Polynomials are a concept that I always have a difficult time working on with my students, largely because it is difficult to find a real-world context relevant to them that is polynomial based. Students have an easy enough time memorizing polynomial rules, but they often have a difficult time conceptualizing these rules or understanding where they came from. While my Algebra 1 students aren't going to be manipulating the function form of the polynomial expression anytime soon, I really like this tool because it helps them to visualize the changes that occur within a polynomial expression and how that change affects the graphic version of the polynomial.

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__Overview__

Name & Link to Tech Tool or Tool homepage: GeoGebraTube: Polynomial Explorations

Brief Description of Tech Tool: With this tool, students use sliders to manipulate coefficient values for constant through quintic polynomial functions to analyze the graphic changes. Students are prompted with a series of questions to guide their explorations.

Technical & Cost considerations: This tool is a free web-based tool that requires internet connection.

__Evaluation__

**Description of Learning Activity**

**This activity would be best used for an individual or pair to explore various changes that can occur within a polynomial function graph for polynomials of various degrees (Constant through quintic). Students would take notes on the graphic changes for the various shifts in coefficient values to come up with generalized rules for graphing polynomial functions.**

## 1. Learning Activity Type:

- LA-Explore - This tool is particularly useful for helping students to explore graphic changes in polynomial functions of various degrees (constant through quintic) as different coefficient values are manipulated.

## 2. What mathematics is being learned?

### NCTM Standards

### Proficiency Strands

Students develop a conceptual understanding of changes in a graphical context to develop generalized rules for these changes.*PS-conceptual understanding*

### Additional comments on what is being learned

This tool allows students to view multiple function graphs at the same time allowing them to compare/contrast between even and odd degree functions or between various functions of the same type of degree. Students can make generalizations based on the various coefficient changes for all functions using this feature as well.

## 3. How is the mathematics represented?

This tool is a virtual manipulative that relies on slide values to alter the coefficient for each of the terms within a polynomial function. Students can slide these values to view the changes in the graphic form of the function.

## 4. What role does technology play?

Technology enables the users to view graphic changes automatically, an activity that would eat up a lot of class time if done by hand, or even if done by graphing calculator. The slide values are essential to allowing the student to work on developing their conceptual understanding without being bogged down by the procedures of choosing specific values or graphing the function.

### Affordances of Technology for Supporting Learning

**Computing & Automating**- This tool enables the user to manipulate the coefficient values within the terms of the polynomial and automatically changes the graph accordingly.

## 5. How does the technology fit or interact with the social context of learning?

This tool can be used by students independently or in pairs, but I would not suggest using it for anything larger than a group of two. Students exploring individually may need to be assigned a talking partner that they can bounce their ideas off of after having experimented with the tool for a while. This use of this tool to develop a conceptual understanding of graphic changes could foster good discussion between a pairing of students to compare and contrast their ideas and ultimately come to a conclusion or generalization that is more sound than if they were working solely independently.

6. What do teachers and learners need to know?

Users do not need much technical prowess to successfully use this tool in the classroom. Teachers may want to demonstrate the type of changes that occur while using one of the sliders, but would likely wish to leave the actual exploration to the individuals.

__How it Supports & Supplements PBL/PrBL__

This tool is very helpful for me in the classroom as it allows for my students to work on a more inquiry driven exploration, similar to the style they use when working on a problem or project. This type of exploration allows for the student to reach their own conclusions about the concept and then check with others for validation. I particularly like this tool because of the automatic changes that it produces within each graph as the coefficient values are being changed, as it frees up the students to think solely about the shifts that are occurring and not about what value they should try next. This also helps students to better analyze graphs of data in later projects as they are more familiar with the type of graphic changes that occur within polynomial functions.