30-3 November's Journal

The Animation Activity

https://www.geogebra.org/m/jtmeyyrt

The Homework from the Variables and Relationships Chapter,

Collaboratively done with Süeda Bozkurt;

 Exploring Variables 1:

When A is being dragged, I automatically move, too. When we tried to move point I, neither point I nor A moved. So, I act as a constant in this relation, and A acts as a variable. A and I move inversely to each other.

It is the same relationship for J as a variable and B as a constant, but this time, they move in the same direction.

When C, a variable, is being moved, B, a constant, also moves; however, their distances remain constant. 

E and H do not move when dragged first, but when G is being moved, both of them move accordingly. They are points of the same linear line. The distance between E and G is about two times the distance between G and H. So, G is a variable that affects both E and H.

F does not affect any of the points, and it acts as a free variable—something like y=x.

Exploring Variables 2: 

K and L are linear expressions that take value according to values of x linked to it. Also, K and L are inverses of each other.

Exploring Variables 3:

M and N are variables, and M+N and N+M are expressions whose values change according to N and M.

Exploring Variables 4: 

The variable here is the distance between the center C and the point on the circle P.

Exploring Variables 5:

Here, a is a variable, making the functions sin(a) and cos(a) take different values as it changes.

Exploring Variables 6:

A, h, k, and x are variables here. While a, h, and k values change, the shape or the position of the graph changes. They make the graph move. X is also a variable, but it does not impact the graph but changes the value of f(x).

In the Exploring_Variables_6.ggb file, there are two types of variables: independent/dependent variables and parameters. Parameters like 'a,' 'b,' and 'c' in a quadratic equation allow for generaling any quadratic function. The dependent variable 'y' relies on these parameter values, but the parameters remain constant as the independent variable 'x' changes. In this case, 'a,' 'b,' and 'c' are parameters because they don't change with varying 'x.'

So, when comparing 'a' and 'x' in Exploring_Variables_6.ggb, the similarity lies in their role as variables in mathematical expressions. However, the difference is that 'x' is an independent variable that can change, while 'a' is a parameter that remains constant once its value is set.

My Reflection/ Journal

As a student who studied and implemented these concepts in my class, I found the transition from arithmetic to algebra to be a significant shift in my mathematical thinking. The introduction of variables and the exploration of relationships beyond basic operations like addition, subtraction, multiplication, and division were indeed challenging but eye-opening. Using variables in algebra was a pivotal moment in my mathematical journey.

The idea that variables are not just placeholders but powerful tools for representing and solving complex problems became apparent as I delved into the subject. The ability to use spreadsheets, graphs, and dynamic geometry software such as GeoGebra to visualize and work with variables provided a practical and intuitive way to understand their role in mathematics. These tools made the abstract concept of variables more concrete and accessible.

The text we read mentions the common joke about algebra being the study of the 24th letter of the alphabet, "x," which I could relate to as it often seemed like we were constantly tasked with "finding x." However, I realized that it's not just about finding a single unknown value; it's about understanding how variables can represent various numbers, objects, or even entire sets, depending on the context. Variables are indeed versatile and allow for generality and flexibility in mathematical modeling.

In my private lesson, I made an intentional effort to clarify and define concepts like variables to help my students grasp their significance. I emphasized the idea that variables are not just random letters but tools that can unlock the power of algebra. I encouraged my students to use multiple representations, such as graphs and spreadsheets, to gain a deeper understanding of the relationships between variables.

Overall, I believe that the transition to algebra, with its focus on variables and their diverse uses, is a fundamental step in a student's mathematical journey. It not only challenges students but also equips them with the tools to tackle complex problems and represent vast amounts of information efficiently. My experience with these concepts in the classroom reinforced the importance of clear explanations and practical applications to help students appreciate the role of variables in mathematics.