The Athens of The Arctic

You are an international treasure hunter.  

You have plundered the ancients in search of a lost city known only as the Athens of The Arctic - somewhere, beneath a mile of ice is a city of unfathomable riches, archeological wonder and mysteries held in the icy dark for a thousand years.  

It was well hidden, and now you find yourself in a hidden mansion in some strange forest in Northern Finland, braving the cold while searching the house for a vault that may contain the next piece of the puzzle to find the Athens of The Arctic: a gold plate that holds the code to an as yet unreadable map.  

You've come a long way and there's no turning back.  

The ancient arctic peoples have hidden their secrets behind geometrical puzzles you will need to solve to find the city.  Be careful, the city is well-defended and you will need to be on the watch for traps.  A wrong answer may end your venture.  Permanently...

Good luck.  

Try not to freeze to death.  

The ancients hid this treasure behind mathmateical knowledge few people had in the world at that time. Lucky for you, you have the vast world wide web to seek out answers to the puzzles before you.  As you go about your journey, keep a journal of what you have learned - including the solutions of the puzzles you must solve.  Remember, if you don't make it, you can always hope that someday, another adventurer will find your journal, and continue on to the Lost City. 

The Task

In this WebQuest, you are going to learn all about regular polygons.  You are going to search through various websites to find information about various regular polygons.  The things that you are to be looking for:

• Definition of a polygon and regular polygon
• Names of several types of regular polygons
• The sum of the angles of certain regular polygons
• For a regular polygon, the measure of each angle
• The number of diagonals that can be drawn from the vertices of any regular polygon
• The number of triangles formed by drawing diagonals from the vertices.

At the end of this WebQuest, you will create a MAP of your findings on a GRAPHIC ORGANIZER of your choosing.  It will contain all you have learned, including the answers to the puzzles you will need to solve.  

It will make an impressive section in your inevitable bestselling memoir.  

It is twenty below zero outside, but the cave you are in is a balmy twenty eight degrees.  In front of you are the gaping mouths of three cave entrances blocked by massive stone doors.  There are inscriptions on the door of various shapes: you recognize them as POLYGONS - enclosed figures.  

TO BEGIN YOUR ADVENTURE, you start HERE

http://www.mathsisfun.com/geometry/polygons.html

READ through the WEBSITE and ANSWER all the questions at the bottom.  

Using THE SNIPPINGTOOL, cut out your score and email it to jlovero@richmond.k12.va.us

When you are finished, you are ready for your first TRAP! 

1.  To get past this first TRAP, you will need to find the correct figure out of three figures.  Only one is correct. You will need to identify the CORRECT image.  YOU WILL HAVE THREE TRIALS, and they will get progressively more difficult.

(INSERT PUZZLE HERE) 

2.  To get past the second trap, you will need to identify the INCORRECT image.  

(INSERT PUZZLE HERE) 

3. To get past the third trap, you will need to find the pattern in the images, and choose the NEXT image.  

(INSERT PUZZLE HERE) 

NOW: if you've gotten this far.... and you haven't plunged into dark, frigid water...and you avoided the giant spider covered in white polar fur.... and the the ice stalagtites haven't fallen from the cieling and crushed you.... THEN you are ready for the next step... 

GO HERE: 

http://www.math.com/tables/geometry/polygons.htm

Before you move on, fill in this chart so you can get by the next boundary to your destiny.... SOME of it you are already an expert in, but don't get cocky... do your research, your life depends on it.  

The following instructions are going to guide you through defining the words polygon and regular, finding the names of the regular polygons and the sum of their interior angles.

Step 1.        The following link contains a lot of information about polygons.  However, all I want you to get from this website is the definition of polygon and regular polygon.  The site can be used to help you on the other parts of this WebQuest.  Both of the definitions are listed on this page and these are the definitions that I would like you to know.  Write the definitions here and in your journal:

Polygon:               ___________________________________________

                            ___________________________________________

                            ___________________________________________

Regular Polygon:   ___________________________________________

                            ___________________________________________

                            ___________________________________________

Step 2.        Please click here.  In this link, find the name of the figures with the given number of sides and the sum if their interior angles (also called sum of angles).  Once you find this information, put it in the chart.  You must scroll down to view the names of what you are looking for.

