Exponents inquiry

The prompt

Mathematical inquiry processes: Extend patterns and find connections; test different cases; generalise and prove. Conceptual field of inquiry: Laws of exponents; factorisation.

Jacob Storback, a teacher at École Cariboo Hill Secondary School in Burnaby (BC, Canada), devised the prompt for his grade 9 class. The class had studied the laws of exponents. Jacob wrote the prompt as 23 - 22 = 22 after noticing the following patterns:

The prompt 33 - 32 = 2 x 32 has an extra level of intrigue and might be more suitable for students with experience of mathematical inquiry.

After exploring the patterns, Jacob's year 9 students noticed that the constant is one less than the base number. They made this generalisation:

January 2024

Lines of inquiry

1. Change the difference between the exponents

What happens to the pattern if the difference between the exponents is increased? 

How do you calculate the constant in each case?

2. Subtract more terms

Is there a pattern if we subtract more than one power of the same base number? 

This time the constant is given by a quadratic expression.

3. Proof

Students can deduce a generalisation by factorisation using the laws of exponents.

Alternatively, if the generalisation has come inductively from pattern-spotting, students can prove their general statement is always true in the same way.