Squares hiding inside triangular numbers

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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.

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What to assume they know

What makes a number a perfect square — square numbers and the notation n²
Triangular numbers from Class 6: the dot staircases 1, 3, 6, 10, 15 and how each is built from the one before
Adding a short run of consecutive whole numbers
Reading a dot diagram and counting rows rather than dots

What they should be able to do

Recognise and extend the triangular numbers as dot staircases
State how each triangular number is obtained from its predecessor
Add two consecutive triangular numbers and identify the total as a square
Extend the chapter's printed pattern by drawing the next term in the empty box
Explain, by pairing rows, why two consecutive triangular numbers always fill a square
Say why two copies of the same triangular number do not give a square
Split a given square into two consecutive triangular numbers, running the relation in reverse

Where it usually goes wrong

"Triangular numbers are another list to memorise." They are a construction: each one is the last plus a row. If a student can rebuild them they never need the list.
"Two instances worked, so the rule holds." The chapter shows three and stops at a blank box. Three instances are a reason to look for an argument, not a substitute for one.
"Two triangles make a square, so two of the same size will do." They make a rectangle. The relation needs consecutive triangular numbers, and showing the failed case is the fastest way to make that land.
"You need the formula for triangular numbers to prove it." You do not, and the chapter never prints one. The pairing of rows settles it with no algebra.
"A square splits into two equal halves, so the two triangles are equal." They differ by exactly one row. That is precisely what makes them fit.
"The stepped line in the picture is a diagonal." It is a staircase. A straight diagonal would cut dots in half and the count would not work.

Questions to check understanding

Extend the triangular numbers and draw the next dot figure
Add a stated pair of consecutive triangular numbers and identify the square
Given a square, name the two consecutive triangular numbers it splits into
Complete a pattern of the chapter's printed form and state the rule in words
"Explain why the sum of two consecutive triangular numbers is a square" — the reasoning form, answerable with the pairing picture and no algebra
Decide whether a given number is triangular, and justify the decision

Examples worth working on the board

The five printed triangular numbers (Part I p.7). Five dot triangles in a row, each labelled beneath: 1, 3, 6, 10, 15. Counted on the printed page, the figures have one, two, three, four and five rows respectively, each row one dot longer than the one above it, and the dots are set as solid discs. The chapter introduces them with a question that assumes they are already familiar.
The three printed instances (Part I p.7), each a square array of dots split by a stepped line into two staircases, with a black square outline drawn round it and the arithmetic beneath:
- 1 + 3 = 4 = 2² — a 2-by-2 array
- 3 + 6 = 9 = 3² — a 3-by-3 array
- 6 + 10 = 16 = 4² — a 4-by-4 array Verified on the printed page. In each picture the smaller staircase is the one sitting in the corner, and the stepped line between the two is what makes the split visible.
The fourth panel is empty (Part I p.7). To the right of the three instances the page prints an empty square outline with an empty caption box beneath it. The instruction above asks the student to extend the pattern and draw the next term. The two triangular numbers the student needs are the next pair in the printed list — 10 and 15.
The row-pairing argument, to build. This is an added reconstruction of what the chapter's three pictures show; the chapter states no general argument. Lay the larger staircase with rows of 1, 2, …, n dots. Lay the smaller one, of rows 1, 2, …, n − 1, upside down beside it so its longest row meets the shortest row of the first. Row 1 of the first now sits with row n − 1 of the second, giving n dots; row 2 sits with row n − 2, giving n again; and so on. The last row of the larger staircase, of n dots, has no partner and needs none. Every completed row holds n dots and there are n rows, so the total is n × n.
Checks against the printed instances, arithmetic added here on the chapter's data. For the 4-by-4 panel: the staircases are 10 (rows 1, 2, 3, 4) and 6 (rows 3, 2, 1 when turned round); the rows pair as 1 + 3, 2 + 2, 3 + 1 and the 4-row stands alone — four rows of four dots, 16 in all, matching the printed 6 + 10 = 16 = 4².
Why two equal triangles fail. Two copies of the same staircase interlock into a rectangle n dots by n + 1 dots, not a square. Worth showing beside the correct construction, because it is the mistake the picture invites.
What the chapter does not give you. It prints no formula for the nth triangular number, and no general argument for the relation — checked against every printed page of Part I pp.1–18. Both instances of "why" in this topic are added here to supply.

Figures to have open

The five dot staircases 1, 3, 6, 10, 15 (Part I p.7). Standard schematic.
The three square panels with the stepped dividing line, and the empty fourth panel with its empty caption box (Part I p.7). Redraw as a schematic; keep the empty panel empty until section 5.
The rotate-and-interlock movement: two consecutive staircases in different colours sliding together into a square. This is an added figure and it carries sections 6 to 8.
The failed case — two equal staircases forming an n by n + 1 rectangle. 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 "Perfect Squares and Triangular Numbers", Part I p.7. The subheading occupies the lower half of that page only, between the hundreds-block table and the "Square Roots" subheading.
The chapter treats triangular numbers as recalled rather than introduced; the parallel Class 6 callback is stated explicitly for the odd-number picture one page earlier (Part I p.6).
Related topic in the same chapter: Why the first n odd numbers add up to n², whose L-shaped border is the other way the chapter builds a square out of smaller pieces.