crime scene decoders rational functions

15
© 21 st Century Math Projects Crime Scene Decoders: Rational Functions Standard Focus: Patterns, Algebra & Functions Time Range : 1-3 Days Topics of Focus : - Multiplying and Dividing Rational Expressions - Solving Equations with Rational Functions - Simplifying Complex Fractions - Asymptotes - Graphing Rational Functions Quantities N-Q 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Creating Equations A-CED 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Creating Equations A-CED 2. Create equations in two or more variables to represent relationships between quantities; graph equations on axes with labels and scales. Reasoning with Equations and Inequalities A-REI 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Interpreting Functions F-IF 7. Graph functions expressed symbolically and show key features of the graph, by hand and using technology for more complicated cases.Interpreting Functions F-IF 7d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Procedures: A.) Student will be given the letter from Chief Harris, the possible suspects, and crime scene puzzles. B.) Students will work in pairs or individually to solve the crime. C.) Students will determine which suspect should be arrested. D.) Students will create a portfolio of evidence proving that they have arrested the right person and will demonstrate their understanding of their mathematical content present in the problem. *Extensions.) Students can present their evidence to the class as an oral presentation.

Upload: others

Post on 21-Jan-2022

70 views

Category:

Documents


3 download

TRANSCRIPT

Page 1: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Crime Scene Decoders: Rational Functions

Standard Focus: Patterns, Algebra & Functions Time Range : 1-3 Days

Topics of Focus :

- Multiplying and Dividing Rational Expressions

- Solving Equations with Rational Functions

- Simplifying Complex Fractions

- Asymptotes

- Graphing Rational Functions

Quantities N-Q 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Creating Equations

A-CED 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Creating Equations

A-CED 2. Create equations in two or more variables to represent relationships between quantities; graph equations on axes with labels and scales.

Reasoning with Equations and

Inequalities A-REI

2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Interpreting Functions

F-IF 7. Graph functions expressed symbolically and show key features of the graph, by hand and using technology for more complicated cases.★

Interpreting Functions

F-IF 7d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

Procedures:

A.) Student will be given the letter from Chief Harris, the possible suspects, and crime scene puzzles.

B.) Students will work in pairs or individually to solve the crime.

C.) Students will determine which suspect should be arrested.

D.) Students will create a portfolio of evidence proving that they have arrested the right person and will demonstrate their understanding of their mathematical content present in the problem.

*Extensions.) Students can present their evidence to the class as an oral presentation.

Page 2: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

CSD: The eviDenCe

name:_______________________________

1.

Clue

___________ 2.

Clue

___________ 3.

Clue

___________

Page 3: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

4.

Clue

___________ 5.

Clue

___________ 6.

Clue

___________ CrypTiC TexT meSSage

SuSpeCT

___________

Page 4: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Crime Scene Decoders Algebra 2 & Pre-Calc: Rational Functions

Detectives,

The boys and girls have once again come out to play. This time, the international evil genius terrorist group the Mathemagicians are running amok over Central America. Although we have apprehended a number of their associates, we have yet to recover any pieces of their ill-conceived world conquering device that they soon may deploy to – conquer the world. From the intelligence we have gathered, the anonymous henchman “Asymptote Swaggster” is the culprit behind these robberies.

As with other crimes related to the Mathemagicians, they have left behind a trail of mathematical puzzles and a cryptic text message that needs to be deciphered. When substituted correctly, the text message will calculate to the Swaggster’s favorite number. So far there are six suspects that police have questioned. It is hoped that someone with a relatively strong number sense can crack some codes that have puzzled the detectives on the case so far.

Your job is to bring Asymptote Swaggster to justice and save the planet. You need to be prepared to state your case and demonstrate your understanding of the following skills that Asymptote Swaggster is known to use in the notes.

- Multiplying and Dividing Rational Expressions

- Solving Equations with Rational Functions

- Simplifying Complex Fractions

- Asymptotes

- Graphing Rational Functions

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

Page 5: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Who is Asymptote Swaggster?

