STUDENT S BOOKLET. Geometry 1. Contents. Meeting 6 Student s Booklet. May 17 UCI. 1 Finding Clovers 2 Angles 3 Circular Mountains

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1 Meeting 6 Student s Booklet Geometry 1 Contents May 17 UCI 1 Finding Clovers 2 Angles 3 Circular Mountains STUDENT S BOOKLET UC IRVINE MATH CEO

2 1. FINDING CLOVERS 2 1. FINDING CLOVERS A robocar explores different perimeters moving at a constant speed during a given time. Find the exact time at which the robocar meets the clover. Note: In each of the pictures, similar objects have the same length. For example, in figure 2, both logs have the same length and both straws have the same length. B 0 60 A EXAMPLE 0 60 C Solution: Each log takes 60 / 3 = 20 minutes to travel, therefore at time T = 40, robocar will meet the clover.

3 1, FINDING CLOVERS 3 C D 0 600

4 1, FINDING CLOVERS 4 CLOSED PATHS Now the car travels through the perimeter of each shape, making a loop, still at constant speed. Find the exact time at which the robocar meets the clover. E F 0 180

5 1, FINDING CLOVERS 5 G I H 0 60

6 2, ANGLES 6 ANGLES Suppose that we have a round watch only measuring minutes, so every 60 minutes we return to our original state. In other words, every complete turn of the minute hand (360 degrees) returns the watch to its original state. 1 : ONE TURN = 360 DEGREES = 60 MINUTES = 0 TURNS 1/2 : HALF A TURN = 180 DEGREES = 30 MINUTES 1/4 : ONE QUARTER OF A TURN = 90 DEGREES = 15 MINUTES 1/12 : ONE TWELFTH OF A TURN = 30 DEGREES = 5 MINUTES A Complete the following: 2/4 of turn equals to degrees. 3/4 of turn equals to degrees. Example: 1/4 turn (45 degrees, 15 minutes) 1/3 of turn equals to minutes. 2/3 of turn equals to degrees.

7 2, ANGLES 7 ANIMAL ANGLES B Aliens from the planet Glucif have come to earth. They have developed a language, and they are using animal sounds to communicate angles. Each word represents certain angle and if they say a phrase with several words, we add the angles. Using the information below, can you discover the meaning of each word? ribbit ribbit baa ribbit baa moo quack quack moo quack

8 2, ANGLES 8 SHORTENING PHRASES C For each sequence of animals: (1) Read the sequence in alien language (so you need to sound like an animal!) (2) Find the shortest possible phase (least number of words) that has the same meaning as the original phrase (3) Read the new phrase and draw the angle in the whiteboard for others to see. Sequence 1: Sequence 2: Sequence 3:

9 3 CIRCULAR MOUNTAINS 9 3 CIRCULAR MOUNTAINS In the Orange Mountain game, we use a carbot to climb a quarter of a mountain of height 100 meters, called "Orange Mountain". This mountain has the shape of a perfect half-orange, so the path to climb it is a quarter of a circle of radius 100m. The score of each carbot is equal to its height (distance to the ground). The maximum possible score is equal to 100, which means reaching the top (climbing 90 from the top). A Estimate the scores of different car-bots, depending on the angle that they climbed, by measuring the heights. Note: Angle measured counterclockwise from the ground. Note: for an angle x. the score divided by 100, is called sine of x: sin(x ) : Score of 100. sin(90 ) = : Score of. sin(60 ) = 45 : Score of. sin(45 ) = 30 : Score of. sin(30 ) = 15 : Score of. sin(15 ) =

10 3 ORANGE MOUNTAIN 10 The Orange Mountain Game The goal of the game is to score points according to the height that your cars reach at the end of the game in 6 different circular mountains of the same height 1 unit (or 100, however you want). The game is played as follows: There are 6 different mountains, numerated 1 to 6. Place those mountains in the center of the table, reachable to all players. Each player controls six different cars (coins or tokens), one per mountain. All cars start at the ground (zero degrees). The game consists of 4 rounds, and in each round every player plays one turn, clockwise. Each turn: Roll 2 dice. Let X be your smaller value and Y be your larger value (or X=Y if they are equal). Choose exactly one: Your car climbs 15Y degrees in one mountain of your choice. Separate X as a sum of two numbers A+B; climb 15A degrees in one mountain, and 15B degrees in another mountain. Example 1: rolled X=2 and Y=6. You decide use Y=6 to advance 90 in Mountain 1, reaching the top since your car was in the ground. Example 2: rolled X=4 and Y=5. You decide to separate X as 3+1, and advance 45d in Mountain 2 and 14 degrees in mountain 6. Example 3: rolled X=3 and Y=3. You decide to separate X as 2+1, and advance 30d in Mountain 4 and 15d in mountain 1. Note: if you advance more than the top, it is fine, you just stay on the top of the mountain. For example, if your car is at 60 degrees in Mountain 1 and you advance 45 degrees, you move your car to the top (90 degrees). You waste 15 degrees. End of the game: each player collects their score. The score is the sum of the heights (not the angles, but the heights!) reached in each mountain. There is a catch, though: in order to win, you need to have won (or tie first place) in at least one mountain. In case of a tie, whoever dominated more mountains wins the game. If tie persist, tied players share the victory.

11 3 ORANGE MOUNTAIN After playing the game B Reflecting on the game: What was your strategy to play this game? Would you change your strategy? Why? C If you have a total of 6x15=90 degrees to distribute however you wish to make your cars advance in the 6 mountains, how would you choose to do this in order to maximize your total score? What would that score be equal to? D Suppose that you now play the following variant of the game: for each mountain, you measure your horizontal distance d to the vertical axis of the mountain, and that gives you negative points. Whoever gets the highest score wins (so the one closest to zero). So for example, if you move 60 degrees in one mountain, your score is -50, and if you move 90 degrees in a mountain, your score in that mountain is 0. question C with this game in mind, after trying various distributions of degrees. d