Standard units, compound measures and inverse proportion
Converting between related standard units, solving speed/density/best-value problems, and inverse proportion word problems
About three lessons of 45 to 60 minutes
Some quantities need MORE than 1 unit at once
'Speed' is not measured in km, and it is not measured in hours, it is measured in km PER HOUR, a COMPOUND unit built from 2 simpler units divided together. Density (g per cm cubed) and unit pricing (pence per gram) work the same way. Before any of that makes sense, you need to be fluent converting between related standard units in the first place, millimetres to centimetres, grams to kilograms, minutes to hours.
A different but related idea is INVERSE proportion: sometimes, as one quantity goes UP, a connected quantity goes DOWN by the same factor (more workers on a job means fewer days needed), the opposite of the direct proportion relationship (more items bought means a bigger total cost) you have already met.
- 2.5 m of ribbon = 250 cmconverting between related standard units of length
- A car travelling 120 km in 2 hours: 60 km/hspeed, a compound unit built from distance divided by time
- 500 g of butter for £2.00: 0.4p per gramunit pricing, used to compare which pack is better value
- 4 workers finish a job in 6 days; 8 workers would take only 3 daysinverse proportion, doubling the workers halves the time
What students will be able to do
Students will convert between related standard units of length, mass, capacity, time and money, solve problems using compound units (speed, density, unit pricing), and solve inverse proportion word problems by recognising that the product of the 2 quantities stays constant.
- I can convert a measurement between related standard units (e.g. cm to mm, kg to g, hours to minutes).
- I can find speed, distance or time using speed = distance / time.
- I can find density, mass or volume using density = mass / volume.
- I can compare 2 packs by their unit price to find the better value.
- I can solve an inverse proportion problem by finding the constant product and dividing.
Standards this unit teaches
- KS3 Maths: Number; Ratio, proportion and rates of changeUK National Curriculum (England)Number; Ratio, proportion and rates of change
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "use standard units of mass, length, time, money and other measures, including with decimal quantities" (Number strand); "change freely between related standard units [for example time, length, area, volume/capacity, mass]"; "solve problems involving direct and inverse proportion, including graphical and algebraic representations" (the inverse-proportion half; UK KS3 batch 1, lib/content_uks3math.ts, already built direct proportion); "use compound units such as speed, unit pricing and density to solve problems" (all Ratio, proportion and rates of change strand).
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Standard unit
- an agreed, fixed unit of measurement, such as a metre, kilogram or second
- Compound unit
- a unit built from 2 (or more) simpler units combined, such as speed (distance per time) or density (mass per volume)
- Speed
- how fast something travels, calculated as distance divided by time
- Density
- how tightly packed a material's mass is, calculated as mass divided by volume
- Unit price
- the price of a single unit of something (per gram, per item), used to compare value between different pack sizes
- Inverse proportion
- a relationship where 1 quantity increases as another decreases, in such a way that their PRODUCT stays constant
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Converting between related standard units
ConcreteEvery family of standard units is connected by a fixed conversion factor: 10 mm in a cm, 100 cm in a m, 1000 m in a km, 1000 g in a kg, 1000 ml in a L, 60 seconds in a minute, 60 minutes in an hour, 100 pence in a pound. To convert to a SMALLER unit, multiply by the factor; to convert to a BIGGER unit, divide by it.
Convert 3.5 kg to grams, and convert 2400 ml to litres.
- 3.5 kg to g: 1 kg = 1000 g, so 3.5 kg = 3.5 x 1000 = 3500 g.
- 2400 ml to L: 1 L = 1000 ml, so 2400 ml = 2400 / 1000 = 2.4 L.
Answer: 3.5 kg = 3500 g. 2400 ml = 2.4 L.
- Why do you multiply when converting to a smaller unit, and divide when converting to a bigger unit?
- Convert 90 minutes to hours.
2. Compound units: speed and density
PictorialSpeed = distance / time, and density = mass / volume. Both formulae can be rearranged the same way any formula can: to find distance, multiply speed by time; to find mass, multiply density by volume.
A journey of 210 km takes 3 hours. Find the average speed. A block of density 8 g/cm3 has volume 5 cm3. Find its mass.
- Speed = distance / time = 210 / 3 = 70 km/h.
- Mass = density x volume = 8 x 5 = 40 g.
Answer: Speed = 70 km/h. Mass = 40 g.
- If a car's speed graph (distance vs time) is a straight line through the origin, what does the gradient represent?
- A cyclist travels 45 km in 3 hours. Find the average speed.
3. Unit pricing and best value
AbstractTo compare 2 differently-sized packs fairly, find the UNIT PRICE of each (price divided by quantity), then compare those unit prices directly. The pack with the LOWER unit price is the better value, even if its total price is higher.
Pack A: 6 apples for £1.80. Pack B: 10 apples for £2.50. Which is better value?
- Pack A unit price: £1.80 / 6 = £0.30 per apple.
- Pack B unit price: £2.50 / 10 = £0.25 per apple.
- £0.25 is lower than £0.30, so Pack B is better value.
Answer: Pack B is better value (£0.25 per apple vs £0.30 per apple).
