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The dial has three different circles. I printed mine each in a different color to add contrast and to help know the difference between each one. Each one has a slit in it so it can overlap and spin. One circle has 6 sections and includes the fractions with three and six in the denominator. One circle is divided into 8 sections and includes fractions with four and eight in the denominator. The large one is divided into 24 sections and has 24 as the common denominator.
When you spin the dial, you match up the lines on the right side. So for instance in the image at the top of the post, I was adding 1/6 + 1/6. I matched up the right line of 1/6 in the center with the right line of 1/6 on the large circle. Then look to the left and the edge of the center circle marks the point where the answer is. The answer here is 2/6 or 1/3.
See more in the related blog post.