Overview
Play with linear functions while you ponder art history, explore geometric transformations, create your own equation, or have fun figuring out the mystery functions! (Paraphrased from site)
Curator: Darrell Glover
Brief Description of Tech Tool:
This tool allows users to explore the basic concepts of linear functions using the popular notion of a function as a machine that yields a specific output for each value that is input. Users can choose from a variety of representations starting with the extremely basic representation of a function using symbols (instead of numbers) to the function builder giving a multistep mystery equation that the user must solve for using the various inputs and their generated outputs. Start by picking the type of representation (pictures, numbers, or equations) and the difficulty level (how many transformations an input will undergo). Place the tile with the marked input value into the machine, watch the transformation(s) happen and see what value comes out the other side. Along with the machine, a table of values, a graph, and (depending on which portion of the activity is selected) the equation being used are all available for display in order to link the concepts in the mind of the user. This variety of representation and difficulty makes it an interesting activity for lessons ranging from the introduction of the concept of a function to the more advanced ideas of utilizing the various representations of data (graphical, tabular, algebraic) to relate to the concept of a function.
Technical & Cost considerations:
PhET simulations are free to use from the website. This particular tool is coded using HTML 5, making it usable on any classroom PC with an internet connection. The tool can also be downloaded or embedded in other web pages (more specific, platform-based software requirements are available on the site). The graphics are clear and colorful and should be easy to see from any classroom vantage point.
Evaluation
Description of Learning Activity
The utilization of this tool would depend on the type of lesson that is being provided; it is useful for many different levels. The most likely utilization would be during the introductory lessons on the idea of functions. After a brief introduction on this idea, small groups of students could use the pattern section to explore and make observations about what is happening to their inputs. With some guiding questions and whole class discussion, student observations could be confirmed or reshaped. Subsequent lessons could utilize the tool in the same way, progressing from the pattern section to the number and equation sections. Once in these sections, the graphical, tabular, and algebraic equation tabs could be integrated to fuse the representations together as different ways to explain functions.
1. Learning Activity Types
2. What mathematics is being learned?
Standards
NCTM:
Common Core (non-specific):
Proficiency Strands
This tool is useful for learning about ordered pairs, functions, transformations of functions, and linear equations.
3. How is the mathematics represented?
The mathematics of functions are represented in various ways in this tool. The tool as a whole is a virtual manipulative, allowing users to create their own function machine as well as pick which values (to a point) are input into it. The pattern section looks at the basic idea of a function using a symbolic representation. As the user steps into the number and equation sections, graphical and numerical (in the form of a table of ordered pairs) representations are available for viewing. The idea of a function as a machine that yields a specific output for a given input is a popular way to introduce the concept and this tool makes it easy to make that connection. Furthermore, by dynamically showing what happens at each point of the function and having the many representations available for viewing at one time, the tool does a good job of giving the user the necessary tools to make connections among those representations to better their understanding of functions as a whole.
4. What role does technology play?
The greatest advantage that I feel this tool provides is the ability to simultaneously view the many representations of a given function all at once. The ability to view these multiple representations (that automatically update) all at once is what makes the tool useful for activities that range from basic introductions to those aimed at providing a more conceptual understanding of the idea of functions.
Affordances of Technology for Supporting Learning
5. How does the technology fit or interact with the social context of learning?
This tool can be used by individuals or small groups (probably no more than three people). The graphics are good enough that it can easily be used by the teacher for whole classroom exploration and discussion. While the tool is useful for individual work and only one person at a time can interact with the tool, the nature of the exercise modeled by the tool lends itself well to use in small group exploration and discussions.
6. Additional Comments
One small issue is the fact that the functions have small numerical values (-3 to 3) but the graphical display's y-value range in from -50 to 50. This large scale can make it a little difficult to note how the graph of a function is transformed when these numbers are tweaked. But, overall, I think that this is a great tech tool to utilize with lessons regarding functions. It ranges from easy to challenging, is fun, and does a great job linking the various representations of a function together.
Play with linear functions while you ponder art history, explore geometric transformations, create your own equation, or have fun figuring out the mystery functions! (Paraphrased from site)
Curator: Darrell Glover
Brief Description of Tech Tool:
This tool allows users to explore the basic concepts of linear functions using the popular notion of a function as a machine that yields a specific output for each value that is input. Users can choose from a variety of representations starting with the extremely basic representation of a function using symbols (instead of numbers) to the function builder giving a multistep mystery equation that the user must solve for using the various inputs and their generated outputs. Start by picking the type of representation (pictures, numbers, or equations) and the difficulty level (how many transformations an input will undergo). Place the tile with the marked input value into the machine, watch the transformation(s) happen and see what value comes out the other side. Along with the machine, a table of values, a graph, and (depending on which portion of the activity is selected) the equation being used are all available for display in order to link the concepts in the mind of the user. This variety of representation and difficulty makes it an interesting activity for lessons ranging from the introduction of the concept of a function to the more advanced ideas of utilizing the various representations of data (graphical, tabular, algebraic) to relate to the concept of a function.
