LastUpdate 3/4/2023 |

It was about 37 years ago that I got the book "〜Microcomputer・Classroom〜 Future
Classroom Challenge to CAI Education written by 中山和彦 & 東原義訓 （筑波出版会）" The following was written there. @The most important thing in this educational reform is to establish the principle of emphasizing individuality. AIn the future society,not only the ability to acquire knowledge and information,but also the ability to make full use of it appropriately,think,create,and express with one's own mind must be emphasized. BIn future school education,on top of the basics,the development of thinking ability such as creativity,logical thinking ability,abstract ability,imagination,and expressive ability should be emphasized. CIt is necessary to shift from the conventional uniform education to an education that respects individuality , and to develop human beings who have creativity and the ability to think ,express and act by themselves. DＣＡＩ（Computer Asisted Instruction） enables the conversion from one-sided lessons to lessons / learning that follow a voluntary free couese according to the needs of the leaner. At that time,above all,I imagined the title of this book "Classroom of the Future",dreamed of "classroom of the future",and devoted myself to the study of CAI in mathematics. Today,computers are far more powerful than they wewe at that time,are easy to use , and are failiar to everyone.In math lessons,I think it is an effective and realistic way to use a computer as a simulation for a part of an hour lesson to make students think and comfirm. However ,even if I used a computer as part of the lesson,I had to secure a computer classroom and move around ,or had to carry a computer and a projector to the classroom and set it up.（In Gifu Prefecture,in the first year of Reiwa,black boards became white boards that could be used as screens in all ordinary classrooms ar all prefectural high schools,and projectors were also installedin all ordinary classrooms. In the second year of Reiwa , tablets were distributed to each students.） With smartohones becoming so popular with students , it is expected that it will not be long before every student owns a smartphone. Then smartphones and tablets make it easier for students to see , think and check the simulation.（Nowadays,teachers can move freely in the classroom with their smartphones , enter the students , and explain while projecting the screen of the smartphone on a wirelessly connected projector.） Dreaming of the arrival of future classrooms , I created an "Android app" that can be used in math classes and introduced it on this web page. STAI（SmartPhone Tablet Asisted Instruction）. This was the "classrooms of the future" for me , but it has become quite realistic. 2021.04.06 Fixed （2012.11.17 Note） |

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【１】 Experiment of Faling Pachinko Balls |

［１］App overview You can observe how the pachinko balls are falling. The pachinko balll will fall to the left and right with a 50-50 chance of splitting and will fit into one of the 11 containers. There are 10 left and right branches before entering the container. Which container is easy for pachinko balls to fit in ? Also,let's mathematically find the probability that pachinko balls will fit in each of the 11 containers. ［２］Screen example spachinkoeng.pdf ［３］Source program list lpachinkoeng.pdf ［４］Instaliation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【２】 Scatter Sesame Seeds to Find Pi （Monte Carlo method） |

［１］App overview Scatter the sesame seeds and and find the approximation of pi. Draw a square and a circle inscribed in it,and scatter sesame seeds randomly from above. The approximation of pi can be obtained from the ratio of the number of sesame seeds in the square to the number of sesame seeds in the circle. Let's explain the reason mathematically. ［２］Screen example sgomaeng.pdf ［３］Source program list lgomaeng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【３】 Buffon's Needle |

［１］App overview Scatter the needles to find the approximate value of pi. Draw parallel lines at equal intervals and randomly scatter the needles from above. The approximate value of pi can be obtained from the ratio of the number of needles that intersect the parallel lines. Let's think mathematically about the reason. ［２］Screen example sharieng.pdf ［３］Source program list lharieng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【４】 Three-Rock-Paper-Scissors |

［１］App overview You can observe how A,B and C play rock-paper-scissors. In one rock-paper-scissors game,which case is more likely to occur , such as when only A wins,when only A loses,or when a tie is drawn ? Also,let's find the probability that only A will win , the probability that only A will lose , the probability of a draw , the probability that only one will win , and the probability that only one will lose. ［２］Screen example szyannkeneng.pdf ［３］Source program list lzyannkeneng.pdf ［４Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【５】 Playing Cards that Are Multiples of 3 |

