Chinese rod numerals: base-10 place value, one symbol short

What to assume they know

• Mesopotamian base-60: place value in a sexagesimal system — position instead of landmark signs, and the trouble a blank causes
• What "base n" means, and why ten is a choice not a law — base-10, and why regrouping is uniform in a base
• Reading a number as a sum of digits times powers of ten
• Reading a table whose rows are two alternative drawings of the same values

What they should be able to do

• Write 1 to 9 in both rod forms and say which form belongs to which place
• Read a rod numeral by pairing each group with its power of ten
• Explain what the alternation of the two forms is for
• Show, with the chapter's own 41, what goes wrong if only one form is used
• Compare the Chinese blank with the Mesopotamian blank and say why one is easier to read
• State exactly what the system lacks, in the chapter's own terms
• Distinguish the two Chinese number systems the chapter mentions and say which one this topic is about

Where it usually goes wrong

• "The two rows are two different number systems." They are two drawings of the same nine digits. Which drawing you use is decided by the place, not by the value.
• "The alternation is decoration." It is the boundary marker. Without it a run of strokes cannot be cut into places, which is exactly what the 41 question demonstrates.
• "With alternation they did not need a zero." They did, and the chapter says so. Alternation tells you where one place ends and the next begins; it cannot tell you that a place is there but empty.
• "Reading right to left is the rule." The plate reads the numeral left to right with the highest power first, exactly as we write. Only the drawing of the digits alternates.
• "Rod numerals are the Chinese characters for numbers." Those belong to the other system, the written one for recording quantities. The chapter separates them in its first paragraph and this topic follows the rods.
• "Six is a new symbol." Six is five's worth of horizontal stroke plus one upright. The chart is built, not memorised, and showing the construction saves a student from learning eighteen shapes.

Questions to check understanding

• Write 1 to 9 in both rod forms
• Convert a four- or five-digit number to rod numerals, choosing the correct form at each place
• Read a rod numeral back into Hindu digits, showing the expansion
• Explain the purpose of the alternation and give a numeral that would be ambiguous without it
• Compare the Chinese and Mesopotamian treatments of an empty place
• State what the chapter says the system would need to be complete

Examples worth working on the board

The plate on Part I p.77 was read off the printed page; its rod drawings do not extract.

• The two systems (Part I p.76): one of them written, for setting quantities down on the page, and one built from rods, for calculating with. The chapter takes the rod system, saying it is the more efficient of the two both for writing and for computing. Name the other; do not teach it.
• Dates (Part I p.76): the rods were in use in China from the 3rd century AD at the latest, and stayed in use until the 17th. Base-10.
• The digit chart (Part I p.77, on a scroll-styled plate headed with the system's name and "Base-10 or Decimal"). Two rows against the digits 1 to 9. Counted: the Zong row draws 1 to 5 as one to five upright strokes; from 6 it puts a horizontal stroke on top and hangs one, two, three or four uprights beneath it, so 6, 7, 8, 9 are the horizontal plus one to four. The Heng row is the same construction turned through a right angle: 1 to 5 are one to five horizontal strokes, and 6 to 9 are a vertical stroke with one, two, three or four horizontals under it — 6 is the vertical over one horizontal and 9 is the vertical over four, exactly mirroring the Zong row's one to four uprights.
• The printed note (Part I p.77): the Zong forms carry units, hundreds and ten-thousands onward; the Heng forms carry tens, thousands and hundred-thousands onward. Read that as: Zong at the even powers, Heng at the odd ones, counting from 10⁰.
• The worked numeral (Part I p.77): read left to right as 2 (Heng), 6 (Zong), 3 (Heng), 4 (Zong), sitting over landmark positions 10³, 10², 10 and 1. The plate prints the expansion as (2) x 10³ + (6) x 10² + (3) x 10 + (4) x 1, and the total as 2634. Every part of this is on the page and the arithmetic is sound.
• The blank (Part I p.77): a skipped place was left as a gap, exactly as in Mesopotamia — but because the nine digit signs are closer to one another in size, a reader could locate the gaps more easily.
• The verdict (Part I p.78): the rod numerals are noticeably similar to the Hindu system, and the chapter says that adding a sign for zero would have made the Chinese system complete as a place value system.
• The exercise (Part I p.80, Figure it Out item 1), three linked questions: why do you think the Chinese alternated between Zong and Heng; if only the Zong forms were used, how would 41 be written; and could the resulting numeral stand for some other number, given that nothing guarantees a visible gap between neighbouring places. Hand over all three parts together — they only work as a set.
• Why 41 is the number they chose. Worked by me, not printed: in Zong form 4 is four uprights and 1 is one upright, so writing 41 with Zongs alone gives five uprights in a row — which is the Zong drawing of 5. One numeral, two readings, and no gap can be relied on to separate them. That collision is the entire answer to all three parts of the question, and it is why alternation exists.
• Numbers worth showing: 2634 with its four places labelled; 41 written both ways; and 5, the number it collides with.

Figures to have open

• The two-row digit chart, redrawn cleanly with the construction visible — five strokes, then the crossing stroke plus one to four (Part I p.77). Section 3 needs it and the printed plate is small and textured.
• The place strip with alternating forms for section 4. An added construction.
• The 41 collision for section 6: five uprights, first read as four-then-one and then as five. This is an added movement and it is the best thirty seconds in the topic.
• A blank-comparison frame for section 8 with a Mesopotamian numeral and a rod numeral both missing a place. An added construction, built from Part I p.73 and Part I p.77.

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 III. The Chinese Number System, Part I pp.76–78. The plate carrying the digit chart and the worked numeral is at Part I p.77; the closing verdict is the first paragraph of Part I p.78.
• The exercise belongs to the Figure it Out set at Part I p.80, item 1.
• Back-reference: the Mesopotamian blank is at Part I p.73, and the chapter draws the comparison itself at Part I p.77.
• The chapter's SUMMARY (Part I p.81) lists the Chinese among the four civilisations that used place value representations.