It was an interesting experience to work with ancient Babylonian problems involving the circle. As we explored them, I found myself taking notes to return to later when planning my lessons. I appreciated how these problems were formulated in ways that naturally encourage thinking and curiosity.
In particular, the task about the relationship between chords and the diameter of the circle stood out to me. It feels like a rich, almost limitless resource for introducing trigonometry to students. There is something powerful about showing how ideas that emerged thousands of years ago can still spark understanding today.
Throughout history, the circle has held a kind of magical attraction for humans, and engaging with these ancient problems reminded me of that. It was inspiring to see how timeless mathematical ideas can support meaningful learning in the classroom.
I am fascinated and at a loss for words by the rediscovery of Iran.
I remember studying Persia and the wars in my high school back in Colombia, but I never heard of that marvellous place again, which lived in my memory like a fantasy tale of history.
Thank you, Maliha, for bringing a piece of memory that enriches our journey to teaching. I couldn't watch the video before your class, but I watched it today, and I spent the following 2 hours googling facts and stories about Iran. Now Iran is in my list of places to visit. Fascinating!!
Euclid and Euclidean geometry history reminds me of the history
of Jesus Christ and the Catholic church. Some men imparted knowledge
and leadership. Those men left lessons in different fields, but their dedication
and ideas turned on the flame of curiosity and inspiration in others who
continued their work. On one hand, we have Euclid with a scientific approach and on
the other hand, Jesus with a spiritual approach. Both are remembered and followed
by people to our day.
I think the beauty of Euclidean postulates roots in its
validity throughout time and the simplicity of the proof procedures. Those
postulates invite us to think mathematically in a transcendental way.
A market vendor sells dried cooking herbs in whole-number amounts
from 1 to 40 grams. The vendor has an old-fashioned two-pan weigh scale,
and has exactly four weights of different amounts that allow them to
weigh out any of these amounts of herbs -- without using the herbs or any other
object as an auxiliary weight.
One of the weights should be 1 g.
My first insight was to find the number of combinations for
4 integers that can add 40, but I realized that the number of combinations is
huge. I need to add the constraint that I can place the weights on both sides of
the scales. Every time the scale should be calibrated to zero with both sides weighing
the same, including the herbs.
If the weights are on the side of the herbs, they rest, but
if they are on the opposite side of the herbs, they add. Because the minimum number
of herbs to sell is 1gr, there should be one weight of 1gr. Then I need to find
how to calibrate the scale to sell 2 gr. I could use the 1gr. along the herbs
and place a 3gr on the other side. It could be a 2g weight too, but remember that
we have that single weight to balance any even number.
If I need to sell 3 gr, I would use the 3-gr weight
If I need to sell 4 gr,
Note: I think the other weights should be odd because I have a 1 g weight to balance.
I would need more time to find the rest of the weights. For now, I am sure, 1 is 1gr and the second should be 3gr.
25gr, 15gr, 10gr, 5gr,1gr is like counting in base 5
After answering this second part, I can re-think the
previous problem. If I use base 3, the weight would be 1gr, 3gr, 9gr, 27gr
I don’t know yet, but these problems have taught me that it
is important to dedicate time to think and think and think.
The problem pushed me to go and think about
alternatives away from the decimal system. This part was difficult because it
is what we use daily. I like the connection between the different
base systems and the diversity of paths to find the solution to a problem.
The most surprising aspect of the ancient
mural at Menna’s tomb is the storytelling capacity of the illustrations. I truly
spent 15 minutes re-creating the scene in my mind and trying to discover all
the symbolism represented.
Humans have developed technologies
according to the development of the economy and society. This wall represents the
vibrant commerce activity of the time. Reading Hans Barnard's document about “Surveying
in Egypt” helped me to identify measurement representations like the “knotted rope”
on the upper level and the angle suggestions on the lower level.
I could almost hear the noise of
the place where the image shows how the men on the left offer the measurement of
the wheat field plus fruits, birds represented with the three trees and fish, which
is carried by one of the characters with a lower hierarchy (if smaller size
represents lower category). Encountering these men, there are the people on the
right who are offering some grains and animals.
The two questions I have are:
-
Do they represent actual quantities in this
mural? You can estimate the number of knots on the rope and the grains inside the containers. Are these containers representing quantities, too? Like weight
or volume?
-
The other question is also regarding the symbols
and the measurement. What does it mean that in the bottom image, 3 men on the
left of the grains, holding 3 white devices. 4 men on the bottom holding 4
devices, and 5 men holding 5 devices, as the men on the left. Does this relate
to the 4-3-5 proportions?
-
The last question is about the old man in the
bottom right side. What is he doing? He looks older because he is depicted with
grey hair and a little belly.
In conclusion, this image represents a complex system of measurements
that includes: distances, angles and weights. Nevertheless, it shows only the exchange of commodities; it doesn’t show any symbol that represents money as an
element of exchange.
Babylonian Algebra.
The reason that Babylonians could formulate
mathematical principles without the development of algebra or algebraic notation
is because of the practical uses of math allowed them to create hypothetical situations,
problems and solutions based on their reality.
Babylonians and other ancient civilizations developed mathematical concepts in response to the need to measure land, register wealth, and track transactions.
This practical understanding is derived from the ability to envision scenarios where math can be applied. But this practical usage limited their imagination. The fact
that almost all the calculations were made based on real quantities prevents
them from thinking about concepts like zero or negative numbers.
When the Babylonians were solving problems, the method didn’t use a second-order polynomial but several versions of simplified quadratic equations,
replacing higher orders with combinations of powers of two.
