AC9M7N01 • Year 7 Maths • Number • Learn
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.
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.
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
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
Final answer: The square root of 70 lies between 8 and 9.
Check: Squaring both bounds gives 64 < 70 < 81.
Example 3
Final answer: 43² = 1849.
Check: The two 40-by-3 rectangles explain the middle term (2\times40\times3).
Example 4
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.
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.
| (n) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| (n^2) | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
| (n) | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| (n^2) | 121 | 144 | 169 | 196 | 225 | 256 | 289 | 324 | 361 | 400 |
Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.
1. Write 9² as multiplication and calculate it.
Answer: 9 × 9 = 81.
Hint: Multiply 9 by 9.
Why: Squaring uses the same factor twice.
2. Find √121 and justify it.
Answer: √121 = 11 because 11 × 11 = 121.
Hint: Think of 11 squared.
Why: The principal square root is non-negative.
3. Which of 36, 40, 49 and 50 are perfect squares?
Answer: 36 and 49, because 36 = 6² and 49 = 7².
Hint: Test nearby whole-number squares.
Why: Perfect squares are products n × n.
4. A square tile has area 169 cm². Find its side length.
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. Place √50 between consecutive whole numbers.
Answer: 7 < √50 < 8 because 49 < 50 < 64.
Hint: Compare 50 with 7² and 8².
Why: Adjacent perfect squares bound the root.
6. A square courtyard has side 18 m. Find its area.
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. A square has area 256 cm². Find its perimeter.
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. Use 43 = 40 + 3 and an area model to calculate 43².
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. The squares 9, 16, 25, 36 have first differences 7, 9, 11. Predict the next square and explain.
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. A square tiled floor has perimeter 72 tile lengths. How many tiles cover it?
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.
The content description and elaborations below show the curriculum ideas taught in this unit. Items marked as teaching context support lesson planning.
This table gives closest-topic mapping for search and planning. The Australian Curriculum code is exact; overseas entries are broad equivalents because each jurisdiction structures outcomes differently.
| Region | Curriculum | Closest mapping |
|---|---|---|
| Australia | Australian Curriculum v9.0 | 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 |
| Victoria | Victorian Curriculum F-10 | Year 7 Maths: closest match in Number. Use this page as a VIC-aligned practice and homework reference. |
| NSW | NSW Curriculum | Stage 4 Maths: closest content focus for 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 and related outcomes. |
| United States | Common Core / NGSS | Grade 7 Common Core Mathematics/ELA closest topic match for 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. |
| England / UK | National Curriculum | Key Stage 3 / Year 7: closest programme-of-study match for 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. |
| Canada | Provincial and territory curricula | Grade 7 closest topic match. Canada varies by province, so use this as a broad Ontario/BC-style learning outcome reference. |
| New Zealand | New Zealand Curriculum | Level 4 Maths: closest achievement-objective topic for 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. |
| India | NCERT / CBSE | Class 7 closest NCERT/CBSE topic match for 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. |
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
Khan Academy — Connect a perfect square with the number whose square gives that value.
As you watch: How does finding a square root reverse squaring a number?
Load video player Loads YouTube in this lesson. See the video notice below.
Try it: Draw a square containing 49 equal tiles and explain how its side length relates to the square root of 49.
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Mapped skill: 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
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M7N01 · Year 7 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M7N01 · Level 7 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA4-IND-C-01 · Stage 4 |
| United States (USA) | Common Core State Standards for Mathematics | Grade 7 |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 7 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 8, Key Stage 3 |
| India | NCERT / CBSE — Mathematics | Class 7 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M7N01 — Perfect Squares and Square Roots
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