Laws+of+Exponents

Laws of Exponents



This resource provides users with definitions of a base and an exponent, and examples of the many laws of exponents. Bases to the first power, bases to the zero power, multiplication of powers with the same bases, and a power raised to an exponent are all discussed in detail. The laws are presented with variables, examples are given with numbers and there are written explanations as well. Negative exponents and fractional exponents are also explained in detail. Quiz questions are presented at the end of the lesson. The applet that accompanies this resource allows students to increase and decrease the base of a power to a positive or negative integer, as well as increasing the exponent of a power to either a positive or negative. The applet uses the base and exponent to produce the answer as well as the long version of all of the terms either being multiplied or divided. This enables students to better understand how increasing and decreasing exponents and bases affects the overall result of a power.


 * Grade Level:** 7-9
 * PSSM Content Standard:** Algebra
 * Math Content:** Exponents

Evaluation & Annotations

 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

This resource focuses upon exponents. There are two major components to this resource. There is a webpage, which details the laws of exponents. Rules are provided for multiplying powers with the same base, dividing powers with the same base, negative exponents, fractional exponents, and for other situations involving exponents. The rules are given as laws with variables, and as examples with numbers. For many of the rules there is also written text, which further explains the rules. This is a great reference tool for students. The second component of this resource is the applet. The applet allows students to change the base and exponent of a power and view the numerical answer along with the long version of how to get to the answer (ie: 2x2x2x2=16). This applet allows students to explore many relationships with exponents. Students could use the applet so better understand why an exponent with a negative base will yield a negative answer if the exponent is odd and yield a positive answer if the exponent is even. Students could explore the difference between positive and negative exponents, or analyze the answers to a power with an exponent of zero. This tool, if utilized correctly, could foster exploration and enable students to find certain exponent laws for themselves. This resource definitely presents students with an opportunity to create relational understanding. The website provides a detailed explanation of the laws through multiple representations. Yet the real richness of the resource comes from the applet which enables students to discover certain laws for themselves.


 * How does learning take place? What are the underlying assumptions (explicit or implicit) about the nature of learning?**

This resource presents students with a unique opportunity to discover select laws of exponents. For many of the laws of exponents, learning is taking place through discovery. Many of the more complicated rules of powers will not be able to be tested on the applet; therefore students will learn through reading the webpage. The webpage provides students with the laws in variable form, examples in numerical form, and written explanations. Through these multiple representations and the basic laws, which can be built through exploring with the applet, students will be able to forge a deep understanding of exponents. The underlying assumptions of this resource are twofold. As previously mentioned, this applet operates on the assumption that students need to explore powers and determine some of the relationships and rules for themselves. The other assumption is that possessing a relational understanding of the rules of powers is essential. The certain rules that will not be discovered due to limitations of the applet will be explained deeply on the adjoining web page.


 * What role does technology play? What advantages or disadvantages does the technology hold for this role? What unique contribution does the technology make in facilitating learning?**

The dynamic software of the applet allows students to manipulate the base and exponent of a power and witness the effect on the answer of the power. Additionally, the applet provides students with the drawn out multiplication or division (if the exponent is negative) of the power, which assists students in understanding the basis of powers. Without this technology students would not be given a visual representation of how powers are either multiplied or divided. Furthermore, without the technology present in the applet, students would not be able to discern rules as effectively. With a few clicks students can witness the effect of altering the base or of altering the exponent. The one disadvantage of this applet is that there is not a way to test the more complex rules, such as the power to product rule or the product sum rule.


 * How does it fit within existing school curriculum? (e.g., is it intended to supplement or supplant existing curriculum? Is it intended to enhance the learning of something already central to the curriculum or some new set of understandings or competencies?)**

This applet would be a great tool for introducing a lesson on exponents. Students could use the applet to determine a rule for an exponent of zero and an exponent of one. Students could also explore the difference between negative and positive exponents, as well as the alternating result when the exponent of a negative base is increased. The web page could be used as the rules of powers become more complex. Teachers would most likely wish to go through the rules with the students in order to take questions and clear up misunderstandings. However, this web page would be a great tool to refer students to as a reference for the more complex rules, given the multiple representations present on the page.


 * How does the technology fit or interact with the social context of learning? (e.g., Are computers used by individuals or groups? Does the technology/activity support collaboration or individual work? What sorts of interaction does the technology facilitate or hinder?)**

This technology would be a great tool for collaborative work. It would be highly beneficial for students to work together while discovering the rules of powers and while attempting to search for patterns as the bases and exponents are altered. This would promote a greater reflection and discussion about possible laws and the rationale behind the laws. The web page would be best used individually as a reference for students while working on problems. For more independent learners the web page could serve as the first exposure to the concepts, but this would be problematic for most students.


 * How are important differences among learners taken into account?**

The resource provides multiple representations of the rules of exponents. They are presented with variables as laws, with numbers as examples, and written with explanations. Even students with visual learning styles are able to use the applet to understand the multiplication or division that is taking process with powers. Given that the applet is focused more on exploration and the web page is used as a reference, there is no need to create different tracks for learners with varying abilities. Teachers should provide practice at varying difficulties to meet student needs.


 * What do teachers and learners need to know? What demands are placed on teachers and other "users"? What knowledge is needed? What knowledge supports does the innovation provide (e.g., skills in using particular kinds of technology)?**

Learner need to have an understanding of multiplication and division prior to entering into a lesson on exponents with this resource. Teachers need to be prepared to create supplemental material for practice with the laws of exponents. The applet allows students to discover and the web page can be used to teach students or as a reference, yet students still need sufficient practice with powers. Teachers need to be prepared to continue to fine tune the use of the laws of exponents with students after the use of this resource. To use the applet there are no technological requirements. There are brief instructions listed above the applet if needed, but students should have no problem with its features.