Zain Rehmtulla

 The first connection I made to this problem was that I most likely would not discover a formula that would work, yet I figured there would be a pattern that should be able to be written as a formula. That being said, the first problem was of higher complexity given that you could use both scales to get from 1-40 so I tried guess and check. I actually started working my way back and forth by increasing from 1 and decreasing from 40. Obviously, we needed a 1. Then for 2 either we would have to pick a 2 or we could use a 3. Given that we can use both scales I think it would be more beneficial to use the 3 since it would also satisfy the next term and we can generate 2 from 3-1. At this point I looked at 40 and thought about two options, first 30 & 6 and then 27 & 9. The reason 30 & 6 failed is because I realized that the gap was significant so while it works for numbers close to 40, it fails late teen numbers (like 19) since 30 - (6 + 3 + 1) only gives 20. So then I conti...

 Personally, I do actually think I've had a slight shift in my approach to word problems. Before looking at the word problems of Egyptians I would think that students would be hesistant to attempt word problems. Part of this is because students are taught math from a numerical standpoint, instead of as it being a "science". It is similar to the scientific method in that we make a decision after making our observations. For example, even in the ancient times, the Babylonians would make observations of the passage of the moon or planets. Moreover, the Egyptians would try to make estimations using for example the false position technique and they would use guess and check. I think that before attempting the algebra, now when I teach word problems, it would be important to get the students to also interpret the problem using words just as the Egyptians did with the loaf interpretation word problem. In a way, this code be pseudocode for math.  .Then shortly thereafter, I would...

  I found it surprising that the the Egyptians would have two systems of cubits “a royal cubit” and a “short cubit”. I also question how they came to decide on 7 palms? Why would they elect to choose a prime number for counting.   Moreover, I interpret the relationship between Remen, cubit, and double-remens as what we use with mm, centimeters, and meters. It also seems like the calculation of root 2 is the result of the diagonal of a square. My question is whether there is some significance to this idea of diagonals it is ancient Egyptian mathematics and many structures did involve diagonals such as the pyramids. It is also quite convenient that t he remen, cubit, and double-remen all amount to a nice number of fingers with 19.5 being the most irrational of the numbers so to speak. Moreover, it seems as if the idea of fingers preceded cubits, yet when calculating the remen as half the diameter of a square with sides of a royal cubit, we once again arrive at a nice number ...

  Unquestionably, the structure of the word problems from the Babylonian era does mirror the ones we have today. While they attempt to apply to the real world, one would argue that some of the problems are more hypothetical in nature. One of the points that stood out to me, was that the Babylonian problems were labelled as “practical”, yet many of the problems extended beyond the nature of a habitual task that the Babylonians or other civlizations would partake in. For example, a grain pile of 36-48 cubits (18-24 m) high is difficult to imagine let alone actually create.   Moreover, I found it highly interesting that the Babylonians would compromise precision for solutions that were more appealing. For example, when solving right-angled triangles and trying to find the length of one of the legs or the diagonal, they would opt for the integer answer so as to avoid fractions. Perhaps, it was due to the literacy of other people and keeping it in integers would make it easier to...

  One thing that stood out to me   was that the st. andrews article mentioned that the sexagesimal system when using decimals has 3 prime factors to 60 (2, 3, and 5) which when expressed as a decimal   will be finite if the denominator has no prime divisors other than 2, 3, and 5. Therefore, in terms of time, this would be more convenient if we needed to write a fraction of time or it’s decimal as there would be more possibilities than if we had denoted an hour as 100. I too find it interesting that some theories suggest a base 60 system based on astronomical events such as the number of planets multiplied by the number of months in the year . One of the major inconsistencies between the two articles was that the sun moves through its diameter 720 times during a day and with 12 Sumerian hours per day, we obtain 60 as a result of 720/12. However, the concept   of fixed hours in a day did not exist supposedly as mentioned in the scientific American article. The divis...

  The article highlights the extent to which the history of math recognition is misrepresented as their were many political, economic, and relational factors that influence the spread of information. One things that surprised me was the extent to which the Arabs helped propagate information in the post-Alexandria era. For example, there were many caliphs that Indians and Chinese came to to translate their works. As a result, resources could be more easily distributed in Europe as a result of the Arabs connections and ability to translate. The article mentioned that the Arabs in some way saved the Greeks which is fascinating.   The second thing I found interesting was the constant reference to the Egyptians. It seems that they are not given the credit they merit. For example, the Greeks actually developed a lot of their mathematics from the Egyptians.  One example of this is how Pythagoras’ teacher (who was also Greek) spent years learning under the Egyptians and even...

  1)       I believe the Babylonians chose a base-60 system as it would be significantly easier to communicate findings on time if each space represented a new place value of 10. Perhaps it’s a bit of an antiquated thought, but I feel before GPS navigation systems were created, people would try and calculate the time it would take them to travel across a river or sea in order to gain access to precious materials or to trade with other colonies. It would assist them because there are 60 seconds in a minute and 60 minutes in an hour. 2)       I don’t know too much about the use of 60 in our daily lives other than the idea of time as well. That being said, I know that when we look at measuring angles, 60 degrees is very useful as it fits into 180 degrees which would be a flat line and 360 degrees making a full circle. There are also 24 hours in a day and that does not seem to fit evenly into this system   3)   ...

  I think that there are elements of Math history that should be introduced into the classroom. For example, I believe it was the Greeks that invented the base 10 system and they also played a role in terms of how we operate our multiplication today. So many students can “game” the educational system today simply by memorizing concepts, formulas, or processes in order to attain a high grade. However, I think we should emphasize more on the process than the product and helping students to understand why they are making certain decisions in solving a problem. Incorporating elements of math history is essential to expand students understanding of basic mathematics such as arithmetic. Moreover, having an understanding of some of the proofs such as Euclid proofs can help build problem solving skills while also reinforcing learning in current mathematics. In fact, I believe there are ancient proofs that show a prior understanding of proof by induction, of course this is a concept taught ...