  How did we find those answers?  On the clock:            3  =  3   12  +  3  =  3  24  +  3  =  3  36  +  3  =  3  48  +  3  =  3  60  +  3  =  3Now let us take one third of each expression:   3  / 3  =  1   (12  +  3) / 3  =  5  (24  +  3) / 3  =  9  (36  +  3) / 3  =  13  =  13 - 12  =  1   (48  +  3) / 3  =  17  =  17 - 12  =  5  (60  +  3) / 3  =  21  =  21 - 12  =  9As you can see, one of the same three numbers (1, 5, or 9) will be the answer for one-third of any number of hours equal to three modulo twelve.  Not only did we find the three answers, we also learned that there are only those three.  The same reasoning would show that if we took 1/4 of 4 (or 1/4 of any number on the clock) there would be four answers. And, as the text noted, when you take the square root of a number you get two answers (for example, 4 x 4 = 16 and -4 x -4 = 16), and when you take the fourth root of a number you get four answers, as for example:        2  x  2  x  2  x  2 =  16           -2  x  -2  x  -2  x  -2   =  16    2i  x  2i  x  2i  x  2i  =  16        -2i  x -2i  x  -2i  x  -2i  =  16                        __where i = v-1.That third graders can comprehend these math concepts suggests that the way math is currently taught is limiting.  Third graders are capable of understanding advanced concepts such as taking the root of a number.The explanation that zero is nothing boxes in both a child's and a teacher's true comprehension of the basics of mathematics.  A more robust explanation fosters both greater creativity and more complete solutions as students apply that knowledge. Unfortunately, there are too many supplementary texts that fill children's heads with mathematical nonsense when they describe zero only as having a value of nothing and of being a place holder.  Main Menu