Quest for the Fortune

Cookie of

KnowledgeGeometry: Chapter 3 ReviewMathematical Reasoning and Problem Solving

Introduction

! Our team of three adventurers have em-barked on a mission to find the Fortune Cookie of Knowledge. This rare wonder of the world has been found by few and corrupted many. The Cookie will answer any one question regarding the fortune of the person asking. Our three travelers--Paula, Al-ladin, and Chester--are de-termined to find the Cookie and each ask of it one question. Their jour-ney will not be easy, and they will run into prob-lems along the way (prob-lems similar to those we have covered in chapter 3). Solve the problems that arise in this book and help our wandering friends find what they search for. You will need a sheet of paper in front of you on which to work some of the problems. Use the tools in the book to remind yourself of some of the concepts we have covered. If you have further questions, bring them to me. Good luck and Godspeed.

Here are our learning objectives:

! •! Apply problem-solving strategies to diagrams, equations, and problems

! •! Correctly write and interpret “if ” statements

! •! Correctly judge the validity of differ-ent statements using logic

! •! Construct logical arguments and use counterexamples to disprove statements

! •! Complete geometric proofs using proven theorems

i

On the surface of the mountain are written numbers. Next to each number is written what the number represents. After that, the mountain gives a question. Our explorers need to work backwards to solve these prob-lems. If the problems are all solved cor-rectly, the mountain will open a tunnel that leads us one step closer to the fortune cookie of knowledge.

2

Can’t go through it... Or can you?

1

Our adventurers have come across a moun-tain, but Alladin knows something special about this mountain...

3

Volume of an equilateral triangle: 60What is the length of one of the triangle’s sides?

Surface Area of a Troll’s eye (a sphere): 50π in^2What is the radius of a Troll’s eye?

The arc length of a circle with radius 3in: 5π/2What is the measure of the interior angle made by the arc (with vertex = center of circle)?

Watch this video if you need a reminder or hint.

Educreations Video: Working Backward

Here is what he says: “I guard this bridge, and I like to play games with those who seek to cross. I will begin a statement with the word ‘if ’, and you must complete a true state-ment beginning with ‘then’. IF you succeed, THEN you are free to cross my bridge. IF you lose, THEN you will all make a fine din-ner for me. Do we have a deal?”

Paula felt confident that she could complete this troll’s riddles, so she stepped up to the task.

4

Tricks of the Troll

2

After passing through the secret tunnel, our adventurers run into a Troll at the entrance to a bridge. Why does he have such a devious smile across his face?

If a polygon is a square, then what two things are true about the interior angles?

If two lines intersect, what is true about the opposite angles formed?

If two lines intersect, what is true about the adjacent angles formed?

If the sine of angle A is .3045, what is true about the cosine of angle (90-A)?

?

? A

After besting the Troll and walking quite a ways, the adventurers find themselves at a fork in the road. Fortunately, while deliberat-ing which path to follow, a friendly owl landed on a nearby tree and greeted the jour-neyers. Similar to the Troll, the owl enjoyed playing games; however, not with the threat of being eaten for dinner. He told the adven-turers that he would reveal to them which path leads to the Fortune Cookie of knowl-edge if they could answer correctly questions regarding inverses, converses, and contraposi-tives. The adventurers all accepted.

6

Who gives a Hoot!

3

1. If you are my sibling, then I am your sibling. Is the converse of this statement true or false?

2. If an animal is a cat, then it has four legs.Is the inverse of this statement true or false?

3. If a car is driving 60 mph, then the car is not parked.Is the contrapositive of this state-ment true or false?

ReviewStart with a conditional statement“If p, then q.”

Converse: “If q, then p”Inverse: “If not p, then not q.”Contrapositive: “If not q, then not p.”

He knows where the Fortune Cookie of knowledge is kept! Our adventurers track him down and find him in the local pub. He sits by himself, and our adventurers ap-proach him. They explain their adventure thus far to the wizard, and he laughs. “Games with Trolls and owls are child’s play! In order to earn my knowledge of the For-tune Cookie’s whereabouts, you must match my wit.” Our travelers accept without hesita-tion. The wizard explains that he will give our travelers four statements. One will be true, and three will be false. They must iden-tify the true statement and give one counter-example for each of the false statements.

8

Wizardly Wit4

After traveling the path, our adventurers find themselves in a small town. Chester heard a rumor that in this town lives a wizard with a secret...

If two angles of a triangle are congruent, these angles are complimen-tary.

If three angles of one triangle are congruent to three angles of an-other triangle, then the triangles are congruent.

Corresponding parts of congruent triangles are congruent.

If two angles form a linear pair, these angles are complimentary.

As they journey closer to the outer shell, they see writing on it. What is the writing? A Proof! The given state-ment and the conclusion are

written with a large blank space between. A brand new piece of chalk lies on the floor in front of the shell. This could be it! If our ad-venturers can complete the proof, maybe the shell will open revealing the treasure after which our adventurers search. Chester grabs the chalk and approaches the mighty shell.

10

Proof Yourself Worthy

5

Our Adventurers followed the wizard’s map to a beau-tiful, hidden sanctuary. In the center of the sanctuary is a large, magical, protec-tive shell. We believe the Fortune Cookie of Knowl-edge is hidden within. How do they open it...

Given: Triangle ABC is a right triangle with angle C a right angle

Conclusion: a^2 + b^2 = c^2

A

B

C

a

b

cx

y

Here is a video to remind you of the main ideas be-hind proofs. Watch if you need help.

Khan Academy Video: Proofs

What Question would YOU ask the Fortune

Cookie of Knowledge?

13

ReferencesTitle PageFortune Cookie photo (also on the last page): http://www.agencypost.com/wp-content/uploads/2013/08/fortune-cookie.jpgIntroductionExplorers photo: http://rickr-d.com/wp-content/uploads/2012/12/cartoon-character-designs-concept-art-archaeologists-and-explorers-by-Rick_Ruiz-Dana.jpgChapter 1Mountain photo: http://3.bp.blogspot.com/-RWITeQLhc-E/UQfJ5Shcg2I/AAAAAAAA$k/2_MFvVSBtXQ/s1600/mountain_cartoon_style_ice_snow_mountains_snowy_cliff_mountian.jpgScientific calculator created by BookryChapter 2Troll photo: http://i.istockimg.com/file_thumbview_approve/15365716/1/stock-illustration-15365716-cartoon-troll.jpgBridge photo: http://vector-magz.com/wp-content/uploads/2013/07/bridge-clipart.pngChapter 3Splitting Path photo: http://www.thelifechurchnow.com/sites/default/files/imagecache/580x235/images/post/the-fork-in-the-road_0.jpgOwl photo: http://www.pamsclipart.com/clipart_images/cartoon_owl_with_big_yellow_eyes_0515-0908-1500-2437_SMU.jpgChapter 4Wizard photo: http://cdn.vectorstock.com/i/composite/53,39/wizard-vector-265339.jpgMap photo: http://3.bp.blogspot.com/_6YkVvbMYgQM/TP2dmNbc6SI/AAAAAAAAAGU/N70Hr6QRQWA/s1600/treasure+map.jpgChapter 5Green photo: http://th04.deviantart.net/fs70/PRE/i/2010/064/1/0/Emerald_Gates_by_Garret_B.jpgVideo on Proofs: http://www.khanacademy.org/math/geometry/basic-geometry/perimeter_area_tutorial/v/triangle-area-proofs?v=YOYQys52sPs