Joe + Kieran, Hiu Fung TAI

We chose to work on a piece by Hiu Fung TAI, linked here and shown below.

Hiu Fung TAI - Bridges 2026 Exhibition of Mathematical Art, Craft, and Design

An initial glance may not grasp the depth to this rather simple submission. The common factor between these shapes is that they share the same integer value of area and perimeter. Further interestingly, each side/arc length is also an integer value, which seems unintuitive when we look at the pie piece. Tai ends his article with a somewhat obvious statement that two unique 2D shapes can share an area and perimeter value. To the more analytic reader, this raises a few questions: Are there infinite unique area/perimeter pairs? Can circular objects be paired with linear objects? Does this statement generalize to 3D? To higher dimensions?

We aimed to answer the penultimate question and examine the relationship between volume and surface area in 3-dimensional objects through a similar lens. It was quickly decided between us that we should create a few shapes and working out a lesson through them. However, glancing at an object in your hand and determining the volume is significantly less obvious than looking at a shape on graph paper. How can we make the demonstration visually intuitive then? 

Joe initially proposed that we 3D print these objects to make them watertight and submerge them within a clear container of water, clearly showing volume via displacement and careful measurements. Surface area would then have to be calculated by hand, which we thought wasn't too much of a hassle, both for us and the audience. 

A cube and sphere, sliced and ready to print

However, due to logistical complexity and issues with the printer, we moved to a demonstration revolving around Gabriel's horn and paper mache.

We chose Gabriel's horn to highlight the complex relationship between surface area and volume, since, of course, this famous example has an infinite surface area paired with finite volume. We chose a scale for the horn and set bounds of y=20/x from x=1 onward, rotated about the x-axis to find a workable volume of 200π. Kieran had an idea to create a hanging display that we could leave in our classroom, and so decided on a final design involving three objects with a volume summing to 200π suspended above the horn. The dimensions of the shapes, a cube, sphere and octahedron were carefully calculated before we got hands-on.

Profile of horn sketched onto cardboard

The finished product

After the outstanding work to create such a beautiful display, we moved onto the interactive component. There was a realization at this point that our demonstration would likely involve a few calculations from the audience. To avoid a boring talk about geometry, we aimed to engage the audience in open-ended, discussion-prompting questions (some unanswered!) with a goal to hold some real estate each students' curiosity after we end our lesson.

Introspective writing: Favourite and least favourite math teachers

I will begin by writing about my least favourite math teacher from math 10. 

It was an attitude problem. That roughly sums it up. She was a female educator, somewhat on the older side. I remember thinking was quite mean to her 15 year old students, like she had something to prove to them. There was one unit that went very poorly for the entire class. It was very rushed, the lessons dragged the class through the content despite us struggling quite a bit, and I believe we did not get any worksheets or quizzes returned to us. She had one of those moments where she stood in front of the class and did the whole "I've never had a class of students perform this bad," and pointed the blame back at us. I believe she called out one of my classmates by name and stated his grade for everyone to know, citing his 'inability to pay attention her classroom.'

This nearly killed my passion for mathematics. However, I had an incredible teacher for the next two years. 

Grade 10 was my COVID year. My pre-calculus 11 course was taken in a quarter system, meaning I had math for a full half of the school day. The classroom environment was completely different. My teacher, Mrs. O, lets call her, took much of the pressure of learning off the students. She planned time during her lessons for her students to work on their own and discuss with one another. The first few minutes of the block were a time for her to chat with the kids and set a relaxed tone. In hindsight, she made the classroom comfortable for all students such that speaking up and asking questions felt natural. This can be very easily contrasted against other classrooms I have been in. Impressive to me now as a teacher candidate is how she was able to do this while retaining an air of formality and respect to the process of education. Despite the above, the students were all focused and intentional with their work on mathematics.

Locker problem

A school has 1,000 students and 1,000 lockers. On the first day of school, all the lockers are open. 

Student #1 closes each locker.

Student #2 opens each second locker.

Student #3 changes the state of each third locker (i.e. opens them if they are closed, or closes them if they are open)

...and so on, until all 1,000 students have had their turn.After all 1,000 lockers are done, which lockers are open? Which are closed? Why?

This is the puzzle we have been assigned to work on. Now, I was shown this problem before in grade 11, so I can't say exactly how I figured it out this time around.

A very common strategy for solving complex math problems is to somewhat tone down the complexity of the problem. In this case, we have 1000 lockers to work with. Quite a few. Instead, I worked with a smaller case to try and notice any patterns. Go through just ten lockers and run the algorithm. What was the result? It turns out that locker 4 and locker 9 are closed. Interesting. What about twenty lockers? Then lockers 4, 9 and 16 are closed. I hypothesise that perfect square numbers remain open in this puzzle scenario. 

Why is that? If we break down the problem a bit more, we note that the lockers are in a binary state, open or closed. This is further determined by how many students have interacted with each locker; an even number of interactions opens the door, while odd closes. 

The arithmetic suggests that each n'th student will interact with each np'th locker, where p is another natural number and np≤1000. Or perhaps simpler, any student assigned a number will only interact with lockers that are a multiple of their assigned number. 

After breaking down the problem like this, the question becomes much more concise: which numbers 1-1000 have an odd number of factors? Which have an even number?

The factors of a number will always be a set of pairs, two distinct numbers that multiply to give the original - except when the number is a perfect square! In the case of 4, 9, 16, etc. one of their factoring pairs will have two of the same number.

Hence, by only working out the first 20 lockers, we have determined the pattern for all 1000 lockers.

 this is a test blog to learn how to create a blog

here is a photo of me and my friends in china this summer