# Golden Angle, Factoring 360, Proper Fraction Shading

Continuing our exploration of the Golden Ratio of 1.618….., the children learned how to use a protractor to measure an obtuse angle of 137.5 degrees. We then discussed the difference between an obtuse angle (between 90 and 180 degrees), an acute angle (between 0 and 90 degrees) and a right angle (90 degrees). Finally, we discussed a reflex angle which is between 180 and 360 degrees. Specifically, we subtracted 137.5 from 360, the degrees of a full circle to get 222.5 degrees. When we divided that reflex angle of 222.5 degrees by the 137.5 obtuse angle, we get the Golden Ratio. This Golden Angle of 137.5 degrees is found in nature in plants looking at the angle of measurement of leaves (see page one of the attached pdf).

We then looked at factoring 360 to find all 12 factor pairs. Each factor pair multiplies to 360:

1 360

2 180

4 90

8 45

3 120

6 60

12 30

24 15

5 72

10 36

20 18

40 9

For example, if you split a circle into 8 equal sectors, each of the central angles is 45 degrees. Conversely, if you split a circle into 45 equal sectors, each of the central angles is 9 degrees.

I created a game in which the children roll two dice to create a proper fraction (between 0 and 1 where the numerator is smaller than the denominator); then find the degree measurement of that fraction and finally shade each equivalent fraction on the game sheet with 12 circles with sectors cut up to represent each factor pair. The attached pdf uses a fraction of 3/4ths as an example.

In class we used dodecahedral dice as well as six-sided dice. Twelve sided dice have a great advantage in creating more complex fractions like 5/12 which is equal to 10/24, and 15/36 and 25/60. I provided a net for the children to create their own dodecahedral dice.

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