Year 7 Mathematics · AC9M7N01

Perfect Squares and Square Roots

We are learning to describe the relationship between perfect square numbers and square roots, and use squares and square roots of perfect square numbers to solve problems

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Learning goalsSay it simply

We are learning to describe the relationship between perfect square numbers and square roots, and use squares and square roots of perfect square numbers to solve problems.

A perfect square is the product of a natural number multiplied by itself. A square array makes this structure visible: 12 rows of 12 contain 144 units, so 12² = 144.

The principal square root reverses squaring by identifying the non-negative side length. Therefore √144 = 12, while a non-perfect root such as √70 can be bounded between nearby perfect squares.

In square contexts, units distinguish quantities: area uses square units, side length uses linear units and perimeter is four times the side. A complete solution follows the requested quantity beyond the intermediate square root.

Success criteria

  • I can recognise perfect squares and represent them with square arrays.
  • I can use a principal square root to recover a square's side length.
  • I can bound a non-perfect root and solve related area or perimeter problems.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Find the side and perimeter of a square

  1. Identify the area: 196 cm².
  2. Use side = √area, so side = √196.
  3. Because 14² = 196, the side is 14 cm.
  4. Calculate perimeter: 4 × 14 = 56 cm.

Final answer: The side is 14 cm and the perimeter is 56 cm.

Check: Multiply 14 × 14 to recover the given area, 196 cm².

Example 2

Bound a non-perfect square root

  1. Find the perfect square below 70: 8² = 64.
  2. Find the next perfect square: 9² = 81.
  3. Write 64 < 70 < 81.
  4. Take roots to obtain 8 < √70 < 9.

Final answer: The square root of 70 lies between 8 and 9.

Check: Squaring both bounds gives 64 < 70 < 81.

Example 3

Use an area model to calculate 43²

40 × 40 = 160040 × 33 × 403 × 3403403
  1. Split 43 as 40 + 3 along both sides.
  2. Calculate all four regions: 40², 40 × 3, 3 × 40 and 3².
  3. Add: 1600 + 120 + 120 + 9 = 1849.

Final answer: 43² = 1849.

Check: The two 40-by-3 rectangles explain the middle term (2\times40\times3).

Example 4

Explain a square-number pattern

  1. Write consecutive squares: 1, 4, 9, 16, 25.
  2. Find first differences: 3, 5, 7, 9.
  3. Find second differences: 2, 2, 2.
  4. Each new square array adds an L-shaped border containing the next odd number of units.

Conclusion: Consecutive square numbers have increasing odd first differences and constant second difference 2.

Check: The next difference is 11, so the next square is 25 + 11 = 36.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7N01 - Perfect Squares and Square Roots
Example 1

Example 1 Find the side and perimeter of a square Identify the area: 196 cm². Use side = √area, so side = √196. Because 14² = 196, the side is 14 cm. Calculate perimeter: 4 × 14 = 56 cm. Final answer: The side is 14 cm and the perimeter is 56 cm. Check: Multiply 14 × 14 to recover the given area, 196 cm².

Example 2

Example 2 Bound a non-perfect square root Find the perfect square below 70: 8² = 64. Find the next perfect square: 9² = 81. Write 64 &lt; 70 &lt; 81. Take roots to obtain 8 &lt; √70 &lt; 9. Final answer: The square root of 70 lies between 8 and 9. Check: Squaring both bounds gives 64 &lt; 70 &lt; 81.

Example 3

Example 3 Use an area model to calculate 43² 40 × 40 = 1600 40 × 3 3 × 40 3 × 3 40 3 40 3 Split 43 as 40 + 3 along both sides. Calculate all four regions: 40², 40 × 3, 3 × 40 and 3². Add: 1600 + 120 + 120 + 9 = 1849. Final answer: 43² = 1849. Check: The two 40-by-3 rectangles explain the middle term (2\times40\times3).

Example 4

Example 4 Explain a square-number pattern Write consecutive squares: 1, 4, 9, 16, 25. Find first differences: 3, 5, 7, 9. Find second differences: 2, 2, 2. Each new square array adds an L-shaped border containing the next odd number of units. Conclusion: Consecutive square numbers have increasing odd first differences and constant second difference 2. Check: The next difference is 11, so the next square is 25 + 11 = 36.

Curriculum examplesCopied content

The content description and elaborations below show the curriculum ideas taught in this unit. Items marked as teaching context support lesson planning.

