Lesson 21
Sums and Products of Rational and Irrational Numbers
Lesson Narrative
This is the second of two lessons that explore the properties of rational 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 and irrational numbers are one type or the other (MP6). Constructing logical arguments encourages students to pay attention to precision (MP6).
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.
Learning Goals
Teacher Facing
 Explain (orally and in writing) why the product of a nonzero rational and irrational number is irrational.
 Explain (orally and in writing) why the sum of a rational and irrational number is irrational.
 Explain (orally and in writing) why the sum or product of two rational numbers is rational.
Student Facing
 Let’s make convincing arguments about why the sums and products of rational and irrational numbers always produce certain kinds of numbers.
Learning Targets
Student Facing
 I can explain why adding a rational number and an irrational number produces an irrational number.
 I can explain why multiplying a rational number (except 0) and an irrational number produces an irrational number.
 I can explain why sums or products of two rational numbers are rational.
CCSS Standards
Glossary Entries

quadratic formula
The formula \(x = {\textb \pm \sqrt{b^24ac} \over 2a}\) that gives the solutions of the quadratic equation \(ax^2 + bx + c = 0\), where \(a\) is not 0.