［１］App overview You can observe the situation when pulling one of the 52 playing cards excluding the joker. What is the percentage of playing cards that are multiples of 3 ? If you have conducted many expriments to find the ratio of playing cards that are multiples of 3 ,what can you say if you draw more playing cards in each expriment than when you draw fewer cards ? Also,let's mathematically find the probability of drawing a playing card that is a multiple of 3. ［２］Screen example storampeng.pdf ［３］Source program list ltorampeng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【６】 Random Walks |

［１］App overview You can observe the drunken gait. A drunk person wanders indiscriminately from side to side. A drunk person may come back to the same place many times while walking around trying to reach his destination. Now suppose a drunk person is at the origin on the number line.Now when a drunk person starts from the origin and wanders 30 times,where is he on the number line ? However,suppose that a drunk person sways in only two directions,left and right,and moves only one with one sway. Also,assume that a drunk person sways to the left or to the right in half. Let's mathematically find the probability that the position on the number line when a drunk person staggers 30 times is 0,the probability that it is 1,and the probability that it is 2. ［２］Screen example srandomwalkseng.pdf place ［３］Source program list lrandomwalkseng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【７】 Two Coins Toss |

［１］App overview You can observe how the front and back appear when two coins are thrown at the same time. What is the ratio that both are tables ? If you have conducted many experiments to throw two coins at the same time ,and if you increase the number of experiments,what can be said about the ratio of both coins in the table compared to when the number of experiments is small ? Also,let's mathematically find the probability that both are front, and the probability that both are back, and the probability that one is front and one is back. ［２］Screen example scointosseng.pdf ［３］Source program lcointosseng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku" （The initial letter "S"is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【８】 Two Dice with an Odd Product |

［１］App overview You can observe how the number comes out when two dice are thrown at the same time. What is the percentage of odd-numbered products of two dice ? If you have conducted many experiments to throw two dice at the same time,what can be said about the ratio of the product of two dice to odd number when the number of experiments is increased compared to when the number of experiments is small ? Also,let's mathematically find the probability that the product of the two dice will be odd. ［２］Screen example ssaicoroeng.pdf ［３]Source program list lsaicoroeng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【９】 Encounter Experiment |

［１］App overview Dr.Boro sometimes dates women. However,the other day,the meeting place was crowded and it was getting dark ,so it took her an hour to find the doctor,and it became a big quarrel. In fact,Dr.Boro,who has considerable myopia and astigmatism,thought that it was better to stay still than to wander around,so he did not move a certain place within the range designated as "wait around here". But she didn't like it. She argues that it should be faster for each other to find each other and move around. The rational is that if both don't work,both will never find the other. Of course,as the saying goes,"If you have a distress in the mountains,don't move a step and wait for help", Dr.Boro thinks he is absolutely correct mathematically.However,her anger has shaken Dr.Boro's self-confidence a bit. Which do you agree with ? And why ? Can you explain the reason mathematically ? (From the 4th Japan Mathematics Competition) In this app , a dog is used as a substitute for her , and Momotaro is used as a substitute for Dr. Boro. ［２］Screen example sdeainoeng.pdf ［３］Source program list ldeainoeng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａy Store】 Please search by "Seitoku" （The initial letter "S"is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【１０】 If You Shoot a Lot of Bad Guns , You Will Hit ! |

［１］App overview There is the word "If you shoot a lot of bad guns , you will hit !" A person with a gun skill who hits only once out of 10 shoots a gun. He shoots 20 guns in one expriment. What is the percentage of experiments that hit at least one of the 20 shots ? Also,let's mathematically find the probability of hitting at least one of the 20 shots. ［２］Screen example steppoueng.pdf ［３］Source program list lteppoueng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku" （The initial letter "S" is uppercase,all haif-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【１１】 Napier's Constant ｅ （The Base of Natural Logarithm ｅ） |

［１］App overview （Let n be a natural number） The above limit is called "the base of natural logarithm" or "Napier's constant" and is represented by "e". Let's observe the situation when n is increased steadily. ［２］Screen example snumberofeeng.pdf ［３］Source program list lnumberofeeng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【１２】 Wallis's Formula |

［１］App overview Use the following "Wallis's formula" to find the approximate value of pi. Lert's observe how the approximate value of pi can be obtained as the number of numerator and denominator numbers increases. ［２］Screen example swourisueng.pdf ［３］Source program list lwourisueng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【１３】 Gregory･Leibniz's Formula |

