Sequences that turn out to be the same sequence in disguise

What to assume they know

• Why adding the odd numbers gives the squares — running totals of the odd numbers are the squares, and the reason is a cut of a dot square
• Drawing a sequence makes its rule visible — the drawings of Table 1's first seven rows, and the hexagonal dot arrangement
• Every number sequence is a rule, not a list — Table 1's ten rows and their rules
• Multiplying a small whole number by 6, and adding 1

What they should be able to do

• Form the running totals of a given sequence and identify the sequence they produce
• State the six relations §1.4 asks about and say which row of Table 1, or of the §1.3 Figure it Out, each one lands on
• Explain at least three of those relations with a drawing rather than a calculation
• Fit two consecutive triangular arrays together and read off a square
• Build a hexagonal arrangement out of six triangular arrays and a centre dot, and say which value that construction misses
• Read the cube-and-hexagon figure on p.9 and use it to explain why running totals of the hexagonal numbers are the cubes
• Draw the map of Table 1 — plus the hexagonal row §1.3 adds — as a set of arrows, and say what the map is evidence for

Where it usually goes wrong

• "These are a handful of unrelated tricks to memorise." Four of the six are one move — take running totals — applied to different rows, and the two that are not (pairing consecutive triangular numbers; six times a triangular number plus one) are still moves you can draw. Teaching them as a list throws away the only thing that makes them memorable.
• "Adding up the powers of 2 gives the powers of 2." It gives one less than a power of 2 every time. The +1 in the question is not decoration; it is the whole content.
• "Six times a triangular number plus one gives the hexagonal numbers." It gives them from the second one onward. The first hexagonal value, 1, is the lone centre dot with no triangles around it, and a student who lines the two sequences up carelessly will be off by one position.
• "The cube picture on p.9 shows the answer." It shows the decomposition. The reader still has to see that the visible layer of a cube is a hexagonal number, which is exactly what the matching colours are there to suggest.
• "An answer is the sequence you land on." §1.4 asks for the sequence and the reason, usually as a picture. Naming the landing sequence is half a mark.
• "Two triangular numbers always add to a square." Only two consecutive ones. 1 + 6 is 7, which is not a square.
• "A relation between two rows is a coincidence in the numbers." Each one here is a statement about how the drawings fit together. Once you see the fit, the numerical coincidence stops being one.

Questions to check understanding

• Add a given sequence up as you go and name the sequence you land on
• Add two consecutive triangular numbers and explain the result with a picture
• Add the powers of 2 as you go, then add 1 to each total. Explain what happens
• Take each triangular number, multiply by 6, add 1. Which row of Table 1 or of §1.3 appears, and starting from which position?
• Add the hexagonal numbers as you go and explain the result using a cube
• Find and explain a relation between two rows of Table 1 that the chapter does not ask about
• The solutions appendix bound with this chapter gives the landing sequences — squares for the paired triangular numbers, cubes for the hexagonal totals, 7, 19, 37, 61, 91 for six-times-triangular-plus-one — and supplies no picture of its own for any of them. It sends the reader back to three places already in the chapter: Table 2 on p.4 (for the counting-number totals, the paired triangular numbers and the hexagonal totals), the doubling drawing on p.6 (for the powers of 2), and the hexagonal arrangement on p.5 (for six-times-triangular-plus-one). Its answer for the All 1's question is left as unevaluated expressions. Checked against the appendix pages footered [3] and [4].

Examples worth working on the board

Everything below is asked in §1.4 on pp.8–9; the chapter prints no answers to any of it, so the values given are the inputs, not the book's conclusions.

• All 1's, totalled up. 1; 1 + 1; 1 + 1 + 1; … The book also asks what happens when the All 1's row is added up and then back down.
• Counting numbers, totalled up. 1; 1 + 2; 1 + 2 + 3; 1 + 2 + 3 + 4; … The book asks for a small picture with it. The triangular arrays of Table 2 are that picture.
• Consecutive triangular numbers in pairs. 1 + 3; 3 + 6; 6 + 10; 10 + 15. The picture: two consecutive triangular arrays, one turned round, fit together into a square array with no dot left over.
• Powers of 2, totalled up. 1; 1 + 2; 1 + 2 + 4; 1 + 2 + 4 + 8; … Then add 1 to each of the totals. The picture the book points at is its own doubling drawing on p.6.
• Triangular numbers, multiplied by 6 with 1 added. 6 × 1 + 1; 6 × 3 + 1; 6 × 6 + 1; 6 × 10 + 1; 6 × 15 + 1. The picture: one dot in the middle with six triangular arrays packed around it, which is exactly how the p.5 hexagonal arrangement is built. Note the offset carefully — this construction starts from the second hexagonal value, because Table 1's triangular numbers begin at 1, while the first hexagonal value is a lone centre dot with no triangular arrays around it at all.
• Hexagonal numbers, totalled up. 1; 1 + 7; 1 + 7 + 19; 1 + 7 + 19 + 37. The book asks for this to be explained with a picture of a cube, and prints one on the same page.
• The p.9 figure, described from the printed page. Two drawings side by side. On the left, a cube built of unit cubes, four along each edge. The unit cube at the corner nearest the reader is yellow; the unit cubes running along the three edges that meet at that corner are red, three in each direction; every other visible unit cube is blue. On the right, a hexagonal arrangement of dots with one yellow dot at the centre and three red arms of three dots each, radiating from it at equal angles, the rest blue. The two colourings match: 1 yellow, 9 red, 27 blue in both. Checked against p.9 — this figure extracts as nothing at all.
• What that figure is doing. Looking at the cube from the yellow corner, the unit cubes you can see are one corner, three edges of three, and three faces of three-by-three: 1 + 9 + 27. Peel that visible layer off and a smaller cube is left, three along each edge, whose own corner layer is 1 + 6 + 12. Keep peeling.
• The book's closing invitation (p.9). Find your own relation among the rows of Table 1, and explain why it happens. It is the only question in the chapter that asks the student to supply the relation itself rather than to check one the book has already named — the two prompts in §1.1 on p.2 are open too, but they are about mathematics in general and not about Table 1 — and it is the one that tests whether the student has taken the chapter's point.

Figures to have open

• The p.9 cube-and-hexagon pair, redrawn: a cube four unit cubes along each edge with corner, edges and faces coloured distinctly, beside the matching hexagonal dot arrangement with the same three colours. This figure carries the section-10 argument and it is drawn art in the book, so it must be rebuilt, not lifted.
• Two consecutive triangular dot arrays rotating together into a square array. Standard schematic; the book does not print it.
• Six triangular arrays plus a centre dot assembling into the hexagonal arrangement. Standard schematic; the book does not print it either — it prints the finished hexagonal arrangement on p.5 and the cube on p.9.
• An arrow map of Table 1's rows plus the hexagonal row §1.3 adds, which Q7 lands on and Q8 starts from. Entirely an added construction.

Where this sits in the book

• NCERT Ganita Prakash, Class 6, Chapter 1 "Patterns in Mathematics", §1.4 "Relations among Number Sequences", pp.7–9 — the "Adding up and down" passage on p.7, the Figure it Out on pp.8–9, and the cube-and-hexagon figure on p.9
• Back references inside the same chapter: Table 1 (§1.2, p.3), Table 2 (§1.3, p.4), the hexagonal arrangement and the doubling figures (§1.3, pp.5–6)
• Solutions appendix bound with this chapter file, pages footered [3] and [4], for the §1.4 Figure it Out