Technical & Cost considerations:
PhET simulations are free to use from the website. This particular tool is coded using HTML 5, making it usable on any classroom PC with an internet connection. The tool can also be downloaded or embedded in other web pages (more specific, platform-based software requirements are available on the site). The graphics are clear and colorful and should be easy to see from any classroom vantage point.
Evaluation
Description of Learning Activity
The utilization of this tool would depend on the type of lesson that is being provided; it is useful for many different levels. The most likely utilization would be during the introductory lessons on the idea of functions. After a brief introduction on this idea, small groups of students could use the pattern section to explore and make observations about what is happening to their inputs. With some guiding questions and whole class discussion, student observations could be confirmed or reshaped. Subsequent lessons could utilize the tool in the same way, progressing from the pattern section to the number and equation sections. Once in these sections, the graphical, tabular, and algebraic equation tabs could be integrated to fuse the representations together as different ways to explain functions.
1. Learning Activity Types
- LA-Present - presentation of the concept of a function as a machine that yields a specific output for each input
- LA-Present-Demo - demonstrate that, for each input, the machine gives a specific output or demonstrate how certain factors (negative slopes, for example) transform the graph of a function
- LA-Present-Explain - explain the different ways to represent a function and how these representations form the concept of a function
- LA-Explore - allow students to explore what a function does and what the various representations and transformations of functions look like
2. What mathematics is being learned?
Standards
NCTM:
- Understand relations and functions and select, convert flexibly among, and use various representations of them
- Interpret representations of functions of two variables
- Use symbolic algebra to represent and explain mathematical relationships
Common Core (non-specific):
- Create equations that describe numbers or relationships
- Understand the concept of a function and use function notation
- Analyze functions using different representations
- Build a function that models a relationship between two quantities
Proficiency Strands
- conceptual understanding - by having the ability to view the various representations of the data on screen at the same time, the user can better link them together with the overall concept of a function
- strategic competence - in the mystery equation format, by having the ability to view the various representations of the data, the user can use whichever representation best helps to figure out the equation
- adaptive reasoning - the tool's utilization of the concept of a function as a machine enhances the users ability to compare the various representations and/or various transformations of functions
This tool is useful for learning about ordered pairs, functions, transformations of functions, and linear equations.
3. How is the mathematics represented?
The mathematics of functions are represented in various ways in this tool. The tool as a whole is a virtual manipulative, allowing users to create their own function machine as well as pick which values (to a point) are input into it. The pattern section looks at the basic idea of a function using a symbolic representation. As the user steps into the number and equation sections, graphical and numerical (in the form of a table of ordered pairs) representations are available for viewing. The idea of a function as a machine that yields a specific output for a given input is a popular way to introduce the concept and this tool makes it easy to make that connection. Furthermore, by dynamically showing what happens at each point of the function and having the many representations available for viewing at one time, the tool does a good job of giving the user the necessary tools to make connections among those representations to better their understanding of functions as a whole.
4. What role does technology play?
The greatest advantage that I feel this tool provides is the ability to simultaneously view the many representations of a given function all at once. The ability to view these multiple representations (that automatically update) all at once is what makes the tool useful for activities that range from basic introductions to those aimed at providing a more conceptual understanding of the idea of functions.
Affordances of Technology for Supporting Learning
- Computing & Automating - this tool automatically calculates the output based on the equation that is entered (which can be viewed step by step if chosen to do so) as well a automatically updating the table of ordered pairs and the representative graph
- Representing Ideas & Thinking - this tool represents the idea of a function as a machine which transforms the given input and which yields a specific output for each input
- Accessing Information - this tool allows the user to simultaneously access the multiple representations of a given function
- Capturing & Creating - this tool allows the user to create their own equations using various parameters
5. How does the technology fit or interact with the social context of learning?
This tool can be used by individuals or small groups (probably no more than three people). The graphics are good enough that it can easily be used by the teacher for whole classroom exploration and discussion. While the tool is useful for individual work and only one person at a time can interact with the tool, the nature of the exercise modeled by the tool lends itself well to use in small group exploration and discussions.
6. Additional Comments
One small issue is the fact that the functions have small numerical values (-3 to 3) but the graphical display's y-value range in from -50 to 50. This large scale can make it a little difficult to note how the graph of a function is transformed when these numbers are tweaked. But, overall, I think that this is a great tech tool to utilize with lessons regarding functions. It ranges from easy to challenging, is fun, and does a great job linking the various representations of a function together.