Square Roots and Perfect Squares

Warm-up

Hand out 25 square tiles per pair: "Arrange all 25 into a perfect square. Now try 26. Now 36." The 26 attempt fails visibly — a row always sticks out.

Name what worked: 25 and 36 are PERFECT SQUARES ($5^2$, $6^2$), and the side length of the square is the SQUARE ROOT. The tiles make both words literal: the root is the side that grew the square.

Explore

Square-building census: pairs chart every perfect square from $1^2$ to $15^2$ and mark them on a number line to 225 — noticing the gaps widen (squares thin out as numbers grow; the gap between consecutive squares is the odd numbers: 3, 5, 7, …).

Then root-trapping: where does $\sqrt{50}$ live? Between $\sqrt{49} = 7$ and $\sqrt{64} = 8$, much nearer 7. Pairs trap $\sqrt{20}$, $\sqrt{90}$, $\sqrt{140}$ between consecutive integers and refine one of them to a decimal guess, checked by squaring ($4.5^2 = 20.25$ — a hair high, so $\sqrt{20} \approx 4.47$).

Formalize

Formalize the inverse pair and the trapping method:

$\sqrt{n^2} = n \;\; (n \ge 0) \qquad 49 < 50 < 64 \;\Rightarrow\; 7 < \sqrt{50} < 8$

Squaring and square-rooting undo each other — the same inverse-operation relationship as add/subtract and multiply/divide, extending the family. Vocabulary: radical sign, radicand, perfect square. Calculator roots are approximations for non-perfect squares; the trapped estimate is the sanity check on the calculator, not vice versa.

Practice

Practice: perfect-square retrieval sprint ($\sqrt{81}$, $\sqrt{144}$, $\sqrt{169}$); trap four roots between integers with the nearer end named; refine two to one decimal; one applied problem (a square garden of 90 m² — fence length?).

Exit ticket: between which two consecutive whole numbers is $\sqrt{75}$, which is it nearer, and why? ($8$ and $9$; nearer 9 since 75 is nearer 81 than 64.)

Exit ticket

Practice: perfect-square retrieval sprint ($\sqrt{81}$, $\sqrt{144}$, $\sqrt{169}$); trap four roots between integers with the nearer end named; refine two to one decimal; one applied problem (a square garden of 90 m² — fence length?).

Exit ticket: between which two consecutive whole numbers is $\sqrt{75}$, which is it nearer, and why? ($8$ and $9$; nearer 9 since 75 is nearer 81 than 64.)