Step 3.        Next, click here.  This link provides the names of the polygons with a given number of sides.  Directly below the chart containing this information, there is a little JavaScript quiz, called “How Many Sides or Angles are in the Polygon”, to test your knowledge on the names of the polygons with 3-10 sides.  Play with this interactive quiz until you get at least 20 questions with 100% accuracy.  This quiz is also timed, so see how fast you can answer 20 questions.  

Having taken the correct path, you find yourself facing a massive ice statue of a anthropomorphic polar bear, wearing a suit of armor, including a helmet on it's regal head and a sceptor in each of its great paws.  The entrance to the next leg of your journey lies atop one of these sceptors.  Move the correct arm and you activate an age old switch, and the polar warriors mouth will open and you will begin the next phase.  

Choose the incorrect sceptor and you should be sure that something bad will happen.  Don't let that happen... study hard.... 

THE FIRST SCEPTOR: in its right paw, the sceptor is topped with a figure you recognize.  It is a flat, concave shape with 26 sides.  

This next set of instructions are designed for you to find the number of diagonals that can be drawn from the vertices of any regular polygon, the number of triangles created when you draw these diagonals, the formula for finding the sum of the interior angles of a polygon, and the measure of each of the angles in a regular polygons.

Step 1.        Click here to determine how to find the sum of interior angles of a regular polygon.  [Note: in Step 2 of the last process, you found the sum of the interior angles.  Now, however, you are finding the equation on how to solve for the sum of the interior angles.]  Once you find the equation, write it here and in your journal:­­­­____________________..

Step 2.        Using the above website (again, found here), interpret from the example given how to determine how many lines can be drawn from a given vertex for a given regular polygon.  The example this website deals with a pentagon.  With a pentagon, choose a vertex O (the definition is on the website) and then connect this vertex to all of the other vertices using a single line to connect two vertices.  Doing so will result in a pentagon having 4 lines connecting all of the vertices: OA, OB, OC, OD and OE (where A, B, C, D and E are the labels for the exterior angles).

Step 3.        Using the following website, find the Java applet that shows how many triangles are in each polygon.  Use the more and less buttons which are found under the Number of Sides located on the right of the page.  Do you see a formula to determine how many triangles are in a given polygon?  For example, how many triangles will a 20-gon have?_________________A 49-gon?_____________________.

Step 4.        Using this site, scroll to the bottom of the page to discover the measure of the interior angles in any polygon.  You must follow the formula listed. What is the formula for finding the measure of each interior angle (also called the vertex angles) of any regular polygon?  Write the formula in your journal and here:______________________.

After you have completed the activities required, you can play with this website to create your own regular polygon.  All you have to do is either enter an angle size between 0o and 180o or enter the number of sides and then click on draw.  The regular polygon will be drawn in the box below the information.  Also, you should note that if you enter an angle measure, then the number of sides fills in automatically or if you enter the number of sides, then the angle measure automatically fills in.  Click here to enter the website.

You should use the table above to organize the information that you find on this WebQuest.  You should also be placing the formulas that I mention in your journal so that you will have in to study from for a test.  If there is any interesting information that you come across, write it down so that you can share it with the class.  Now, use the following checklist to make sure that you have everything completed:

            Completed regular polygons chart

            Blanks filled in with the appropriate information.

            Formulas written in your journal

Questions to Consider:  Do you think that the information you have learned in this WebQuest will hold true for irregular polygons?  If so, will all of the formulas hold?

Evaluation

The chart that you complete should be turned in and you will be graded on it.  The chart below shows the point distribution.  Be for you turn in your chart, make sure that all of the blanks are filled and that the information is correct.  You are to be graded individually, but working in groups is allowed.  However, each person is to turn in a completed chart.

Congratulations!  You have just completed the WebQuest.  You have learned all of the names of regular polygons with 3 to 10 sides.  You have also learned how many triangles are contained in these regular polygons, what their interior angle totals are, as well as what each vertex angle is. 

But most importantly, you have found the next location and with your destination in mind, you are one step closer to discovering the Athens of the Arctic.