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# Developing Expert Voices

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### Developing Expert Voices

1. 1. Chrycel Buenviaje Remyshire Raymundo Michelle Santos
2. 2. Unit Contributor Exponential Functions Chrycel Probability Michelle Trigonometric Identities Michelle Transformation Remyshire Combinatorics Remyshire Reflection Chrycel Michelle Remyshire
3. 8. <ul><li>San Antonio is to play Denver at their court. 23% of the game, Iverson tries to make a shot and 18% of the game, Parker tries to make a shot. The probability a shot goes in is 75% when Iverson makes a shot, and 64% when Parker makes a shot. Julian and Michael makes a bet. Julian betting on Denver and Michael betting on San Antonio. What is the probability Parker can make the winning shot? </li></ul>
4. 10. <ul><li>STEP 1 : Draw a tree diagram of the sample space . </li></ul><ul><li>STEP 2 : Multiply the connecting branches to find the outcome. </li></ul><ul><li>EX.) Parker ( 0.18 ) x Point ( 0.64 ) </li></ul><ul><li>STEP 3 : Since we are looking for the probability of Parker making the winning shot. You take the outcome of Parker | Point ( 0.1152 ) and divide it from the outcome of Points made ( 0.1725 + 0.1152 ) </li></ul><ul><li>STEP 4 : The answer will determine the probability of the winning shot. </li></ul>P ( Parker | Point ) = P ( Parker and Point ) P ( Point ) P ( Parker | Point ) = 0.1152 0.1725 + 0.1152 P ( Parker | Point ) = 0.4004
5. 12. <ul><li>Prove this Identity : </li></ul><ul><li>sinx + cosx + 1 = 2cscx </li></ul><ul><li> cosx + 1 sinx </li></ul>
6. 21. <ul><li>In a beach, high and low tide occurs at least once a day. The average level of water is 5.25 and the lowest level is 1.5m. The first low tide occurred at 6:30am and the high tide occurred halfway before the next low tide occurred. The period is 12 hours and 30 minutes. </li></ul>
7. 22. <ul><li>What is the exact time the low tide would occur and what time is the next high tide? </li></ul><ul><li>What is the highest level of water on high tide? </li></ul><ul><li>Draw the graph </li></ul><ul><li>Determine the maximum and minimum value, sinusoidal axis and amplitude </li></ul><ul><li>Write the sine and cosine equation. </li></ul>
8. 23. <ul><li>Transform minutes to decimal : </li></ul><ul><li>0:15 = 0.25 0:30 = 0.5 0:45 = 0.75 </li></ul><ul><li>Answers: </li></ul><ul><li>12:30=12.50 (period) 6:30=6.50 (time of first low tide) </li></ul><ul><li>12.50 + 6.50 = 19.00 </li></ul><ul><li>19.00 – 12.00 = 7.00 </li></ul><ul><li>7.00 = 7:00pm **the time the nest low tide will occur </li></ul><ul><li>6:30=6.50 (time of first period) 6.25 (half of period ** 12.5/2) </li></ul><ul><li>6.50 + 6.25 = 12.75 </li></ul><ul><li>12.75 = 12:45pm ** the time the high tide will occur </li></ul>
9. 24. <ul><li>Amplitude is the height from the min or max to the sinusoidal axis. Given the lowest level or the min we can determine the amplitude. </li></ul><ul><li>Average level – lowest level = x </li></ul><ul><li>5.25m – 1.5m = 3.75m </li></ul><ul><li>Amplitude = 3.75m </li></ul><ul><li>To determine the Max or the Highest level of water: </li></ul><ul><li>Amplitude + average level = max </li></ul><ul><li>3.75m + 5.25m = 9m </li></ul><ul><li>Max = 9m = Highest level of water </li></ul><ul><li>  </li></ul>
10. 25. y x
11. 26. <ul><li>Amplitude = 3.75m </li></ul><ul><li>Average level – lowest level = x </li></ul><ul><li>5.25m – 1.5m = 3.75m </li></ul><ul><li>  </li></ul><ul><li>Min = 1.5m </li></ul><ul><li>the lowest value of water is the same as the min. value of the question </li></ul><ul><li>Max = 9m </li></ul><ul><li>Amplitude + average level = x </li></ul><ul><li>3.75m – 5.25m = 9m </li></ul><ul><li>Period = 12h 30m or 12.50 in decimal </li></ul><ul><li>30 is in min so to get the decimal value we will divide 30/60 </li></ul><ul><li>60 is the total number of minute in one hour </li></ul><ul><li>  </li></ul><ul><li>Sinusoidal axis = 5.25m </li></ul><ul><li>the average level of water in this question is the same as the sinusoidal axis </li></ul>
