Map scale as a ratio, and what it lets you compute

What to assume they know

• What a ratio claims, and that equivalent ratios are scaled copies of each other — What a ratio claims, and why it is not a difference, Direct proportion restated: the quotient that stays constant
• Unit conversion treated as a proportion — Unit conversion is a proportion in disguise
• The metric ladder for length: 100 cm in a metre, 1000 m in a kilometre
• Measuring a straight-line distance with a ruler, to the nearest millimetre
• Multiplying and dividing by powers of ten, and reading a number written in the Indian grouping such as 60,00,000

What they should be able to do

• State what a representative fraction asserts about a drawing and the ground it represents
• Explain why the two terms of a representative fraction are written without units, and what that lets you do with any ruler
• Convert a ground distance given in centimetres to metres and to kilometres, and back
• Reduce a stated representative fraction to a working rate of the form "one centimetre on the page stands for so many kilometres"
• Measure a straight-line separation between two points on a map and convert it to a ground distance using the map's own ratio
• Distinguish geographical distance from road distance, and say which of the two a map's ratio delivers
• Predict that maps at different scales must give the same ground distance, and explain why
• Build a scale drawing of a real space at a stated ratio, working the conversion in the other direction

Where it usually goes wrong

• "The bigger the second number, the bigger the map." It is the other way round. A larger second term means the ground was shrunk harder, so more country fits on the page and less detail survives. Put 1 : 50 and 1 : 60,00,000 side by side and ask which one could show a chair.
• "1 : 60,00,000 means one centimetre to sixty lakh kilometres." The second term inherits whatever unit you measured the first in. Sixty lakh centimetres, not kilometres — and the whole reason the ratio is written bare is that it refuses to commit to a unit.
• "So the RF only works in centimetres." Measure in inches and the same ratio tells you the ground distance in inches. The number does not change; the unit you feed it does.
• "Map distance is how far I would drive." The chapter flags this itself. A straight line between two cities crosses whatever lies between them.
• "A map drawn at a different scale will give a different real distance." If it did, one of the two maps would be wrong. The ground distance is what both maps are reporting; the scale is only the shrinking factor.
• "If a ratio is printed on a map, the map is drawn to it." Not on this page: the same artwork carries an RF in one corner and a disclaimer in the other. See section 9. This is a genuinely useful lesson about trusting a printed figure.
• "A map ratio is a different kind of object from a recipe ratio." Both are two same-unit quantities compared by relative size. The map ratio just happens to compare a drawing with the thing drawn.

Questions to check understanding

• Convert a stated RF into "1 cm represents … km"
• Given a measured length on a map and the map's RF, find the ground distance
• Given a ground distance and an RF, find how long the line would be on the map
• Choose which of two RFs belongs to the more detailed map, and justify it
• Explain in one or two sentences why the same pair of places gives the same ground distance on maps of different scales
• Produce a scale drawing of a stated room at a stated ratio, and mark one object in its correct scaled position
• Say which of geographical and road distance a map's RF yields, and why the other cannot be read off the map

Examples worth working on the board

Values marked printed appear on the page. Values marked not in the book are an added measurement or arithmetic and must be presented as such.

