Part of being a Mathematician is being able to solve, represent, or understand a problem in different ways. After solving the Diophantus riddle algebraically in class, I wondered if there was a way that I could represent the problem visually so that I could better understand it. From a teacher viewpoint, I know that each student learns and understands differently, so some of my future students might better understand a problem like this if it is represented visually. In some of my teaching classes, when dealing with fractions, we have worked with visual models of fractions like arrays. I worked through the Diophantus riddle again and found a way to represent each part of the riddle visually in an array.
'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'
The first thing that I noticed when I worked through the riddle again was that I could establish a common denominator of 84 for the fractions 1/6, 1/7, and 1/12, since the least common multiple of 6, 7, and 12 is 84:
Once I established a common denominator of 84, I knew that I could represent each of these fractions visually in a 7 x 12 array, which makes 84 equal-sized units. In the model below, I represented each value in the problem in a different color:
Since Diophantus was 84 years old when he died, based on this model, we know that: