Fractional indices inquiry
The prompt
Mathematical inquiry processes: Generate examples; reason and prove. Conceptual field of inquiry: Fractional indices; powers and roots; laws of indices; surds.
Andrew Blair designed the prompt to introduce fractional indices to his year 10 class. The students had learnt about the laws of indices and raising fractions to an integer power.
The class found the prompt intriguing. In the question, notice, and wonder phase of the inquiry students attempted to understand its meaning. They made the following contributions:
What does a half and a quarter mean as powers?
The power of a half means the square root. Does a quarter mean the root four times?
What is the square root of a half?
The power of a quarter means you need a number that you multiply together four times to get a quarter.
How can they be equal? A quarter to the power of a quarter must be smaller than a half to the power of a half.
The product of a half and a half is a quarter? Does that help?
If you work them out on a calculator, they are equal. They both equal 0.7071 ...
Guided discussion
As the students were close to a full understanding of the prompt, Andrew drew upon their responses to show that the two terms are equal.
He explained that the square root of a half is one over root two. As one over root two raised to the power four is a quarter, the fourth root of a quarter is one over root two.
Andrew then presented the chains of reasoning below that use one of the law of indices to show that the two terms are equal. Students discussed the chains in pairs before taking turns to feed back to the class on how to get from one term to the next.
Andrew encouraged the class to speculate about the index required for one-eighth if the term were equal to those in the prompt. Some students argued for one eighth; others proposed one-sixth. After checking on a calculator, the class consensus was one-sixth - a result justified in the following way:
The structured inquiry proceeded on a pre-planned course. Students created more chains with unit fractions before attempting to generalise their results. They then used other types of fractions (see the lines of inquiry below).
Reflecting on the course of the inquiry
At the end of the inquiry, Andrew invited the students to reflect on the course of the inquiry by considering the regulatory cards. The class agreed that the inquiry had followed the following sequence.
September 2024
Lines of inquiry
1. Create chains with unit fractions
Students create their own chains with fractional indices. The teacher could use slides 10 to 12 in the PowerPoint to demonstrate how to extend the two terms in the prompt into a chain. There are also starting points for other chains in which the unit fraction used for the base and index are the same and others where the unit fractions are different.
The teacher encourages students to generalise from their examples. The aim is to express the relationship between the terms in a chain either with words or algebraically.
In words, a new term in the chain is created by raising the base in the first term to a power (a) and multiplying the denominator of the fraction in the index by a. You can explain why the process works algebraically (see illustrations).
Base and index: same unit fraction
Base and index: different unit fractions
2. Create chains with other fractions
Creating chains with fractions other than unit fractions is more challenging. It requires a knowledge of squares, cubes, and higher powers (see the table in the PowerPoint).
To support this line of inquiry, students can follow the instructions for making a chain. In the example, the index is multiplied by two to create the next term (step 5) and so the square roots of the numerator and denominator - that is, to the power of a half - are required for the new base fraction.
