On October 14, 2024, I took three classes as part of my teaching schedule at St. Joseph HSS. The day's lessons were centered on two main topics: "Problems of Similar Triangles in Real-World Applications" for 9C and the "Sum of Squares Identity" for 8A. My goal was to ensure that the students grasped the essential principles of these mathematical concepts and could apply them confidently. However, as I worked through the lessons, I quickly realized that I needed to use simpler methods to make these topics more accessible to the students. The problems were complex, and I aimed to find ways to break them down into smaller, more understandable parts.
For 9C, the focus was on applying similar triangles to solve real-world problems. We had previously discussed the basic concept of similar triangles, including the conditions for similarity and how to identify them. This time, I wanted to emphasize how these concepts could be applied in practical scenarios. To start, I presented a problem that involved finding the height of a building using the shadow method—a common example in geometry that helps students visualize how similar triangles can be used to measure distances indirectly.
As I introduced the problem, I noticed that while some students were familiar with the idea, others seemed confused about how to set up the proportions between the triangles. The challenge was not just in recognizing which triangles were similar, but also in applying the ratios correctly. I realized that my initial explanation might have been too abstract for some students, so I decided to pause and revisit the fundamentals of similar triangles. I broke down the problem step by step, emphasizing how the properties of similarity allowed us to relate the sides of one triangle to the corresponding sides of another.
To make the concept more relatable, I used visual aids and real-life examples. I drew a diagram on the board showing two triangles—one representing the building and its shadow, and the other a smaller triangle formed by a stick and its shadow. By relating the problem to everyday objects, I aimed to create a more intuitive understanding of how similar triangles work. I then asked the students to come up with examples of their own, encouraging them to think of situations in which they might use similar triangles to solve problems in their daily lives. This approach helped engage the students more actively, but I still noticed that some of them were struggling to connect the dots.
One of the challenges I faced was maintaining the students’ focus throughout the lesson. The complexity of the real-world application problems required sustained attention, which was difficult to achieve for some students. To address this, I tried to make the class more interactive by posing questions to the students and encouraging them to discuss their answers with their peers. I found that involving them in the problem-solving process, rather than just presenting solutions, helped to keep their interest alive.
For the class with 8A, the topic was the "Sum of Squares Identity," specifically the formula (a + b)^2 = a^2 + b^2 + 2ab. Algebraic identities can be challenging for students because they require not just a conceptual understanding, but also the ability to manipulate algebraic expressions. I started by introducing the formula and explaining each term in detail, highlighting how the expansion works. Despite my efforts to clarify the process, I could see that many students were still finding the algebraic manipulations overwhelming.
To simplify the concept, I decided to use a geometric approach. I drew a square on the board, showing how it could be divided into smaller regions that represented each term in the expanded formula. By visualizing the areas corresponding to a^2, b^2, and the 2ab term, I hoped to give the students a concrete representation of the identity. This method seemed to resonate with a few more students, as they could see the connection between the geometric representation and the algebraic expression. Visual aids often help in making abstract concepts more tangible, and I could see that this approach was starting to make a difference.
However, even with this approach, there were still some students who found it difficult to follow. I realized that I needed to slow down the lesson's pace and simplify my explanations even further. Breaking the formula down into smaller, more manageable steps became my priority. I went over each term individually, showing how (a + b)^2 is not just a straightforward multiplication but involves distributing each term in the parentheses. By taking this incremental approach, I was able to clarify the reasoning behind each step, helping students to see the logic behind the algebraic identity.
I also included some practical problems where the identity could be applied, such as calculating areas in different scenarios or finding relationships between quantities in algebraic equations. I wanted the students to see that this identity is not just a formula to memorize, but a tool that can be used in solving various mathematical problems. I encouraged them to practice with simpler examples before moving on to more complex problems. This step-by-step method helped to build their confidence and made the process of learning feel more achievable.
Reflecting on the day's teaching experience, I recognized the importance of adapting my methods to the students' needs. The complexity of both the similar triangle problems and the sum of squares identity required a more focused and patient approach. It became clear that rushing through the material would not benefit the students in any way. Instead, I needed to create an environment where they felt comfortable asking questions and expressing their doubts without feeling overwhelmed by the difficulty of the topic.
One of my key takeaways from the day was the need to continuously find simpler, more relatable methods to explain difficult concepts. Mathematics can often seem abstract and intimidating to students, especially when they are confronted with complex problems that require a deep understanding of the principles involved. To help them, I need to be flexible in my teaching style, using a variety of strategies to make the material more accessible. This might include more visual representations, practical examples, or breaking the problems into smaller, more manageable parts.
In the future, I plan to integrate more interactive elements into my lessons, such as group activities and discussions that encourage peer learning. I believe that students often learn best when they can talk through problems with their classmates, share ideas, and explain concepts to each other. This collaborative approach not only helps to reinforce their understanding but also builds their confidence in tackling mathematical challenges.
Overall, the experience on October 14, 2024, reinforced the idea that teaching is a dynamic process that requires constant adaptation and reflection. Understanding my students' needs and finding ways to simplify complex ideas are crucial aspects of being an effective educator. I am committed to continually improving my teaching methods to ensure that all students, regardless of their initial understanding, can grasp the concepts and apply them confidently in both academic and real-world context.
