Rounding to a nearest neighbour, and choosing which one

What to assume they know

• When an approximate answer is the better answer — that approximation is a choice, and that rounding up and rounding down are directions
• Crores and millions: one number, two naming systems — reading an 8- or 9-digit number in the Indian system, and the places thousand, ten thousand, lakh, ten lakh, crore
• Addition and subtraction of 6- and 7-digit numbers
• Comparing two large numbers to decide which is nearer to a third

What they should be able to do

• Find the nearest thousand, ten thousand, lakh, ten lakh and crore of a given 8-digit number
• Explain why the five neighbours of one number need not run in the same direction, and give an example where two of them disagree
• Show that each neighbour must be computed from the original number rather than from another neighbour
• Describe all the numbers whose five neighbours are all the same value, and count them
• Bracket a sum or difference between two round numbers without computing it
• Tighten a bracket by replacing one of the two numbers with a nearby round number, and say which way the replacement pushed the answer
• Judge which of two estimates of the same sum is the closer, and justify it

Where it usually goes wrong

• "Round the number, then round the answer again to get the next place." The neighbours are computed from the original. The 6,74,90,000 case in the worked examples shows chaining giving the wrong ten-lakh neighbour.
• "The five neighbours get steadily bigger, or steadily smaller." They do not. In the chapter's own table the nearest lakh is above the number and the nearest ten lakh is below it.
• "Rounding to a bigger place always moves the number further from the truth." It usually does, but not always — in 3,87,69,957 the nearest thousand and the nearest ten thousand are the same value.
• "An estimate is a guess." Roxie and Estu both make claims that can be checked and both turn out true. What distinguishes them is closeness, not correctness.
• "If two estimates disagree, one of them is wrong." Neither is wrong here. Near-and-above 8,00,000 and near-and-below 9,00,000 are both true of 8,82,810.
• "You must compute the sum to know it is under 8,83,128." You must not — the point of the question is the one-line argument: one addend is short of 4,20,000, so the total is short of 8,83,128.
• "Nearest means chop the digits off." Chopping gives the neighbour below, every time. Three of these five answers are above.

Questions to check understanding

• Write the five nearest neighbours of a given 8- or 9-digit number — the chapter's own form
• Round a stated figure to the nearest lakh and to the nearest crore, and say which moved it further
• Given two estimates of a sum, decide whether each is correct and which is closer
• Decide whether a sum lies above or below a stated value, justified without computing the sum
• Compute the exact value and state the error of a given estimate
• Given a target range, name two numbers whose sum falls inside it — the inversion the teacher's note asks for
• Competency-style: explain why the nearest lakh and the nearest ten lakh of one number can lie on opposite sides of it

Examples worth working on the board

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

• The worked number (Part I, §1.4, p.11, table). 6,72,85,183, with all five of its neighbours printed in a five-row table. Rounded at the thousands place it reads 6,72,85,000. One place higher, at ten thousands, it becomes 6,72,90,000. The closest lakh is 6,73,00,000; the closest ten lakh is 6,70,00,000; the closest crore is 7,00,00,000. This is the only fully worked case in the section. Verified: the directions run down, up, up, down, up. Rows three and four are the whole argument — the nearest lakh sits above the number and the nearest ten lakh sits below it.
• Two numbers set for the student (Part I, §1.4, p.11): 3,87,69,957 and 29,05,32,481, each for all five neighbours. Verified for the first: 3,87,70,000; 3,87,70,000; 3,88,00,000; 3,90,00,000; 4,00,00,000 — the first two coincide, which is worth showing. Verified for the second: 29,05,32,000; 29,05,30,000; 29,05,00,000; 29,10,00,000; 29,00,00,000 — here the direction changes twice.
• The Math Talk puzzle (Part I, §1.4, p.11). A number all five of whose neighbours are 5,00,00,000; how many such numbers are there? Verified reasoning: the nearest-thousand condition is the tightest, and it alone pins the number to within five hundred of 5,00,00,000; the other four conditions then follow. So the answer is a run of a thousand consecutive whole numbers centred on 5,00,00,000. Exactly which thousand depends on how an exact half is settled, and the chapter states no rule for that.
• An added caution about chaining. Verified construction, not the chapter's: 6,74,90,000 is nearer to 6,70,00,000 than to 6,80,00,000, so its nearest ten lakh is 6,70,00,000. But its nearest lakh is 6,75,00,000, which sits exactly halfway between the two ten-lakh marks, and any tie-breaking habit sends it to 6,80,00,000. Round the rounded copy and you get the wrong neighbour. Present this as a warning built, clearly not as the book's example.
• Estimating a sum (Part I, §1.4, p.11). 4,63,128 + 4,19,682. Roxie says the sum is near 8,00,000 and above it; Estu says it is near 9,00,000 and below it. The chapter then asks whether the sum is above or below 8,50,000, whether it is above or below 8,83,128, and finally for the exact value. Verified: the sum is 8,82,810; both children are right; Estu is much closer (17,190 away against 82,810); the sum is above 8,50,000 because 4,63,128 already exceeds 4,50,000 and 4,19,682 exceeds 4,00,000; and it is below 8,83,128 because 4,19,682 is below 4,20,000 — by 318, which is exactly the gap.
• Estimating a difference (Part I, §1.4, pp.11–12). 14,63,128 − 4,90,020. Roxie says the difference is near 10,00,000 and below it; Estu says near 9,00,000 and above it. Then: above or below 9,50,000, above or below 9,63,128, and the exact value. Verified: the difference is 9,73,108; both are right; Roxie is closer (26,892 against 73,108); it exceeds 9,63,128 because taking away 5,00,000 would leave exactly 9,63,128 and we take away less; and 9,63,128 already exceeds 9,50,000.
• The teacher's note (Part I, §1.4, p.12) turns the exercise round and asks what the two numbers could have been if the sum had to stay under 8,50,000. That inversion is the natural closing beat of section 12.

Figures to have open

• The five-row neighbour table for 6,72,85,183 (Part I, §1.4, p.11). Redraw it; it is the topic's central evidence.
• A number line that can be zoomed five times — thousands, ten thousands, lakhs, ten lakhs, crores — with the same number marked on each. This is the picture the whole topic rests on and the chapter does not print it, so it must be built. Standard schematic.
• A shaded-window diagram around 5,00,00,000 for section 6. Standard schematic.
• Two overlapping estimate regions on one line for section 7. Standard schematic.
• No photograph or textbook art is required.

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.11–12, under the unnumbered subheading "Nearest Neighbours". The subheading carries no number and cannot be cited as one.
• The Math Talk prompt about a number with five identical neighbours sits on Part I, p.11.
• The Note to the Teacher inverting the estimation exercise sits on Part I, p.12.
• The remainder of §1.4 — its opening pages (pp.9–10) and "Populations of Cities" (pp.12–13) — belongs to When an approximate answer is the better answer.