Name: Hazel

Occupation: Pharmacist

Favorite Number: 8

Name: Enrique

Occupation: Pro Ping Pong Player

Favorite Number: -59

Name: Quinn

Occupation: Cake Decorator

Favorite Number: 17 ¼

Name: Waldo

Occupation: Private Investigator

Favorite Number: 911

Name: Stillman

Occupation: ATV Salesman

Favorite Number: -77

Name: Ting Fen

Occupation: Speech Therapist

Favorite Number: -90

Page 6: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Scene #1 Television Manufacturer –- Juárez, Mexico Asymptote Swaggster broke into an international TV manufacturer and the inventory of unused cathode ray tubes was stolen. Investigators found this note in the parking lot.

Hello all! It’s Swaggster here and I’m gonna perplex you with puzzles. While I’m stealing stuff I’m going to look for some rational mathematics. Just like this one I pulled off the CEO’s desk.

Giga Electronica Proposal: Add computer manufacturing to business. Build 650 computers as trial run.

Estimated Start-Up (Fixed) Costs: 250,000 pesos

Itemized Cost: each computer from scratch is 2,000 pesos.

Necessary Function: 𝑐𝑐̅ = (𝑓𝑓𝑓𝑓𝑥𝑥𝑓𝑓𝑓𝑓 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)+𝑐𝑐𝑥𝑥𝑥𝑥

, where 𝑐𝑐 is the cost of each item, x is the number of items produced and 𝑐𝑐̅ is the average total cost of each item.

Implementation Restrictions: We will only do this if the average total cost for each computer is under 2,400 pesos.

Should we do this?

Yes No

m = 1000 r = -1000

Scene #2 Instituto Tecnologica de Costa Rica –- San José, Costa Rica

After a campus tour, Swaggster snuck away from the tour group and into the Department of Engineering. From there, they discovered the laser cutter was taken.

While I robbed the local college, I peeked in a math class. I can’t help myself.

Isabella and Santiago graph functions on a calculator. Isabella graphs 𝟔𝟔𝟔𝟔+𝟐𝟐𝟔𝟔

and Santiago graphs + 𝟐𝟐𝟔𝟔 . They both have vertical asymptotes at x

= 0 and horizontal asymptotes at y = 6. They are confused. What is the correct explanation?

They have the same asymptotes , but are different functions

They have the same asymptotes because they’re the same function They are incorrect

a = 0 c = 6 i = -10

Page 7: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Scene #3 Mayan Ruins –- Zaculeo, Guatemala

Conservationists were stunned to find out that the large stucco mask of Temple 33 was taken. It is unclear to investigators how the mask may be used in the World Conquering Device.

The Mayan could build cities with rocks, mud and their hands. Can you solve five rational function questions in a row? I have my doubts.

The excluded value of g(x) equals b

𝑔𝑔(𝑥𝑥) = 𝑥𝑥2+4𝑥𝑥−5(𝑥𝑥−4)(𝑥𝑥+𝑎𝑎)

𝑑𝑑(𝑥𝑥) =𝑏𝑏 + 1

𝑥𝑥−6𝑎𝑎 + 2

𝑥𝑥3

The excluded value of f(x) equals a

𝑓𝑓(𝑥𝑥) = 𝑥𝑥2− 6𝑥𝑥+ 17𝑥𝑥+1

𝑦𝑦(𝑥𝑥) = 𝑑𝑑(𝑥𝑥) ∙3𝑥𝑥3 + 1𝑥𝑥4

What are the correct horizontal and vertical asymptotes of z(x)

Horizontal at y = 0 Vertical at x = -8, x = 0, x = 2 e =3 Horizontal at y = 0 Vertical at x = -2, x = 0, x = 4 i =3 Horizontal at y = 1 Vertical at x = -4, x = 0, x = 8 r =3

𝑧𝑧(𝑥𝑥) = 𝑦𝑦(𝑥𝑥) ÷𝑥𝑥2 + 6𝑥𝑥 − 16𝑥𝑥(𝑥𝑥 + 4)

START HERE

Page 8: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Scene #4 Hospital CIMA –- San José, Costa Rica

One of the nicest hospitals in Central America, Hospital CIMA was broken into last night and their radiology equipment was stolen. Later, nurses found this note from Asymptote Swaggster.

