Wednesday, September 23, 2026 The market scales puzzle & connections with ancient Egyptian mathematics: Homework for Monday Sept 28

For a two pan scale, the weights you can use are 1g, 3g, 9g, 27g. Essentially, we can formulate the equation like this:

                                                  weight_wanted = a(1) + b(3) + c(9) + d(27)

where:

                                                                    weight_wanted ∈ {1, 2, ... , 40}

                                                                            a, b, c, d ∈ {-1, 0, 1}

TLDR: -1 = putting the weight on the left side of the scale, 0 = leaving the weight out, 1 = putting the weight on the right side of the scale, and every integer weight between one and forty can be made from some combination using these four weights. 

For a one pan scale, the weights you can use are 1g,  2g,  4g,  8g,  16g. Essentially, we can formulate the equation like this:

                                                  weight_wanted = a(1) + b(2) + c(4) + d(8) + e(16)

where:

                                                                    weight_wanted ∈ {1, 2, ... , 31}

                                                                            a, b, c, d, e ∈ {0, 1}

TLDR: 0 = leaving the weight out, 1 = putting the weight on the scale, and every integer weight between one and thirty-one can be expressed from some combination of these five weights.

A good extension for this problem would be finding the pattern. Say the two-pan problem's parameter weight_wanted becomes bigger. Students might notice that "hey, the weights are three times the previous weight", so the most logical weight to add next would be eighty-one. Students would need to explain why. I think this is a decent activity for Foundation Math 10, since one of the curricular competencies is pattern matching. 

The one pan problem is somewhat related to binary numbers? That problem is essentially just flipping bits in a way? Could be an interesting activity to do before introducing binary numbers in computer science classes or math classes! 

Homework for Monday Sept 21-- Read and respond : Some interesting resources on the (Babylonian-based) history of time calculations, base 60 and base 12

When I think about a year, January is not my top pick as the beginning. I go crabbing every May, so the month of May feels like the start of my personal year cycle. I picture the year as a circle, beginning in May, then moving through summer, fall, winter, spring, and eventually May again. Turns out, the way we experience time can be related more to personal experiences rather than official holidays or calendars!

The articles made me reconsider how arbitrary some measurements of time around the world are. For instance, dividing hours into 60 minutes and minutes into 60 seconds originates from ancient Babylonian mathematics, while other cultures have their own takes on the topic of "time" based on observing weathers, seasons, and daylight. Take the Chinese 24 Solar Terms as an example. 

One inconsistency between the articles is that Scientific American suggests that base 60 is useful because it can be divided nicely in many ways, while the MacTutor article suggests that historians are not quite sure on why base 60 was originally chosen. I found it surprising that rather natural / normal looking measurements actually dated back to historical and cultural origins that might be completed unrelated to us in some sense.

**EDIT: Blog post due Wed Sept 23: Did Mesopotamian scribes have algebra? (And how can we recognize 'rhetorical' and 'syncopated' algebra?)

There are four questions listed on the blog, I will answer each question independently.

1. A general principle could be expressed as a verbal procedure (or like a recipe?) rather than a formula with symbols. For instance:

                                                                    x^2 + 7x = 60

can be expressed as:

                   Take a quantity, multiply it by itself, then add seven times itself, and the result is 60

The Babylonians often used terms like "length" or "breadth" or "area" to represent unknown quantities / relationships, showing that algebraic expressions can exist without x or y:

                    Take a length, multiply it by itself, then add seven times itself, and the result is 60

2. Not really? Mathematics HAS generalization and abstraction, but it can still start with things like measurements, visual diagrams, or patterns. Again, the examples shown in the text (say example 4.7) were still solved using practical ideas like length and width, with no signs of  "let this be x and let that be y" anywhere in the solutions.

3. Said topics can certainly be expressed through words only. Take this example from geometry:

                                                                      a^2 + b^2 = c^2

the words equivalent would be:

           the squared of longest side of a right triangle has the same length as the 2 other sides' squared

Although one thing is definitely noticeable: words are a lot more cumbersome, where a few symbols might translate to multiple sentences in words.

4. Our students 100% do! Take this example:

                                                                        x + 5 = 12

I think both us teachers & students might first express this equation like this:

                                                       A number plus five equals twelve

which can be viewed as:

                                                                        ? + 5 = 12

that looks like some kind of syncopated algebra. 

And another small bit of homework for Monday September 21:Create your own Babylonian-style base 60 multiplication table for the number forty-five

3 15

4 11,15

5 9

6 7,30

8 5,37,30

Homework reading and response for Wednesday: Crest of the Peacock introduction

One thing that surprised me was definitely China's place in the history of mathematics. One theorem that I remember from undergrad is the Chinese Remainder Theorem: 

which I have never really thought about historically. Upon further research, an early version of this theorem appeared in the math text Sunzi Suanjing, written approximately between the 3rd - 5th centuries CE. The actual text gives a problem involves a number that has different remainders when divided by 3, 5, 7, and provides a method of finding the solution:

It is pretty cool that a "modern" math theorem that I learned from university can be traced back to my home country more than 1500 years ago!

Another thing that surprised me was how important researchers in the Islamic world were. The book explains that mathematic ideas were combined from Greek, Indian, and Persian traditions, where Baghdad actually became a central hub for learning math in the world. 

Lastly, the book points out that historians should not assume that two civilizations using similar mathematics mean that they borrowed / stole (?) the idea from each other. The actual development of the same mathematic idea could have happened in multiple places, and evidence of contact or translation must be found before making a conclusion of "civilization xxx borrowed set theory from civilization yyy"

Research: https://mathshistory.st-andrews.ac.uk/HistTopics/Ten_classics

Sept 14 Entrance Ticket

Prior to reading the article, I think mathematics history could be incorporated into lessons by explaining historic contexts of where certain techniques came from, how similar techniques originated, or just general fun facts to enhance the lesson? From personal experiences, I have not been exposed to much mathematics history in the past, so I am unsure of how I can add mathematics history into my lessons. 

After reading the article, two ideas stood out to me. First, the authors explained that textbooks often present polished results while hiding the mistakes made behind the scenes. Showing the mistakes could help students recognize that making mistakes is very normal in mathematics. In addition, introducing a historical problem before teaching its modern solution is useful because it can help students understand why the modern solution is needed. In the future, I think my goal with mathematical history is to choose examples that support mathematical understanding, while also showing that math is developed through real life problems. 

Lastly, the article mentioned several key factors on why history of mathematics is often neglected in classrooms: lack of time, lack of resources, lack of expertise, lack of assessment. I think teachers could use short and accessible (i.e., can be googled easily) historical examples in lessons instead of treating history as an entirely separate topic, which I will try to do during practicum.

 Hello world

I have a pen~~