Activity

Proportions in reality

Characters:

Theano of Crotone

Theme: Equations, proportion

Competencies

Mathematical competence in science, technology and engineering

Personal, social and learning to learn competence

Competence in cultural awareness and expressions

Subjects and year by Educational System

Spain > Mathematics > 2nd ESO > Number sense

Spain > Mathematics > 2nd ESO > Algebraic sense

Spain > Mathematics > 2nd ESO > Socio-affective sense

Spain > Mathematics > 3rd ESO > Number sense

Spain > Mathematics > 3rd ESO > Algebraic sense

Spain > Mathematics > 3rd ESO > Socio-affective sense

Enunciation

The golden number might have been the first irrational number known to the Greeks. When the Pythagoreans discovered that irrational numbers existed, i.e. that they could not be written as the quotient of two whole numbers, they were dismayed, as this fact broke many of their philosophical theories. That is why they decided to keep this discovery a secret.
Theano of Crotone, a Pythagorean mathematician, was the first woman to carry out these divisions, confirming thus the existence of irrational numbers. As a good Pythagorean, she believed and defended that all material objects were composed of natural numbers, so that the measure of anything could be expressed with an exact measure. However, she was also the first to posit the existence of the golden ratio as the essence of the universe.
The golden number, or golden ratio, is represented with the Greek letter &Phi; (Phi), honouring Phidias. Remember that Phi is a proportion in which "The whole is to the part as the part is to the remainder". The following image shows an example:
<img title="The whole is to the part as the part is to remainder" src="http://inclusion.womenslegacyproject.eu/uploads/resume/attachment/4009/The_whole_is_to_the_part_as_the_part_is_to_the_remainder.png" alt="The image shows the algebraic expression need to calculate the golden number" width="120" height="49" />
<img title="Segment showing the hole and the part" src="http://inclusion.womenslegacyproject.eu/uploads/resume/attachment/4008/segmento1.png" alt="The image shows a segment with the hole and the part to build the golden number" width="450" height="132" />
1. Let's find now the golden number. We assume that the smaller segment is 1 cm, and that we do not know the value of the larger one, i.e. a = x and b = 1. Then, the whole segment measures x+1, and we can state the above relationship as:
<img title="The whole is to the part as the part is to the remain when the part is 1" src="http://inclusion.womenslegacyproject.eu/uploads/resume/attachment/4010/The_whole_is_to_the_part_as_the_part_is_to_the_remainder_when_the_part_is_1.png" alt="The image shows the algebraic expression of the &quot;The whole is to the part as the part is to the remain when the part is 1&quot;" width="120" height="54" />
If we multiply crosswise, we obtain the following quadratic equation:
<img title="Equation to find out the golden number" src="http://inclusion.womenslegacyproject.eu/uploads/resume/attachment/4011/Equation_to_calculate_Phi.png" alt="The image shows the equation that permits to calculate the golden number." width="350" height="42" />
Solve the equation to get x, which will be the golden number, also called the "divine proportion''.
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2. Use your ruler to obtain the measurements of a credit card and an ID card (8.6 cm long, 5.4 cm wide) and check that they are in the same proportion as the golden ratio.&nbsp;
3. In ancient times, beauty was related to this proportion and this is still used today as it is thought that it makes objects more attractive. Bearing in mind that all bodies are valid in their different shapes and sizes, check if your height is in this proportion to the distance from the ground to your belly button.&nbsp;

Observations and context

- The first part of the activity may be more abstract for students in 2nd of ESO. It is recommended to represent the drawing on the blackboard, and then move on to explaining the proportion and changing the values a and b to those that will later form our equation a = x, b = 1.

- Crotone was in Theano's time a colony of Magna Graecia.

- Enheduanna (25th century BC) was a predecessor of Theano of Crotone, considered the first recorded woman in the history of science and the first to sign her works, in cuneiform script.

- Some of Theano's contemporaries are other women in the Pythagorean school that were born around 500 BC, such as Damo, Myia and Arignote of Crotone, considered to be daughters of Theano and Pythagoras by several authors. Even though there is not much information about them, some other women belonging to this group were Babelica of Argos, Beo of Argos, Quilonis, Echecrates of Phlius, Ecellus and Ocellus Lucanus, Habrotelia of Tarento, Cleecma, Cratesiclea, Lastenia of Arcadia, Pisirroda of Tarento, Filtis, Teadusa, Timica and Tirsenis of Sibaris.

- After Theano we can mention Aglaonice, or Aganice of Thessaly, (3rd century BC, known for her ability to predict eclipses) and Hypatia (4th century AD).

Description

Solving second degree equations using proportion. The aim of this task is to find the golden number in a segment by solving a second degree equation. In addition, we will study if this proportion exists in different common objects and in our own bodies' measurements.

Proportions in reality

Enunciation

Observations and context

Description

Answer

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