Lesson 20
Rational and Irrational Solutions
Lesson Narrative
Students have seen both rational and irrational solutions when solving quadratic equations in this unit. Before this lesson, minimal emphasis was placed on reviewing the meaning of rational and irrational numbers (which students first learned in grade 8), or on the fact that certain solutions are irrational. Students may have noticed, however, that it can be tricky to get a sense of the value of irrational solutions—even if they are exact—because irrational solutions are expressed with the square root symbol and their decimal values need to be approximated.
This is the first of two lessons in which students look closely at whether a solution to a quadratic equation is rational or irrational. Distinguishing solutions as rational or irrational does not necessarily impact students’ ability to solve applied problems about quadratic functions. In those cases, we deal mostly with finite decimal approximations. The work here extends students’ understanding of the real number system. Reasoning about the properties of rational and irrational numbers also offers opportunities to construct logical arguments and attend to precision in reasoning (MP3, MP6). Along the way, students also practice solving quadratic equations and finding zeros of the corresponding functions.
For some solutions to quadratic equations, it is relatively straightforward to classify them as rational (for example, \(\frac14\) or 7) or irrational (for example, \(\sqrt 8\) or \(\text \sqrt {\frac12}\)). Other solutions are harder to classify, however, based on what students have learned up until this point. For instance, is \(\text2 + \sqrt5\) rational or irrational? What about \(\frac13 \sqrt2\)? Students begin wondering about these questions and making conjectures about what category each of these numbers belongs to. In an upcoming lesson, they will reason about the sums and products of rational and irrational numbers more generally and justify them as being one type of number or the other.
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 consider the kinds of numbers we get when solving quadratic equations.
Required Materials
Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
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.