Unit conversion is a proportion in disguise

What to assume they know

• Solving a proportion problem, and the Trairasika rule of three — modelling a situation as a proportion, and cross multiplication
• Simplest form, and using it to test whether two ratios are proportional — reducing a ratio to compare it with another
• Metric prefixes: gram and kilogram, millilitre and litre, metre and kilometre
• Area as a product of two lengths, and square units
• Multiplying and dividing by a decimal such as 3.281, and by a fraction
• Reading a two-way formula and substituting into it

What they should be able to do

• Rewrite a ratio's terms in a common unit before comparing two ratios
• Show that the simplest form of a ratio changes if one term's unit is changed, and explain why that is not a contradiction
• Treat a stated conversion as one ratio of a proportion, and use it to convert a measurement in either direction
• Decide which of two purchases is cheaper by pricing a shared quantity
• Convert areas, and explain why the area factor is the square of the length factor
• Use the chapter's acre, hectare and square-foot entries to convert a plot size and then apply a per-acre rate
• Convert between millilitres, cubic centimetres and litres, and say why these conversions are definitions rather than measurements
• Convert a temperature with the chapter's formulas, and demonstrate that proportional reasoning gives the wrong answer for temperature
• Compare two places or two products by a rate per unit rather than by totals

Where it usually goes wrong

• "A ratio's simplest form is a property of the two quantities." It is a property of the two numbers, and the numbers depend on the units. The chapter's tea packet has two different simplest forms depending on whether its weight is written in grams or kilograms.
• "Bigger packet, cheaper per unit." The chapter's tea data says the 1 kg packet is the better buy and its shampoo table (Part I p.171) says the 6 mL sachet is. Compute, do not assume.
• "Convert at the end." Convert before forming the proportion. The chapter's own first attempt at the car problem is the cautionary example.
• "If 1 metre is 3.281 feet then 1 square metre is 3.281 square feet." The commonest area error in the chapter's exercises. The factor is squared, because both of the lengths being multiplied have to be converted.
• "An acre is a metric unit", or "a hectare is the same as an acre." A hectare is nearly two and a half acres by the chapter's own list.
• "A millilitre is roughly a cubic centimetre." It is exactly one, by definition. The volume entries in the list are agreements, not measurements, and that is worth saying out loud.
• "Temperature converts by a factor, like everything else." It does not, and the reason is not that the factor is awkward — it is that the two scales put their zeros in different places. Run the wrong method and let it fail.
• "Two totals are enough to say which place is more crowded." Bigger city, bigger population, no conclusion. Divide first.

Questions to check understanding

• Convert a measurement between two units given a stated conversion
• Compare two prices for different quantities by reducing both to a common amount
• Given two ratios whose terms are in different units, decide whether they are proportional
• Convert an area, and explain the relationship between the length factor and the area factor
• Convert a plot size to acres and apply a per-acre rate
• Convert between litres and millilitres inside a rate problem
• Convert a temperature both ways using the chapter's formulas
• Show by counter-example that a temperature cannot be converted by cross multiplication
• Rank two places or products by a rate per unit and justify the ranking
• Solve a problem that needs a share in a ratio followed by a unit conversion

Examples worth working on the board

Values marked verified are worked out here. Where the chapter does the work itself, that is stated.

