fall at this rate, how many meters will it fall during the 8th second?

You might like this equation... an = a1 + (n – 1)d e=73.5

Scene #3 Remains of the Berlin Wall, Germany

It is believe that after his raid in France, the Diophantus hitchhiked to Germany. Our researchers seems chunks of the mortar from the remains of the Berlin Wall was chiseled away and stolen.

Brava, Brava, Bravissima. I seem to have reached the Point of No Return. Ironically where there once was a point of no return. Is this yours?

f(x) = 3x - 2 g(x) = x2 – 4 q(x) = PHANTOMx

P = f(2) + 4 = 8 H= 2[g(3)] = 10 A= f(0) + 5 = 3 N= g(-2) + 3 = 3

T= 5[f(4)] – 49 = 1 O= -3[g(1) + 9] = -18 M= f(1) + g(1) = -2

q(x)? = 6480

(8)(10)(3)(3)(1)(-18)(-2)(X) = 6480

25,920X = 6480

X = ¼ X=_______

t (seconds) 1 2 3 4 5 6 7 8

meters fallen during that second 4.9 14.7 24.5 e


Scene #4 Aristotle University of Thessaloniki -- Greece

A Greek manuscript filled with philosophical writings was found on a table next to a dozen empty cans of Mountain Dew. Investigators believe the Diophantus was attempting to translate the document.

My sweet Christine, tis best if we can improve our relations. Remember that relations may take many different forms and look different ways -- masks of a sort. Unfortunately only one of our famous Greek friends could translate their relation correctly every time. Who’s right?

R = {(1, 4), (-2, 5), (5, 5), (2, 3), (-1, -1)}

Greek Mathematician Domain & Range Table, Mapping

Relation, or Graph Is it a

Function? Inverse

Aristotle

x = 10 D: {-2, -1, 1, 2, 5} R: {-1, 3, 4, 5}

Yes {(4,1), (2, -5), (5, 5), (3, 2), (-1, -1)}

Euclid

e = -5 D: {-2, -1, 1, 2, 5} R: {-1, 3, 4, 5}

Yes {(4,1), (5, -2), (5, 5), (3, 2), (-1, -1)}

Plato

u = -4 D: {-2, -1, 1, 3, 5} R: {-1, 2, 4, 5}

Yes {(4,1), (5, -2), (5, 5), (3, 2), (-1, -1)}

Ptolemy

ω = 0 D: {-2, -1, 1, 2, 5} R: {-1, 3, 4, 5}

Yes {(4,1), (5, -2), (5, 5), (3, 2), (-1, -1)}

Pythagoras

k = 20 D: {-2, -1, 1, 2, 5} R: {-1, 3, 4, 5}

No {(4,1), (5, -2), (5, 5), (3, 2), (-1, -1)}

Who’s right?

____=___


Scene #5 Stonehenge -- England

The curators of the mysterious ruins found that a mythic satchel was missing. They believe the contents are jewels from a departed alien life form. This could potentially be the fuel for a world conquering device. Atop one of the mammoth stones, investigators found nearly a hundred half eaten apple cores and this note written with sidewalk chalk.

The final act is near. I thought it’d be best to honor the locals by paying respects to their most significant mathematical figure – Isaac Newton. I toyed with some

different apple tossing strategies to inspire the genius I’ll soon need to complete this world conquering device.

I leave you with my last puzzle. A bit of calculus. Here are the models for my two different apple tosses.

Complete the tables. Graph them.

f(t) = 2t g(t) = t2

If you shade the region between the intersection points of the two functions, which shape will be drawn? This is the final piece of the puzzle.

y= 9 z= 9 e= 9 x= 9


Scene #6 Gondola Dock -- Venice, Italy

A pair of gondoliers came upon a surprise when the found the following carved in the floor of their boats. Soon after, they discovered some famous Fibonacci mystical artwork was missing. Later, investigators were sent a cryptic text message.

Blaise Pascal and Fibonnacci join together for a duet aria for the ages. Who will steal the stage? The Triangle or the Golden Spiral?

A + B + C +D = Z

4 + 10 + 10 +13 = 37

Z = 37 CRYPTIC PUZZLE SOLVER TEXT MESSAGE

It’s golden! Good luck cracking this code. ~The Diophantus of the Opera Z=37 e= 73.5 UK=-8 x = ¼ ω =0 y=9 ANSWER IS

D

))(( xUKyUKze

−

−+ω

Crime Scene Decoders Algebra: Relations, Functions & Patterns

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