Rational Number Mastery

Lesson Overview

This 40-minute lesson is designed for Year 10 students in New Zealand and tightly aligns with the New Zealand Curriculum (Te Mātaiaho 2025 draft) for Mathematics and Statistics. The focus is on deepening understanding of rational numbers—specifically, fractions, decimals, and percentages—as well as their representations, conversions, and applications.

Curriculum Links

• Strand: Number
• Level: Year 9 and Year 10 (Phase 4 in Te Mātaiaho)
• Achievement Objective: Identify, read, write, represent, compare, order, and convert between fractions, decimals, and percentages.
• Focus Skills: Representations on number lines and 100s squares; use of flow diagrams; simplifying fractions; conversions; calculating percentage increases/decreases; real-life applications.

Learning Objectives

By the end of this lesson, students will be able to:

1. Accurately represent fractions, decimals, and percentages on number lines and 100s squares.
2. Convert between fractions, decimals, and percentages fluently.
3. Simplify fractions and convert improper fractions to mixed numbers.
4. Calculate percentage increases and decreases in practical contexts.
5. Explain and justify conversions and calculations using appropriate mathematical vocabulary.

These objectives reflect the emphasis on mathematical reasoning, communication, and proportional reasoning within the NZ Curriculum's Number strand for Years 9 and 10 .

Key Competencies Included

• Thinking: Analyze and apply numeric concepts to practical problems.
• Using Language, Symbols, and Texts: Use mathematical vocabulary and notation precisely.
• Relating to Others: Engage in dialogue to clarify and justify reasoning.

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Lesson Breakdown

1. Getting Started (5 minutes)

• Activity: Quick interactive warm-up using a double number line.
• Task: Teacher displays pairs of numbers like 0.5 and 50%. Students identify and discuss the equivalence.
• Purpose: Activate prior knowledge of fractions, decimals, and percentages.
• Teacher Prompt: “How can we represent these numbers differently but still meaning the same value?”

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2. Exploring Representations and Conversions (15 minutes)

• Activity: Hands-on with number lines and 100s squares.

  a. Representation

  • Use a printed 100s square and number lines to represent fractions (e.g., 3/8), decimals (e.g., 0.375), and percentages (e.g., 37.5%).
  • Students place cards with fractions, decimals, and percentages on the number line and 100s square.

  b. Conversion through Flow Diagrams

  • Guide students through flow diagrams to convert fractions to decimals and decimals to percentages.
  • Example: Convert 3/8 → 0.375 → 37.5%.

• Teacher Support: Explain that ‘improper’ fractions are just ‘top-heavy’ and encourage use of simplification (e.g., convert 9/4 to 2 1/4).

• Materials Needed: 100s squares, double number line printed sheets, fraction/decimal/percentage flashcards, whiteboard for flow diagrams.

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3. Applying Percentage Increase and Decrease (10 minutes)

• Activity: Practical calculations with context.

  a. Problem: "If an item’s price increases from $50 to $75, what is the percentage increase?"

  b. Process:

  • Use decimal multipliers to calculate (percentage increase = ((75−50)/50) × 100 = 50% increase).
  • Reverse problem: “20% of an amount is $30. What is the original amount?”

  c. Representation: Use bar models or flowcharts to illustrate “whole to part” and “part to whole” relationships.

• Teacher Questions:

  • “Why might businesses use this type of calculation?”
  • “Can you think of examples where discounts use these same ideas?”

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4. Consolidation and Discussion (7 minutes)

• Activity: Student-led explanation and justification.
• Each student picks a different example from the lesson and explains their reasoning to peers.
• Encourage use of mathematical vocabularies such as numerator, denominator, equivalent, simplify, percentage, multiplier, increase, decrease.
• Teacher monitors understanding and addresses misconceptions immediately.

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5. Reflection and Next Steps (3 minutes)

• Wrap-up Discussion:
  • Highlight connections between fractions, decimals, and percentages.
  • Emphasise real-life relevance (e.g., shopping discounts, sports statistics).
  • Pre-teach the idea of ratios and rates, the next step in proportional reasoning.
• Feedback: “What is one new thing you learned today that you found interesting or useful?”

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Assessment and Evidence of Learning

• Observation during activities: correct placement on number lines and accurate conversions.
• Student verbal explanations demonstrating understanding.
• Completed quick calculations of percentage increases/decreases.
• Use of correct vocabulary and notation in explanations.

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Teacher Notes and Tips

• Use clear, concise mathematical language supported by visual aids.
• Scaffold tasks initially, then gradually encourage independence.
• Encourage students to visualise numbers on number lines for better conceptual understanding.
• Prompt students to use their first languages if helpful to connect terminology.
• Reinforce the idea that fractions, decimals, and percentages are multiple ways to describe the same part–whole relationship.

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Resources Needed

• Printed number lines and 100s squares.
• Fraction, decimal, and percentage cards.
• Whiteboard and markers.
• Bar model templates or flowchart templates.
• Calculators for quick checks (optional).

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This lesson design draws directly from the 2025 Mathematics and Statistics curriculum draft showing appropriateness for Year 9 and 10 rational number learning. It balances conceptual understanding, procedural fluency, problem-solving, and communication — key strands of the NZC Mathematics framework.