• The two teas (Part I, §7.4, Example 10, pp.169–170, worked on the page). A small Himachal Pradesh farmer sells 200 g packets at ₹200 each; a large Meghalaya estate sells 1 kg packets at ₹800 each. The question asks whether the weight-to-price ratios are proportional and which tea is dearer. The page writes the Himachal ratio as 200 : 200, considers writing the Meghalaya one as 1 : 800, rejects that because the first was in grams, converts to 1000 : 800, reduces the two to 1 : 1 and 5 : 4, and concludes they are not proportional. It then answers the price question separately: 1 kg from Meghalaya is ₹800, and for Himachal it sets the price of a kilogram as x, notes that 200 g is one fifth of a kilogram, solves one fifth of x equal to 200, and gets ₹1,000 — so Himachal's tea is dearer. A Note to the Teacher on Part I p.170 asks for a class discussion of why it might be dearer.
• The unit trap made explicit. Verified: the same Meghalaya packet is 1 : 800 in kilograms-to-rupees and 1000 : 800 in grams-to-rupees, and those reduce to 1 : 800 and 5 : 4 — two completely different names for one packet of tea. Nothing about the tea changed; only the unit of the first term did. This is the single most important slide in the topic and the chapter gives it in one sentence.
• The car (Part I, §7.4, Example 9, p.169). 90 km in 150 minutes; the distance in 4 hours at the same speed. The page writes 150 : 90 :: 4 : ?, asks whether that is the right formulation, answers no because one time is in minutes and the other in hours, and rewrites it with 240 minutes. This brief owns that repair step; the cross-multiplied solution and the answer belong to Solving a proportion problem, and the Trairasika rule of three.
• The chapter's conversion list (Part I, §7.6, p.176), grouped under four headings. Under length, one metre is given as 3.281 feet. Under area, a square metre as 10.764 square feet, an acre as 43,560 square feet, a hectare as 10,000 square metres and also as 2.471 acres. Under volume, one millilitre as one cubic centimetre, and a litre as 1,000 of either. Under temperature, 0 °C as 32 °F, then Fahrenheit as nine fifths of Celsius plus 32, and Celsius as five ninths of Fahrenheit less 32, with the worked instance 25 °C = 77 °F.
• The list is internally consistent, and checking it is a section of the explanation. Verified: 3.281 squared is 10.765, which is the area entry to within a thousandth — so the area factor is the length factor squared, and that is a fact about what area is, not a second measurement. Verified: a hectare is 10,000 square metres, which at 10.764 square feet each is 107,640 square feet, and dividing by 43,560 gives 2.4711 acres — so the fifth entry follows from the other four and is not independent information. Worth saying which entries are definitions and which are roundings: the acre-to-square-feet and the litre-to-millilitre figures are exact by definition, while 3.281, 10.764 and 2.471 are rounded decimals.
• Manure for a tomato plot (Part I, p.177, exercise item 8). Good practice is 10 tonnes of cow manure per acre; the plot is 200 ft by 500 ft. The item points the student back at the conversion list. Verified: the plot is 100,000 square feet, which is about 2.3 acres, so a little under 23 tonnes. The interesting step is that the area has to be computed in square feet first because that is the unit the acre is given in.
• The cost of land (Part I, p.177, exercise item 10). One acre costs ₹15,00,000; what does 2,400 square feet cost? Verified: 2,400 out of 43,560 is about 5.5% of an acre, so about ₹82,600. A good place to show that the answer need not be a round number even when the inputs are.
• A litre of gold (Part I, p.177, exercise item 7). Equal volumes of gold and water have masses in the ratio 37 : 2, and a litre of water has a mass of 1 kg. Verified: a litre of gold comes to 18.5 kg. Note the given ratio does the work of a conversion here, which is the topic's thesis arriving from the other side.
• Filling a bucket (Part I, p.177, exercise item 9). A tap fills a 500 mL mug in 15 seconds; how long for a 10-litre bucket? Verified: the bucket is twenty mugs, so 300 seconds, which is five minutes. The whole difficulty is the litre-to-millilitre step, and once it is done there is nothing left.
• The ₹10 coin (Part I, p.177, exercise item 12, marked "Try This"). The coin is cupro-nickel, copper and nickel in the ratio 3 : 1, with a mass of 7.74 grams. Copper costs ₹906 per kg and nickel ₹1,341 per kg; the item asks for the cost of the metals in one coin. Verified: the copper is 5.805 g and the nickel 1.935 g, and at those prices the metal in the coin comes to about ₹7.85 — under the coin's face value, which is the punchline worth building the section around. Note that the problem is two topics stacked: share a mass in the ratio 3 : 1, which is Sharing a whole in a given ratio's method, then convert grams to kilograms to use the prices.
• Which city is more crowded (Part I, p.176, exercise item 3). Delhi's area is 1,484 sq km and Mumbai's is 550 sq km; the populations are given as approximately 30 million and 20 million. The item asks which is more crowded and why. Verified as an error-check: the two totals rank one way and the two rates per square kilometre rank the other way, so this item is the topic's argument in civic form — you must divide before you compare. An illustration of a crane standing beside a girl sits in the right margin of this exercise block.
• The temperature counter-demonstration (not in the book, built from the chapter's own entries on Part I p.176). Verified: the chapter's formula turns 25 °C into 77 °F. If instead you treat the two scales as proportional and use the pair 100 °C and 212 °F as a rule-of-three, you get 25 times 212 divided by 100, which is 53 °F — wrong by 24 degrees. The error comes from the offset of 32 that the proportional model ignored; at 25 °C it shows up as 24 rather than the full 32 because the factor 212/100 has already absorbed part of the offset, and at 0 °C the proportional model is wrong by the full 32. Verified also: what is proportional is Celsius against Fahrenheit-minus-32, in the ratio 5 : 9 — at 25 °C the second quantity is 45, and 25 : 45 reduces to 5 : 9. So the fix is to move the origin first and only then use a factor. This is the section that earns the topic a thesis rather than a table.

Figures to have open

• The two tea packets, each labelled with weight and price, and the same packet's ratio written in two units. An added figure; this is the topic's centre.
• The conversion list, redrawn as a grouped reference card. The chapter's own content (Part I p.176) and worth showing in full.
• A square metre subdivided into square feet, showing why the factor is squared. Standard schematic and essential to section 8.
• Two thermometers side by side, Celsius and Fahrenheit, aligned at the freezing point of water so that the 32-degree offset is visible, with a second pair of scales showing Celsius against Fahrenheit-minus-32. An added figure and the one that carries section 11.
• Two city rectangles, drawn to their stated areas, carrying their populations. Standard schematic.
• No textbook artwork or photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 7, "Proportional Reasoning-1", §7.6 "Unit Conversions", Part I pp.175–177 — the four-heading reference list and the twelve-item exercise that follows it.
• Part I pp.169–170, §7.4 Example 10 (the two teas) and the Note to the Teacher below it. Example 10 sits in §7.4 but is the chapter's own demonstration of §7.6's point, and §7.6 opens by referring back to having met the need for conversions, which is why it is carried here.
• Part I p.169, §7.4 Example 9, for the unit-repair step only.
• Exercises carried here: Part I p.176 item 3; Part I p.177 items 7, 8, 9, 10 and 12. Items 1 and 4 of the same block are carried in Simplest form, and using it to test whether two ratios are proportional and Sharing a whole in a given ratio; items 2, 5 and 11 in Solving a proportion problem, and the Trairasika rule of three; item 6 in What a ratio claims, and why it is not a difference.
• Part I pp.171–172, Activity 2 and its shampoo table, for the price-per-volume contrast in the misconceptions list. That activity is owned by Solving a proportion problem, and the Trairasika rule of three.
• The chapter's SUMMARY (Part I p.177) says nothing about unit conversion. Its three bullets cover the reading of a ratio and its terms, proportionality with ad = bc, and the two parts of a share — so §7.1's similarity argument, §7.3's simplest-form test and §7.6's unit conversions are all absent from it. Worth knowing before a revision video is planned: the summary is not a checklist of the chapter.