Classify Numbers as Rational and Irrational Lesson Plan

Introducing the real number system

Start by giving each student a blank graphic organizer. They’ll fill in the labels for whole numbers and integers.

Whole numbers

Review what a whole number is and write some examples. I like to explain whole numbers as all the counting numbers and `0`. Have students fill in some examples in their graphing organizer. 

Integers

Review what an integer is and some examples. Integers are all the whole numbers, and their opposites. So we’re starting to get into negative numbers now too. 

What types of numbers have we not yet discussed?

Once students review whole numbers and integers, ask what kind of numbers seem to be missing from the graphic organizer. Allow students some time to come up with examples of numbers that are not included as a class.

Rational numbers

If students readily give fractions or decimals as the missing values, you can let them know that they are a part of the next category: rational numbers. You may consider converting some of the fractions, like `1/4` and `1/3`, to show that the decimals may be repeating or terminating. Be sure to also emphasize that whole numbers and integers are also rational numbers.

Irrational number examples

If students did not include irrational numbers, ask them if a number like `\pi` or `\sqrt{12}` would fit in the same category as rational numbers. Students should ideally recognize the separate sections of the organizer. You may still want to explicitly explain that irrational numbers do not fit into the same category as the others, which is why it is separated from the others.

How to classify radicals

It’s important to explicitly call out where radicals fall into the organizer. You’ll have already mentioned that `\sqrt{12}` is an irrational number, but make sure students understand that that does not mean all radicals are irrational.  

Show `\sqrt{16}` and ask students whether it is rational or irrational. Some students may see radicals and assume they’re always irrational. Tell them `\sqrt{16}` and ask them to figure out why. Quite likely some student will point out that since `\sqrt{16} = 4`, it is a rational number.  

Classifying activity

Once students have filled in the graphic organizer with examples, definitions, and nonexamples as needed, they can review with a classifying activity. You can have students work independently or with a partner. Students can record their answers underneath their graphic organizer.

Possible misconceptions

Once students have an opportunity to classify each number, review students’ reasoning to help catch any misconceptions, such as:

• Identifying `\sqrt{25}` as irrational because they saw a radical but didn’t realize it was a perfect square.
• Not identifying any numbers that include `\pi` as irrational.