Drawing a sequence makes its rule visible

What to assume they know

• Every number sequence is a rule, not a list — a sequence is the rule, and Table 1's ten rows are ten rules
• Mathematics is the search for patterns *and* for why they hold — a pattern needs its reason
• Counting dots in a rectangular array without recounting
• Recognising a triangle, a square and a cube by sight

What they should be able to do

• Match each of the first seven rows of Table 1 to its picture in Table 2
• Draw the next picture in any of those seven, and read the next value off it
• Explain from the drawing why the triangular numbers, the squares and the cubes carry those names
• Show that 36 can be built both as a triangle of dots and as a square of dots, and say what that means about the number
• Read the hexagonal numbers off a picture and continue the sequence, given that the book names them but does not list them in Table 1
• State which rows of Table 1 are left undrawn in Table 2, and where one of them is picked up later
• Describe the doubling picture the book offers for the powers of 2, and say why it runs out of ordinary space at the fifth figure

Where it usually goes wrong

• "The picture is there to make the page look nice." In Table 2 the picture is what the name means. Take the picture away and "triangular number" is an unexplained label.
• "Odd numbers are drawn as one row with a gap." In Table 2 the book draws them as two rows differing by one, and that is the arrangement to copy here. It is not, though, the picture §1.4 later argues from: there each odd number is drawn afresh as a bent band wrapped round the corner of a square. Same counts, different arrangement.
• "36 being triangular and square is a fluke worth one line." It is the book's illustration that a number is not one thing — and drawing both arrangements is a real exercise, not a remark.
• "Hexagonal numbers are in Table 1." They are not. They arrive only as a picture on p.5, which is the point: the picture came first and the name after.
• "You cannot draw the powers of 2, so §1.3 skips them." §1.3 asks for them in its Figure it Out and then supplies one answer on the very next page.
• "The fifth powers-of-2 figure is just two cubes." Drawn as two cubes, yes — but the joining lines make it a single object with sixteen corners, and it is the same doubling step applied once more.
• "Cubes are square numbers in disguise." Table 2 draws them as solids: 1, 8, 27, 64, 125 count unit cubes in a stack, not dots on a flat page.

Questions to check understanding

• Draw the next picture in a given row of Table 2 and state the value
• Why is 1, 3, 6, 10, 15 called the triangular numbers? Answer with a drawing
• Show 36 as a triangle of dots and as a square of dots
• What is the next hexagonal number after 37, and how does the picture tell you?
• Which rows of Table 1 have no picture in Table 2?
• Draw a way of picturing the powers of 3
• The solutions appendix for §1.3 answers the "why these names" question by sending the student back to Table 2 rather than restating it (page footered [1]); on the following appendix page it gives 61 for the hexagonal numbers and supplies drawings of its own for the powers of 3 — a dot, a triangle, three joined triangles and a 27-point figure — and for the powers of 2, carried one step further than the chapter does, to 32. Checked against the appendix pages footered [1] and [2].

Examples worth working on the board

All descriptions of Table 2 below come from rendering p.4, since the drawings themselves extract as nothing.

• Table 2 (p.4), seven rows, five drawings each, every drawing labelled with its value.
  • All 1's — one dot each time; the labels read 1, 1, 1, 1, 1
  • Counting numbers — a single row of dots, one longer each time: 1, 2, 3, 4, 5
  • Odd numbers — two rows of dots whose lengths differ by one, the shorter above the longer: 1; then 1 over 2; then 2 over 3; then 3 over 4; then 4 over 5 — labelled 1, 3, 5, 7, 9
  • Even numbers — two rows of equal length: 1 and 1; 2 and 2; 3 and 3; 4 and 4; 5 and 5 — labelled 2, 4, 6, 8, 10
  • Triangular numbers — dots in a triangular array: 1, 3, 6, 10, 15
  • Squares — dots in a square array: 1, 4, 9, 16, 25
  • Cubes — solid cubes drawn in three dimensions, built of unit cubes: 1, 8, 27, 64, 125
• What Table 2 does not draw. Table 1 has ten rows; Table 2 draws the first seven. The Virahānka numbers, the powers of 2 and the powers of 3 have no drawing in Table 2. Checked against p.4. One of the three is picked up on p.6.
• 36, twice (p.5). 36 is the eighth triangular number and it is 6 × 6. A triangle with rows of 1, 2, 3, 4, 5, 6, 7, 8, and a 6-by-6 square. Do not present this as a coincidence to be admired and dropped — the book's point is that a number carries whichever role its arrangement gives it.
• Hexagonal numbers (p.5). Four dot pictures are printed, labelled 1, 7, 19 and 37 — a centre dot with rings of dots around it. The book names them after the picture and then asks for the next value. It does not print the answer in the chapter; the solutions appendix bound with the same file gives 61 on the page footered [2].
• Powers of 2 as shapes (p.6, above §1.4). Five figures, labelled 1, 2, 4, 8 and 16: a single dot; two dots joined by a segment; a square with four corners; a cube with eight corners; and a fifth figure drawn as two cubes with matching corners joined, giving sixteen. Each figure is two copies of the one before it with matching corners joined — which is exactly "double". Checked against p.6.
• The book's framing of that last figure. §1.3 offers this as one way to think about the powers of 2, not the way. The fifth figure has no home in ordinary space; §1.5 will say outright that shapes can live in more than three dimensions.

Figures to have open

• Table 2 as printed on p.4, rebuilt: seven rows, five drawings each, values labelled. The square and triangular rows must match the book's arrangement, because §1.4 and its Figure it Out work from those two. Redraw; do not lift the image.
• The hexagonal dot arrangement from p.5 for 1, 7, 19, 37, with the rings distinguishable so the +6, +12, +18 structure is visible. This one must be drawn faithfully to the book's arrangement, since the name is derived from it.
• The powers-of-2 shape sequence from p.6: dot, segment, square, cube, and the doubled cube with joining lines. Redraw. The joining lines are load-bearing — without them the fifth figure reads as two separate cubes.
• 36 dots arranged twice, as a triangle and as a 6-by-6 square. Standard schematic.

Where this sits in the book

• NCERT Ganita Prakash, Class 6, Chapter 1 "Patterns in Mathematics", §1.3 "Visualising Number Sequences", pp.3–6 — the opening on p.3, Table 2 on p.4, the Figure it Out on p.5, and the powers-of-2 drawing at the top of p.6
• Back reference inside the same chapter: Table 1, §1.2, p.3
• Forward reference inside the same chapter: §1.5, p.9, for shapes in more than three dimensions
• Solutions appendix bound with this chapter file, pages footered [1] and [2], for §1.3