The Doctor is in! The Dr. of Swagg, that is. While robbing the hospital, I decided to impersonate an Anesthesiologist for a few hours. I had to administer some anesthesia, did I mess up?

Anesthesiology Report

Local anesthesia was administered to the patient at 1:58 prior to surgery.

The concentration of the drug in the bloodstream (in mg/L) is modeled by the

function where t is minutes since receiving the drug: 𝐶𝐶(𝑡𝑡) = 8𝑐𝑐0.3𝑐𝑐2+4.1

Note: When the concentration drops below 0.1 mg/L the patient may become conscious. They

will need the additional anesthesia administered one half hour prior to reaching this level.

The surgery expects to last 6 hours. Based on this information, when would I need to administer more anesthesia?

5:48 5:54 6:02

a = 548 i = 554 m = 602

Scene #5 Sinkhole –- Guatemala City, Guatemala

One of the largest sinkholes on record, the Guatemala City Sinkhole was looted last night. It is believed that Asymptote Swaggster stole the underground piping.

I’ve got some bad news. The earth is about to fall out from under you. Luckily, my fortune teller has given me an equation that models its destruction. Graph it!

𝑆𝑆𝑖𝑖𝑆𝑆𝑆𝑆ℎ𝑜𝑜𝑜𝑜𝑜𝑜(𝑥𝑥) = − 3𝑥𝑥4−12𝑥𝑥3+22𝑥𝑥2+84𝑥𝑥+49

+ 1

How many Sinkholes are going to form?

One A=1

Two R=2

Three M=3

Page 9: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Scene #6 Caribbean Sea –- Belize City, Belize

A figure parachuted from a helicopter and landed on a yacht in the middle of the Caribbean Sea. The passengers were sent to shore in lifeboats, but the yacht was taken. Victims claimed the figure kept yelling “My name is Asymptote Swaggster! Go tell police! I’m Swagg!”

I’m floating off into the sunset on my brand-new stolen yacht. There are a lot of vessels out on the Caribbean. Jet Skis are flying around circles around me and I’m just waiting for one of them to crash.

The routes of three Jet Skis are modeled by a rational function. Graph the function and determine how many collisions (if any) there will be.

𝑓𝑓(𝑥𝑥) =2𝑥𝑥2 + 3𝑥𝑥2 − 9

How many yachts will get crashed into?

CRYPTIC PUZZLE SOLVER TEXT MESSAGE

Good luck. Lol. Stay Swaggin.

C • A + (M ÷ (E + R) – i) ÷ C – A

Asymptote Swaggster

0 a = 0

3 c = 3

5 r = 5

Page 10: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Who is Asymptote Swaggster?

Name: Hazel

Occupation: Pharmacist

Favorite Number: 8

Name: Enrique

Occupation: Pro Ping Pong Player

Favorite Number: -59

Name: Quinn

Occupation: Cake Decorator

Favorite Number: 17 ¼

Name: Waldo

Occupation: Private Investigator

Favorite Number: 911

Name: Stillman

Occupation: ATV Salesman

Favorite Number: -77

Name: Ting Fen

Occupation: Speech Therapist

Favorite Number: -90

Page 11: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Scene #1 Television Manufacturer –- Juárez, Mexico Asymptote Swaggster broke into an international TV manufacturer and the inventory of unused cathode ray tubes was stolen. Investigators found this note in the parking lot.

