For today's Entry Ticket: Rewriting Radical and Rational Expressions students complete a review of simplifying expressions with exponents. I want to activate student's prior knowledge around working with exponents.
The reason for focusing on this skill is to give students practice with a foundational skill that is crucial to today's lesson. The lesson goes much smoother if students are comfortable with working with exponents and this entry ticket has the intent of getting students to that point.
After reviewing the entry ticket, I pass out a reference sheet that summarizes the rules of working with exponents. One such source can be found here: Exponent Rules Reference
Another great resource is the Monterey Institute website on Rewriting Radical and Rationals. This website is clear-cut and clearly shows a number of examples of how to write radical expressions as rationals and vice versa. I also like the site because it includes the expressions as an integer when possible (aka the cube root of 27 is equal to 27 to the 1/3 power which is equal to the integer 3) to help students make the connection between what they already know and the day's lesson.
I give out a reference because most of my students are familiar with working with exponents and I want to focus on the Common Core Standards for rewriting radical and rational expressions using exponential properties, not just the properties.
Students take Two-Column Notes on the topic of rewriting radical and rational expressions (See Class Notes). I typically make copies of the class notes for my students. During the notes I am asking questions to encourage students to not only write, but also engage in other domains of language - namely listening, reading and speaking. I also want students to use these notes in the future as a reference and as a beginning tool/skill set to "learn how to learn" something that I value as an educator and a person.
After today's explicit instruction, I review a number of problems with my students using Khan Academy. I show the entire video for the first example without pausing. During this time students are taking notes. After showing the video, I give students a couple of minutes to ask clarifying questions. After answering any questions I give students another minute to revise and add additional details to their notes.
I like to use video demonstrations for two main reasons. First, I like to have students hear alternate methods for solving solutions and to hear from people other than myself on how to solve different types of math problems. I also like to make students aware of different resources and technologies in the hopes that they will be able to strategically identify appropriate tools to solve and make meaning of problems.
After students watch the first video, I have students explain the steps back as a way to gauge their understanding of the process of rewriting radical and rational expressions using rules of exponents.
For the next example, I typically will show the first few seconds of the video and then pause the video and give students a chance to complete a Think-Pair-Share to begin to peel away the scaffolding for today's lesson.
The class engages in a Jigsaw: Rewriting Radical and Rational Expressions for the next two sections of class. Students begin the work in “expert” groups diving deep into the set of problems rewriting radical and rational expressions.
In this segment, students are all assigned to an Expert Group. One way to arrange students is by their current understanding of the material. For this particular activity I use a teacher-generated worksheet on rewriting radical and rational expressions. I assign different expert groups different sections of the worksheet.
The beauty of the jigsaw activity is students will all have a chance to teach each other. All students have access to the different tasks of this assignment, and get to hear multiple perspectives and strategies to solve the different types of problems. I also thoroughly enjoy running this activity. My role as a teacher shifts to that of a facilitator as the students take on the role of teaching each other.
As an alternative instructional strategy to the jigsaw, teachers can assign particular focus problems for groups to present on the whiteboard and explain to the class.
In this section, there is a transition where students go from their expert groups to mixed groups. I would plan for this transition to take a few minutes, especially if students are not used to the jigsaw format. One way to keep the learning going is to have pre-assigned group names that students can connect to (local sports teams, community hangouts, etc.).
The goal at the close of the mixed jigsaw group activity is for ALL students to have an understanding of how to solve all of the problems and for each student to have a completed worksheet. For problems students are struggling on, they can utilize the experts on those particular types of problems for support and guidance.
When students are in expert groups give them a number so there will be a number 1, 2, 3, …. For each expert group. That way all of the number 1’s can be a mixed group and you can assure each mixed group will have at least one expert from each area. In the mixed groups students are asked to go throught the worksheet as a group, taking the time for peers to ask and answer questions of each other. Note, students are still working on the Jigsaw Activity from the previous section.
During this time, the teacher is making rounds checking in and providing cues and tips to keep each group on track. Once groups are finished I take a few minutes to have the class reflect and engage in some in-process metacognitive skills. This can be as easy as asking students what the big picture task is and where we are in the process. This is also a great opportunity to ask students to reflect on what they have done well and what they need to work on to be successful for the remainder of the class.
To wrap up the assignment, I have students choose one problem from the worksheet on rewriting a radical expression into a rational expression. I ask them to complete an Exit Ticket: Rewriting Radical and Rational Expressions as a way to write about their thinking and reasoning.
The intent of this assignment is to help students consolidate and summarize their thinking. in addition, I want students to critique an argument, in line with Math Practice MP.3, a skill that many of the students I work with have relative difficulties with.
The benefit to this assignment lies in its flexibility. It allows students at various levels of mathematical understanding to access and critique the argument. The assignment is also flexible in how it can be used in the classroom. For example, a teacher might assign the Idea Organizer as an exit ticket to wrap up this lesson or students ca come into class the next day, with the Idea Organizer completed for homework. They can then work on writing up their response in a well-polished multi-paragraph response.
Some students are reluctant to write in the math classroom, even saying they do not think they need to be able to write about math - my response is writing is a skill that might be the best mortar or glue around - it gives each of us the power to connect and communicate to people with various backgrounds and skill sets. Writing also can help students better understand the content because the process requires students to translate their ideas and understanding into another form
For homework I assign a problem set on Delta Math on properties of exponents and to complete the vocabulary for the unit. I typically have students complete ten problems in a row, with a penalty of one for an incorrect response. In other words, if students complete the first 8 problems correctly, but then get the 9th incorrect, then they have credit for 7 in a row and need to complete 3 more problems in a row to get full credit for the assignment.
For an overview video on Delta Math and how to set up an account (it is free!), click here: Delta Math Overview Video
I also have students complete the Vocabulary: Exponential Functions for the unit for homework. I explicitly teach students the difference between Brick and Mortar Words as I firmly believe students need to be taught what types of conversation and language are valued in school AND how to engage in those skills.
Students can complete the vocabulary work on iPad minis (part of a technology grant for Modeling with Mathematics and Universal Design for Learning in the Math Classroom) using a math dictionary to look up the definitions for the terms. As an alternative, students could use an online math dictionary. I recommend Wolfram Alpha as it provides excellent knowledge, is accurate and also provides good visual examples for many terms.
Teachers can also choose to provide students with the definitions of the words (there are two versions of the vocabulary as resources in this section - one with definitions and one without definitions) to allow students to focus more on generating multiple representations of meaning for each word to develop a deeper understanding of the vocabulary terms. I have included two versions of the vocabulary for teachers - one with definitions and one without.
After completing the assignment, I have students file the work in a vocabulary section of their notebook and remind them vocabulary is part of the Notebook Check. We will be using the vocabulary later in the unit for a writing exercise on creating their own functions.