STUDENT s BOOKLET. Zot! Zot! Zot! Part 2. Meeting 18 Student s Booklet. Contents. Sporting Goods 2 Archery 3 Three Fans

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1 Meeting 18 Student s Booklet Zot! Zot! Zot! Part 2 Contents 1 April 1 UCI Sporting Goods 2 Archery Three Fans STUDENT s BOOKLET UC IRVINE MATH CEO

2 1 SPORTING GOODS 1 SPORTING GOODS The three fans Triangela, Squarenest and Circollin are shopping for sporting goods. They are interested in buying hats, scarfs, shirts, balls and racquets, amongst other goods. All hats cost the same price. All shirts cost the same price. And so on... Hat Ball Shirt Scarf Racquet

3 1 SPORTING GOODS Shirts and Basketballs a We know the prices for buying the following goods: $155 $70 So buying 4 shirts plus 1 ball cost $155. Also, buying 2 balls costs $70. How much does it cost to buy two shirts and two balls? $

4 1 SPORTING GOODS Hats and Racquets b We know the prices for buying the following goods: $76 $88 Find the price of one racket, and the price of one hat. $ $

5 1 SPORTING GOODS Challenge: Scarfs, Mugs and Frisbees c We know the prices for buying the following goods: $54 $47 $62 Find the price of each article. $ $ $

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7 2 ARCHERY 2 ARCHERY In the sport of Archery, you shoot an arrow and you gain points depending on where the arrows lands (the landing spot will be marked with an X). We can come up with different scoring systems. We will assume that no arrow can land right on the boundary of a disc.

8 2 ARCHERY A simple target x1 Example: The configuration on the right gives a score of points. Rules: You get 1 point anytime your arrow hits the target (anywhere inside the large disc). In addition, if the arrow lands inside the small disc, you receive a bonus and your score for that arrow gets multiplied by 10, resulting in 1 x 10 = 10 points (instead of 1). Configurations In mathematics, a configuration is a specific situation, usually described by a diagram or a picture. In this activity, the configuration consists of the marks in the targets after an archer s round. For each configuration we can find the corresponding score. Example: The configuration on the right gives a score of 10 points. Example: The configuration on the right gives a score of 22 points. x10 x10

9 2 a ARCHERY Find the score corresponding to the following configuration: c Find two different configurations that get you a score of 5 points, and speak well of your quality as an archer. Score b True or False?: These two configurations give the same number of points... Discuss What is the total number of possible configurations that produce a score of 5? Explain.

10 2 ARCHERY Overlapping targets Every arrow which hits the target is worth at least one point. Sometimes you get a bonus. Example: The following configuration of four marks gives an overall score of = 1102 points. If an arrow lands in the inner right circle, its score is multiplied by 100. If an arrow lands in the inner left circle, its score is multiplied by 100. x1 x100 x10 The points are assigned as follows: 1 pt Discuss Discuss How many points do you gain for an arrow that lands in the overlap of the two inner circles? 100 pt 1000 pt 1 pt

11 2 d ARCHERY Find the score corresponding to the following configuration: e Find a configuration that gives you a score of 421 points. x1 x10 x100 Score Discuss Can you find another one?

12 2 ARCHERY Challenge Find the score corresponding to the following configuration: f Every arrow gives you at least 1 point. There are various ways to get a bonus: if your arrow lands in the large/medium/small inner circle, your score gets multiplied by 2, or 5, respectively. x2 Score x Example: The following configuration of two marks gives an overall score of = 17 points. x5 x x x2 g How many different possible scores can an archer get by shooting a single arrow (depending on where the arrow lands)? x5 x h d How do these various possible scores relate to all the possible divisors of 0? x

13 2 ARCHERY Challenge Three discs are used to build a target: area 6 area 12 area 8 outer ring inner ring inner circle [Not drawn in scale.] An archer shoots an arrow blindfolded. Which event is more likely to happen? (1) The arrow lands inside the inner circle, or (2) The () The arrow lands inside the outer ring. arrow Justify your choice. lands inside the inner ring, or

14 THREE FANS THREE FANS The three fans Triangela, Squarenest and Circollin are getting ready to go to the game. Each will choose if he/she wants to wear a hat, wants wear a nice scarf and/or wants to paint his face to support his/her team. Triangela Circolin Squarenest

15 THREE FANS A puzzle a Based on the following clues, can you discover who was wearing a hat, who was wearing a scarf and who painted his face? At least two friends wore a hat. Circolin told Squarenest: if I paint my face you will not wear a hat. At most one fan painted his face. Triangela did not wear a scarf. Circolin was wearing either a hat or a scarf, but not both. Triangela told Circolin: If you don t paint your face, I will wear a scarf. At least one fan was wearing a scarf. T C S Hat Scarf Paint Triangela (T) Squarenest (S) Circolin (C)

