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* *Reflection: Real World Applications
Rational Exponents - Section 1: Warm Up

This was the very first warm up I wrote for BetterLesson! I think this is a perfect way to start this lesson. It really got the students talking and thinking about exponential models. A few students stuck with the $200 per day, like Warm Up 1. Most however, like Warm Up 2, figured out that the doubling pennies would be much better in the long run. After my students completed this warm up, I lead a class discussion. I asked who would be willing to share their opinion and called on volunteers. I find that this language is powerful as asking for an opinion rather than asking for a correct answer takes some of the pressure off and more students tend to raise their hand. They were excited to share and some expectation for set for what they would see in this unit.

*Real World Applications: A Unit Hook*

# Rational Exponents

Lesson 1 of 15

## Objective: Students will be able to use rational exponents to simplify expressions.

*57 minutes*

#### Warm Up

*7 min*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm up- Rational Exponents which asks students to find the error in a fictional student's work simplifying an expression.

I also use this time to correct and record the previous day's Homework.

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The goal of this unit is to help the students gain a conceptual understanding of rational exponents, relate them back to operations with rational numbers, and then using them in expressions. To begin, students make a list of their prior knowledge of exponents. I give them 2 minutes to make a list in pairs and then we compile a class list.

Students then chart and graph the first seven ordered pairs in the warm up scenario(**Math Practice 2**). The graph should be points without a curve running through them. I remind the students to label both the table and the axes on the graph. These will vary and this is a good opportunity for students to explain why they chose the labels they did (**Math Practice 3**). This is their introduction to exponential functions. We extend the graph to include the full exponential curve rather than just the integer pairs. By analyzing the curve as a class, the existence of rational exponents will become apparent.

Using their prior knowledge of multiplicative inverses and quadratic and cubic equations, the students have an opportunity to "discover" the meaning of the denominator in a rational exponent (**Math Practice 1**).

#### Resources

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#### Rational Exponents

*20 min*

The rest of the lesson will include a Guided Practice session of several applications of Rational Exponents in expressions. I always give students the opportunity to work on problems before we discuss them as a class. After rewriting a few in either radical or exponential form, we simplify basic rational exponents where the base is a number. This is a great place for **Math Practice 3****. **Once students have solved the first problem, I have students discuss in pairs and then as a whole their approach to this problem. It is valuable to see that while a person can take the power or the root first to obtain a correct solution; it is easier to take the root. If a student doesn't bring this up, I will ask the class " Is it easier to take the root or the exponent first?"

The next set of problems has students rewrite and simplify more involved radical expressions (**Math Practice 7**).

The final set of questions has students simplify numerical radical expressions. Scaffolding should be added here for students that struggle with the rules of exponents. First, students may want to combine the bases as well as the exponents. Second, they may not simplify the 8 into 2^{3} or the 4 into 2^{2}. Reminding them that you can only combine exponents when the bases are the same may be enough of a hint for them to figure it out themselves. I like to give them an opportunity to try it and then we discuss the troubling places and then they have a chance to finish. It takes a bit longer than just doing it but the students will be better for it at the end.

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#### Homework

*5 min*

If there is time, the students may also work on their Assignment with their partner. This is also a time I may provide some individual instruction or remediation. The assignment to this lesson includes problems that practice simplifying and rewriting rational exponents. It also includes a problem that asks students to identify equivalent expressions.

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#### Exit Ticket

*5 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

Today's Exit Ticket asks students to explain the meaning of both the numerator and the denominator of a rational exponent.

#### Resources

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Rational Exponents
- LESSON 2: Real Number Exponents
- LESSON 3: Exponential Models Day 1 of 2
- LESSON 4: Exponential Models Day 2 of 2
- LESSON 5: Exponential Functions
- LESSON 6: Exponential Decay Functions
- LESSON 7: Simplifying Logarithms
- LESSON 8: Exponential and Logarithmic Equations
- LESSON 9: Logarithmic Functions
- LESSON 10: Exponential Growth and Interest Day 1 of 2
- LESSON 11: Exponential Growth and Interest Day 2 of 2
- LESSON 12: Natural Logarithms
- LESSON 13: Exponential and Logarithmic Functions Review Day 1
- LESSON 14: Exponential and Logarithmic Functions Review Day 2
- LESSON 15: Exponential and Logarithmic Functions Test