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# Manipulating Rational and Irrational Numbers

Lesson 5 of 10

## Objective: SWBAT... 1. Compare and contrast rational and irrational numbers 2. Reflect on current understanding of solving equations and inequalities and develop a study plan.

To start class, I give students an **Entry Ticket ** to work on classifying numbers as rational or irrational. The definition of a rational number is given at the top of the entry ticket so students have the support and necessary tools to work on the task relatively independently during the first few minutes of class.

After about 5 minutes I give the class a cue to begin talking with a partner on the **Turn and Talk **for the entry ticket if conversations have not yet begun (I find students tend to not begin mathematical conversations but will more once they have a consistent structure like Turn and Talks, Think Pair Shares, etc. where they know the routine and protocol).

The intent of this opening activity is for students to activate prior knowledge and comfort level when talking about rational and irrational numbers. This is important because the lesson is based on working with, and manipulating these types of numbers. I want students to attend to precision (**MP.6) **in this activity as attending to the details of each number, and the definition of rational numbers. is crucial in correctly classifying the numbers.

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Students take **Two-Column Notes** during this section of class on manipulating rational and irrational numbers. I am modeling the notes on the whiteboard in two column form (at this point in the year many students still struggle with translating a powerpoint slide into two-column note form and benefit from a model set of notes on the whiteboard).

One way to **Differentiate Instruction **in this section is to provide students with a typed set of notes, allow students to use a word processor to type notes, and/or assign 2 or 3 students as the class note-takers and make copies of those students notes for the class.

I also differentiate instruction by providing support around math specific (brick) and general (mortar) vocabulary. For example, the use of the term manipulate may be a stretch for many 9th graders, but it is a term that will come up often in math and other fields as well. I typically check in with each class on their understanding of the term and ask them to identify some uses of the term (related to math or another context). This strategy can help students in their understanding and use of more and more sophisticated vocabulary.

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To practice the ideas presented in the active note-taking section of this class, I then have students work in pairs on Practice Problems.

For this practice, I have students use their **Entry Ticket** as a resource. The practice problems ask students to take different combinations of rational and irrational numbers to test the patterns that we found during the note-taking section of class.

For each of the four manipulations (adding rationals, adding rational and irrational, multiplying rationals, and multiplying rational and an irrational) students are asked to generate an example and justify why or why the rule/pattern holds.

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During this section of class, I have the class work on generating a study guide from their notes. I provide a list of objectives that will be covered on the quiz. I also provide a list of class materials where students can find example problems (see Creating a Study Guide on Solving Equations and Inequalities).

For this activity, students work in groups. Each group is assigned the task of creating a problem set for one of the learning objectives on the study guide sheet. My goal is for each group to become expert in one area. My students are comfortable with this jigsaw approach to learning. In an upcoming lesson I put each groups' work together to make a comprehensive study guide for the upcoming quiz. As an alternative, a teacher can give students time to generate their own study guide for an upcoming quiz on the first half of the unit.

The intent of this activity is beyond simply copying class problems down. Most of my students are not ready to independently identify the overarching objectives of a unit. However, with the right peer support, I find that students produce a big picture idea of what we have been studying. While finding the time to fit in this activity can be difficult, I have found it pays off. My students become better at self-monitoring their progress. They also demonstrate stronger meta-cognitive awareness.

**Extension/Scaffold**: One extension to this activity is to have students create a study guide and then give them a teacher-generated study guide. The task involves comparing and contrasting the two guides, and lends itself nicely to a written response by students. This type of extension again allows students to work on picking out relevant details and provides them with a model study guide that clearly sets the expectations of the assignment.

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For today's **Exit Ticket ** I provide an opportunity for students to generalize the rules for working with both rational and irrational numbers through writing by completing an **Idea Organizer**.

For example, I am looking for students to identify that if two rational numbers are multiplied, then the product must be rational because a factor of a rational number has to be rational. I also review their writing using this Exit Ticket. I want to assess how well they back up their claims with examples and/or evidence.

I have included one example of a completed assignment by a student as a resource in this section: Student Work - Exit Ticket: Writing Idea Organizer on Rational and Irrational Numbers

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For homework I assign students to complete the self-generated study guide that they made in class. I typically ask students to complete the problems for the objective that their group focused on. This way, during the next class we can put together a comprehensive study guide for the quiz that can be shared throughout the class.

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- UNIT 1: Thinking Like a Mathematician: Modeling with Functions
- UNIT 2: Its Not Always a Straight Answer: Linear Equations and Inequalities in 1 Variable
- UNIT 3: Everything is Relative: Linear Functions
- UNIT 4: Making Informed Decisions with Systems of Equations
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- UNIT 8: Our City Statistics: Who We Are and Where We are Going

- LESSON 1: Introduction to Linear Equations and Inequalities in One Variable
- LESSON 2: Solving and Proving Equations with One Unknown
- LESSON 3: Solving and Proving Multi-Step Equations with One Variable
- LESSON 4: Solving and Proving Linear Inequalities in One Variable
- LESSON 5: Manipulating Rational and Irrational Numbers
- LESSON 6: Solving and Proving Compound Inequalities
- LESSON 7: I Absolute(ly) Don't Care About Direction: Solving and Proving Absolute Value Equations and Inequalities
- LESSON 8: Estimation, Measurement and Significant Figures
- LESSON 9: Writing in Math Classroom, Part 1: Functions and Linear Functions
- LESSON 10: Unit Assessment: Linear Equations in 1 Variable