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PolySTARS is a unique game in which abstract number relationships are demonstrated through beautiful visual geometric patterns that students of all age may enjoy. Users will be able to discover many mathematical concepts relating to number sense, including counting, addition, subtraction, multiplication, division, complimentary/commutative numbers, fractions, factors, prime, and relatively prime numbers. In Geometry, concepts such as polygons, inscribed angles, However, the game is designed to discover number relationship through visual geometric patterns.
A repetitive addition is the base for multiplication. Have students stand in a circle, the teacher counts 1, 2, 3, and gives a candy to every 3rd one and continues in the same manner, until she/he gives the child who already has a candy and then stops. Then he/she may continue counting, which will result in giving a second candy to each child who already has a candy, and so on...
The question is: Will every child get a candy? It would depend on the number of children in the circle. This special case is an unfair game because some children get and some will never get a piece of candy. The result/outcome of the game depends on the number of students in the circle and the number in which the teacher repeatedly counts. If these two numbers have a common factor, the game will be unfair. For example, common factor of 12 and 8 that is: 12 students and counting 8 and given a candy to every 8th student) will be the same as counting every 4 and will result in only three students will get the candy out of 12.
One can visually discover if a number pair has common factor or not by using the PolySTAR game. Lets test an example by clicking on (HYPERLINK "../../Desktop/MamikonGames/Angie/PolyGoN/PolyGoN.html"PolySTARS). Set up the number of students by clicking on the + or button. Then enter a number representing the jumping or counting number in the box. Let the number of students be 12 and enter the number 8 as the number of jumps or counts. Immediately, students will be able to visually see an equilateral triangle showing that only 3 students will receive a candy, and no matter how many times you continue jumping others will never receive a candy.
Now, try a put in a different counting (jumping) number for the same number of students. Lets say counting by 7s. Wow! The star is complete, that means everyone gets a candy. Why? That is because, 12 and 7 dont have any common factors. This time try counting by 5s by entering the number 5 in the box with the same number of students (12). Again, the star is completed and every student gets a candy (12 and 5 do not have any common factors).
Can you guess what numbers will result in an unfair game? If you guess, 2, 3, 4, 6, 8, 9, and 10 some students will not get a candy in each case. This is because these numbers have common factors with the number 12. Try putting a number 9 in the box with 12 students. The geometric figure shows a square and the students quickly see that only 4 students will get a candy. That is because 9 and 12 has a common factor of 3. Same square you will obtain by entering the complimentary number 3 in the box because 3 and 12 has the same common factor of 3. So, 9 and 3 have the same patters (that means that they have the same factors), because they add up to 12 (the number of students).
A fair game will take place when two numbers, the number of students and the jump count number are relatively prime, that is, they dont have common factors (mathematically called relatively prime for example 12 and 7 are relatively prime although 12 is not prime, but because they dont have common factors and therefore are relatively prime). In this game, common factor is the largest number that both numbers are divisible by. For example, 12 and 8 have the largest common factor 4, no larger number divides both numbers. 2 is a common factor too, but not the largest.
We can check with the PolySTARs game that every factor of the largest common factor we find is a common factor of the initial two numbers as well.
The PolySTARS Game will allow middle school students to experience and discover the concept of common factors, prime number, and relatively prime numbers in a playful manner.
Lets choose a prime number for the number of students, say number 11 by clicking the + or buttons. Now, whatever number you entered less than 11 as a counting number in the box, the resulting polygon or stars will always be completed. That is because 11 is relatively prime to every number less than 11. Enter any number less than 11 in the box, say 8, what do you see? The polygon is complete, passes through every student, which means every student gets a candy. Why? Yes, thats correct! 11 and 8 are relatively prime numbers.
Lets say we want to find if the numbers 21 and 14 have a common factor? Enter, 21 for the number of students by clicking the + or button and 14 in the box for the number of counts. You can visually see that the polygon (triangle) is incomplete. That means they have common factors. In this case, the triangle shows that the common factor fits 3 times into 21. It is 7, which is 21 divided by 3.
Similarly, for the numbers 12 and 9, we got a square, that means their common factor fits 4 times into 12, which is 3. In the case of 12 and 8, we have a triangle, which means the common factor of 4 fits 3 times into 12. This also demonstrates the commutative property of 3 x 4 is the same as 4 x 3 = 12.
As an extension of this piece, students could use the same concept in a puzzle format.
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