Solution to Puzzle Problem

In tackling this problem, I was able to determine that 1 and 3 must be among the four weights, while the other two weights fall between 10 and 30. However, because I was using an elimination-by-exhaustion approach, I didn’t explore all the possible combinations. For a one-pan scale, the five weights would be 1, 2, 3, 6, and 21.

This puzzle can serve as an introduction to combinatorics, demonstrating how different combinations can be assembled to achieve a desired outcome. Students can explore how many possible combinations exist, which work, and which don't. This encourages them to practice elimination and grouping methods, while strengthening their understanding of numerical expressions and the concept of equality.

The puzzle also ties into number theory, illustrating how numbers can be manipulated in various ways. I started to imagine possibilities beyond one- or two-pan scales. Could there be three, four, or even more pans for measuring a larger number of weights? I believe there could be a pattern that forms an arithmetic series.

How would I teach word problem?

Our class recently had a thought-provoking discussion about word problems in math. The general consensus was that we’ve all become so adept at solving them that, over time, they’ve lost their meaning in a real-world context. Instead, word problems now feel like an exercise in identifying keywords and units that can easily be plugged into formulas. While we’ve become proficient at this, the process feels mechanical—less about understanding the problem and more about finding the right numbers to fit a predetermined solution.

However, when I was a child, I had a very different experience with word problems. Growing up in my home country, the math curriculum placed a greater emphasis on storytelling and narrative within word problems compared to the curriculum here in British Columbia. I remember that those word problems often felt like a refreshing break from the gruelling algebraic exercises. While algebra could be tedious and repetitive, word problems brought in a sense of creativity and engagement. 

One of the most rewarding moments came when I got my tests back and saw that I had earned full marks on the word problems. It felt like a validation of my abilities, especially because I was often one of the few students who not only understood the questions but could also work through all the steps to arrive at the correct solution. That sense of mastery was incredibly motivating.

Reflecting on both our class discussion and my personal experiences, I believe that to make word problems more conducive to learning, they need to be more engaging and challenging. Teachers should experiment with how word problems are framed—how the information is presented and how the questions are worded. The goal should be to spark curiosity in young learners, encouraging them to think critically and creatively rather than simply applying formulas. By integrating more context, narrative, and even humour, we can turn word problems into opportunities for exploration. Because after all, children are naturally inquisitive and constantly seeking new information.

Response to Surveying in Ancient Egypt

One thing that truly surprised me in the reading was the significant role religion played in ancient Egypt’s measurement systems. Their measurements, proportions, and even the angles of their building structures carried deep religious and spiritual implications. Temples, pyramids, and other sacred sites were designed to honour their gods and reflect their beliefs. This concept reminds me of the Chinese practice of Feng Shui, which also emphasizes the influence of spatial arrangements on energy and fortune. According to Feng Shui principles, the placement and orientation of objects or structures, such as homes or furniture, can directly affect a person’s well-being, luck, and prosperity. For instance, Feng Shui experts often recommend ensuring that a backdoor is not directly aligned with the front door. This is believed to prevent wealth or good fortune, represented by energy entering through the front door, from flowing straight out through the back door. In this way, people can retain the positive energy within the home. It’s fascinating how these two seemingly distinct cultures shared the same belief about spatial orientations. 

I have two questions: First, who was responsible for educating the surveyors about the various types of measurements? Second, building on that, how did the ancient Egyptians teach the measurement concept? Was there a formal system in place for instruction, or was it passed down through verbal or informal methods?

Multiplication Table of 45 in Base 60

 

Response to Word Problems as Genre in Mathematics Education

When I was working as a barista at a local cheesecake café in high school, one of my responsibilities was ordering various types of cheesecake from the main distributor. I estimated how much of each flavor would be sold based on past sales records and placed the order a week in advance. The calculation seemed simple: a variable, x (representing the projected daily sales for a particular flavor), multiplied by 7 to get the weekly supply. However, managing cheesecake orders became one of the greatest challenges of my short-lived barista career. I struggled to account for all the factors that could influence x. For example, there were times when it seemed like everyone in the city suddenly lost interest in hazelnut, and I’d be left with chocolate hazelnut cheesecakes sitting untouched in the fridge. Other times, I'd get a call from a customer requesting 20 blueberry cheesecakes for a catering event, only to realize I didn’t have enough stock. Missing out on such a large sale was frustrating (especially because it could’ve earned me a promotion to manager!).

Reflecting on this experience reminds me of how different real-life situations are from textbook math. In school, math was my strongest subject, and I excelled at it. Yet, when faced with real-world decisions, my judgment wasn’t always spot on. All the neat tricks I learned in math class—like calculating probabilities—felt almost useless in these unpredictable, practical scenarios. As Robson pointed out, applying textbook knowledge to real life isn’t always straightforward. Sometimes, I wonder: in a world full of endless possibilities and unexpected outcomes, how can my math education truly help me succeed both professionally and practically?

Response to two articles on measuring time

The first article discusses the compounding effect of the 12-hour time intervals and Balylonian’s base 60 system on the division of time the way it is today, while the second article expands on why the sexagesimal concept came into place. The article drew my attention to the proposed theories of the origin base 60, and how a lot of them seemed to be self-contradicting. It was an interesting point because it made me go back to article no1. A question formed in my head-- is it really necessary to go to lengths to find out the “real story” behind this counting system? Could it be possible that base 60 happened to be the most applicable in studying astronomy and geometry (as seen in the first article), and for that reason it has persisted all throughout history. In other words, base 60 has more added significance than the real value it possesses. 

I visualize the passage of time as making a pizza. First, you have pizza dough. It is round and malleable, and one is free to choose any shape and thickness, depending how large they want the pizza to be. Next thing comes to stretching out the dough with the choice of shape. Like how the day and night are divided into equal intervals, we can slice the pizza into 12 or 24 pieces, representing the hours. Pizza toppings (black olives, pineapples, sausage crumbs…) come on top of each slice. They are comparable to minutes because they constitute each slice (each hour). Interestingly, the first article introduced me to different variations of time division, and how the standard way of measuring time took its form after a long evolution. I couldn’t help but wonder how weeks came into place. Perhaps this would be my own inquiry on the side! 

Why base 60?

 My thought was that the Babylonians associated 60 with a special meaning, like how people often say 7 is their lucky number. Having 60 as their base, it is possible that people wished for good fortune. My other theory is about some historical events that resulted in this number system. A modern example would probably be “a baker’s dozen”.

After my research online, I found out that  60 is unique in a way that it is the smallest number divisible by all numbers from 1-6. It is easily divided into portions without fractions.The second reason is because 60 is an overarching theme in time. Every 60 seconds is a minute, and every 60 minutes is an hour. Also, 60 is a multiple of 12,  so it is somehow related to the duodecimal system, base 12.The way the people at the time counted by hand. Here is a video that demonstrates this.

https://www.facebook.com/jain108academy/videos/origins-of-base-60-finger-counting-method/760257301495219/