［１］App overview Use the following "Gregory･Leibniz's Formula" to find the approximate value of pi. Let's observe how the approximate value of pi can be obtained as the number of terms increases. ［２］Screen example sgreraieng.pdf ［３］Source program list lgreraieng.pdf ［４］Installation program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【１４】 Euler's Formula |

［１］アプリの概要 Use the following "Euler's formula" to find the approximate value of pi. Let's observe how the approximate value of pi can be obtained as the number of terms increases. ［２］Screen example sformulaofeeng.pdf ［３］Source program list lformulaofeeng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【１５】 Yoshisuke Matsunaga's Formula 1 |

［１］App overview Use the following "Yoshisuke Matsunaga's formula" to find the approximate value of pi. Let's observe how the approximate value of pi can be obtained as the number of terms increases. You can see that the convergence is very fast. ［２］Screen example smatsunagaf1eng.pdf ［３］Source program list lmatsunagaf1eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【１６】 Yoshisuke Matsunaga's Formula 2 |

［１］App overview Use the following "Yoshisuke Matsunaga's formula" to find the approximate value of pi. Let's observe how the approximate value of pi can be obtained as the number of terms increases. You can see that the convergence is very fast. ［２］Screen example smatsunagaf2eng.pdf ［３］Source program list lmatsunagaf2eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width). ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【１７】 Approximation 1 of Napier's Constant "e" |

［１］App overview Use the following approximate formula to find the approximation of the base of the natural logarithm "e". Lert's observe how the approximation of the base of the natural logarithm "e" can be obtained as the number of terms increases. You can see that the convergence is very fast. ［２］Screen example skinjiofe1eng.pdf ［３］Source program list lkinjiofe1eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【１８】 Approximation 2 of Napier's Constant "e" |

［１］App overview Use the following approximate formula to find an approximation of the reciprocal of Napier's constant . Let's observe how the approximation of the reciprocal of Napier's constant "e" can be obtained , as the number of terms increases. You can see that the convergence is very fast. ［２］Screen example skinjiofe2eng.pdf ［３］Source program list lkinjiofe2eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【１９】 Approximation 3 of Napier's Constant "e" |

［１］App overview Use the following approximate formula to find an aproximation of the base of the natural logarithm "e". Let's observe how the approximation of the base of the natural logarithm "e" can be obtained as the number of tems increases. You can see that the convergence is very fast. ［２］Screen example skinjiofe3eng.pdf ［３］Source program list lkinjiofe3eng.pdf ［４］Installation of program files （ａｐｋ） Please purchase from 【Ｐｌａｙ Stote】 Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【２０】 Approximation 6 of Pi by Infinite Series |

［１］App overview Use the following approximate formula to find the approximation of pi. Let's obseve how the approximate value of pi can be obtained as the number of terms increases. ［２］Screen example skinjiofp6eng.pdf ［３］Source program list lkinjiofp6eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【２１】 Approximation 7 of Pi by Infinite Series |

［１］App overview Use the following approximate formula to find the approximation of pi. Let's obseve how the approximate value of pi can be obtained as the number of terms increases. ［２］Screen example skinjiofp7eng.pdf ［３］Source program list lkinjiofp7eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【２２】 Approximation 8 of Pi by Infinite Series |

［１］App overview Use the following approximate formula to find the approximation of pi. Let's obseve how the approximate value of pi can be obtained as the number of terms increases. ［２］Screen example skinjiofp8eng.pdf ［３］Source program list lkinjiofp8eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【２３】 Approximation 9 of Pi by Infinite Series |

［１］App overview Use the following approximate formula to find the approximation of pi. Let's obseve how the approximate value of pi can be obtained as the number of terms increases. ［２］Screen example skinjiofp9eng.pdf ［３］Source program list lkinjiofp9eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【２４】 Approximation 10 of Pi by Infinite Series |

［１］App overview Use the following approximate formula to find the approximation of pi. Let's obseve how the approximate value of pi can be obtained as the number of terms increases. ［２］Screen example skinjiofp10eng.pdf ［３］Source program list lkinjiofp10eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【２５】 Approximation 11 of Pi by Infinite Series |

