Square roots, and the prime-factor test for a perfect square

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What to assume they know

What makes a number a perfect square — square numbers, perfect squares, area and sidelength
What a perfect square's last digits can and cannot be — the six endings a square can have, and that the test only rejects
Why the first n odd numbers add up to n² — the successive-subtraction test and why it works
Prime factorisation of a number up to four digits, by factor tree or repeated division
Multiplying a negative by a negative
Expanding a product of two two-term sums, at least in the form used for 45²

What they should be able to do

Recover the sidelength of a square from its area
State the definition of a square root and identify both integer roots of a perfect square
Read and write the radical symbol, and state the chapter's convention of keeping only the positive root
Decide whether a given number is a perfect square by prime factorisation, and produce its root from the same working
Explain why the pairing of primes is decisive — that is, why unique factorisation is what makes the test valid
Compare the three tests the chapter gives and say what each does and costs
Estimate the square root of a number that is not a perfect square by bracketing it between known squares
Use the units digit and a midpoint square to narrow a bracket
Apply an estimated root to a practical cutting or fitting problem

Where it usually goes wrong

"√64 has one value." A perfect square has two integer roots. The chapter keeps the positive one by an announced convention, and a student who has not heard the announcement will later be confused by −8.
"The radical symbol means the positive root by definition." In this chapter it is a working agreement stated on Part I p.8, and a few lines earlier on the same page it writes √64 = ±8.
"It ends in 6, so it is a square and its root ends in 6." Two errors in one: the ending never confirms squareness, and even when the number is a square the root could end in 4 or in 6.
"Prime factorisation is just a third way of checking." All three produce the root, so that is not what separates them. Factorisation is the one that does not have to search — the other two take roughly as many steps as the root is large, which is why the chapter calls 729 tedious by subtraction — and it is the only one that also tells you the smallest multiplier that would complete the pairs, which is exactly what item 6 asks for. Make that comparison explicit; it is the point of sections 5 to 9.
"An unpaired prime can be removed." Item 6 asks you to multiply by the missing factor. Taking a prime out changes the number; supplying its partner is what completes the pairs.
"156 fails because 13 is large." It fails because 13 appears once. Size has nothing to do with it.
"45² = 40² + 5²." The cross term 2 × 40 × 5 is exactly what the chapter spells out, and it is the term students drop.
"If a number is not a perfect square, its root is useless." Akhil's problem is answered entirely by a bracket, with no exact root anywhere in it.
"Estimating means guessing." Each of the chapter's five steps narrows a stated interval. Show the interval shrinking.

Questions to check understanding

Find the side of a square given its area
Decide by prime factorisation whether a four-digit number is a perfect square, and give its root if it is
Find the least multiplier, or divisor, that turns a given number into a perfect square, and give the resulting root
Find the smallest square number divisible by a stated set of numbers
Estimate the square root of a non-square to the nearest whole number, showing the bracketing squares
Count a structured array and give the prime factorisation of the count
"Between which two whole numbers does the root lie?" — the estimation form the board asks in place of long division