  • Content description: 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
  • E1: investigating squares of natural numbers from one to 20, and connecting them to visual representations such as dots arranged in a square pattern
  • E2: using the square and square root notation, and the distributive property and area diagrams to calculate the squares of two-digit numbers; for example, 43^2=(40+3)^2=40^2+2\times40\times3+3^2=1600+240+9=1849
  • E3: determining between which 2 consecutive natural numbers the square root of a given number lies; for example, 43 is between the square numbers 36 and 49 so \sqrt{43} is between \sqrt{36} and \sqrt{49} and therefore between 6 and 7
  • E4: generating a list of perfect square numbers and describing any emerging patterns; for example, the last digit of perfect square numbers, or the difference between consecutive square numbers, and recognising the constant second difference
  • E5: using the relationship between perfect square numbers and their square roots to determine the perimeter of a square tiled floor using square tiles; for example, an area of floor with 144 square tiles has a perimeter of 48 tile lengths
Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. Write 9² as multiplication and calculate it.

    Check answer

    Answer: 9 × 9 = 81.

    Hint: Multiply 9 by 9.

    Why: Squaring uses the same factor twice.

  2. 2. Find √121 and justify it.

    Check answer

    Answer: √121 = 11 because 11 × 11 = 121.

    Hint: Think of 11 squared.

    Why: The principal square root is non-negative.

  3. 3. Which of 36, 40, 49 and 50 are perfect squares?

    Check answer

    Answer: 36 and 49, because 36 = 6² and 49 = 7².

    Hint: Test nearby whole-number squares.

    Why: Perfect squares are products n × n.

  4. 4. A square tile has area 169 cm². Find its side length.

    Check answer

    Answer: The side is √169 = 13 cm.

    Hint: Find the number whose square is 169.

    Why: A square's side is the principal square root of its area.

  5. 5. Place √50 between consecutive whole numbers.

    Check answer

    Answer: 7 < √50 < 8 because 49 < 50 < 64.

    Hint: Compare 50 with 7² and 8².

    Why: Adjacent perfect squares bound the root.

  6. 6. A square courtyard has side 18 m. Find its area.

    Check answer

    Answer: 18² = 324, so the area is 324 m².

    Hint: Multiply the side by itself.

    Why: Squaring the 18-metre side gives an area of 324 square metres, so the answer uses square units.

  7. 7. A square has area 256 cm². Find its perimeter.

    Check answer

    Answer: The side is √256 = 16 cm, so perimeter = 4 × 16 = 64 cm.

    Hint: Use √area, then multiply by 4.

    Why: Find the side before the perimeter.

  8. 8. Use 43 = 40 + 3 and an area model to calculate 43².

    Check answer

    Answer: 43² = 40² + 2 × 40 × 3 + 3² = 1849.

    Hint: Include the large square, two rectangles and the small square.

    Why: Both cross rectangles are required.

  9. 9. The squares 9, 16, 25, 36 have first differences 7, 9, 11. Predict the next square and explain.

    Check answer

    Answer: The next difference is 13, so the next square is 36 + 13 = 49.

    Hint: Consecutive odd differences increase by 2.

    Why: Consecutive square arrays grow by successive odd-number borders.

  10. 10. A square tiled floor has perimeter 72 tile lengths. How many tiles cover it?

    Check answer

    Answer: The side is 72 ÷ 4 = 18 tiles, so 18² = 324 tiles cover the floor.

    Hint: Reverse the perimeter first, then square the side.

    Why: Perimeter gives boundary length; squaring the recovered side gives area.

Practice and reviewReady for practice

Common mistake: √a is treated as a÷2.

Correction: A square root asks which non-negative number squares to a.

Common mistake: (a+b)² expanded as a²+b².

Correction: The two cross rectangles contribute 2ab.

Common mistake: Every whole number has a whole-number root.

Correction: Only perfect squares do.

Curriculum alignmentStart here

We are learning to describe the relationship between perfect square numbers and square roots, and use squares and square roots of perfect square numbers to solve problems.

A perfect square is the product of a natural number multiplied by itself. A square array makes this structure visible: 12 rows of 12 contain 144 units, so 12² = 144.

The principal square root reverses squaring by identifying the non-negative side length. Therefore √144 = 12, while a non-perfect root such as √70 can be bounded between nearby perfect squares.

In square contexts, units distinguish quantities: area uses square units, side length uses linear units and perimeter is four times the side. A complete solution follows the requested quantity beyond the intermediate square root.

Success criteria

  • I can recognise perfect squares and represent them with square arrays.
  • I can use a principal square root to recover a square's side length.
  • I can bound a non-perfect root and solve related area or perimeter problems.
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