Tangrams: rearranging pieces without changing the area

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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

Recognising triangles, squares and four-sided figures by their sides and angles
Knowing that turning or flipping a shape leaves its size unchanged
Cutting out and handling paper or card shapes
No construction from §6.1 of this Part II chapter is needed; this topic opens its second half

What they should be able to do

Describe how the seven tangram pieces are obtained from one square
Identify the seven pieces by shape and by their relative sizes
State the two conditions a valid tangram arrangement must satisfy
Explain why every arrangement of the same seven pieces covers the same amount of surface
Rearrange the pieces to match a given outline
Explain why a tangram target is a question with a definite answer
Connect the tangram rules to the definition of tiling that follows

Where it usually goes wrong

"Tangram pieces can be resized to fit." They cannot. The whole point of a fixed set is that only position and orientation are free.
"You can leave a piece out if it does not fit." Leaving one out changes how much surface is covered, so it changes what outlines are reachable. Keep the set intact.
"A close-enough outline counts." The pieces have exact edges and exact angles, and either they close up or they do not. This is the yes-or-no habit the rest of Part II's §6.2 depends on.
"The two large triangles are interchangeable with the medium one." They are not: each large triangle is twice the medium one. Getting the size relations wrong is the commonest reason a solution attempt stalls.
"Flipping a piece over is cheating." Flipping is a normal tangram move, and it matters for the four-sided piece, which is the only one that changes appearance when turned over.
"This is just a game before the real mathematics." It is the training for the real mathematics: it is where covering exactly, with no slack and no doubling, becomes something a student can feel in their hands.

Questions to check understanding

Say how many pieces a tangram has and what single figure they come from
Given the seven pieces, reproduce a stated outline (the shape of the Figure it Out task, Part II, p.156)
Explain why two different tangram figures made from the full set cover the same amount of surface
Identify which two pieces are the largest, and say how the medium piece compares with them
State the two conditions that make an arrangement a tiling (Part II, p.157, and the SUMMARY bullet on p.163)
Make your own outline from the seven pieces and challenge a classmate to solve it

Examples worth working on the board

The tangram square (Part II, §6.2, p.155). Checked against the printed page. A solid black square with white cut lines and seven regions lettered A to G. Read off the printed page: the diagonal from the top-right corner down to the bottom-left corner is drawn in full, while the one from the top-left corner stops part-way, at the lower corner of the small square. A is the large triangle filling the left, B the large triangle filling the top; E and C are the two small triangles; D is the square standing on a corner; F is the medium triangle in the bottom right; G is the four-sided piece along the bottom left with two pairs of parallel sides. The letters are printed on the artwork; the shape names are not.
The relative sizes. A and B are each a quarter of the whole square; D, F and G are each an eighth; C and E are each a sixteenth. Those fractions are not printed in the chapter and are supplied here so the figure can size the pieces correctly — they should be treated as background data, not the explanation.
The arrow (Part II, §6.2, p.156). Checked against the printed page. The head is two large triangles meeting along a horizontal join; the shaft is a rectangle with the square piece at its left end and four further pieces divided out to the right of it — seven pieces in all. Use it as the worked example: one square in, one arrow out.
The ten outlines (Part II, §6.2, "Figure it Out", p.156). Checked against the printed page. Ten solid black silhouettes in two columns. Left column, top to bottom: a blocky letter C, a pinwheel of four blades, a cat-like animal with a pointed ear and a tail, a bird or duck with a raised head, and a running human figure. Right column: a letter M, a wide zigzag, a boat with a slanting sail, a fish with a forked tail, and a standing figure with a raised arm. Pick three or four rather than all ten.
The two rules (Part II, §6.2, p.157). Inputs: nothing left bare, nothing doubled. The chapter states them two lines into p.157, as the definition of tiling; the tangram has been obeying them since p.155.

Figures to have open

The seven-piece square, drawn to correct proportions, able to be shown moving so the pieces can separate and reassemble. This is the topic's key image and must be redrawn rather than reproduced.
A surface bar or counter that stays fixed while the outline changes, for section 6. Built fresh; the chapter has no such figure.
The arrow, with its seven pieces distinguishable.
Two or three of the p.156 silhouettes, redrawn, with a solution movement for at least one.
No photograph, table or dataset from the textbook is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 6 "Constructions and Tilings", §6.2 "Tiling", p.155 — the origin of tangrams, the seven-piece square lettered A–G, and the note that ready-made cut-outs are bound in at the back
Same part, same chapter, §6.2, p.156 — the arrow, and the "Figure it Out" set of ten outlines
Same part, same chapter, §6.2, p.157, the definition of tiling
Same part, same chapter, SUMMARY, p.163, the tiling bullet
Forward pointer: What it takes to cover a region with no gaps and no overlaps, which takes the same two rules and asks whether a region can be covered at all