High School – Statistics and Probability Tetra Dice A game requires each player to roll three specially shaped dice. Each die is a regular tetrahedron (four congruent, triangular faces). One face contains the number 1; one face contains the number 2; on another face appears the number 3; the remaining face shows the number 4. After a player rolls, the player records the numbers on the underneath sides of all three dice, and then calculates their sum. You win the game if the sum divides evenly by three. What is the probability of winning this game? Un juego exige que cada jugador lance tres dados con forma especial. Cada dado es un tetraedro regular (cuatro caras triangulares y congruentes). Una cara contiene el número 1; otra cara contiene el número 2; en otra cara aparece el número 3; la cara que sobra muestra el número 4. Después de que un jugador lanza los dados, el jugador registra los números de los lados cara abajo de los tres dados, y luego calcula su suma. Se gana el juego si la suma se divide exactamente por tres. ¿Cuál es la probabilidad de ganar este juego? В этой игре от каждого игрока требуется бросить три игральных кубика особенной формы. Каждый игральный кубик представляет собой правильный тетраэдр (четыре подобные треугольные грани). На однойграни стоит цифра 1; на одной грани стоит цифра 2; ещё на одной грани стоит цифра 3; на последней грани стоит цифра 4. После броска игрок записывает цифры, выпавшие на нижнейграни всех трёх кубиков, и затем вычисляет их сумму. Выигрывает тот игрок, чья сумма делится на три без остатка. Какая вероятность выиграть эту игру?

1 – Statistics and Probability Tetra dice A game requires each player to roll three specially shaped dice. Each die is a regular tetrahedron (four congruent, triangular faces). One face contains the number 1; one face contains the number 2; on another face appears the number 3; the remaining face shows the number 4. After a player rolls, the player records the numbers on the underneath sides of all three dice, and then calculates their sum. You win the game if the sum divides evenly by three. What is the probability of winning this game? HS – 1 – 1 CU 6 The translation of the key concepts (sample space related to the

shape and number of dice, sums divisible by three, and theoretical probability) is enhanced when each of the translations is thoroughly developed and then s/he shows the sample space in three ways and connects the pieces in the “tree” solution to those in the “matrix”.

PS 6 The process of identifying each of the possible sums when starting

with each possible roll of the dice, finding the sums divisible by three, comparing the number of successful sums to the total number of sums is thoroughly developed in the original approach. The verification provides further evidence of the possible sums, those divisible by three, and the size of the sample space. Together these make the process enhanced.

V 6 The review is related to the task, and enhanced, by using a

different recording process and by the connections s/he made between the parts of the original solution and the same results in the parts of the verification.

C 5 The path connecting the translation of the key concepts to the

strategy of using tree diagrams to show the possible sums and the theoretical probability then connecting to the second look and the identified solution is thoroughly developed.

Acc. 5 22/64 is a mathematically justifiable solution to the task and is

supported by the work.

1 – Statistics and Probability Tetra dice A game requires each player to roll three specially shaped dice. Each die is a regular tetrahedron (four congruent, triangular faces). One face contains the number 1; one face contains the number 2; on another face appears the number 3; the remaining face shows the number 4. After a player rolls, the player records the numbers on the underneath sides of all three dice, and then calculates their sum. You win the game if the sum divides evenly by three. What is the probability of winning this game? HS – 1 – 2 CU 4 The translation of the key concepts (sample space related to the

shape of the dice, sums divisible by three, and theoretical probability) is complete. The translation could have been thoroughly developed with the addition of labels.

PS 4 The process of creating an organized list of all the possible sums

from the three dice making up the sample space and then crossing-out those with a sum not divisible by three and then finding the number not crossed out compared to the size of the sample space is complete.

V 4 The verification provides a complete second solution to the task

and results in supporting the solution, making it complete. C 4 The path connecting the translation of the concepts to the process

used to solve the task to the second look and to the identified solution is complete.

Acc 5 22/64 is a mathematically justifiable solution supported by the work.

1 – Statistics and Probability Tetra dice A game requires each player to roll three specially shaped dice. Each die is a regular tetrahedron (four congruent, triangular faces). One face contains the number 1; one face contains the number 2; on another face appears the number 3; the remaining face shows the number 4. After a player rolls, the player records the numbers on the underneath sides of all three dice, and then calculates their sum. You win the game if the sum divides evenly by three. What is the probability of winning this game? HS – 1 – 3 CU 3 The translation of the key concepts (sample space related to the

shape of the dice, sums divisible by three, and theoretical probability) is only partially useful for this student, since s/he couldn’t connect the results of the translations to the results of the process used to solve the task.

PS 2 The process of starting with a particular roll of the first die and then

finding the sums created by the rolls from the other two dice and identifying those divisible by three is partially recorded and not connected to the sample space size of 64. Finding the successful outcomes compared to the number of outcomes from one starting point -- then simplifying the result with no change when s/he multiplies by 4 is ineffective --- making the overall process used underdeveloped.

V 3 Although the verification is almost identical to the original approach

(except the 1st dice), it would be a complete solution supporting the original one if s/he would have dealt with the 64. Without purposefully deleting the 64, or justifying the difference in sample space size, the verification is only partially completed.

C 3 There is a significant gap between the sample space size of 64 not

being used or eliminated (translation to process) as well as between the process (3/8) and the solution remaining 3/8 although multiplied by 4 (for the 4 different 1st rolls possible)

Acc. 1 3/8 is not a mathematically justifiable solution to this task.