This is the second of two lessons that explore the properties of rational numbers and irrational numbers. In the first lesson, students classified solutions to quadratic equations as rational or irrational. Along the way, they noticed that some solutions are expressions that combine—by addition or multiplication—two numbers of different types: one rational and the other irrational. They began experimenting with concrete examples to find out whether the sums and products are rational or irrational.

In this lesson, students develop logical arguments that can be used to explain why the sums and products of rational numbers and irrational numbers are rational or irrational (MP3). As students look closely at combining rational numbers and irrational numbers, they make use of repeated reasoning to generalize the result as either rational or irrational (MP8).

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.