12. 27. <ul><li>Cosine equation: </li></ul><ul><li>D = sinusoidal axis = 5.25 </li></ul><ul><li>A = amplitude = 3.75 </li></ul><ul><li>B = 2π/period = 2π/12.20 </li></ul><ul><li>C = phase shift = 0 </li></ul><ul><li>F(x) = 3.75 cos (2π/12.20 x) + 5.25 </li></ul><ul><li>  </li></ul><ul><li>Sine equation: </li></ul><ul><li>D = sinusoidal axis = 5.25 </li></ul><ul><li>A = amplitude = 3.75 </li></ul><ul><li>B = 2π/period = 2π/12.20 </li></ul><ul><li>C = phase shift = 9.38 </li></ul><ul><li>F(x) = 3.75 sin (2π/12.20 (x + 9.38)) + 5.25 </li></ul>
13. 29. <ul><li>Tong-its , a famous Filipino card game played by a lot of Filipinos. This game can consist of three players. The dealer deals 12 cards to each player including himself then dealing an additional card to himself before he puts down the rest of the card in the centre of their table. Given a standard deck of 52 cards how many ways are there to draw 12 cards to obtain each hand. </li></ul>
14. 30. <ul><li>Q: Given a standard deck of 52 cards how many ways are there to draw 12 cards to obtain each hand. </li></ul><ul><li>use formula n C r or n! *n! = factorial notation </li></ul><ul><li> (n-r)!r! </li></ul><ul><li>N is the number of object to choose from </li></ul><ul><li>R is the number of object to be arranged or we need. </li></ul><ul><li>n C r is read as n choose r </li></ul><ul><li>A: n=52 r=12 </li></ul><ul><li>52 C 12 = very big number </li></ul>
15. 31. <ul><li>The goal of this game is to finish all the cards in your hand at least before the pile of unseen cards are out in the centre of the table. And in order to finish all you cards faster you can either put down one or more of these: </li></ul><ul><li> ~ Trio ( three of a kind ) </li></ul><ul><li> ~ Quad ( four of a kind ) </li></ul><ul><li> ~ Three or more card straight flush ( 3 or more cards of the same suit in sequence ) </li></ul>
16. 32. <ul><li>How many ways are there to drawn 12 cards with: </li></ul><ul><ul><ul><li>12 cards straight flush </li></ul></ul></ul><ul><ul><ul><li>5 cards straight flush,1 trio, 3 random </li></ul></ul></ul><ul><ul><ul><li>2 trio, 1 quad 6 random </li></ul></ul></ul><ul><li>from a deck of 52 cards </li></ul>
17. 33. <ul><li>4 C 1 * 13 </li></ul><ul><li>4 * 13 = 52 </li></ul><ul><li>-> 52 ways to have a 12 card straight flush </li></ul>12 cards straight flush Number of ways to get card in sequence From the 4 suits we need to choose 1
18. 34. <ul><li>4 C 1 * 13 C 5 * 8 C 1 * 4 C 3 * 12 C 3 * 4 C 1 * 4 C 1 * 4 C 1 </li></ul><ul><li>4 C 1 * 13 C 5 = from 4 suits we pick 1 * number of ways 5 cards in sequence </li></ul><ul><li>8 C 1 * 4 C 3 = from 8(13-5 (5 cards used in straight flush)) face value we pick 1 * number of suit chosen </li></ul><ul><li>12 C 3 * 4 C 1 * 4 C 1 * 4 C 1 = 3 different face value from 12 cards ( less the face value used for the trio) * suit for each of the last 3 cards. </li></ul><ul><li>4 * 1287 * 8 * 4 * 220 * 4 * 4 * 4 </li></ul><ul><li>5148 * 32 * 14080 </li></ul><ul><li>2319482880 </li></ul><ul><li>Ways to have a 5 cards straight flush with a trio and 3 random cards. </li></ul>5 cards straight flush , 1 trio , 3 random
