Overview
Curator: Marissa Hoffman
Brief Description of Tech Tool: The Polynomial Roots Calculator is an excellent tool for students to use to relationally understand the multiple roots of a polynomial function, how they are related to the Rational Roots Theorem, and the behavior of the graph at each root. This tool is best used when students are understanding the big picture of polynomials and patterns in their graph and roots behavior. This tool will support a lesson on multiplicity and the Rational Roots Theorem. Students can input their own polynomial or study one generated by the website (the preferred method).
Technical & Cost Considerations: This tech tool is completely free and available to everyone. Based on my experience, the Polynomial Roots Calculator will work with any kind of internet connection on any internet browser. Although, it is advised that Flash Player is up to date to ensure the calculator and its graphic features will work reliably. Use on a SmartBoard for whole class instruction is possible although it is advised that this tool is best used when students are working together in small groups.
Evaluation
Description of Learning Activity
The use of this tech tool will be different depending on the level of the learners and the goal of the lesson. What this lesson should not be used for is checking a student's manual graph to see what it should look like. The use of the Polynomial Roots Calculator should allow students to develop a relational understanding of the roots and graph's behavior of a polynomial function. The tool may used as an exploratory resource to help students make a connection between where the roots of a polynomial function come from and make conjectures about the multiplicity of each root. As well, it could be used for students to reinforce their understanding of these principles or check their algebraic work to the graphical representation.
1. Learning Activity Types
- LA-Present - The Polynomial Roots Calculator presents to students the graph of a high-degree polynomial as well as the values for all of its theoretical and actual roots.
- LA-Present-Demo - This tech tool presents the graphical representation of each polynomial to demonstrate to students that the roots found using the Rational Roots Theorem connect to the graph of the polynomial as well as proving multiplicity rules accurately predict the behavior of the graph around the roots.
- LA-Explore - If the lesson plan is designed for an exploratory activity, students can make conjectures about the above mentioned polynomial principles of roots and multiplicity.
2. What mathematics is being learned?
Standards
Common Core State Standards Mathematics (CCSSM)
CCSS.MATH.CONTENT.HSA.APR.B.3
- Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
CCSS.MATH.CONTENT.HSA.REI.D.10
- Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
CCSS.MATH.CONTENT.HSA.SSE.A.2
- Use the structure of an expression to identify ways to rewrite it. For example, see x4- y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
CCSS.MATH.CONTENT.HSA.SSE.B.3
- Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*
Principles and Standards for School Mathematics (PSSM)
- Understand relations and functions and select, convert flexibly among, and use various representations for them.
- Analyze functions of one variable by investigating...intercepts, zeros,...and local and global behavior.
- Understand and compare the properties of classes of functions...
- Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations.
- Use symbolic algebra to represent and explain mathematical relationships.
Proficiency Strands
- conceptual understanding - The Polynomial Roots Calculator gives students the chance to relationally understand where the roots come from when using the standard form of a polynomial and how the multiplicity of each factor is related to the graph's behavior as it approaches each root.
- adaptive reasoning - Students must adapt what they know about the Rational Roots Theorem and multiplicity rules to the image of the graph of the polynomial and as it changes with each new polynomial. Students must also understand that how to find all possible roots for each polynomial will be the same steps but the actual roots will vary based on each polynomial.
- productive disposition - Depending on which parts of the problem students solve by hand will determine how much of a productive disposition will be needed by the students. However, if students work with the graph of a polynomial and connect the graph back to the function, students will not have a lot of manual work to complete and therefore a productive disposition may not be of as much importance when compared to the need for reasoning skills.
As was mentioned above, students should not be using this tech tool as a way to check their work after they have gone through the tedious task of finding all of the roots by hand and applying the rules of multiplicity to determine the graph's behavior. Rather, the students should be using the tool to make connections between the polynomial in standard form, its roots, and the graphs behavior without putting in too much time instrumentally understanding and solving for where these key values and features came from.
3. How is the mathematics represented?
Graphical and numeric representations of polynomial functions are the primary features emphasized by the Polynomial Roots Calculator. Especially by generating the graph of every given polynomial students can quickly see the location of roots and the graph's behavior without spending valuable time doing so by hand. The graphical representations change as each polynomial is changed (the student can input their own polynomial or have the computer generate one). The technology makes it possible for students to identify and connect key features of the graph of a polynomial back to its function by providing a verbal explanation that walk students through the process of going from function to graph or vice versa. Again, each explanation changes as each polynomial function changes that was input into the Polynomial Roots Calculator.
4. What role does technology play?
Believe it or not, there are some disadvantages to this tech tool. Like with every other calculator, students can choose to blindly copy answers from the Polynomial Roots Calculator and not put in the effort to make a connection between the graph, the Rational Roots Theorem, and the standard form of the polynomial function. However, when used properly, this tool facilitates a learning opportunity where students can construct their own knowledge between the Rational Roots Theorem, multiplicity rules, and the graphical behavior of a polynomial.
Affordances of Technology for Supporting Learning
- Computing & Automating - Since this tool is, in essence, a calculator, students no longer have to divide endless values from a polynomial function and determine algebraically if these numbers will become the roots of the polynomial. Rather, the Polynomial Roots Calculator will prove this for students and they can focus on what the roots represent and the graph's behavior at these roots.
- Representing Ideas & Thinking - Looking back at the automating affordance, this tech tool allows for the polynomial function students are working with to be represented visually automatically. This way students do not have to take a break in analyzing the function and its behavior to draw out what the function should look like. This latter task is burdensome and is where many students become unmotivated.
- Capturing & Creating - Depending on the needs of the student and goal of the lesson, students can input their own polynomial function to analyze or allow the Polynomial Roots Calculator to generate its own with the click of a button. Students will never run out of examples when using this tech tool no matter how quickly they may work.
5. How does the technology fit or interact with the social context of learning?
Although students working in small groups allows for communication and the sharing of mathematical ideas and problem solving skills, it is possible that the Polynomial Roots Calculator could be effectively used by single students if those students are motivated and engaged in the material. However, partners working together seems ideal when discussing the three main concepts this tech tool presents (Rational Roots Theorem, multiplicity rules, and behavior of the graph). The calculator presents a graph that displays information covering several different mathematical topics which encourages commentary. However, because so much information is presented at once, students will most likely need the structure of a lesson plan and goals from the teacher to guide their thinking and conversations.
6. Additional Comments
The Polynomial Roots Calculator fills in the gaps of understanding that websites such as www.desmos.com leave behind when referring to the Rational Roots Theorem, multiplicity rules, and the behavior of the graph.