L3 · Approximating Square Roots of Non-Perfect Squares (within 144)

Learning goal

“I can approximate the square root of any non-perfect square up to 144 - first by bracketing it between two whole numbers, then by squaring tenths to pin it to the nearest tenth - and I can use that to find the side length of a square from its area.”

Approximation — a value close to the exact number, written with ≈: √50 ≈ 7.1.

Bracketing — trapping a root between two consecutive perfect squares: 49 < 50 < 64 traps √50 between 7 and 8.

Refining — testing tenths by squaring them (7.0^2 = 49, 7.1^2 = 50.41) to see which side of the radicand each lands on, until the closest tenth is cornered.

Worked example

Approximate √50 to the nearest whole number. Step 1 (bracket): find the perfect squares around 50: 49 < 50 < 64, so 7 < √50 < 8. The root is trapped in a one-wide cage. Step 2 (which end?): 50 is 1 away from 49 and 14 away from 64 - it hugs the low end, so √50 is much closer to 7 than to 8. Step 3: To the nearest whole number, √50 ≈ 7. (Its fuller value, 7.0711..., belongs to Chunk B's refinement.) Say it plainly: squaring preserves order for positive numbers - if the AREAS are in order (49 < 50 < 64), the SIDES are in the same order (7 < √50 < 8). Bracketing the area brackets the side.

Check yourself — explain it

Why does 49 < 50 < 64 guarantee 7 < √50 < 8?

Paired problem — example still visible

Approximate √90 to the nearest whole number. -> 9 (81 < 90 < 100 traps it between 9 and 10; 90 is 9 above 81 and 10 below 100, and refining gives √90 ≈ 9.49 - nearer 9).

Faded completion — last step is yours

"√20: 16 < 20 < 25, so ___ < √20 < ___." -> 4; 5

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

Start practice →Take the unit quiz