Every number sequence is a rule, not a list

What to assume they know

• Mathematics is the search for patterns *and* for why they hold — mathematics looks for patterns and for the reasons behind them
• Whole numbers and their order, from primary school
• Adding and multiplying small whole numbers without a calculator
• Knowing what "double it" and "add three" mean as instructions

What they should be able to do

• Distinguish a sequence from a finite list of its early values
• Read each row of Table 1 and state its rule in the student's own words
• Produce the next three values of any row of Table 1 and justify them from the rule rather than by eye
• Show that a shared opening does not determine a sequence, using two rows of Table 1 that begin the same way
• Classify Table 1's rules by how each one gets from a value to the next: a fixed step, a growing step, a multiplier, or a look-back
• Name the whole numbers as the objects §1.2 says these patterns are made of, and number theory as the study of them
• State where the book's Virahānka sequence begins, which is not where most other books start it

Where it usually goes wrong

• "If I can continue it, I know it." Refuted from inside Table 1: after 1, 2, 3 the honest answer is which row? Continuing is evidence for a rule, never a substitute for one.
• "Every sequence goes up by a fixed amount." Only three of Table 1's ten do: the counting numbers, the odd numbers and the even numbers. The All 1's row does not go up at all. Of the remaining six, three change their step every time — the triangular numbers, the squares and the cubes — and two multiply instead of adding. The sixth, the Virahānka row, does neither: its steps run 1, 1, 2, 3, 5, 8, so the step does not change every time, and nothing is being multiplied. It looks back instead.
• "The Virahānka numbers start 1, 1, 2, 3, 5." This book prints them starting 1, 2, 3, 5, 8, 13, 21. The rule is the same — add the previous two — but the starting values differ from the version students meet online, and the examinable row is the printed one.
• "Powers of 2 means 2, 4, 8, 16." The printed row begins at 1. So does the powers of 3 row. Anything multiplied by itself no times is 1, which is why.
• "Squares and cubes are just names." §1.3 will cash both names out as pictures two pages later; treat the names here as promises, not decoration.
• "Whole numbers and counting numbers are the same list." §1.2 starts the whole numbers at 0; Table 1's counting numbers start at 1.

Questions to check understanding

• Write the next three numbers of a given sequence and state its rule in words
• Given the rule in words, produce the first five values
• Two sequences begin with the same three numbers. Give an example from Table 1 and show where they separate
• Which sequences in Table 1 are built by adding, and which by multiplying?
• Is 0 a whole number? Is it a counting number in this book?
• Continue the Virahānka numbers as this book prints them
• The solutions appendix bound with this chapter answers only the first of the two §1.2 questions, and answers it by naming the rules for powers of 2, powers of 3 and the Virahānka numbers and pointing at Table 2 for the rest — so the "write the rule in your own words" half is left open on purpose. Checked against the appendix page footered [1].

Examples worth working on the board

• Table 1 (p.3), all ten rows exactly as printed. Row order matters, because §1.3 draws the first seven and leaves the last three undrawn. Nine rows print seven values; the cubes row prints six.

| printed row | as printed | rule to state in words | |---|---|---| | All 1's | 1, 1, 1, 1, 1, 1, 1, … | never change | | Counting numbers | 1, 2, 3, 4, 5, 6, 7, … | add 1 | | Odd numbers | 1, 3, 5, 7, 9, 11, 13, … | add 2, starting at 1 | | Even numbers | 2, 4, 6, 8, 10, 12, 14, … | add 2, starting at 2 | | Triangular numbers | 1, 3, 6, 10, 15, 21, 28, … | add 2, then 3, then 4, … | | Squares | 1, 4, 9, 16, 25, 36, 49, … | multiply a counting number by itself | | Cubes | 1, 8, 27, 64, 125, 216, … | multiply a counting number by itself twice over | | Virahānka numbers | 1, 2, 3, 5, 8, 13, 21, … | add the two previous values | | Powers of 2 | 1, 2, 4, 8, 16, 32, 64, … | double | | Powers of 3 | 1, 3, 9, 27, 81, 243, 729, … | triple |

The three continuation values are not supplied here, on purpose. Producing them is exactly the task p.3 sets the student.

• The collision inside the table. Counting numbers and Virahānka numbers both open 1, 2, 3. Their fourth values are 4 and 5. Three rows open 1, 2 — counting numbers, powers of 2 and the Virahānka row — and they separate at the third or fourth value. Three more open 1, 3 — odd numbers, triangular numbers and powers of 3 — and all three separate at the third value, which is 5, then 6, then 9. Nine of the ten rows open with 1 at all; only the even numbers do not. This is the whole argument of the topic and it needs no material from outside p.3.
• The whole numbers as §1.2 gives them (p.2): 0, 1, 2, 3, 4, … Note the 0. Table 1's counting numbers start at 1, so the two are deliberately different.
• The task the book sets (p.3). Copy each row into a notebook, extend it by three, then write the rule in your own words. It carries the Math Talk badge, so the intended output is spoken as well as written.
• Two people, one rule, different words. For the triangular numbers, "add one more each time than you added last time" and "add the next counting number" describe the same machine. Use this to establish that the rule, not its phrasing, is the answer.

Figures to have open

• Table 1 as printed on p.3 — ten rows, each with its printed name. Must match the book's rows and order.
• A "collision" graphic: one shared opening, branching into two labelled continuations. Standard schematic.
• No drawn figure from the textbook is required for this topic. Table 1 is set as type and extracts cleanly; the pictures arrive in §1.3 and belong to Drawing a sequence makes its rule visible.

Where this sits in the book

• NCERT Ganita Prakash, Class 6, Chapter 1 "Patterns in Mathematics", §1.2 "Patterns in Numbers", pp.2–3, including Table 1 on p.3 and the Figure it Out on p.3
• Forward pointer inside the same chapter: §1.3, pp.3–6, which draws the first seven rows of Table 1
• Solutions appendix bound with this chapter file, page footered [1], for §1.2