# Rational or Irrational (Day 2 of 2)

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## Objective

SWBAT differentiate between rational and irrational numbers.

#### Big Idea

Every number expands infinitely. The repeating digit may be zero, which means the number terminates, or another digit pattern.

## Launch

10 minutes

For today's lesson, it is important for students to begin where we left off yesterday. As students enter the room, I ask them to take out their Venn Diagram worksheet and to pair up with partners from yesterday's lesson. I also hand each student the Entrance Slip day 2.docx for this lesson. I allow students to discuss the Entrance Slip with their partners for a few minutes. Then, I ask volunteers to share their thoughts.

My focus for this warmup is on identifying the rational set of numbers. Students usually find that the numbers in Set B are either terminating or have another repeating digit pattern. In other words they are rational, based on our work in yesterday's lesson. When I am satisfied that every one is thinking about this idea, it is time to move onto the New Info section of the lesson.

## New Info

10 minutes

As we continue, we are still talking about the Entrance Slip. Having identified Set B as rational numbers, I tell the class that the other numbers, those in Set A, don't eventually have a repeating pattern. We call these numbers Irrational. I ask the class to share their initial ideas: What types of numbers are commonly irrational.?

One common answer is square roots. Students may understand that not all radicals are irrational, yet they cannot say it in words. This is usually the case. So, I proceed by writing a few square root expressions, some rational and some irrational, on the board. Then, I ask students to evaluate the expressions as square roots. I choose students at random to answer.

Students will easily give the answers to √25 or √81, etc, yet they will not be able to give easy answers to the irrational expressions. I then lead them through a conversation where we identify which numbers are, and which are not, rational numbers.

During this conversation, I like to include expressions like √5. I find that many students evaluate this as 2.5, an interesting misconception. When I come across an opportunity like this, I take advantage of it. I ask students to check the answer on a calculator. They quickly see that they are wrong. I then ask, "What was the mistake?"

## Activity

25 minutes

I give students the Activity sheet with a series of numbers to identify as rational or irrational. In groups of twos, I have students work through the list. I ask them to write all fractions in decimal form. They will see that some decimals may take a while before they repeat. For example, Problem 5, has a long period. In number 5, 1/7 is already written in a way that makes it clear it is a rational number, although some students might say it's irrational,  because the repeating part of the decimal is longer than many familiar repeating decimals. But, if there is a pattern the number is rational. Just because a decimal is long, does not mean it is irrational.

Teacher's Note: The Activity Sheet assumes that students are able to express a repeating decimal as a fraction. I shared a Khan Academy video on this topic on Day 1 of this lesson

Opportunity for Reasoning and Proof: By the time that students get to Number 9, I've found that many 8th graders learn that the square root of a prime number is irrational. if time allows, this problem is a good time to ask if there is a good explanation why this is true (MP3, MP7, MP8).

## Closure

10 minutes

I close the lesson with a strategy called Whip Around

I usually have a tennis ball in my cabinet, but you can use another object that is safe to toss around. I take the ball and toss it to a student and quickly ask him or her to say one thing they learned today. That student then tosses it to another and so forth.

Alternative Whip Around: Sometimes I toss the ball to a student, ask this student a question, that student tosses the ball back, and I toss it to another student. Which ever method you chose, make sure it is a trusting environment.

The video below discusses rational and irrational numbers. I will share it if my students can use a resource to review.