Perfect Squares

Overview

Students review and build understanding of prime, composite and square numbers by using their key properties to decide classification and solve problems. The lesson also reinforces precise number reasoning through “test, record, and justify” strategies.

Learning intentions

WALT:

• identify prime, composite and square numbers using their properties
• explain why a number is (or isn’t) prime or composite
• connect square numbers to arrays and recognise square roots as the side length of the array
• use number properties to solve classification and comparison problems accurately

Success criteria

• I can tell if a number is prime or composite and explain my reasoning.
• I can recognise square numbers and describe what makes them squares.
• I can connect a square number to a square root (e.g., (\sqrt{49}=7)) and place non-perfect-square roots between consecutive integers.
• I can use correct notation in explanations (square notation and square root notation).

Curriculum links

• Number — AC9M6N02: identify and describe properties of prime, composite and square numbers and use these properties to solve problems and simplify calculations.
• Number reasoning — AC9M6N02: testing numbers by division and recording results to find patterns.
• Number reasoning — AC9M6N02: representing composite numbers as products of factors (including prime factors when needed).
• Number reasoning — AC9M6N02: identifying square numbers as a number multiplied by itself, linking arrays and square roots.

Lesson structure (60 minutes)

1. 0–8 min · Hook: Classify fast. Teacher displays 8–10 numbers on the board (mix of primes, composites and squares, e.g., 2, 6, 9, 11, 12, 16, 18, 25) and models one example briefly. Students do quick mini-whiteboard sketches/marks answering: “prime / composite / square / not sure” and add a one-sentence reason for any number they’re confident about.

2. 8–18 min · Direct teach: “What property decides?” Teacher think-alouds using three property checks:

• Prime: “Only divisible by 1 and itself.” Show with a short division test.
• Composite: “Has more than two factors.” Show with a factor pair product.
• Square: “It’s an array with equal side lengths.” Build an example array (e.g., 16 as (4\times4)). Students copy a three-column “Property Check” on a template: Prime / Composite / Square → key test and example, using dyslexia-friendly spacing (short bullets, large print option).

3. 18–30 min · Guided practice: Test and record. Teacher introduces a partially completed number chart (or mini table) for numbers students are investigating (e.g., 1–30 or a smaller set like 10–24). Students choose 6 target numbers and:

• test for prime/composite using division (try factor pairs up to the relevant point)
• record outcomes on the chart (divides evenly / does not divide evenly)
• identify squares using equal side array knowledge (e.g., spot 9, 16, 25) Teacher circulates with prompts: “What division result convinced you?” “How does your result show it’s composite?” “How do you know it’s square?”

4. 30–43 min · Focus: Square roots as side length. Teacher models:

• Perfect square: (\sqrt{49}=7) because (7^2=49)
• Non-perfect square: e.g., (43) is between (36) and (49), so (\sqrt{43}) is between 6 and 7. Students complete a short whiteboard set:
• a) (\sqrt{25}), b) (\sqrt{81})
• c) For (\sqrt{43}), (\sqrt{50}), (\sqrt{63}): write “between ___ and ___” Teacher checks for two common misconceptions: confusing (n) with (n^2), and treating (\sqrt{43}) as a whole number.

5. 43–53 min · Independent practice: Use properties to solve. Teacher explains the task (4–6 questions) with sentence frames available:

• “(n) is a square because ( \square \times \square = n). So (\sqrt{n}=\square).”
• “(\sqrt{m}) lies between ___ and ___ because ___ and ___ are consecutive squares.” Students work independently on a practice sheet set containing:
• classify each number (prime/composite/square)
• factor a composite into a product of factors (when required)
• match square notation and square root notation, and place non-perfect-square roots between consecutive integers

6. 53–60 min · Exit ticket: Justify quickly. Students answer on paper (collected immediately): a) (\sqrt{64}=?) b) Is 72 a perfect square? Explain using an array or factor reasoning. c) Between which two consecutive whole numbers is (\sqrt{57})? Write both numbers and a short reason.

Resources

• Mini whiteboards, markers, and erasers (1 per student)
• Number cards for primes/composites/squares
• Prime/composite/square “Property Check” template (dyslexia-friendly)
• Chart/table for testing divisibility results
• Practice sheet: “Prime, Composite & Square” (classification + short justifications)
• “Square roots” quick tasks (perfect squares + non-perfect squares)
• Sentence starter cards for explanations and notation
• Optional: counters or grid paper for building square arrays
• Projector/whiteboard to show worked examples

Assessment

• Formative: hook responses—listen for correct classification reasons (not just answers).
• Formative: guided practice—check students’ recorded division tests and whether squares are identified via equal side arrays.
• Targeted: during independent practice, observe correct use of notation and “between consecutive integers” reasoning.
• Summative for the lesson: exit ticket scored for accuracy and brief justification.

Differentiation

• Support: sentence starters for root reasoning (e.g., “I know ___ is between ___ and ___, so (\sqrt{,}) is between …”); provide a number line with labelled endpoints (6 and 7) for non-perfect squares.
• Support for reading load: chunked worksheet sections, clear short instructions, and optional audio explanation from teacher during the square root focus.
• SEN/EAL: allow oral recording of reasoning using the same frames; provide a worked example card students can mirror.
• Extension (advanced learners): give an extra challenge: “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.”