Since the set of real numbers is so large, it can be useful to have students come up with values that are not real numbers. This can help strengthen and verify their understanding of the principles behind what makes a number real or not. A prerequisite for this activity is teaching about imaginary numbers, as it will be nearly impossible for students to think of a non-real number if they are not aware of imaginary numbers.
You can test and reinforce students' understanding of real numbers by creating flash cards with numbers that represent different classes of numbers, such as irrational numbers, imaginary numbers, negative integers, transcendent numbers, fractions and decimals. Hold up one card at a time and ask whether the number on it is a real number or not, and why or why not. Students can also work in pairs to quiz each other or sort cards individually into piles of real and irrational.
Draw a number line on the board that stretches from -10 to 10. Pick one number at a time from a pre-defined list or a pile of flash cards, or allow students to take turns picking numbers, excluding integers. Have students point to the approximate point on the number line where each number would be found. After warming up with decimals, move on to increasingly difficult numbers like unsimplified fractions, repeating decimals and, if your students can handle them, irrational numbers.
Concrete applications of mathematical theory can help some students make sense of the ideas you are presenting. To apply the concept of real numbers, give students a few examples of places where real numbers appear in daily life, such as money, quantity of gas in a gas tank or the number of students in a classroom. You can point out that real numbers are not limited to numbers of objects (that is, integers), but also include non-discrete ranges of numbers, like distances and volumes.