Examples worth working on the board

The opening question (Part I p.7, foot): a square of area 49 sq. cm; find its side. 7 × 7 = 49, so 7² = 49, so the side is 7 cm and 7 is called the square root of 49.
The general statement (Part I p.8), in the chapter's own algebraic form: whenever one number is the square of another, that other one is its square root.
The sign problem (Part I p.8): 8 × 8 = 64, and also −8 × −8 = 64, so 8² = 64 and (−8)² = 64 and the square roots of 64 are +8 and −8. Printed alongside: √64 = ±8, √100 = ±10, √(8²) = ±8, √(10²) = ±10, and in general √(n²) = ±n. The chapter then states it will keep only the positive root for the rest of the chapter — a convention it adopts, not a fact it proves.
The two candidates (Part I p.8): 576 and 327. The chapter dismisses 327 on its units digit alone and says 576 cannot be settled that way.
Test one, the list (Part I p.8): 20² = 400, 21² = 441, 22² = 484, 23² = 529, 24² = 576. The chapter's verdict is that this becomes inefficient for larger numbers.
Test two, on 81 (Part I p.8): 81 − 1 = 80, 80 − 3 = 77, 77 − 5 = 72, 72 − 7 = 65, 65 − 9 = 56, 56 − 11 = 45, 45 − 13 = 32, 32 − 15 = 17, 17 − 17 = 0. Zero arrives at the ninth step, so √81 = 9. The chapter notes that 729 by this route is possible but tedious.
Test three, on 324 (Part I p.9): 324 = 2 × 2 × 3 × 3 × 3 × 3. Regrouped as (2 × 3 × 3) × (2 × 3 × 3) = (2 × 3 × 3)² = 18². Also written as (2 × 2) × (3 × 3) × (3 × 3), which shows the pairs. Therefore √324 = 18.
When it fails (Part I p.9): 156 = 2 × 2 × 3 × 13; the factors cannot be paired up, so 156 is not a perfect square.
Left to the student (Part I p.9): decide 1156 and 2800 by prime factorisation. Inputs only.
Estimating √1936 (Part I p.9), the chapter's five printed steps: (i) 1936 lies between 1600 = 40² and 2500 = 50², so the root is between 40 and 50; (ii) 1936 ends in 6, so the root's units digit is 4 or 6, making it 44 or 46; (iii) 45² is computed as (40 + 5)(40 + 5) = 40² + 2 × 40 × 5 + 5² = 1600 + 400 + 25 = 2025; (iv) 2025 exceeds 1936, so the root lies between 40 and 45; (v) the root is then guessed and verified as 44.
The guessing game (Part I pp.9–10). Aribam says 25 and Bijou answers 5; Bijou says 81 and Aribam answers 9; Aribam then says 250 and the game stalls. Bracketing: 100 < 250 < 400 with √100 = 10 and √400 = 20, so the root is between 10 and 20; then 15² = 225 and 16² = 256, so the root is between 15 and 16, and since 256 is the nearer of the two the root is close to 16 but below it.
Akhil's cloth (Part I p.10). A square piece of area 125 cm². He would like a 15 cm square cut from it for a handkerchief. But 125 is not a perfect square, and the closest ones either side of it are 11² = 121 and 12² = 144, so 11 cm is the biggest whole-number side he can cut. Printed in full.
§1.1 Figure it Out (Part I pp.10–11), the items that belong to this topic: item 4, the side of a square of area 441 m²; item 5, the smallest square number divisible by each of 4, 9 and 10; item 6, the smallest multiplier that turns 9408 into a perfect square, and the root of the product; item 9, the count of tiny squares in the printed picture together with its prime factorisation.
The tiny-squares picture (Part I p.11, item 9). A green square panel holding a 9-by-9 arrangement of white motifs, alternating in a checkerboard between upright squares and squares stood on a corner. Every motif — upright or tilted — is a 5-by-5 block of 25 tiny squares. Counted on two adjacent motifs, one of each orientation.
Two printing slips to steer around. On Part I p.8 the running text prints "202 = 400" where the display line immediately beneath it correctly sets 20² = 400 — the superscript is missing in the inline copy, verified on the printed page. On Part I p.9 the last step of the √1936 working refers back to "point b", although the steps on that page are labelled (i) to (v); it means step (ii), the units-digit observation. Neither should be shown as printed.

Figures to have open

A two-pile dealing diagram for 324: the six primes dropping alternately into two trays, the trays ending identical. An added figure and the load carrier for sections 7 to 9. Repeat it for 156 so the failure is visible as an odd tray.
A number line carrying 1600, 1936, 2025 and 2500 with the bracket closing in three stages (Part I p.9 is the source of the numbers; the line itself is added here).
The 9-by-9 tiny-squares panel with its alternating upright and tilted motifs (Part I p.11). Redraw as a schematic — the counting only works if the student can see one motif's 5-by-5 grid clearly, so zoom one motif out of the panel.
A square of cloth 125 cm² in area with an 11 cm and a 15 cm square laid over it. An added figure.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 1, §1.1 "Square Numbers", the subheading "Square Roots", Part I pp.7–10, running from the foot of Part I p.7 to the "Figure it Out" heading on Part I p.10.
§1.1 "Figure it Out", Part I pp.10–11, items 4, 5, 6 and 9.
Part I p.17 SUMMARY, third and fifth bullets, restate the root definition and the two-identical-groups test.
The cube counterpart of this topic, using three groups instead of two, is What makes a number a perfect cube, and the three-identical-groups test; the cube root itself is Cube roots, and what successive differences expose.