What a perfect square's last digits can and cannot be

These teaching notes are for members

What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

Create a free account Sign in

What to assume they know

What makes a number a perfect square — what a square number and a perfect square are, and the notation n²
Multiplying two two-digit and three-digit numbers by hand
Knowing that a number ending in 0 is a multiple of 10
Even and odd numbers, and that a product of two odd numbers is odd
Reading a place-value column, and naming the units place

What they should be able to do

Compute the squares of the first thirty natural numbers and tabulate them
State which digits can stand in the units place of a perfect square and which cannot
Explain why the units digit of n² is fixed by the units digit of n
Use the units digit to reject a candidate square, and explain why the same digit can never confirm one
Name the digits whose squares end in 1, and those whose squares end in 6
Predict how many zeros end the square of a number that ends in a given number of zeros, and justify the doubling
State how a number's parity governs the parity of its square
Given four candidate numbers, sort out which the digit test settles and which it leaves open

Where it usually goes wrong

"It ends in 6, so it is a square." 26 is the chapter's own refutation, and it prints it immediately after the pattern so the student cannot draw the wrong conclusion first.
"It ends in 2, but it might still be a square if the number is big enough." Never. The units digit of the square is determined, not merely likely.
"Filling in thirty squares is busywork." The table is the data the whole section reasons from. Show it being filled and then reasoned over.
"Only 6 squares to something ending in 6." Both 4 and 6 do. Two inputs sharing one output is precisely why the test cannot be reversed — build the section on that, not on the list.
"A square can end in a single zero." If 10 divides n then 100 divides n², so the zeros arrive two at a time.
"A square's zeros double, so 3 zeros give 5." They give 6. The count of trailing zeros doubles, it does not increase by a fixed amount.
"Squares are always odd" or "always even." Squares inherit the parity of the number squared; both happen, in strict alternation down the table.
"The digit rule is a fact about squares." It is a fact about how multiplication treats last digits. Squares are just the case where both factors are equal.

Questions to check understanding

"Which of these is not a perfect square?" answered from the units digit alone, with a stated reason
Write several numbers that can be rejected as squares on sight
"Which of these squares ends in 4 (or 6, or 1)?" from the base numbers' endings
Given the count of zeros ending a number, state the count ending its square
Complete rows of a squares table and state two conjectures from it
Explain why the units-digit test can reject but not confirm — the reasoning question the board asks in place of a computation

Examples worth working on the board

The squares table (Part I p.4). Three columns, and the pre-filled cells are not symmetric — read them off the printed page, not from the extraction. Column 1 holds slots 1² through 10², with 1² = 1, 2² = 4, 3² = 9, 4² = 16 and 5² = 25 printed and 6² through 10² left blank. Column 2 holds slots 11² through 20², with only 11² = 121 printed. Column 3 begins 21² = 441 printed, then 22² = blank, and its remaining cells carry no labels at all. The instruction above the table asks for the squares of the first thirty natural numbers, so 23² onwards have no printed slot and the student supplies both label and value.
The endings the chapter reads off that table (Part I p.4): every entry ends in 0, 1, 4, 5, 6 or 9; none ends in 2, 3, 7 or 8.
The counterexample triple (Part I p.4): 16 and 36 are squares ending in 6; 26 ends in 6 and is not a square.
Squares ending in 1 (Part I p.4): 1², 9², 11², 19², 21², 29². The student is asked for the next two in that family. Inputs only — do not print them.
Squares ending in 6 (Part I p.4): 16 = 4², 36 = 6², 196 = 14², 256 = 16², 576 = 24², 676 = 26².
The selection question (Part I p.5): which of 38², 34², 46², 56², 74², 82² end in 6. The text layer drops the superscripts and shows these as 382, 342 and so on; the printed page sets every one as a square. Verified on the printed page.
The trailing-zeros figure (Part I p.5). Two clusters of computed squares, each flanked by a speech bubble on the left and another on the right. First cluster: 10² = 100, 20² = 400, 40² = 1600, with "one zero" on the left and "two zeroes" on the right. Second cluster: 100² = 10000, 200² = 40000, 700² = 490000, 900² = 810000, with "two zeroes" left and "four zeroes" right. The chapter then asks what happens for a number ending in three zeros, whether the doubling always holds, and whether a square can only ever end in an even count of zeros.
The parity prompt (Part I p.5). One line, no worked instance: the student is asked how a number's parity relates to that of its square.
Figure it Out item 1 (Part I p.10): 2032, 2048, 1027, 1089, asked which are not perfect squares. Their endings are 2, 8, 7 and 9. Three of the four end in a digit no square can end in, so the digit test finishes them; the fourth ends in a permitted digit and has to be settled some other way — which is exactly the one-way nature of the test being examined. Do not state which is which before the student has worked it.
Figure it Out item 2 (Part I p.10), and a printing slip in it: the item asks which one among 64², 108², 292², 36² has 4 as its last digit, but two of them do. The bases end in 4, 8, 2 and 6, so the squares end in 6, 4, 4 and 6 — 108² and 292² both qualify. Pose it as "which of these" and land both answers; that is not a workaround but the better lesson, since it shows two different digits (8 and 2) driving one ending, which is section 8's whole point. The units-digit reasoning still does all the work.
The SUMMARY lines (Part I p.17) restate both rules: the six permitted endings, and that a square's trailing zeros come in an even count.

Figures to have open

The chapter's three-column squares table with its uneven pre-filled cells (Part I p.4). Reproduce which cells were given and which were blank — the asymmetry is pedagogically deliberate and a redrawn "complete" table loses it.
A ten-row digit map: 0 to 9 down one side, the last digit of each square down the other, with the four impossible endings greyed out. An added figure, and it carries the argument.
The chapter's trailing-zeros figure with its four speech bubbles (Part I p.5). Standard schematic; keep the left/right bubble contrast, which is what makes the doubling visible.
No photograph is needed.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 1, §1.1 "Square Numbers", the subheading "Patterns and Properties of Perfect Squares", Part I pp.4–5, running as far as the next subheading.
Exercise items drawn on here sit in the §1.1 "Figure it Out" set at Part I p.10, items 1 and 2.
Part I p.17 SUMMARY, second bullet.
The chapter marks two prompts on Part I p.4 with the Math Talk badge; both are discussion questions rather than computations, and both are load-bearing here.