Indices laws (AASL 1.5/1.7)

Author:Cliff Packman

Factual Questions	Conceptual Questions	Debatable Questions
1. What is the law of multiplication for indices (exponents)?	1. Explain why .	1. Is the law of indices intuitive, or does it require a deeper mathematical understanding to grasp fully?
2. Simplify the expression .	2. Discuss the reasoning behind the law when .	2. Debate the relevance of teaching the laws of indices in an era dominated by digital calculators and computational software.
3. What is the law of division for indices (exponents)?	3. How does the power of a product law (xy)^n = x^n \cdot y^n follow from the basic laws of indices?	3. Can the mastery of indices laws be seen as foundational for higher mathematics, including calculus?
4. Simplify the expression .	4. Explain the significance of the negative exponent law .	4. Discuss the statement: "Understanding the laws of indices is more critical for mathematical theory than for practical application."
5. State the zero exponent law and simplify .	5. Compare and contrast the laws of indices with the laws of logarithms.	5. Evaluate the impact of indices laws on the development of algebraic skills in students.

Mini-Investigation: Exploring the Laws of Indices Objective: To understand and apply the laws of indices by expanding and simplifying given expressions and then generalizing the patterns observed. Activity Steps: 1. Observation: Look at the examples provided in the applet. Notice how the expressions are simplified or transformed. 2. Expansion Exercise: Take each example and write out the expansion explicitly. For instance, for

, write out

and then

six more times, and simplify. 3. Simplification: After expanding, simplify the expressions to see how they reduce back to the given simplified form in the applet. 4. Identify the Laws: As you simplify, note down the patterns or 'laws' that are applied. For example, when you multiply terms with the same base, you add the exponents. 5. Create New Examples: Using the laws identified, create your own set of examples. For instance, what would

simplify to? Write out the expansion and then simplify to check if the law holds. 6. Challenge Problems: Introduce more complex examples involving division and negative exponents. For example, what is

, and how does it look when expanded and simplified? 7. Reflection: Discuss or write about why these laws make sense. Can you explain why

is always 1, without just saying 'it's the rule'? 8. Generalization: Write down the general laws of indices that have been observed. For example,

. 9. Application: Use these laws to solve more complex problems or real-life examples where exponents are used. 10. Presentation: Encourage students to present their findings to the class, explaining the laws of indices and demonstrating with their examples. Extension: -Consider how these indices laws relates to the logarithm rules. - Exploration with Different Bases: Extend the investigation by exploring the laws of indices with different bases, such as

and

. Real-World Application: Discuss how the laws of indices are used in scientific notation, especially in fields like astronomy and physics where very large or very small numbers are common.By the end of this mini-investigation, students should have a better understanding of the laws of indices and how to apply them. They should be able to explain these laws and use them in a variety of mathematical contexts.