［１］App overview Use the following approximate formula to find the approximation of pi. Let's obseve how the approximate value of pi can be obtained as the number of terms increases. ［２］Screen example skinjiofp11eng.pdf ［３］Source program list lkinjiofp11eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】 Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【２６】 Approximation 12 of Pi by Infinite Series |

［１］App overview Use the following approximate formula to find the approximation of pi. Let's obseve how the approximate value of pi can be obtained as the number of terms increases. ［２］Screen example skinjiofp12eng.pdf ［３］Source program list lkinjiofp12eng.pdf ［４］］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【２７】 Approximation 13 of Pi by Infinite Series |

［１］App overview Use the following approximate formula to find the approximation of pi. Let's obseve how the approximate value of pi can be obtained as the number of terms increases. You can see that the convergence is very fast. ［２］Screen example skinjiofp13eng.pdf ［３］Source program list lkinjiofp13eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku"（The initial letter "S" is uppercase,all half-width）. ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【２８】 Collatz Problem （３ｘ＋１ Problem） |

［１］App overview No matter what natural nuber you start with , if the number is odd , multiply by 3 and add 1, and if it is even , divide by 2 and you will always get 1. Is this really true ? This is called the"3x+1 Problem" or the "Collatz Problem" and is a difficult problem that has not yet been solved. For example , 11→34→17→52→26→13→40→20→10→5→16→8→4→2→1. Let's check from various natural numbers. ［２］Screen example sp3xplus1eng.pdf ［３］Source program list lp3xplus1eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width). ［５］Supported OS Android 4.1 (Jelly Bean) 〜 ［６］How to run See "Running the app" below. |

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【２９】 Amidakuji |

［１］App overview Randomly make Amidakuji with 10 vertical lines and 50 horizontal lines. Suppose that the hit is in the third position from the left. Is Amidakuji the same ease of winning no matter where you draw it ? Or is there a place that is easy to hit and a place that is hard to hit ? Let's draw lots of Amidakuji many times and observe the situation. ［２］Screen example samidakujieng.pdf ［３］Source program list lamidakujieng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width). ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【３０】 Translation of Graph of Quadratic Function |

［１］App overview Let's see that the graph of "y=a(x-b)^2+c" is a trranslation of the graph of "y=ax^2" by +b in the x-axis direction and +c in the y-axis direction. First enter integers in a,b,and c to determine the quadratic function y=a(x-b)^2+c. Observe how the graph of "y=ax^2" moves in parallel by +b in the x-axis direction and +c in the y-axis direction and overlaps with the graph of "y=a(x-b)^2+c". At this time,note that these two graphs have the same shape and spread,but differ only in position. ［２］Screen example stwokansueng.pdf ［３］Source program list ltwokansueng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width). ［５］Supported OS Android 4.1 (Jelly Bean) 〜 ［６］How to run See "Running the app" below. |

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【３１】 Translation of Graph of Quadratic Function （Convex downward） |

［１］App overview Let's see that the graph of "ｙ＝(ｘ＋７)^2−９" is a translation of the graph of "ｙ＝ｘ^2" by -7 in the x-axis direction and -9 in the y-axis direction. Observe how the graph of "ｙ＝ｘ^2" moves in parallel by -7 in the x-axis direction and -9 in the y-axis direction and overlaps with the graph of "ｙ＝(ｘ＋７)^2−９". At this time , note that these two graphs have the same shape and spread , but differ only in position. ［２］Screen example stwokansu1eng.pdf ［３］Source program list ltwokansu1eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width). ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【３２】 Translation of Graph of Quadratic Function （Convex upward） |

［１］App overview Let's see that the graph of "ｙ＝-2(ｘ-6)^2+10" is a translation of the graph of "ｙ＝-2ｘ^2" by +6 in the x-axis direction and +10 in the y-axis direction. Observe how the graph of "ｙ＝-2ｘ^2" moves in parallel by +6 in the x-axis direction and +10 in the y-axis direction and overlaps with the graph of "ｙ＝-2(ｘ-6)^2+10". At this time , note that these two graphs have the same shape and spread , but differ only in position. ［２］Screen Example stwokansu2eng.pdf ［３］Source program list ltwokansu2eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width). ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【３３】 Spread of Graph of Quadratic Function |

