Shapes come in sequences too, with rules of their own

What to assume they know

• Every number sequence is a rule, not a list — a sequence is its rule, not the values shown
• Drawing a sequence makes its rule visible — a drawing can carry a definition
• Recognising and naming a triangle, a square and a circle
• Drawing straight line segments with a ruler
• No formal work on angles is needed; §1.6 defers angles to the chapter that follows this one

What they should be able to do

• State what a shape sequence is and how it differs from a number sequence
• Name geometry as the branch of the subject these patterns belong to
• List the five shape sequences printed in Table 3 and describe how each is built
• Continue four of the five sequences by one more shape, drawn accurately
• Say why the fifth sequence resists the same request, and what a fair answer to "why or why not" looks like
• State the construction rule for the Koch snowflake in the student's own words
• Recognise that the book allows shapes to live in more than three dimensions

Where it usually goes wrong

• "A shape sequence is just a decorated number sequence." It is a sequence of objects with a construction rule. The numbers appear only in §1.6, when you choose something to count.
• "Regular means it looks neat." §1.6 gives it a meaning: the sides are all one length and the corners are all alike. P.11 hands angles on to the chapter that follows, without naming it.
• "A quadrilateral is a square." In the regular row it is — a regular quadrilateral is a square — but the general word covers any four-sided closed figure, and the book is careful to say regular.
• "K5 has five lines." K5 has a line for every pair of its five dots, and the drawing shows a pentagon with all its diagonals. Counting them is the business of the next topic, not this one; here the point is only the rule "join every pair."
• "Stacked triangles are made of upward triangles only." They are not. Table 3 draws five of them, of one to five rows, so a student who counts only the upward ones gets 1, 3, 6, 10, 15 across the row instead of the 1, 4, 9, 16, 25 that §1.6 is after.
• "The Koch snowflake cannot be continued." It can — the rule never stops. What stops is your pencil. Saying "you cannot draw it" without saying "the rule still works" gets the lesson exactly backwards.
• "Shapes only exist in 1D, 2D and 3D." §1.5 explicitly leaves the door open past three, which is what the fifth powers-of-2 figure on p.6 was already hinting at.

Questions to check understanding

• Describe in your own words how each sequence in Table 3 is built
• Given one of the five sequences, draw what comes after the last shape shown
• For which of the five sequences is drawing the next shape a problem, and why?
• What does "regular" mean for a polygon in this book?
• Explain the rule that turns one Koch snowflake shape into the next
• Which shape sequence would you use to explain the word "iteration"?
• The solutions appendix bound with this chapter answers only the first of the two §1.5 questions, and answers it with number sequences rather than with descriptions of the constructions — so the "draw the next shape, why or why not" question, which is the one this topic turns on, has no printed answer to check a student against. Checked against the appendix page footered [4].

Examples worth working on the board

Table 3 is drawn art and extracts as nothing but its labels, so everything below comes from rendering p.10.

• Table 3, row 1 — Regular Polygons. Eight solid coloured shapes in two rows of four, each captioned. Reading across the top: triangle, then quadrilateral, then pentagon and hexagon. Across the bottom: heptagon, then octagon, and then nonagon and decagon. Each is drawn filled, in a different colour, and each is drawn regular — all sides one length.
• Table 3, row 2 — Complete Graphs. Five figures captioned K2, K3, K4, K5, K6. Each is a set of small dots with a straight line drawn between every pair. K2 is two dots and one line; K3 is a triangle; K4 is a square with both diagonals; K5 and K6 are five and six dots on a circle with all connections drawn.
• Table 3, row 3 — Stacked Squares. Five square blocks, ruled into little squares: one cell, then two cells along each side, then three, four and five.
• Table 3, row 4 — Stacked Triangles. Five triangles, each ruled into little triangles, with one, two, three, four and five rows. Note: the rows contain both upward-pointing and downward-pointing little triangles, and the downward ones are easy to miss. §1.6 depends on them.
• Table 3, row 5 — Koch Snowflake. Five outlines: a plain triangle, then a six-pointed star, then three progressively more crinkled closed curves. Every straight piece of the previous outline has been replaced.
• The construction rule, as the book gives it (p.12). Each straight piece is replaced by a bump — the book prints a small drawing of the bump and calls it a speed bump. Repeating this makes the changes smaller and smaller.
• The book's two questions (p.11). Recognise the pattern in each row of Table 3; then redraw each row, draw the next shape, say why or why not, and describe the forming rule in your own words. The block carries the Math Talk badge.
• Working the "why or why not". For the first four rows the next shape is drawable: an eleven-sided regular polygon, K7, a six-by-six block, a six-row triangle. For the Koch row the next shape exists and is perfectly well defined, but each round leaves four times as many straight pieces and every one of them is shorter than the piece it replaced, so at some point the drawing stops being possible on paper even though the rule has not stopped. That distinction — a rule that keeps working past the point where the pencil can follow it — is the payoff of section 10.

Figures to have open

• Table 3 as printed on p.10, rebuilt: five labelled rows with the shapes the book draws. This is the central figure of the topic and it is entirely artwork, so it must be redrawn from the printed page rather than lifted.
• A close-up of one stacked triangle with upward and downward little triangles distinguished. Not in the book; the book prints the triangles undifferentiated.
• The Koch replacement shown on a single straight piece, enlarged. The book prints this bump inline on p.12 at text size; the explanation needs it big.
• An eleven-sided regular polygon, K7, a six-by-six stacked square and a six-row stacked triangle — the "next shapes" for section 9. Standard schematics; none of them is printed.

Where this sits in the book

• NCERT Ganita Prakash, Class 6, Chapter 1 "Patterns in Mathematics", §1.5 "Patterns in Shapes", pp.9–11 — the opening on p.9, Table 3 on p.10, and the Figure it Out on p.11
• §1.6, p.11, for the meaning of "regular", and p.12 for the Koch replacement rule and the Summary's list of shape sequences
• Forward pointer: §1.6, p.11, hands angles on to the chapter that follows, which it does not name. That chapter is Chapter 2, "Lines and Angles" — a locator supplied by this brief, not by p.11
• Solutions appendix bound with this chapter file, page footered [4], for §1.5