16 THREE FANS Three fans go to the game The three fans Triangela, Squarenest and Circollin are getting ready to go to the game. Each will choose if he/she wants to wear a hat, wants wear a nice scarf and/or wants to paint his face to support his/her team. Draw a picture representing the following scenario: Traingela and Squarenest wear a hat; everybody except Circulin has a painted face. Representing the situations with pictures We can represent each scenario with a picture. For example, if Triangela decides to wear a hat, Squarenest decides to wear a scarf and Circolin decides to both paint his face and wear a hat, we can draw the following picture: Rephrase the instructions... DISCUSS Together with your group, find other sets of instructions that give rise to the exactly the same picture you drew above.

17 THREE FANS Mathematical codes This code can be replaced by shorter codes, like the following one: HAT (triangela + squarenest + circolin ) We can also represent the various situations using mathematical codes. Suppose, for example, that all three friends to wear a hat (and nothing else). (Put a hat to Triangela, Squarenest and Circollin) Here is another way to say it: HAT ( all ) We can express this with a mathematical formula which we will call a code, using +, and parenthesis: (Put a hat to everybody) As another example, we describe the picture below HAT (triangela) + HAT (squarenest) + HAT (circolin) The code above, reads: put a hat to Triangela, put a hat to Squarenest and put a hat to Circolin. with a code, for example: HAT (triangela + squarenest) + SCARF (all - triangela)

18 THREE FANS More examples Another example: (we give two valid codes) Suppose that we want to write a code for the following picture: PAINT ( triangela + squarenest ) PAINT ( ALL - circolin ) b The first thing that we discover is that Triangela wears a hat, so we can write: HAT (triangela) The next thing we discover is that all fans wear a scarf, so this can be expressed as: SCARF ( all ) Putting these two together, we obtain our code: HAT (triangela ) + SCARF ( all ) Write a code for the following picture:

19 c 1 THREE FANS For each given code, complete the picture to match the description. Exercise 1 is solved. SCARF (triangela + circolin) + PAINT(all) (Put a scarf to triangela and circollin, and paint everyone) 2 HAT (circollin) + SCARF (all - squarenest) + PAINT ( triangela ) (Put a hat to circollin, put a scarf to everyone except squarenest, and paint triangela) (PAINT + HAT) (squarenest) + SCARF (all -triangela - squarenest) 4 (SCARF + PAINT) (triangela + circolin) + HAT ( squarenest )

20 d THREE FANS For each given picture, write a code to match the picture. Exercise 1 is solved as an example. HAT ( triangela ) + (SCARF + PAINT) (all triangela ) 1 2 4

21 THREE FANS In the example illustrated below, Triangela (T) has decided to wear a hat, Squarenest (S) has decided to wear a scarf and Circlollin (C) has decided to both paint his face and wear a hat: (S) Squarenest Fans with a scarf Fans with a hat Using Venn diagrams (T) Triangela T S C (C) Circolin Fans with a painted face We can represent this situation by means of a Venn diagram. We draw three regions, each representing one property: wear a hat, wear a scarf and paint the face. We then put the fans T, S and C inside or outside the regions, according to the picture: For example, because Squarenest (S) has a scarf, but no hat and no painted face, we place S inside the set of fans with scarfs, but outside the set of fans with painted face and outside the set of fans with hat. Similarly, because Circolin (C) has both a painted face and a hat, but no scarf, it will be in the intersection of the sets of fans with painted face and fans with hats, but outside the set of fans with scarfs.

22 THREE FANS Another example: Yet one more example: This time every fan is wearing a hat (and nothing else). This time Tiangela and Squarenest are wearing a hat, Squarenest and Circolin have a scarf, and no one has a painted face. (C) Circolin (S) Squarenest (T) Triangela HAT HAT HAT We can represent a situation using a Venn diagram: Fans with a scarf Fans with a hat T We can represent a situation using a Venn diagram: Fans with a hat Fans with a scarf T S C S C Fans with a painted face Fans with a painted face

23 e THREE FANS For each given picture, complete the Venn diagram to match the picture. Exercise 1 is solved as an example. Fans with a scarf Fans with a hat 1 T C S Fans with a painted face Fans with a hat Fans with a scarf 2 Fans with a painted face Fans with a hat Fans with a scarf Fans with a painted face Fans with a hat Fans with a scarf Fans with a painted face 4