Perfect Squares Exploration

Lesson Overview

This 45-minute lesson is designed for Year 7 students to explore the relationship between perfect square numbers and square roots, using the Australian Curriculum v9 content description AC9M7N01. This lesson will enable students to understand and use squares and square roots of perfect square numbers to solve problems, integrating visual patterns, algebraic reasoning, and real-world applications.

Curriculum Alignment

• Content Descriptor: AC9M7N01
  Describe the relationship between perfect square numbers and square roots, and use squares of numbers and square roots of perfect square numbers to solve problems
• Achievement Standard: Year 7 students describe and use perfect squares and their roots, apply properties in problem solving

Learning Objectives

By the end of this lesson, students will:

1. Recognise and describe the relationship between perfect square numbers and square roots.
2. Use visual dot patterns and algebraic expressions to explore squares of natural numbers up to 20.
3. Solve problems involving squares and square roots, including estimating square roots of non-perfect squares between two integers.
4. Apply knowledge of squares and square roots to context-based problems, such as calculating area and perimeter of square tiles.

Lesson Duration

45 minutes

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Teaching and Learning Activities

1. Introduction and Engagement (5 minutes)

• Begin with a question: “What do you think a perfect square number is?”
• Use dot patterns on the board (or projector) showing squares of 1, 4, 9, 16 dots arranged in a square shape to visually demonstrate perfect squares (AC9M7N01_E1).
• Ask students to identify any patterns they notice about these numbers.

Materials: Grid paper or digital dot array, visualiser or projector.

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2. Guided Exploration: Squares and Square Roots (10 minutes)

• Display the notation for squares and square roots, e.g., (4^2=16), (\sqrt{16}=4).
• Demonstrate the distributive property for calculating squares of 2-digit numbers using area diagrams. Example: (43^2=(40+3)^2=40^2 + 2 \times 40 \times 3 + 3^2) (AC9M7N01_E2).
• Give students 5 practice questions to calculate the squares of given numbers using this method (e.g., (12^2), (15^2), (37^2)).

Materials: Whiteboard, worksheets, calculators (optional).

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3. Investigating Square Roots Between Numbers (10 minutes)

• Demonstrate how to determine between which two consecutive natural numbers the square root of a non-perfect square lies (AC9M7N01_E3).
• Example: (\sqrt{43}) is between (\sqrt{36} (=6)) and (\sqrt{49} (=7)), so between 6 and 7.
• Have pairs of students work through a set of 5 numbers (e.g., 50, 65, 80, 90, 120) to estimate square roots.

Materials: Number cards or worksheet, number line poster.

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4. Pattern Recognition and Problem Solving (10 minutes)

• Guide students to generate a list of perfect squares from 1 to 400 and observe patterns (last digits, differences between consecutive squares) (AC9M7N01_E4).
• Discuss the second difference being constant and what this means (link to quadratic patterns).
• Present a contextual problem: “A square-tiled floor has 144 tiles. What is the perimeter of the floor in tile lengths?”
• Students solve using the relationship between area and perimeter (AC9M7N01_E5).
• Discuss and explain answers as a class.

Materials: Calculators, problem worksheet.

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5. Consolidation and Reflection (5 minutes)

• Summarise key ideas: perfect squares, square roots, notation, properties, and real-world applications.
• Ask students to write a short explanation in their own words describing the relationship between perfect squares and square roots.
• Collect or review responses informally as formative assessment.

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Assessment

• Observation of student participation and accuracy during pair activities.
• Responses in problem-solving and pattern recognition tasks.
• Written reflection explaining the relationship between perfect squares and square roots.

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Differentiation Strategies

• Provide calculators for students who need computational support.
• Extension challenge: Explore squares of larger numbers using algebraic expressions or digital tools.
• Visual and hands-on learners engaged through dot patterns and area diagrams.

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Resources

• Grid paper or digital dot arrays
• Number line displays
• Calculators
• Worksheets with practice questions and real-world problems

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Teacher Notes

• Emphasise connecting visual patterns to numeric notation to deepen conceptual understanding.
• Encourage students to verbalise their reasoning behind estimating square roots between integers.
• Relate mathematical concepts to everyday contexts familiar to students, such as tiled floors or area computations.
• Incorporate cultural perspectives where possible, e.g. exploring patterns in traditional First Nations tile or weaving patterns.

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This lesson plan follows the Australian Curriculum v9 Mathematics standards for Year 7 in the specified content code AC9M7N01, ensuring students develop a solid foundational understanding of perfect squares and square roots with engaging, purposeful activities.