Mathematics 7–10 · Year 7
Square numbers and square roots with tiles
Number and algebra
The idea
A number is a perfect square when its tiles form a square array, its square root is the side of that array, and a number with tiles left over has a square root between two whole numbers.
What you need
- 50 square tiles (2 cm) per pair
- 1 cm grid paper and ruler
How to do it
- Build the largest square possible from 16, 25 and 36 tiles and record the side length each time.
- Try 40 tiles: build the largest square, count the tiles left over, and state the two whole numbers the square root of 40 lies between.
- Estimate the square root of 40 to one decimal place by reasoning about how much of the next row and column the leftover tiles would fill, then check with a calculator.
- On grid paper draw a 1 by 1 square and its diagonal; measure the diagonal to the nearest millimetre and compare with sqrt(2).
- List the perfect squares up to 100 and describe the pattern of gaps between them (3, 5, 7, ...).
What you should see
16, 25 and 36 tiles make squares of side 4, 5 and 6. Forty tiles make a 6 by 6 square with 4 left over, so sqrt(40) lies between 6 and 7; the calculator gives 6.32. The diagonal of a 1 cm square measures 1.4 cm, and sqrt(2) = 1.414. The gaps between consecutive perfect squares are the odd numbers, because (n + 1)² − n² = 2n + 1. The learner knows it worked when the leftover-tile reasoning places the root in the correct whole-number interval every time.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The square root of a number is half of it.
- Every whole number has a whole-number square root.
- sqrt(40) is 6 because 6 × 6 is closest.
Safety card
Hazards
No hazard is listed.
Controls
No control is listed.
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 4 Indices; page read 2026-09-22MA4-IND-C-01
- Australian Curriculum v9AC9M7N01
Sources
The pages the author read to write this activity.