19. 35. <ul><li>13 C 1 * 4 C 3 * 12 C 1 * 4 C 3 * 11 C 1 * 4 C 4 * 10 C 2 * 4 C 1 * 4 C 1 </li></ul><ul><li>13 C 1 * 4 C 3 = (first trio) face value of 3 cards * 3 of 4 suit on one face value </li></ul><ul><li>12 C 1 * 4 C 3 = (second trio) face value of 3 cards (chosen value in the 1 st trio cant be chosen again) * 3 of 4 suit on one face value </li></ul><ul><li>11 C 1 * 4 C 4 = face value of 4 cards (value chosen for the trios cant be use again) * all the 4 suits </li></ul><ul><li>10 C 2 * 4 C 1 * 4 C 1 = 2 different face value (cant use the value used in quad nor trios) * suit for each of the last cards. </li></ul><ul><li>13 * 4 * 12 * 4 * 11 * 1 * 45 * 4 * 4 </li></ul><ul><li>52 * 48 * 11 * 720 </li></ul><ul><li>19768320 </li></ul><ul><li>Ways to have 2 trio a quad and 2 random cards </li></ul>2 trio, 1 quad, 2 random
20. 36. CHRYCEL B.
21. 37. <ul><li>Why did you choose the concepts you did to create your problem set? </li></ul><ul><li>I chose these concepts because Trigonometric Identities was one of the units I was having a hard time on. I’m fine with doing the algebra, it’s the part where you have to be familiar with the corollaries. Before we had started our unit on Probability, my group had already decided to do a question from this unit. Probability is quite simple, but could be confusing at times. But, since we are on this unit now it helped me understand how to approach these types of questions. </li></ul><ul><li>How do these problems provide an overview of your best mathematical understanding of what you have learned so far? </li></ul><ul><li>These problems provided an overview of my mathematical understanding of what I have learned in those units. In Trigonometric Identities, I tried to understand where I was going wrong and why I wasn’t able to prove the Identities. I was then able to create my own problem, and work out a solution that would equal the other. Probability on the other hand, was a question put together from the current things we have just learned. </li></ul><ul><li>Did you learn anything from the assignment? Was it educationally valuable to you? </li></ul><ul><li>Doing this project has allowed me to have a better understand in a unit I wasn’t able to figure out a few months ago. It has made me realize where I was making my mistakes, and from that I learned. Not only did I learn from those mistakes, but from those mistakes I was able to create a problem and find a solution to it. Creating the problem was quite difficult, not having the numbers to work out the way you want. But, I think this was valuable for me. It was a good review for the exam. </li></ul>MICHELLE S.
22. 38. <ul><li>• Why did you choose the concepts you did to create your problem set? </li></ul><ul><li>I choose these concepts because these units, especially transformation, are one of the units I did pretty bad. I choose to do it because it had been a really long time since we looked at the unit and learned it. And by this I want to review what we learned and studied before. I think by using these problems it will help me improve my understanding in the units I’ve chosen to use. </li></ul><ul><li>• How do these problems provide an overview of your best mathematical understanding of what you have learned so far? </li></ul><ul><li>The problem I choose did not really prove what I am capable of in solving these types of problem. I mostly did everything straight to the point and as short as possible. My ability to solve these problems was shown but to show how I did everything is not specified. </li></ul><ul><li>• Did you learn anything from this assignment? Was it educationally valuable to you? </li></ul><ul><li>I learned how to improve my skills in solving such problems. I also learned how to analyze a problem correctly. This assignment not only helps me improve my ability or knowledge in the units I did poorly but it also helped me review the past lessons we had. It is made me realize that I really need to start reviewing for the coming exam because I think almost half or more of what we learned were already gone inside my head. This project helped me realize that I need to put more effort in what I need to do in order to finish it. </li></ul>REMYSHIRE R.