Algebra II : Unit #7 - Rational Functions : Lesson #1

Quotients of Monomials

Objective: Students will be able simplify quotients of monomials.
Standards: HSA-SSE.B.3 HSA-APR.D.7 MP2 MP3 MP8
Subject(s): Math
60 minutes
1 Warm up and Homework Review - 10 minutes

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.  

Number Sense in Algebra 2
Diverse Entry Points

This warm up highlights an extremely important strategy for increasing both number sense and conceptual understanding in students.  When students are trained to test real numbers into an algebraic identity, it helps them “prove” the identity's accuracy.  It also helps them associate the algebra back to the computational mathematics that they are comfortable with.  This is an excellent scaffolding method for dealing with student misunderstanding in any sort of algebraic situation. 

I would say 20% of my students missed this problem, like in Warm Up 1.  They got stuck on the subtraction and forgot their order of operations.  Most students submitted something like Warm Up 2.  The conversation that we had after the students solved this problem was really good.  I’d say almost all of the students caught on quickly when they heard their peer’s explanations (Math Practice 3).  

2 Rational Numbers - 17 minutes

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.

Exponent Brain Dump
Connection to Prior Knowledge

I have introduced the idea of a quick write into my classroom this year.   I have been very pleased at its overall effectiveness. I asked my student to write for two minutes without stopping and describe everything they could remember about exponents.  I request that they continue to write even if they run out of things to say.  Since this is one of the first times my students have done this activity with me this year,  I got varied results, but most wrote for the entire two minutes.   While the information varied from paper to paper, we were able to compile a complete list as a class.  Here are several samples taken from student papers (1, 2, 3, 4, and 5).  The fifth sample is my favorite.   Overall, I feel like this was an excellent way to access prior knowledge and I am going to find ways to incorporate a quick write into other places in my curriculum.

3 Quotients of Monomials - 17 minutes

Next, I have the students investigate the pattern xa/xb = xa-b (Math Practice 8) using several sample problems.  After trying a couple of examples, the students look at reasoning behind a0=1. 

Our next task is looking at 00.  After consulting with their partner, I have the students raise their hands if they think 00 = 0 or if 00 = 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. 

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

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 00 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 00=1.  The a lesser extent, students submitted statements supporting 00=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.  

4 Exit Ticket - 3 minutes

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.

5 Homework - 0 minutes

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.