Counting the parts of a shape sequence produces a number sequence

What to assume they know

• Shapes come in sequences too, with rules of their own — the five shape sequences of Table 3 and how each is constructed
• Why adding the odd numbers gives the squares — the running totals of the odd numbers are the squares, and the reason is a picture
• Every number sequence is a rule, not a list — Table 1's rows, especially the triangular numbers and the squares
• Careful counting of parts in a drawing without double-counting

What they should be able to do

• Choose a countable feature of a shape sequence and produce the resulting number sequence
• Count the sides of the regular polygons and identify the row of Table 1 the counts belong to
• Explain why counting corners instead of sides changes nothing
• Count the lines in each complete graph and justify the triangular numbers from the construction
• Count the little squares in a stacked square and give the reason
• Count the little triangles in a stacked triangle, including the inverted ones, and give the reason
• State that the two stacks yield the same number sequence and give the two different reasons
• Produce the Koch segment counts and describe them as repeated multiplication by 4

Where it usually goes wrong

• "Each shape sequence has a number sequence." It has as many as there are things worth counting. On the regular polygons alone you can count sides, corners, or diagonals, and only the first two agree.
• "Sides equal corners because both happen to be 3, 4, 5, 6." It is not a numerical coincidence. Walking round any closed figure, every side ends at a corner and every corner starts a side, so they pair off. That argument works for figures that are not in Table 3 at all.
• "K6 has 6 lines." It has 15. The count is over pairs of dots, not over dots, and the drawing on p.10 shows all fifteen.
• "A stacked triangle with 4 rows has 1 + 2 + 3 + 4 little triangles." That counts only the upright ones and gives 10. Including the inverted ones gives 1 + 3 + 5 + 7 = 16. This is the single most common error in §1.6 and the book's printed hint exists because of it.
• "The two stacks give the same sequence, so they are really the same problem." They are not. Two different constructions can meet at the same numbers; noticing that they do is interesting precisely because the reasons are unrelated.
• "Every sequence you find must be in Table 1." The Koch counts are not, and §1.6 says so on the page. Table 1 was a sample, never a catalogue.
• "The Koch count doubles." It is multiplied by 4 each round, because one straight piece becomes four.

Questions to check understanding

• Count a chosen feature of each shape in a given sequence and name the number sequence you get
• Why do the sides and the corners of a polygon give the same count?
• How many lines does K7 have? Explain from K6
• How many little triangles are in a stacked triangle with 6 rows? Justify it
• Give two different reasons why two different shape sequences both produce 1, 4, 9, 16, 25
• Continue the Koch straight-piece counts by two more values and say what rule you used
• The solutions appendix bound with this chapter answers all five §1.6 questions: the counting numbers from 3 for both sides and corners with the closed-figure reason, the triangular numbers for the complete graphs, 1, 4, 9, 16, 25 for both stacks, and 3, 12, 48, 192, 768 for the Koch snowflake. For the stacked triangles it reaches the squares by the up-and-down route rather than by the odd-numbers route the printed hint points at — both are correct. Checked against the appendix page footered [5].

Examples worth working on the board

• The regular polygons, name against count (Table 3, p.10; §1.6, p.11).

| printed name | how many sides | how many corners | |---|---|---| | triangle | 3 sides | 3 corners | | quadrilateral | 4 sides | 4 corners | | pentagon | 5 sides | 5 corners | | hexagon | 6 sides | 6 corners | | heptagon | 7 sides | 7 corners | | octagon | 8 sides | 8 corners | | nonagon | 9 sides | 9 corners | | decagon | 10 sides | 10 corners |

§1.6 states that this side count is the counting numbers begun at 3, and says that this is the reason behind the names. The book uses "square" as the plain word for the regular quadrilateral.

• What "regular" means (p.11). All sides one length, and all corners alike. §1.6 states plainly that angles — the corner half of that — are taken further in the chapter that follows, which it does not name. That chapter is Chapter 2, "Lines and Angles", a locator this brief supplies. Either way the corner half is a promissory note here.
• Complete graphs (Table 3, p.10). Count the lines: K2 has 1, K3 has 3, K4 has 6, K5 has 10, K6 has 15.
• The reason for those counts. Go from K5 to K6 by putting one more dot down and joining it to each dot already there — five new lines. Going from K4 to K5 added four. From K3 to K4, three. The line counts therefore grow by 2, then 3, then 4, then 5, which is exactly how the triangular numbers grow.
• Stacked squares (Table 3, p.10). Count the little squares: 1, 4, 9, 16, 25. The reason: the nth block is n rows of n little squares.
• Stacked triangles (Table 3, p.10). Count the little triangles: 1, 4, 9, 16, 25. The reason is not rows of equal length. Take the four-row stack: the top row holds 1 little triangle, the second holds 3, the third holds 5, the fourth holds 7 — because each row after the first contains inverted triangles as well as upright ones. Adding 1, 3, 5, 7 gives 16, and §1.4 has already shown why running totals of the odd numbers are the squares. The book's own hint on p.12 points the student at the rows.
• Koch snowflake (Table 3, p.10; §1.6, p.12). Count the straight pieces in each shape: the book prints 3, 12, 48 in the question itself and says these are 3 multiplied by successive powers of 4, and that this row does not appear in Table 1. The solutions appendix bound with the same file continues it to 192 and 768 (page footered [5]). The reason: each round replaces every straight piece with four.
• The two-reason collision. Stacked squares and stacked triangles both give 1, 4, 9, 16, 25. One reason is a rectangle argument, the other is an odd-numbers argument. Same landing place, genuinely different routes — this is the strongest single example in the chapter of why the count is not the explanation.

Figures to have open

• Table 3 recalled in reduced form, one row at a time, matching Shapes come in sequences too, with rules of their own. Redrawn on the printed page.
• A four-row stacked triangle with upright and inverted little triangles in different shades and per-row counts. Not in the book; the book prints the triangles undifferentiated and this is where students go wrong.
• K5 gaining a sixth dot, with the five new lines appearing one at a time. Standard schematic.
• A closed figure with sides and corners paired off around it, for the section-6 argument. Standard schematic; the book states the fact only in the solutions appendix.
• The first three Koch shapes with straight pieces individually visible enough to count 3 and 12. The third shape's 48 pieces should be counted by rule rather than by eye.

Where this sits in the book

• NCERT Ganita Prakash, Class 6, Chapter 1 "Patterns in Mathematics", §1.6 "Relation to Number Sequences", pp.11–12 — the worked example and the meaning of "regular" on p.11, the Figure it Out spanning pp.11–12, and the Summary on p.12
• Table 3, §1.5, p.10, for the shapes being counted
• Table 1, §1.2, p.3, and the §1.4 odd-numbers argument, pp.6–7, for the rows the counts land on
• Forward pointer: p.11 sends angles on to the chapter that follows without naming it; that chapter is Chapter 2, "Lines and Angles"
• Solutions appendix bound with this chapter file, page footered [5], for §1.6