• The stated ratio (Part II §3.2, p.56). The example the section works with is 1 : 60,00,000, and the same ratio is printed inside the map artwork in the lower right corner, set out digit by digit as R F = 1 : 6 0, 0 0, 0 0 0.
• The conversion. Printed: the page asks the reader to turn 60,00,000 cm into kilometres and gives the answer as 60 km, then asks for it to be checked. The two steps: 60,00,000 cm ÷ 100 = 60,000 m, and 60,000 m ÷ 1000 = 60 km.
• The working rate. An added phrasing of a printed fact: 1 cm measured on such a map stands for 60 km on the ground. That single line is the whole of what the RF gives you, and it is the number a student should write down before touching the ruler.
• The map (Part II p.56, artwork above §3.2's text). A shaded relief map of peninsular India, framed by longitudes labelled every two degrees from 72°E to 86°E along both the top and the bottom edge, and latitudes labelled every two degrees from 8°N to 18°N along both sides. The degree labels sit outside the frame lines rather than on them, which is what the measurement note below turns on. Thirteen cities are marked with a ringed dot and named: Hyderabad, Vishakapatnam, Tirupati, Chennai, Bengaluru, Mangaluru, Mysuru, Kannur, Puducherry, Coimbatore, Kochi, Madurai and Thiruvananthapuram. The two seas are lettered ARABIAN SEA and BAY OF BENGAL. A compass rose sits top right; a LEGEND block with the single entry CITY sits right of centre; the words Map not to scale sit in the lower left corner; the RF sits in the lower right. All of this was read off the printed page and a close-up rather than off extracted text, and every label but one is artwork lettering — the exception is the disclaimer, which does reach the text layer (see Notes).
• The two tasks set. Find the geographical distance from Bengaluru to Chennai, and from Mangaluru to Chennai. The printed hint is to measure with a ruler and then apply the map's ratio. No answers are printed.
• What a ruler actually gives on this page (an added measurement). On the printed page of Part II p.56, taken as true page size, 14 degrees of longitude span about 12.6 cm and 10 degrees of latitude about 9.2 cm — measured degree label to degree label, not frame edge to frame edge. The frame is the wider box, about 13.6 cm across and about 9.5 cm down, because the graticule is inset within it; measure the frame and call it 14 degrees and you understate the rate by about eight per cent. Taken from the graticule, both axes agree on about 120 km for every centimetre of page, an actual scale near 1 : 1,20,00,000 — almost exactly twice the printed RF. The straight-line gap from Bengaluru to Chennai measures about 2.41 cm on the printed page, which the printed RF turns into about 145 km. See the note in section 9 — this is the single most important practical fact about the figure.
• Reference values for the two tasks (not in the book, computed from the two cities' coordinates, not printed and not obtainable from this map): Bengaluru to Chennai is about 290 km in a straight line; Mangaluru to Chennai is about 590 km. These are the true separations, and the printed RF on this page will not deliver them — a class working the page correctly lands on about 145 km and about 293 km, almost exactly half of each, and that is the right answer to the book's question. So do not hold 290 and 590 up as the target a class should reach; hold them up as the thing the measurement fails to recover, which is what section 9 exists to explain. Say where they came from either way.
• Geographical against road distance (Part II p.56). The chapter states plainly that what the RF converts is the direct separation, not the length of any road. Not in the book: the road distance between the same two cities is always the larger of the two, and by an amount no map ratio can tell you.
• The different-scales question (Part II p.57, top). Repeat the same two measurements on other maps drawn at other scales and check whether the ground distances agree, at least approximately. The section then carries a Note to the Teacher asking for atlases in the classroom and for students to compare their answers with each other and account for any large disagreement.
• The map-making activity (Part II p.57, a boxed activity). Sketch the classroom to a stated ratio of 1 : 50, placing at their scaled positions the blackboard, the teacher's desk, the ceiling fans and the light fittings, with symbols of the student's own choosing standing in for each kind of object. An added worked instance: at 1 : 50 a wall 6 m long is drawn 12 cm long, because 6 m is 600 cm and 600 ÷ 50 = 12. The conversion runs the opposite way from the map problem, and that is the point of including it.

Figures to have open

• The chapter's own map (Part II p.56) is close to unavoidable for sections 1, 6 and 9, because the argument of section 9 is about this figure carrying two contradictory pieces of information. Rather than reproducing the printed relief art, redraw a simplified outline of peninsular India carrying the same thirteen city dots, the same degree frame, a LEGEND block, the RF in the lower right and the disclaimer in the lower left. The redraw must be honestly scaled if students are going to measure on it.
• A single figure showing one printed length with a centimetre ruler above it and an inch ruler below, and the two matching ground distances. Standard schematic, and it carries section 3 by itself.
• A metric ladder strip for cm → m → km with the two divisions marked. Standard schematic.
• A classroom floor plan at 1 : 50 with one wall dimensioned in both metres and centimetres. Standard schematic; no textbook art needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 3, "Proportional Reasoning-2", §3.2 "Ratios in Maps", Part II pp.56–57. The map artwork and the RF example are on Part II p.56; the different-scales question, the Note to the Teacher and the boxed map-making activity are on Part II p.57, above the start of §3.3.
• Backward pointer: Part I printed Chapter 7, §7.6, for unit conversion read as a proportion.
• Forward pointer inside this chapter: §3.3 begins immediately below the map-making activity on Part II p.57 and is covered by Ratios of three or more quantities at once.