［１］App overview Let's see the spread of the graph of the quadratic function "ｙ＝ａｘ^2". Observe the graph of "ｙ＝ａｘ^2" when ａ＝１,when ａ＝２,when ａ＝３,when ａ＝４,and when ａ＝５. Next , observe the graph of "ｙ＝ａｘ^2"when ａ＝−１,when ａ＝−２,when ａ＝−３,when ａ＝−４,and when ａ＝−５. Let's see that when ａ＞０ , the spread of the graph of "ｙ＝ａｘ^2" becomes narrower as the value of "a" becomes larger. And let's see that when ａ＜０ , the spread of the graph of "ｙ＝ａｘ^2" becomes narrower as the value of "a" becomes smaller. ［２］Screen example stwokansu3eng.pdf ［３］Source program list ltwokansu3eng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width). ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

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【３４】 Throw Coins to Find the Pi （Monte Carlo Method） |

［１］App overview Throw coins to find an approximation of pi. Draw parallel lines of vertical and horizontal lines at equal intervals , and throw coins randomly from above. However , the distance between the parallel lines should be equal to the diameter of the coin. The approximate value of pi can be obtained from the ratio of the number of coins rhrown to the number of coins that overlap the grid points （The intersection of vertical and horizontal lines is called a grid point）. Let's explain the reason mathematically . ［２］Screen example stenencoineng.pdf ［３］Source program list ltenencoineng.pdf ［４］Installation of program files (apk) Please purchase from 【Ｐｌａｙ Store】. Please search by "Seitoku" （The initial letter "S" is uppercase,all half-width). ［５］Supported OS Android 4.1 (Jelly Bean) 〜 |

■ There are three ways to install the app on your smartphone.［１］Attach the apk file to the email and and send it to install.@ Open this page on your computer. A Check the ［４］ program file of each application on this homepage to save the apk file on the desktop or the like. B Attach the apk file to the email and send it to the smart phone that you want to install. C On the sent smartphone , tap the attachedfile to install it. ＜For GalaxyS9＞ Galaxy → MyFile → APKInstallFile → Download → TapApkFile → Install ＜Caution＞ Security UnknownApp Must be set to "Allow installation." ［２］Open the homepage on your smartphone and install it.@ Open the homepage on your smartphone. A You can dowmload the apk file by tapping the [4] program file of each application on the homepage. B Tap "Install" to install. ［３］Install the apk file in the "Download" folder.@ Click the [4] program file of each application on this homepage to save the apk file on the desktop or the like. A Connect your computer and smartphone with a USB cable , and copy (transfer) the apk file to the download folder of your smartphone. ＜For GalaxyS9＞ GalaxyS9\ Phone\Download B On the smartphone to which the apk file was transfered , tap it to install it. ＜For GalaxyS9＞ Galaxy → MyFile → APKInstallFile → Download → TapApkFile → Install As mentioned above , it has been confirmed on the smartphone GalaxyS9 (2021/04/05). ■ There are two ways to install the app on your tablet.［１］Open the homepage on your tablet and install it.@ Open this homepage on your tablet. A You can download the apk file by tapping the [4] program file of each application on the homepage. B Tap the apk file that was downloaded to install. ＜For aiuto＞ Files → Download → TapApkFile → Install ［２］Install the apk file in the "download" folder.@ Open this homepage on your computer. A Click the [4] program file of each application on this homepage to save the apk file to USB. B Insert the USB into the tablet and copy the apk file to the tablet's internal storage. C On the tablet , tap the apk file that you want to install in the internal storage to install it. ＜For aiuto＞ Files → ＵsbFlashDisk → CopyApkFile → InternalStorage → CopyHere → Return → InternalStorage → TapApkFile → Install As mentioned above , it has been confirmed on the tablet aiuto (2021/04/05). |

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@ Tap the "frog" icon on the screen of your Android smartphone
or Android tablet. A Tap anywhere on the app screen. However , in the "【２８】Collatz Problem" and in the "【３０】Translation of Graph of Quadratic Function", tap the input position on the screen to display the character / number keys . As mentioned above , it has been confirmed on the smartphone Galaxy S9 (2021/04/06) , it has been confirmed on the tablet aiuto (2021/04/06) . |

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Android Smart Phone or Android Tablet Android 4.1（Jellly Bean）API16 〜 Screen resolution 1480 × 720 or more |

please contact Email：wuenaegi@yahoo.co.jp