Rational and Irrational Solutions

5 min

Narrative

This Warm-up refreshes students’ memory about rational and irrational numbers. Students think about the characteristics of each type of number and ways to tell that a number is rational. This review prepares students for identifying solutions to quadratic equations as rational or irrational, and thinking about what kinds of numbers are produced when rational and irrational numbers are combined in different ways.

Launch

Remind students that a rational number is a number that can be written as a positive or negative fraction and that numbers that are not rational are called irrational. (The term “rational” is derived from the word “ratio,” as ratios and fractions are closely related ideas.)

Student Task

Numbers like -1.7, 16\sqrt{16}, and 53\frac53 are known as rational numbers.

Numbers like 12\sqrt{12} and 59\sqrt{\frac59} are known as irrational numbers.

Here is a list of numbers. Sort them into rational and irrational.

  • 97
  • -8.2
  • 5\sqrt5
  • -37\text-\frac{3}{7}
  • 100\sqrt{100}
  • 94\sqrt{\frac94}
  • -18\text-\sqrt{18}

Sample Response

Rational: 97,-8.2,-37,100,9497, \text-8.2, \text- \frac{3}{7}, \sqrt{100}, \sqrt{\frac94}. Irrational: 5,-18\sqrt5, \text-\sqrt{18}

Activity Synthesis (Teacher Notes)

Ask students to share their sorting results. Then, ask, “What are some ways you can tell that a number is irrational?” After each student offers an idea, ask others whether they agree or disagree. Ask students who disagree for an explanation or a counterexample.

Point out some ways to tell that a number might be irrational:

  • It is written with a square root symbol, and the number under the symbol is not a perfect square, or not the square of a recognizable fraction. For example, it is possible to tell that 94\sqrt{\frac94} is rational because 94=(32)2\frac94=\left(\frac32\right)^2. However, 18\sqrt{18} is irrational because 18 is not a perfect square.
  • If we use a calculator to approximate the value of a square root, the digits in the decimal expansion do not appear to stop or repeat. It is possible, however, that the repetition doesn’t happen within the number of digits we see. For example, 149\sqrt{\frac{1}{49}} is rational because it equals 17\frac17, but the decimal doesn’t start repeating until the seventh digit after the decimal point.
Standards
Building On
  • 8.NS.1·Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
  • 8.NS.A.1·Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Building Toward
  • HSN-RN.B.3·Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
  • N-RN.3·Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
  • N-RN.3·Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
  • N-RN.3·Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

15 min

15 min