Overview
Curator: Darrell Glover
Brief Description of Tech Tool: This tool allows the user to practice identifying the graphs of different functions by matching a manipulative graph to that of a given graph. The user can choose from a variety of types of functions to try matching, from linear to polynomial to trigonometric to exponential and logarithmic, and is able to manipulate their graph through changes to its equation by entry boxes. Once the graph matches, it automatically turns green indicating a correct answer. Because of all the different functions available to choose from, this tool can be used throughout middle and high school mathematics to practice identifying graphs and their functions.
Technical & Cost considerations: This tool is free from the NCTM illuminations website. It utilizes Adobe Flashplayer which might limit its use on certain networks depending on security protocols or if the installed version is up to date.
Evaluation
Description of Learning Activity
I think that this tools best use is for exploration of the connections between different types of functions and their equations and for practice identifying these objects.
1. Learning Activity Types
- LA-Practice – this tool is made to practice identifying functions and their equations
- LA-Present – teachers could utilize this tool to present what a general graph for a new type of function being taught looks like
- LA-Present-Demo – this tool can be used to demonstrate how manipulating certain variables in different function’s equations will affect that function’s graph
- LA-Explore – users can explore how manipulating certain variables in different function’s equations will affect that function’s graph
2. What mathematics is being learned?
Standards
NCTM:
- Generalize patterns using explicitly defined and recursively defined functions
- Understand relations and functions and select, convert flexibly among, and use various representations for them
- Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior
- Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions
- Use symbolic algebra to represent and explain mathematical relationships
Common Core (non-specific):
- Interpret the structure of expressions
- Understand the relationship between zeros and factors of polynomials
- Create equations that describe numbers or relationships
- Represent and solve equations and inequalities graphically
- conceptual understanding – this tool allows users to link the algebraic form of an equation with its graphical form allowing for connections between them to be seen in real time
- strategic competence – this tool allows users to formulate the algebraic equation based on its graph giving users the ability to see how each portion of the equation affects the graph
- adaptive reasoning – by linking the algebraic and graphical representations, users are able to justify how manipulating certain portions of their equation will result in the correct graph being displayed
3. How is the mathematics represented?
This tool utilizes graphical and numerical/symbolic representations by asking the user to use one to match the other. This is done through a virtual manipulative interface with the algebraic equation. The changes to a given equation are dynamically displayed by the graph as the user increases or decreases different values within the equation.
4. What role does technology play?
One advantage of this tech tool is that the graph moves in real time as the user increases or decreases different coefficients and constants within the algebraic equation for the graph – the graph as a whole walks along the screen as the user makes the changes. This provides the user instant feedback on how each portion of the algebraic equation affects the graph (horizontally, vertically, etc.). By internalizing these connections, the user will be better able to quickly identify potential equations of a graph and vice versa.
Affordances of Technology for Supporting Learning
- Computing & Automating – this tool automatically computes and displays the graph of a given equation as it’s manipulated
- Representing Ideas & Thinking – this tool represents equations both algebraically and graphically allowing for a connection between the two
- Accessing Information – there are a few guided learning activities associated with the tool that are available on the “Exploration” tab
- Capturing & Creating – users can create their own graphs and equations
5. How does the technology fit or interact with the social context of learning?
This tool would likely be better for small group or individual use v. whole class discussion. It is an exploration tool and as such would not provide much benefit to a teacher standing in the front of a classroom. Small groups, however, can manipulate the tool and discuss what effects they observed when the different functions are changed.
6. Additional Comments
Some of the graphs that are generated can be quite difficult to match, even with good knowledge of the tool and how each portion of an equation affects its graph. There are zooming abilities on the graphical display but, depending on the type of function, some of the constants and coefficients can have a wide range (-100 to 100, for example) which contributes to the difficulty that sometimes could occur.