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Grade 10 · Core Mathematics · Lesson 4 of 11

Justifying Real-Number Classifications

A Grade 10 Core Mathematics CBE lesson from the sub-strand Real Numbers. In it, learners classify perfect-square roots and non-perfect-square roots.

Strand Numbers and AlgebraSub-strand Real NumbersLesson code 10CM-11-L440 minutes · 2 slides

KICD learning outcome (b)

Classify real numbers as rational and irrational in different situations.

By the end of this lesson, learners should be able to

  1. Classify perfect-square roots and non-perfect-square roots.
  2. Use denominator structure and decimal patterns as supporting evidence.
  3. Communicate a classification with a precise reason.

Key vocabulary

Surd
An exact root expression whose value is irrational.
Square root of eleven is a surd because eleven is not square.
Perfect square
A whole number produced by multiplying an integer by itself.
Forty-nine is a perfect square because seven times seven equals forty-nine.
Justify
Support a mathematical conclusion with relevant evidence and reasoning.
Justify √11 by noting eleven has no integer square root.

Lesson preview

Slide 1 · Roots and Perfect SquaresOpening · 18 min
A root ladder demonstrates how perfect squares simplify to integers, while non-perfect squares fall into the irrational tray as non-terminating, non-repeating decimals.
Figure 1: A root ladder demonstrates how perfect squares simplify to integers, while non-perfect squares fall into the irrational tray as non-terminating, non-repeating decimals.

A root can be rational when it simplifies to an integer or fraction, but a root of a positive non-perfect square remains irrational. Check the exact radicand to guarantee a precise classification, rather than trusting a rounded calculator approximation. Use Figure 1 to visualise how simplifying exact roots determines their rational or irrational classification.

  • Perfect squares have exact whole-number square roots.
  • A positive whole number that is not a perfect square has an irrational root.
  • Simplify exact roots before assigning classifications confidently.
  • Rounded calculator displays cannot establish exact rationality.
Worked example. Since nine times nine equals eighty-one, its square root equals nine, which is rational. Twelve lies between nine and sixteen and is not a perfect square. Therefore square root of twelve cannot simplify to an integer and is irrational. Answer: Square root of eighty-one is rational; square root of twelve is irrational..

Check question: Why is 25 classified as a rational number?

  1. It simplifies exactly to 5, and every integer can be written as a fraction over 1.
  2. Because 25 is a positive integer, and the square roots of all positive integers are rational.
  3. It results in a non-terminating decimal, which is a defining property of a rational number.
  4. Any number written with a square root symbol is automatically considered a rational number.
Show answer

It simplifies exactly to 5, and every integer can be written as a fraction over 1. The root simplifies exactly to the integer 5. Since 5 can be written as 51, it satisfies the definition of a rational number without relying on a decimal approximation.

Slide 2 · Build a Classification ArgumentCheck · 22 min
A flowchart demonstrates how to classify numbers by identifying their form, applying a criterion, and reaching a rational or irrational conclusion.
Figure 2: A flowchart demonstrates how to classify numbers by identifying their form, applying a criterion, and reaching a rational or irrational conclusion.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

  • Both slides with diagrams and class activities
  • Teacher notes, SNE adaptations and assessment checks
  • Homework and the other 10 lessons in this sub-strand
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All 11 lessons in Real Numbers

  1. 1Sorting Whole Numbers by StructureClassify whole numbers as odd, even, prime and composite in different situations
  2. 2Classifying Numbers in SituationsClassify whole numbers as odd, even, prime and composite in different situations
  3. 3Recognising Rational and Irrational NumbersClassify real numbers as rational and irrational in different situations
  4. 4Justifying Real-Number ClassificationsClassify real numbers as rational and irrational in different situations
  5. 5Finding Reciprocals by DivisionDetermine the reciprocal of real numbers by division
  6. 6Reciprocals with Signs and DecimalsDetermine the reciprocal of real numbers by division
  7. 7Using Tools to Find ReciprocalsDetermine the reciprocal of real numbers by use of mathematical tables and calculators
  8. 8Reciprocals in Fraction ComputationApply reciprocal of real numbers in mathematical computations
  9. 9Reciprocal Methods for RatesApply reciprocal of real numbers in mathematical computations
  10. 10Real Numbers in Daily DecisionsPromote the use of real numbers in day-to-day activities
  11. 11Communicating Real-Number SolutionsPromote the use of real numbers in day-to-day activities