Math Practice 1: Sensemaking, Problem Solving, and Perseverance

This project enables students to investigate how the change of one variable impacts the resulting graph, while coordinating the relationship throughout the entire function. This experience is critical for students to discern functions as a whole. In the example of y=3x, students need to learn that when computing y-values for corresponding x-values, the value of y increases as the input x increases, and this increment is consistent due to the linear relationship of the function.

Through this project, students have opportunities to explore or investigate the functional relationship and to practice higher-level conceptual understanding without being confined by the burden of tedious graphing procedure. With the technology providing the computation result instantly, students with low computational skill can now investigate the structural nature of function without algorithm.

Students are willing to take rise as the real-life consequences were lowered. In fact, students accept failure and saw their mistakes as a necessary process of finding the correct position of the function. 

The project enables students to look for multiple entry points and find solutions through multiple routes to make progress without fixed approaches or solutions. For example, a V-shaped graph may be formed by two linear functions or a single absolute value function.

Math Practice 2: Reasoning and Math Practice 4: Modeling

In this project, students take a real-world picture and think abstractly about what function may be the most appropriate to produce the intended shape.  They are also thinking quantitatively about reasoning with relationships between entered data and their effect on the graph.

The idea of creating an image or picture with algebraic functions intrigues students.  Students take everyday items, such as a snowman, house, mountains, or their names, and think of how mathematic functions can reproduce those shapes. These projects blend students’ artistic view with mathematics.

Math Practice 5: Strategic Tools

Desmos friendly user interface allows students to input, duplicate, edit and manipulate (add constraints) an unlimited number of functions to join the pieces of graph and produce an image in the same coordinate plane. Students have the freedom to show/hide any functions, or to adjust the color or the thickness of the function graphs.  The joint screen of algebraic and graphical interface allows students to directly observe the effect of the functions, allowing them to focus on their conjectures.

Desmos offers students the opportunity to interact with instant and precise feedback, enabling students to establish and test their conjectures through problem-solving. 

Constructing a graph is possible without technology as teachers and students were doing so with paper and pencil before the recent intervention of graphing technology.  Nevertheless, without assistance, students with limited computational skill may focus on or be hindered by the execution burden and miss the opportunity to observe key features of this mathematical terrain, such as investigating the relationship among multiple input-output pairs—an essential concept for understanding the continuous feature within the entire function.  Constructing a graph requires complex and precise procedures, and the instant feedback of technology provides students greater opportunity to develop higher-level conceptual understanding, including engaging in reasoning, and conjecture.

Math Practice 6: Precision

The artistic nature of the project enforces students to be precise in their algebraic input in order to produce an image at the exact location, such as the effort to make all circles the same size, or to connect all graphs.

Because students see this as their own artwork, this project creates the ownership or pride ingredient where students are often commitment to detail and precision in their creations.  Student precision in this project is often self-motivated instead of teacher-driven.

Math Practice 7: Structure

The ample opportunity of practice allowed students to transfer what they had learned earlier to later tasks, such as after enough practice, they would realize that increasing the slope of a linear equation makes the graph steeper, and could apply that knowledge if they intended to modify the graph in other situations. 

A student may be trying to make a linear function’s graph steeper for an intended location, and discovered the need to increase value of m in a linear function. As this student inputs various values of m, he or she is systematically varying the variables for determining and investigating the nature of the essential function structure. 

Math Practice 8: Patterns and Repeated Reasoning

The project encouraged students’ pattern thinking and reasoning ability. While students observed the change of the graphic form as they altered the variable in the algebraic function, they attend to the generalization of between multiple function representations.

The learners received ample opportunities to practice the content.  While adjusting the shape, domain, and steepness of the function to the correct location, students’ experienced considerable amounts of attend, in addition to the need of multiple functions to create an image.  

Students now have the opportunity to make direct comparisons between their anticipated result and what actually happens, criticize their own assumptions and organize their growing knowledge of functions, then build up empirical evidence for interpreting the result of their conjectures or experiments. As students observe the result of their experiment in the forms of multiple functional representations instantaneously and simultaneously, they can witness the range of impact of a variable on the entire behavior of the function. Students can thus focus their attention toward sense-making by examining the continuous behavior of the function beyond an individual point.