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* *Reflection: Writing Across the Disciplines
Quotients of Monomials - Section 3: Quotients of Monomials

A great way to extend an engaging conversation or lesson is to offer extra credit. It amazes me that students will spend all sorts of time on extra credit while forgetting to do their actual homework. This lesson provided an excellent place to give students just this opportunity.

The conversation around 0^{0} went really well. The students got fired up and defended/critiqued their own and each other’s opinions (**Math Practice 3**). By the end of the discussion, we talked about how this is one of those places where math isn't always black or white, and that experts in the field have differing opinions. I then offered them extra credit for submitting a position on this topic. The better researched they were, the more extra credit I would offer.

I received many interesting submissions. Many, like Extra Credit 1 and Extra Credit 3, supported the stance that 0^{0}=1. The a lesser extent, students submitted statements supporting 0^{0}=0 like Extra Credit 4. Only one student, Extra Credit 2a, 2b, and 2c, really did her research. This young lady obviously has strength in writing. She even cited her references. In the future, I may add a full essay (or two) to my course. What a great way to cross over into English while supporting mathematical habits of mind.

*Writing Across the Disciplines: Extra Credit- A Great Way To Extend A Lesson*

# Quotients of Monomials

Lesson 1 of 15

## Objective: Students will be able simplify quotients of monomials.

## Big Idea: Investigate the concepts behind, as well as the numerical limitations of, the simplest form of a rational expression.

*47 minutes*

#### Warm up and Homework Review

*10 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. Please watch my Video Narrative which explains this lesson’s Warm Up- Quotients of Monomials.

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

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#### Rational Numbers

*17 min*

This lesson in a support/scaffolding lesson for my unit on rational functions. Though these concepts have been mapped in the Common Core into the Algebra 1 standards, they are fundamental to my students’ success in the upcoming lessons. Since this is a skills based lesson, it does include guided practice. To balance this, I have included conceptual activities that provide an opportunity to deal with the WHY as we talk about the HOW.

The first activity asks students to spend two minutes writing down everything they know about exponents. The goal of this is to activate all their prior knowledge of exponents which will increase their success in this lesson.

Next, we discuss the concept behind rational numbers as well as their major limitations, namely a zero in the denominator. We look at two reasons why this can't happen (**Math Practice 2**).

Please see the PowerPoint for detailed presentation points.

#### Resources

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#### Quotients of Monomials

*17 min*

Next, I have the students investigate the pattern x^{a}/x^{b} = x^{a-b} (**Math Practice 8**) using several sample problems. After trying a couple of examples, the students look at reasoning behind a^{0}=1.

Our next task is looking at 0^{0}. After consulting with their partner, I have the students raise their hands if they think 0^{0} = 0 or if 0^{0} = 1. This discussion provides an opportunity to see how mathematics isn’t always black and white and gives the students an opportunity to defend their own reasoning (**Math Practice 3**). Here is a website with some great information on this topic.

Finally, we discuss negative exponents and look at several examples combining all of these concepts.

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#### Exit Ticket

*3 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 simplify a quotient of monomials that include negative exponents.

#### Resources

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This homework begins with eight problems which provide a small amount of practice to reinforce the day’s lesson. Next, the students identify a quotient that has a variable with a constraint not equal to zero. This is a preview into the next lesson which includes binomial denominators. Finally, they determine whether a quotient has been simplified properly (**Math Practice 3**).

This assignment was recreated with Kuta Software, an amazing resource for secondary mathematics teachers.

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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: Quotients of Monomials
- LESSON 2: Modeling Rational Functions
- LESSON 3: Transformations of Rational Functions
- LESSON 4: Properties of Rational Functions Investigation Day 1 of 2
- LESSON 5: Properties of Rational Functions Investigation Day 2 of 2
- LESSON 6: Rational Functions Quiz
- LESSON 7: Modeling and Simplifying Rational Expressions
- LESSON 8: Multiplying and Dividing Rational Expressions
- LESSON 9: Adding and Subtraction Rational Expressions
- LESSON 10: Complex Rational Expressions
- LESSON 11: Equations with Rational Coefficients
- LESSON 12: Rational Equations
- LESSON 13: Rational Functions Review Day 1
- LESSON 14: Rational Function Review Day 2
- LESSON 15: Rational Functions Test