Squares, roots, and arrays

Overview

Students connect perfect square numbers to square roots by using visual arrays, then represent squares using notation and solve problems that require interpreting square roots as values between consecutive integers. This lesson builds on their idea of arrays as multiplication facts and extends it to Year 7 square and square root reasoning.

Learning intentions

• Students will identify perfect square numbers and explain why they are square.
• Students will represent a perfect square using an array and link it to multiplication.
• Students will describe the relationship between perfect squares and square roots.
• Students will use square and square root notation to solve problems.

Success criteria

• I can build an array for a number and tell if it forms a perfect square.
• I can state whether a number is a perfect square and find its square root.
• I can find between which two consecutive whole numbers the square root of a non-perfect square lies.
• I can use correct notation (e.g., (7^2), (\sqrt{49}), (\sqrt{43})) in my explanations.

Curriculum links

• Number — AC9M7N01: describe the relationship between perfect square numbers and square roots, and use squares and square roots of perfect square numbers to solve problems.
• Number — AC9M7N01: investigate patterns in square numbers and apply reasoning to locate square roots between consecutive integers.
• Mathematical representation: use arrays, square notation, and square root notation to model relationships between numbers.

Lesson structure (60 minutes)

1. 0–8 min · Hook: array warm-up (retrieve). Teacher displays 4–6 numbers (e.g., 4, 6, 9, 12, 16, 18) and asks: “Which of these would make a square array?” Students quickly sketch a rectangle on mini whiteboards and justify with one sentence: “It has equal sides / it doesn’t.”

2. 8–18 min · Direct teach: what makes a perfect square. Teacher think-alouds building arrays for one number (e.g., 8 is not square) then for a perfect square (e.g., 9). Prompts: “If the side length is (n), the total counters are (n \times n), so the total is (n^2).” Students copy a worked example: array → multiplication sentence → square notation (e.g., (3 \times 3 = 9) so (3^2 = 9)). Offer dyslexia-friendly option: provide printed worked examples with short labels and plenty of spacing; students may underline key steps.

3. 18–32 min · Guided practice: identify square numbers up to 100. Students use “Square numbers: Practice” sheets. Task set includes: drawing arrays for given numbers, and checking/correcting arrays. Teacher circulates with prompts: “How do you know the side lengths match?” “What square notation matches your array?” Formative check: collect 3–4 samples of student reasoning (not just answers).

4. 32–43 min · Focus: square roots as the side length. Teacher models the link: “If (n^2) is a perfect square, then (\sqrt{n^2} = n). For example, ( \sqrt{49} = 7).” Then teacher introduces a non-perfect square: “(43) is not a perfect square. It is between (36) and (49). So (\sqrt{43}) is between (\sqrt{36}=6) and (\sqrt{49}=7), therefore between 6 and 7.” Students complete a short set on whiteboards: find (\sqrt{25}), (\sqrt{81}), and then state between which two consecutive integers each non-perfect square root lies (e.g., (\sqrt{43}), (\sqrt{50}), (\sqrt{63})).

5. 43–53 min · Independent practice: use square notation and root relationship. Students use “Square notation: Practice” to: write the calculation for each square, find values, and fill missing numbers linked to squares. Teacher targets misconception checks:

• confusion between (n) and (n^2)
• treating (\sqrt{43}) as a whole number Provide number sentence frames for support: “(\sqrt{ \ \ \ }) equals __ because __ squared is __.”

6. 53–60 min · Exit ticket: demonstrate relationship. Students answer:

• a) ( \sqrt{64} =?)
• b) Is 72 a perfect square? Explain using arrays or square comparison.
• c) Between which two consecutive whole numbers is ( \sqrt{57})? Teacher collects immediately for quick review.

Resources

• Mini whiteboards and markers (1 per student)
• Counters (optional if using a hands-on moment during guided practice)
• “Square numbers: Practice” sheet (Resource 2) for each student
• “Square notation: Practice” sheet (Resource 3) for each student
• “Square numbers: Slides” (Resource 1) for display
• Printed dyslexia-friendly copies of key worked examples (short steps, large spacing)
• Sentence starters and notations card (e.g., “(n^2) means (n \times n)”)

Assessment

• Formative during array decisions in the hook (teacher listens for correct justification)
• Guided practice checks: whether students can justify “square vs not square” using equal side lengths and matching multiplication
• Targeted whiteboard checks for (\sqrt{n^2}) and “between” reasoning for non-perfect squares
• Exit ticket marked for: correct perfect-square roots, correct “between consecutive integers” claim, and clear notation use

Differentiation

• Support: provide sentence starters for root reasoning (“I know (43) is between 36 and 49, so (\sqrt{43}) is between 6 and 7.”); allow using a number line with labelled endpoints 6 and 7.
• Support for reading load: dyslexia-friendly reading options such as audio explanation from teacher, chunked worksheet sections, and colour-coding steps (array → multiplication → square notation → root).
• Extension (advanced learners): give a challenge set: “List three non-perfect squares between (6^2) and (7^2) and place their square roots between 6 and 7. Explain why none are whole numbers.”
• Extension (advanced learners): ask for pattern reasoning: “What stays the same about the step from (n^2) to ((n+1)^2)?” (students can investigate examples even if not deriving a full rule yet)
• Scaffolding for SEN/EAL: offer a partially completed table template with columns “perfect square?”, “square root”, “array side length”, and one worked example completed.