The Mayan system, and a placeholder symbol for zero

What to assume they know

• Mesopotamian base-60: place value in a sexagesimal system — position instead of landmark signs, and why an empty place must be visible
• What "base n" means, and why ten is a choice not a law — the two-clause definition of base-n, and the closure property that a base gives
• Multiplication of a two-digit number by 20 and by 360
• Reading a numeral written vertically, bottom place first

What they should be able to do

• Build any number from 1 to 19 out of dots and bars
• List the Mayan landmark numbers as printed and check them against the base-n definition
• Read a vertical Mayan numeral by pairing each row with its landmark
• Explain why 20 x 20 is not a landmark here, and what that costs
• Write a given number in Mayan form, including one that needs a shell in the middle and one that needs it at the end
• State what the Maya contributed that Mesopotamia did not
• Spot an arithmetic slip in a printed worked example by checking it against the stated total

Where it usually goes wrong

• "The Mayan system is base 20." The plate itself says almost. Read the landmark list: 20 x 18 is 360, not 400, so the ratio between the second and third landmarks is 18. One irregular rung breaks the ladder.
• "So they just made an arithmetic mistake." No — the third landmark is deliberate, and the chapter connects it to their calendars. It is a design choice with a cost, not an error.
• "A shell for zero means they had the number zero." It marks an empty place. Whether zero counted as a full number there is a separate question, and the chapter reserves that step for the Indian system.
• "Two systems this alike must share an origin." The chapter states that the Mayan design owed nothing to the Asian ones. That independence is the most interesting fact in the section.
• "Vertical writing is just a style." It is the same idea as writing left to right — position fixes the landmark. Only the direction differs, which is worth saying because students read the stack the wrong way round on first sight.
• "Without a base you cannot compute at all." You can; it is just that multiplying no longer reduces to landmark bookkeeping, because a product of two landmarks need not be a landmark. 20 x 20 = 400 is the cheapest counterexample.

Questions to check understanding

• Write 1 to 19 in dots and bars, and read a given group back
• Convert a number below 7200 to Mayan form and back
• State the Mayan landmark numbers and say which clause of the base definition fails
• Give a product of two Mayan landmarks that is not a landmark
• Write a number requiring an empty place, and say what would happen without the shell
• Check a printed expansion against its stated total and report any inconsistency

Examples worth working on the board

Everything from the full-page plate was read off the page image; that page carries no extractable text at all.

• The framing (Part I p.74): a Central American civilisation which the chapter dates to the 3rd through 10th centuries CE and credits with major intellectual and cultural achievement; its place value system was devised with no input from the Asian ones, and it used a placeholder for what we write as 0, shaped like a seashell.
• The plate (Part I p.75), a full-page figure styled as a carved stone slab, headed with the system's name and the line that it is almost a base-20 system. It prints the landmark numbers as 1, 20, 20 x 18 = 360, 20² x 18 = 7200, 20³ x 18 = 144000, and the three marks: shell is 0, dot is 1, bar is 5. It states that symbols are placed vertically to write a number.
• The worked numeral on the plate (Part I p.75). Counted: the top row is four dots; the middle row is one dot above two bars; the bottom row is a shell. The plate's own reading strip gives 360 against 4, 20 against 11, and 1 against 0 — so four 360s, eleven 20s and an empty ones place. The plate then prints the expansion with an error; see Notes. The stated total, 1660, is correct.
• The three marks in use (Part I p.76): a dot standing for 1, a bar standing for 5, and between them everything from 1 to 19. So each place holds up to three bars and up to four dots.
• The reading order (Part I p.76): the lowest set of marks counts the ones, the set above counts the twenties, the set above that counts the 360s, and so on upward.
• The open question (Part I p.76): why the third landmark is 360 rather than 400. The chapter reports only that some scholars connect it to Mayan calendars, and leaves it there.
• The verdict (Part I p.76): since the landmark list is not the powers of 20, the system forfeits the computing advantages a genuine base confers — yet its place value notation and its placeholder for zero are an important advance.
• The exercise (Part I p.76): write 77, 100, 361 and 721 in the Mayan system. Hand over the four numbers. As a check only, as background: the four are chosen to exercise the awkward cases — one needs two full places, one ends with an empty place, and two need an empty place in the middle with the 360s occupied. That is the set a placeholder exists for.
• The loose end (Part I p.76): traces of counting by twenties survive in how a few European languages name their numbers. The chapter names none of the languages, so neither should the explanation.
• Numbers worth showing: the landmark list 1, 20, 360, 7200, 144000; the ratios between consecutive landmarks, which run 20, 18, 20, 20; and 400, the number that is missing from the list and would have to be there for the system to be base 20.

Figures to have open

• The landmark ladder with ratios marked (built from the list on Part I p.75). An added construction, and it carries section 2 and section 8.
• The 1-to-19 chart in bars and dots. The chapter states the rule but prints no such chart, so this is added here to build, and section 4 needs it.
• The plate's worked numeral, redrawn as three stacked rows with a landmark label beside each. Redraw rather than reproduce; the printed plate is a textured stone graphic and the marks are small.
• A side-by-side of the printed expansion and the corrected one for section 7. An added construction. This must show the total agreeing in both, because that agreement is how the error is detected.
• The map of Part I p.80 can supply the Mayan location if section 1 wants one.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 3, "A Story of Numbers", §3.4 "Place Value Representation", subsection II. The Mayan Number System, Part I pp.74–76. The full-page plate at Part I p.75 belongs to this subsection and carries no printed folio-level caption of its own.
• The exercise for this subsection sits inline at Part I p.76 and is not headed Figure it Out.
• The placeholder discussion that motivates this subsection is at Part I p.74, in the Mesopotamian subsection.
• The chapter's SUMMARY (Part I p.81) names the Mayan civilisation among the four that used place value representations.