Reflection: Student Ownership Exponential Equations - Section 2: Discussion and Guided Instruction


Students in my classes have a wide variety of learning styles and ways of approaching unfamiliar math content.  It is often the case that a student will be able to get to a solution of an equation through an excellent, intuitive approach, but then has difficulty explaining their process to others.  I like to highlight these situations because I believe it helps everyone get better at the logic of mathematics.

The discussion of how students went about solving these (currently unfamiliar) exponential equations usually provides an excellent time to instruct students in explaining their thought process. It is usually the case that students who were able to solve some of the more difficult problems like 16^x=1/8 will provide an explanation that involves lots of intuition.  Perhaps the student will share a thought process like "I knew that if x were 1/4, I'd be back at two. I really wanted to be at 1/8, though, so I made the exponent 3 times bigger and negative so that the 2 would turn in to 1/8.  The answer is x=-3/4"   This is excellent thinking. My challenge as a teacher is to celebrate this student's intuitive approach while at the same time helping to translate it into a more systematic approach.  This is important for the students who don't understand the intuitive approach but also for the student who provided the clever solution.  Everyone should be able to explain their process in a way that other people understand.  

For this particular explanation I would ask the student to stay at the board with me and repeat her process to me slowly while I try to translate into symbols.  The process may go something like this:

"I knew that if x were 1/4, I'd be back at two"          

 That is, 16^(1/4) = 2, because 2^4 = 16.  We can write the left side as (2^4)^x or 2^(4x)

"I really wanted to be at 1/8, though, so I made the exponent 3 times bigger and negative so that the 2 would turn in to 1/8."

That is, 2^3 = 8 and 8^(-1) = 1/8, so 2^(-3)=1/8.  We can rewrite the right side as 2^(-3).

"The answer is x=-3/4" 

We can rewrite the equation as 2^(4x)=2^(-3).  If we raise two to two different powers and end up with the same number, the exponents must be equal.  So 4x=-3 and x=-3/4

  Helping Students Explain Intuitive Approaches
  Student Ownership: Helping Students Explain Intuitive Approaches
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Exponential Equations

Unit 2: Exponential Functions
Lesson 2 of 11

Objective: SWBAT solve exponential equations by using properties of exponents.

Big Idea: We can solve some exponential equations by rewriting both sides of the equation as a power of the same base.

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