How to explain it
At this standard, students classify a real number as rational or irrational by testing whether its decimal form terminates, repeats in a fixed block, or does neither.
The anchor students hold onto: Look at the decimal form. If it TERMINATES or REPEATS in a fixed block, the number is RATIONAL. If it never ends and never repeats, it is IRRATIONAL.
Students will need this skill to interpret outputs of 8.EE.A.2 square root and cube root operations, and to classify length values from 8.G.B.7-8 Pythagorean theorem applications.
Worked examples
Example 1
Rational (Terminating)
Is 0.625 rational?
Step 1Check the decimal: does it terminate or repeat?
Step 2It ends after 3 digits (terminates).
Step 3Terminating decimal → RATIONAL. (= 5/8)
AnswerRational
Example 2
Rational (Repeating)
Is 0.4444.. rational?
Step 1Check the decimal: does it terminate or repeat?
Step 2The digit 4 repeats forever (0.4444..).
Step 3Repeating decimal → RATIONAL. (= 4/9)
AnswerRational
Example 3
Irrational
Is √7 rational?
Step 1Check the decimal: does it terminate or repeat?
Step 2√7 = 2.6457513.. never terminates, never repeats.
Step 3Non-terminating non-repeating → IRRATIONAL.
AnswerIrrational
Common mistakes
What students write
Thinking every square root is irrational, so √16 must be irrational.
The fix
Check for a perfect square first: √16 = 4 and √0.25 = 0.5 are rational.
Try this
Cole and Diego classify √81. Cole says √81 must be IRRATIONAL because of the square root sign. Diego says √81 = 9, which is RATIONAL. Who is correct? Find and correct the error.
What students write
Calling any long or complicated-looking decimal irrational.
The fix
A decimal that ends, or has a fixed repeating block, is rational no matter how long.
Teacher tip
Head off the two predictable errors before they happen. First: Check for a perfect square first: √16 = 4 and √0.25 = 0.5 are rational. Second: A decimal that ends, or has a fixed repeating block, is rational no matter how long.