Euclid Poems

Euclid is often considered the Father of Geometry. However, given some quick research I did about Euclid, he did not invent all concepts of geometry but did develop the discipline of geometry by creating a textbook called the Elements. This textbook was filled with his organized ideas and acted as a source of geometric reasoning which future mathematicians used to expand upon. As well, little is known about Euclid's life aside from his mathematical contributions. 

While analyzing the poem by Edna St.Vincent Millay I noticed there was a large focus around beauty. From my interpretation I believe Millay was suggesting that Euclid's mathematical work had beauty. When Millay refers to the sandal set in stone, I imagine she is referring to Euclid's work in mathematics and the beauty of it being a long lasting and integral part of mathematical history as it is set in stone.

While analyzing the poem by David Kramer, I had a bit of a difficult time trying to figure out what he was trying to portray throughout the poem. Again, like the previous poem there was a focus on beauty, but this time it seemed more geared towards the beauty of poetry and the poem itself. However, I also think this poet is trying to bring attention to Euclid’s mathematical contributions and their lasting impressions as he again refers to what I imagine is Euclid’s mathematical work and its beauty being set in stone. The last line of this poem made me wonder if the poet is making reference to the idea that Euclid’s mathematical work is beautiful, being controversial.

References: 

Who is Euclid and What Did He Do? Retrieved October 22, 2020, fromhttps://biography.yourdictionary.com/articles/who-is-euclid-and-what-did-he-do.html 

Eye of Horus and unit fractions in ancient Egypt

The most interesting thing I found while researching was the fact that in Egyptian times the symbol for the ‘whole eye’ only summed to 63/64. The last 1/64 to sum to 1 was thought to represent the magic which was used to reassemble the eye to make it whole. While today this specific collection of numbers represents the first 6 terms of a geometric series whose sum converges to 1. I found this interesting given that today we usually try and always make things a whole number if possible. And personally I would prefer to work with whole numbers, however I found the potential reasoning behind the last 1/64 being left out very interesting! 

In my life there aren’t any numbers that I can think are specifically connected to stories. However, my whole life the number 9 has been my favourite number and I will choose it should I ever be in a situation (sports, pick a number between 1-....) where I can choose a number. This number is my favourite as it is the number my dad played with during all of his time playing baseball. Baseball is my favourite sport and the ballpark is my favourite place to be so the number 9 has a special connection to baseball and my dad.

Magic Square

My first step was figuring out how many ‘addition combinations’ each individual box in the square is associated with: 

• Middle square  - involved with 4 different ‘addition combinations’ to get to 15

• Top Left/ Top Right/ Bottom Left/ Bottom Right - all involved with 3 different  ‘addition combinations’ to get to 15

• Middle Top/ Middle Bottom/ Middle Left/ Middle Right - all involved with 2 different  ‘addition combinations’ to get to 15

Once I figured this out I started listing different combinations of 3 numbers which add to 15. As I did this I noticed I had come up with 4 different combinations of 3 which added to 15 that all included 5. Thus I decided to put 5 in the middle square. I then placed the other number combinations in different sequences until I had used all of them in a way that resulted in all of the rules of the magic square being satisfied. 

In the photo attached is the magic square and the ‘addition combinations’ I used in the magic square.

Was Pythagorus Chinese?

 I definitely think it makes a difference to students learning if we acknowledge or don't acknowledge non-European sources of mathematics. As teachers it is our job to give a true and accurate explanation of material. I believe including and acknowledging where sources of mathematics came from, can act as a way for students to gain a greater connection to the material and it applies context to material, both of which I think will positively contribute to the students learning. As well, acknowledging all sources of mathematics, not just European, provides an opportunity for international students or students from diverse cultures to feel more connection to their cultures.

In terms of the naming of the Pythagorean Theorem, I do not think Pythagoras should be given full credit for it, especially as it is unknown if he was the one who performed the proof. As well Babylonian’s and Egyptians showed understanding of this theorem much earlier, they just were not the first to prove it. Looking at the history of Pascal’s Triangle it seems as though Pascal got credit for the concept, but many others studied ‘Pascal’s Triangle’ long before Pascal in other areas such as India, China and Italy. Thus, I'm unsure if Pascal’s name should be the only one associated with this concept. I have to wonder if naming mathematical concepts and theorems after a person is something that should be done. It seems as though theorems and concepts are named after those who are able to provide proof first, however given the geographical isolation of some mathematicians and lack of proof passed on throughout centuries, many are left behind in the recognition process.

False Position

 Problem: At the end of the week a farmer has a total amount of 100 pounds of grain. The farmer started the week with an unknown quantity of grain and sold a third of that quantity. What quantity of grain did the farmer start the week with? 

Try x=75

75-(75/3)=50

Since we need the answer to be 100, which is twice as big as 50, x must then be 2 times larger than the trial number 75(75x2=150).

Thus if x=150

150(150/3)=100

Therefore the farmer started the week with 150 pounds of grain.

Assignment #1 Reflection

 Problem (from the text): 

Solution to the Problem: 

Given the problem, it appears as though it may be a specific example for the Pythagorean triple (3,4,5). To solve the problem we used geometry, using the picture below. Given the picture, we can see that the area of the overall square with side a+b, minus the area of each rectangle with the sides a,b, equals the small square in the middle which has sides a-b. Thus we can solve for the formula (a+b)2 - 4A = (a - b)2. The algebraic solution is also below.

Extension to the Problem: 

For our extension to the problem we decided to derive the Pythagorean formula with the use of geometry using the picture below. We thought deriving the Pythagorean theorem could help high school students gain a deeper understanding of the formula and help them identify when this formula can be applied in complex situations. Now we can see that the area of the square with sides d is equal to the square of sides a+b minus the triangles of sides a, b. This gives us the formula d2 = (a + b)2 - 4(a · b)/2. We can then use algebra to derive the Pythagorean formula. 

d2 = (a + b)2 - 4(a · b)/2

d2  = a2 + 2ab + b2 - 2ab

d2  = a2 + b2 

A link to our presentation is here - https://docs.google.com/presentation/d/1zqvY38kiCLM6I0fIv7PpQY2Y8IaLpx1ZO_f1124WV7s/edit?usp=sharing