##
* *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*

# Exponential Equations

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.

*95 minutes*

#### Warm Up

*30 min*

As a warm-up, I ask my students to complete Warm up Exponential Equations.docx, which is a collection of equations in which the variable is an exponent. Although they have probably not seen exponential equations before, they can solve this specialized set by working backwards from the properties of exponents. I like this this warm up because it provides much-needed practice on applying the properties of exponents and also helps students see why they needed to learn new properties!

#### Resources

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When my students have completed the warm-up, I ask volunteers to come to the board to demonstrate their solution to some of the more difficult equations. We discuss the different approaches students used and take time to record some strategies for solving exponential equations. Most importantly, I want them to see that writing both sides of an equation as a power of the same base can be a very efficient method of solving exponential equations.

I will ask my students to include in their notebooks 4 problems, three of which can be solved with the strategy outlined above. The fourth problem cannot be solved in this way (because the solution is irrational) and I ask them to identify the problem they cannot solve.

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#### Practice and Assignment

*20 min*

Students will take time to work individually on WS Solving Exponential Equations.docx. This is a collection of equations that are slightly more challenging than the ones we discussed in the warm up. I assign this set in class so that students have the opportunity to ask questions of each other or me.

For homework students will complete WS Solving Exponential Equations.docx.

#### Resources

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- LESSON 1: Properties of Exponents
- LESSON 2: Exponential Equations
- LESSON 3: Logarithms
- LESSON 4: Quiz and Graphing Exponential Functions
- LESSON 5: Multiple Representations of Exponential Functions
- LESSON 6: Quiz and Comparing Linear, Exponential Models
- LESSON 7: Drinking and Driving Activity
- LESSON 8: Exponential Models in the Sciences
- LESSON 9: Exponential Functions in Finance
- LESSON 10: Review Workshop: Exponential Functions and Logarithms
- LESSON 11: Unit Test: Logarithms and Exponential Functions