My students have not yet encountered expressions that include fractions as exponents. It is usually the case that my students are not completely proficient with all of the properties of exponents they learned in Algebra 1, so I like to take some time to present the logic of the properties. Biology Intro to Rational Exponents ask students to complete a table about the number of fruitflies in a doubling scenario. By considering this concrete situation, they work backwards to discern what a quantity raised to a power 1/2 must represent.
As a warm up, I ask students to work together in table groups to complete the activity. Then as a whole group we discuss the results and the implications for how we work with fractional exponents.
To review properties of exponents, I ask students to work with assigned partners on Property of Exponents Match Up.docx. This is essentially the 8 properties of exponents separated into 3 parts each (name, left side of = and right side of =). The goal of the activity is for students to discuss how to reassemble the properties (MP3). I tell students that if there is discrepancy amongst the team members, they should use numerical examples to resolve their disputes. For example, to determine whether a^m*b^m is equal to a^(m+n) or a^(m*n), they should think about 2^3*2*4.
The first time I use this activity, I print the document on colored cardstock and cut along all the solid lines so that the name of the property, the left side of the property and the right side of the property are all separated. I address misconceptions about the existence of an "addition property of exponents" by providing this name, a card with a^m + a^n, and a third card that says "this is not a property of exponents."
When students have completed their match up, I ask them to consult with another group to see if their matches are the same. Once they have verified their matches with another group, I check their work and ask them to copy the properties into their notebooks.
As they finish, I provide a property of a puzzle or worksheet like Rational Exponents Mini Quiz, HW, CW.doc to practice simplifying with the 8 properties of exponents.
In order to unearth confusion about the properties of exponents and help students assess their own understanding, I set up a "fishbowl" activity in which 6 students watch 4 other students work together to simplify an exponential expression. The students who are "doing the math" are seated at a table with a copy of Exponential Expressions for Discussion.doc. The students who are observing stand with a copy of Discussion Notes Properties of Exponents.doc on a clipboard. I initially allow students to self select into these groups so that those who feel confident can serve as a model for students who are feeling less confident.
The expressions on Exponential Expressions for Discussion.doc require students to use multiple properties to simplify an expression. The observers take note of which properties were used, whether students approached the same task in different ways and whether or not there was confusion about whether they were "done" simplifying. I selected these questions because these are the things that tend to cause my students confusion regarding the properties of exponents.
When the students have completed their fishbowl discussions, we engage in a whole group discussion about what was recorded on the observation sheets. Going through each question, I ask volunteers to share out what they noticed [MP3]. Through this discussion, I'll make sure students understand the conventions of "simplest form" of exponential expressions. These conventions are really about examining the structure of expressions and making them easy to understand [MP7].
The exit ticket Exit Ticket Rational Exponents.doc focuses on simplifying expressions that contain rational exponents. I ask students to simplify 4 expressions so that I can determine whether to provide direct instruction in rational exponents in the next day's lesson.
The homework is to complete a worksheet like Simplifying Rational Exponents on simplifying expressions using the properties of exponents.