When an approximate answer is the better answer

What to assume they know

• What it actually takes to make one lakh — comparing a 5- or 6-digit number with one lakh
• Crores and millions: one number, two naming systems — reading a 7- or 8-digit number in the Indian system
• Subtraction and comparison of large whole numbers
• Reading a two-column table and matching a row to a value

What they should be able to do

• Distinguish a quantity, which can sensibly be approximated, from a label, which cannot
• Decide from a described situation whether to round a number up or down, and justify the choice by what would go wrong the other way
• Report a large figure to a stated level of roughness and say what has been given up
• Answer questions about a large data table by approximation rather than by exact arithmetic, and say when that is legitimate
• Compare growth in two ways — as a difference and as a ratio — and say which question each answers
• Identify, from a table of figures, the entry that moved most and the entries that roughly doubled

Where it usually goes wrong

• "Approximating means you did the sum badly." It means you answered the question that was asked. Nobody wants a book-fair crowd reported to the last person.
• "Round up if the next digit is 5 or more — that is what rounding means." That is one rule for one job. §1.4 is not about that rule at all; it is about choosing a direction from the situation. Sweets go up, prices come down, and neither is decided by a digit.
• "75,000 is what you get by rounding 76,068." Not by any standard place — the nearest thousand is 76,000 and the nearest ten thousand is 80,000. The reporter chose a convenient step of five thousand.
• "All numbers can be rounded." Not phone numbers, not account numbers, not house numbers. Section 6 exists for this, and the cartoon makes it memorable.
• "The city that grew the most is the biggest city." Mumbai is the biggest and grew by less than five lakh; Bengaluru grew by over forty-one lakh.
• "Grew most and almost doubled are the same question." Vadodara more than doubled while adding under nineteen lakh; Mumbai added under five lakh on a base of over a crore. Difference and ratio rank the table differently.
• "Every population in the table went up." Kolkata's did not.

Questions to check understanding

• Given a situation, state whether to round up or down and justify it
• Round a given figure to a stated place and to a stated convenient step
• "Answer by approximation" questions on a supplied data table — the chapter's own form at Part I, §1.4, p.13
• Compute an increase from two census figures and express it to the nearest lakh
• Decide from two figures whether a quantity has roughly doubled
• Identify a situation in which an exact number is required, and say why
• Competency-style: criticise a reported figure as too precise or too rough for its stated purpose

Examples worth working on the board

Values marked verified are worked out here against the printed page.

• The book-fair headline (Part I, §1.4, p.9, artwork). Three children; one reads out that a lakh of people came to the fair; the third points out that had she stayed away it would have been reported as 99,999. The joke is that 99,999 would never have been reported at all. Use it to open: a headline reports a size, not a count.
• Chintamani twice. About 75,000 at Part I, §1.1, p.3; exactly 76,068 at Part I, §1.4, p.10. Verified: the difference is 1,068, about one and a half per cent. Note that 75,000 is not the nearest thousand (76,000) nor the nearest ten thousand (80,000) — it is the nearest five thousand. The roughness itself was a choice, and that choice is section 3.
• Rounding up (Part I, §1.4, p.10). A school of 732 people; the principal orders 750 sweets rather than 700. Verified: 750 leaves 18 spare; 700 leaves 32 people with nothing. The chapter defines rounding up as landing above the actual figure.
• Rounding down (Part I, §1.4, p.10). An item costing ₹470 described as around ₹450 rather than around ₹500. Verified: ₹450 understates by ₹20; ₹500 overstates by ₹30. The chapter defines rounding down as landing below the actual figure.
• The phone call (Part I, §1.4, p.10, six-panel cartoon — read as an image). A father asks Bappi to ring and check whether the Toofan Express is running on time. A poster of emergency numbers is drawn in the panel: Police 100 or 112, Fire 101, Ambulance 102, Disaster Management Services 108, National Emergency Number 112, Railway Enquiry 139. The doorbell goes and a police officer is at the door; the boy explains cheerfully that he dialled the rounded number. Verified by inspection of the artwork: the punchline is that 139 rounded to the nearest hundred is 100, the police number. That reading is an added inference from the drawn panels — the chapter does not spell it out in type, and the poster's numbers, which the joke turns on, do not survive text extraction.
• The city table (Part I, §1.4, pp.12–13). Twenty rows, ranked by the 2011 figure, with 2011 and 2001 populations. Mumbai 1,24,42,373 and 1,19,78,450; New Delhi 1,10,07,835 and 98,79,172; Bengaluru 84,25,970 and 43,01,326; Hyderabad 68,09,970 and 36,37,483; Ahmedabad 55,70,585 and 35,20,085; Chennai 46,81,087 and 43,43,645; Kolkata 44,86,679 and 45,72,876; Surat 44,67,797 and 24,33,835; Vadodara 35,52,371 and 16,90,000; Pune 31,15,431 and 25,38,473; Jaipur 30,46,163 and 23,22,575; Lucknow 28,15,601 and 21,85,927; Kanpur 27,67,031 and 25,51,337; Nagpur 24,05,665 and 20,52,066; Indore 19,60,631 and 14,74,968; Thane 18,18,872 and 12,62,551; Bhopal 17,98,218 and 14,37,354; Visakhapatnam 17,28,128 and 13,45,938; Pimpri-Chinchwad 17,27,692 and 10,12,472; Patna 16,84,222 and 13,66,444.
• What the table asks (Part I, §1.4, p.13): a general observation, a title for the table, Pune's rise, the largest riser, the near-doublers, and the multiplier taking Patna to Mumbai — all to be answered by approximation. Verified: Pune rose by 5,76,958, about six lakh. Bengaluru rose most, by 41,24,644, about forty-one lakh. Roughly doubled: Bengaluru (about 1.96 times), Hyderabad (about 1.87), Surat (about 1.84) and Vadodara (about 2.10, so past double). Patna needs multiplying by about 7.4 to reach Mumbai.
• The one that fell. Verified against the printed pages: Kolkata is the only city in the table whose 2011 figure is lower than its 2001 figure, down by 86,197. Any statement that every city grew is contradicted by row 7.

Figures to have open

• The six-panel cartoon from Part I, §1.4, p.10. Its argument does not exist in the text — the extracted words give the three speech balloons, but the emergency-number poster the joke turns on is lettering inside the artwork and does not extract at all. Rebuild it as original artwork; do not reproduce the printed strip.
• The twenty-row population table (Part I, §1.4, pp.12–13). Redraw it. The explanation needs at minimum the eight rows named in section 9 and section 10.
• A double bar chart of 2001 against 2011 for six chosen cities, so the difference reading and the ratio reading can be shown on the same picture. Standard schematic; the chapter prints no chart.
• A number line around 76,068 with the three candidate roundings marked. Standard schematic.

Where this sits in the book

• NCERT Ganita Prakash Class 7, Part I, printed Chapter 1, "Large Numbers Around Us", §1.4 "Exact and Approximate Values", pp.9–13. This topic uses the section's opening pages (pp.9–10) and the unnumbered subheading "Populations of Cities" (pp.12–13). The other unnumbered subheading inside §1.4, "Nearest Neighbours" (pp.11–12), belongs to Rounding to a nearest neighbour, and choosing which one.
• The exact 2011 figure for Chintamani (Part I, §1.4, p.10) answers back to Part I, §1.1, p.3, where the same town is given rounded.
• Part I, SUMMARY, p.22 states that rounding up or down is the ordinary treatment of large numbers and that roughly how big something is is often enough.