Hello all! It’s Swaggster here and I’m gonna perplex you with puzzles. While I’m stealing stuff I’m going to look for some rational mathematics. Just like this one I pulled off the CEO’s desk.

Giga Electronica Proposal: Add computer manufacturing to business. Build 650 computers as trial run.

Estimated Start-Up (Fixed) Costs: 250,000 pesos

Itemized Cost: each computer from scratch is 2,000 pesos.

Necessary Function: 𝑐𝑐̅ = (𝑓𝑓𝑓𝑓𝑥𝑥𝑓𝑓𝑓𝑓 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)+𝑐𝑐𝑥𝑥𝑥𝑥

, where 𝑐𝑐 is the cost of each item, x is the number of items produced and 𝑐𝑐̅ is the average total cost of each item.

Implementation Restrictions: We will only do this if the average total cost for each computer is under 2,400 pesos.

Should we do this?

Yes 625 No

m = 1000 r = -1000

Scene #2 Instituto Tecnologica de Costa Rica –- San José, Costa Rica

After a campus tour, Swaggster snuck away from the tour group and into the Department of Engineering. From there they discovered the laser cutter was taken.

While I robbed the local college, I peeked in a math class. I can’t help myself.

Isabella and Santiago graph functions on a calculator. Isabella graphs 𝟔𝟔𝟔𝟔+𝟐𝟐𝟔𝟔

and Santiago graphs + 𝟐𝟐𝟔𝟔 . They both have vertical asymptotes at x

= 0 and horizontal asymptotes at y = 6. They are confused. What is the correct explanation?

They have the same asymptotes , but are different functions

They have the same asymptotes because they’re the same function They are incorrect

a = 0 c = 6 i = -10

Page 12: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Scene #3 Mayan Ruins –- Zaculeo, Guatemala

Conservationists were stunned to find out that the large stucco mask of Temple 33 was taken. It is unclear to investigators how the mask may be used in the World Conquering Device.

The Mayan could build cities with rocks, mud and their hands. Can you solve five rational function questions in a row? I have my doubts.

The excluded value of g(x) equals b

𝑔𝑔(𝑥𝑥) =𝑥𝑥2 + 4𝑥𝑥 − 5

(𝑥𝑥 − 4)(𝑥𝑥 + 𝑎𝑎)

𝑔𝑔(𝑥𝑥) = (𝑥𝑥+5)(𝑥𝑥−1)(𝑥𝑥−4)(𝑥𝑥−1)

B = 4

𝑑𝑑(𝑥𝑥) =𝑏𝑏 + 1

𝑥𝑥−6𝑎𝑎 + 2

𝑥𝑥3

𝑑𝑑(𝑥𝑥) =4𝑥𝑥3 + 𝑥𝑥2

6𝑥𝑥3 + 2

The excluded value of f(x) equals a

a = -1

𝑓𝑓(𝑥𝑥) = 𝑥𝑥2− 6𝑥𝑥+ 17𝑥𝑥+1

𝑦𝑦(𝑥𝑥) = 𝑑𝑑(𝑥𝑥) ∙3𝑥𝑥3 + 1𝑥𝑥4

𝑦𝑦 =4𝑥𝑥3 + 𝑥𝑥2

6𝑥𝑥3 + 2 ∙3𝑥𝑥3 + 1

𝑥𝑥4

𝑦𝑦 =4𝑥𝑥 + 1

2𝑥𝑥2

What are the correct horizontal and vertical asymptotes of z(x)

Horizontal at y = 0 Vertical at x = -8, x = 0, x = 2

e =3

Horizontal at y = 0 Vertical at x = -2, x = 0, x = 4 i =3 Horizontal at y = 1 Vertical at x = -4, x = 0, x = 8 r =3

𝑧𝑧(𝑥𝑥) = 𝑦𝑦(𝑥𝑥) ÷𝑥𝑥2 + 6𝑥𝑥 − 16𝑥𝑥(𝑥𝑥 + 4)

