Discovering Real Numbers

Lesson Overview

Curriculum Area:
Mathematics – Number
Year Level: Year 9
Australian Curriculum Descriptor:
AC9M9N01 – Recognise that the real number system includes the rational numbers and the irrational numbers, and solve problems involving real numbers using digital tools.

Lesson Duration:
50 minutes
Class Size:
12 students

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Learning Intentions

By the end of this lesson, students will:

• Understand the difference between rational and irrational numbers
• Identify examples of both types of numbers
• Place rational and irrational numbers on a number line
• Use digital tools to solve problems involving real numbers
• Appreciate the historical and practical context of irrational numbers

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Success Criteria

Students can:

• Correctly categorise numbers as rational or irrational
• Explain key characteristics of irrational numbers (e.g. non-repeating, non-terminating)
• Use a calculator or digital graphing tool to explore square roots and approximations
• Collaboratively solve and explain real-world problems involving irrational numbers

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Required Materials

• Student laptops or iPads (with internet access)
• Scientific calculators
• Whiteboard and markers
• Sticky notes or coloured cards
• Printable number line (A3 size per student group)
• "Rational or Irrational?" sorting cards
• Digital tool (e.g. Desmos, GeoGebra, or equivalent calculator application) pre-loaded

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Lesson Structure

⏳ 0–5 mins | Warm-Up: The Mystery Number

Objective: Activate prior knowledge and spark inquiry.

• Write on the board:

  "I am a number. I can’t be written as a fraction. I go on forever, and my digits never repeat. Who am I?"

• Encourage students to suggest possible answers. Guide them if needed towards π (pi) or √2.

• Transition into: Today we’re exploring the difference between rational and irrational numbers, and how together they create the real number system.

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📚 5–15 mins | Explicit Teaching: The Real Number System

Objective: Clearly establish definitions and properties.

Using the whiteboard or smartboard, guide students through a breakdown of the real number system:

• Whole numbers ⟶ integers ⟶ rational numbers (fractions, terminating and repeating decimals)
• Irrational numbers (non-terminating, non-repeating decimals such as π, √2, e)

Visual Aid: Use a Venn diagram on the board to illustrate how rational and irrational numbers are both subsets of real numbers, but do not overlap.

Key Points to Emphasise:

• Rational numbers can be written as a ratio (fraction)
• Irrational numbers cannot, and their decimal form never repeats or terminates
• Real numbers = all rational and irrational numbers

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🤹 15–25 mins | Activity 1: Rational or Irrational? Card Sort

Objective: Practise and apply knowledge collaboratively.

• In small groups of 3, students are given 20 cards, each featuring a different number (e.g. 0.333..., √4, π, 5/7, e, -2, √10, etc.)
• Their task is to sort these into "Rational" and "Irrational" piles
• After sorting, students must justify at least 2 of their choices to the class
• Teacher circulates and supports groups; ensures opportunities for formative assessment

Pro Tip: Include some "trick" cards (like √9, which equals 3 and is rational) to stimulate discussion.

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🌐 25–35 mins | Activity 2: Digital Number Line Challenge (Using Desmos or GeoGebra)

Objective: Use digital tools to approximate and locate real numbers.

• Pairs of students use digital graphing tools on their devices
• Provide them with a list of irrational numbers (e.g. √2, √5, π, e)
• Students must:
  1. Approximate each to at least 3 decimal places using a calculator
  2. Place these values on an interactive number line
• Encourage students to experiment: "What if we zoom in further? Will irrational numbers ever ‘end’?"
• Discuss how irrational numbers challenge our understanding of precision

Teacher Prompt:

“Why do we even care about irrational numbers? Where do they ‘exist’ in real life?”

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🔧 35–45 mins | Application: Real-World Problem Solving

Objective: Solve practical problems using rational and irrational numbers.

Students work individually or in pairs on the following problems:

1. A ladder problem – You have a ladder leaning against a wall that is 2 metres high. The base of the ladder is 1 metre from the wall. How long is the ladder? (Use Pythagoras; answer is √5)
2. Find the area of a circle with radius 3 cm (Area = πr²)
3. Estimate the diagonal of a square tile that is 6 cm on each side

Students must:

• Write down calculations
• Approximate irrational answers to 2 decimal places
• Identify whether final answers are rational or irrational

Extension:
Challenge faster students with irrational number trivia or provide a practical estimation activity: "How would a builder approximate the square root of 2 on-site?"

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🚦 45–50 mins | Reflection & Exit Ticket

Objective: Consolidate learning and reflect.

Reflection Prompt (on sticky note or in digital notebook):

In your own words:
a) What makes a number irrational?
b) Why do irrational numbers matter in mathematics?
c) One thing I still wonder is...

Collect these as an exit ticket and use them to inform the next lesson.

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Assessment & Feedback

Formative Assessment Opportunities:

• Observation during sorting task and digital skills application
• Responses in class discussion
• Exit tickets for metacognitive insight
• Problem-solving responses reviewed for depth and accuracy

Opportunities for Feedback:

• Verbal feedback given during group work and digital exploration
• Peer-to-peer discussion and self-assessment during sorting tasks
• Written feedback (highlighting reasoning and misconceptions) on real-world problems

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Differentiation & Inclusion

• Support:

  • Provide visual aids and examples of irrational/rational numbers
  • Use guided questions to scaffold problem-solving
  • Pair students strategically for peer mentoring

• Extension:

  • Challenge students to explore other irrational numbers (φ – the golden ratio, etc.)
  • Investigate why decimal expansions of irrational numbers never repeat

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Teacher Notes

Pedagogical Focus:
Encourage mathematical curiosity and critical thinking through hands-on exploration and guided digital tool use. Highlight the significance of irrational numbers in real-world contexts (engineering, architecture, nature).

Next Steps:
In the following lesson, explore operations involving irrational numbers and investigate limits of calculators and digital approximations.

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Wow Factor ✨

• This lesson blends concrete tasks with digital exploration, helping students connect logic and real-world maths
• The mystery number hook engages curiosity from the outset
• Using interactive number lines transforms abstract numbers into something concrete and manipulable
• The exit ticket provides a student voice to guide responsive teaching

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A great mathematician is not someone who knows all the answers, but someone who keeps asking better questions.
Encourage Year 9s to keep asking — what is a number, really?