𝑧𝑧 =4𝑥𝑥 + 1

2𝑥𝑥2 ÷

(𝑥𝑥 + 8)(𝑥𝑥 − 2)𝑥𝑥(𝑥𝑥 + 4)

𝑧𝑧 =(4𝑥𝑥 + 1)(𝑥𝑥)(𝑥𝑥 + 4)

(2𝑥𝑥2)(𝑥𝑥 + 8)(𝑥𝑥 − 2) =

(4𝑥𝑥 + 1)(𝑥𝑥 + 4)(2𝑥𝑥)(𝑥𝑥 + 8)(𝑥𝑥 − 2)

START HERE

Page 13: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Scene #4 Hospital CIMA –- San José, Costa Rica

One of the nicest hospitals in Central America, Hospital CIMA was broken into last night and their radiology equipment was stolen. Later, nurses found this note from Asymptote Swaggster.

The Doctor is in! The Dr. of Swagg, that is. While robbing the hospital, I decided to impersonate an Anesthesiologist for a few hours. I had to administer some anesthesia, did I mess up?

Anesthesiology Report

Local anesthesia was administered to the patient at 1:58 prior to surgery.

The concentration of the drug in the bloodstream (in mg/L) is modeled by the

function where t is minutes since receiving the drug: 𝐶𝐶(𝑡𝑡) = 8𝑐𝑐0.3𝑐𝑐2+4.1

Note: When the concentration drops below 0.1 mg/L the patient may become conscious. They

will need the additional anesthesia administered one half hour prior to reaching this level.

The surgery expects to last 6 hours. Based on this information, when would I need to administer more anesthesia? 266.6 minutes ->4 hr 26 minutes

5:48 5:54 6:02

a = 548 i = 554 m = 602

Scene #5 Sinkhole –- Guatemala City, Guatemala

One of the largest sinkholes on record, the Guatemala City Sinkhole was looted last night. It is believed that Asymptote Swaggster stole the underground piping.

I’ve got some bad news. The earth is about to fall out from under you. Luckily, my fortune teller has given me an equation that models its destruction. Graph it!

𝑆𝑆𝑖𝑖𝑜𝑜𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜(𝑥𝑥) = − 3𝑥𝑥4−12𝑥𝑥3+22𝑥𝑥2+84𝑥𝑥+49

+ 1

How many Sinkholes are

going to form?

One A=1

Two R=2

Three M=3

Page 14: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

Scene #6 Caribbean Sea –- Belize City, Belize

A figure parachuted from a helicopter and landed on a yacht in the middle of the Caribbean Sea. The passengers were sent to shore in lifeboats, but the yacht was taken. Victims claimed the figure kept yelling “My name is Asymptote Swaggster! Go tell police! I’m Swagg!”

I’m floating off into the sunset on my brand-new stolen yacht. There are a lot of vessels out on the Caribbean. Jet Skis are flying around circles around me and I’m just waiting for one of them to crash.

The routes of three Jet Skis are modeled by a rational function. Graph the function and determine how many collisions (if any) there will be.

𝑓𝑓(𝑥𝑥) =2𝑥𝑥2 + 3𝑥𝑥2 − 9

How many yachts will get crashed

into?

CRYPTIC PUZZLE SOLVER TEXT MESSAGE

Good luck. Lol. Stay Swaggin.

C • A + (M ÷ (E + R) – i) ÷ C – A

Asymptote Swaggster

0 a = 0

3 c = 3

5 r = 5

Page 15: Crime Scene Decoders Rational Functions

© 21st Century Math Projects

From Puzzle 1 m = 1000

From Puzzle 2 c = 6

From Puzzle 3 e =3

From Puzzle 4 i = 554

From Puzzle 5 R=2

From Puzzle 6 a = 0

C • A + (M ÷ (E + R) – i) ÷ C – A

6 • 0 + (1000 ÷ (3 + 2) – 554) ÷ 6 – 0 = -59