- Why can't you compare Pack A and Pack B by their TOTAL price alone?
4. Inverse proportion
AbstractIn DIRECT proportion, 2 quantities increase together (double one, the other doubles too), and dividing one by the other always gives the same constant. In INVERSE proportion, 1 quantity increases exactly as much as the other decreases, so MULTIPLYING them together (not dividing) always gives the same constant.
It takes 5 workers 8 days to complete a job. How many days would 10 workers take, at the same rate?
- Workers x days stays constant: 5 x 8 = 40.
- With 10 workers: days = 40 / 10 = 4.
Answer: 4 days.
- Why does doubling the number of workers HALVE the number of days, rather than also doubling it?
- 3 pipes fill a tank in 20 hours. How many hours would 5 pipes take?
Common misconceptions and how to address them
MisconceptionTo convert 5 m to cm, divide by 100 (mixing up which direction needs multiplying).
Why it happens: Students remember 'there's a 100 involved' but forget which OPERATION matches which DIRECTION of conversion.
How to address it: Anchor the direction with a sanity check: cm are SMALLER than m, so there must be MORE of them, so the operation must make the number bigger (multiply), never smaller.
MisconceptionDoubling the number of workers on a job also doubles how long it takes (treating it like direct proportion).
Why it happens: Direct proportion is met first and more often, so it becomes the default assumption for ANY 'more of one thing' scenario, even when the real relationship is inverse.
How to address it: Ask explicitly before solving: 'if one goes up, does the other ALSO go up (direct), or does it go DOWN (inverse)?' Ground the answer in a real-world sanity check (more workers should finish FASTER, so time must go down).
MisconceptionThe better-value pack is always the one with the bigger total size, or the bigger total price.
Why it happens: Students compare the wrong quantity (total size or total price) instead of the derived UNIT price, which is the only fair basis for comparison.
How to address it: Always compute BOTH unit prices explicitly before comparing, never judge value from the totals or the pack sizes alone.
Guided practice (with answers)
1. Convert 4.2 m to cm.
Answer: 420 cm, because 1 m = 100 cm, so 4.2 x 100 = 420.
2. Convert 5000 g to kg.
Answer: 5 kg, because 1 kg = 1000 g, so 5000 / 1000 = 5.
3. A car travels 300 km in 5 hours. Find its average speed.
Answer: 60 km/h, because 300 / 5 = 60.
4. A material has density 3 g/cm3. Find the mass of 20 cm3 of it.
Answer: 60 g, because 3 x 20 = 60.
5. Pack A: 4 pens for £2. Pack B: 6 pens for £2.40. Which is better value?
Answer: Pack B, because Pack A is £0.50 per pen and Pack B is £0.40 per pen.
6. 6 workers finish a job in 10 days. How many days would 12 workers take?
Answer: 5 days, because 6 x 10 = 60, and 60 / 12 = 5.
Independent practice worksheets
Practise unit conversions, compound units and inverse proportion with computed, never-wrong answer keys.
Differentiation
- Keep a printed conversion-factor reference table (mm/cm/m/km, g/kg, ml/L, s/min/hour, pence/pounds) visible until the factors are memorised.
- For compound units, always write the formula (speed = distance / time) before substituting numbers, every single time.
- For inverse proportion, always compute and label the constant product explicitly before dividing to find the missing value.
- Use consistent, familiar contexts (cars for speed, workers for inverse proportion) before switching to unfamiliar ones.
- Convert between units 2 steps apart (e.g. mm directly to km) by chaining 2 conversion factors.
- Solve a 3-pack best-value comparison, ranking all 3 packs by unit price.
- Investigate the compound unit for fuel economy (miles per gallon, or km per litre) and solve a related problem.
- Explain, using the constant-product idea, why an inverse proportion graph (quantity vs quantity) is a curve, never a straight line.
Assessment: exit ticket
A three-question exit ticket sampling conversions, compound units and inverse proportion.
1. Convert 1500 ml to litres.
Answer: 1.5 L, because 1500 / 1000 = 1.5.
2. A cyclist rides at 18 km/h for 4 hours. Find the distance travelled.
Answer: 72 km, because 18 x 4 = 72.
3. 4 pipes fill a tank in 15 hours. How many hours would 6 pipes take?
Answer: 10 hours, because 4 x 15 = 60, and 60 / 6 = 10.
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 unit conversions (section 1), Lesson 2 compound units and best value (sections 2-3), Lesson 3 inverse proportion (section 4) plus the exit ticket.
- UK KS3 batch 1 (PR #378, lib/content_uks3math.ts) already built DIRECT proportion (scale factors, the unitary method); this unit deliberately covers only the inverse-proportion half of the same curriculum bullet, contrasting the 2 directly in section 4 rather than duplicating batch 1's work.
- Every conversion, compound-unit and inverse-proportion amount in the matching worksheets is constructed BACKWARD from an exact integer result (the divisor is chosen first, then the dividend is built as a clean multiple of it), so every division in this unit's answer keys is exact, never a rounded decimal.
- The section-4 functionGraph figure deliberately uses connect: false (discrete points, not joined by a line), since joining inverse-proportion points with straight line segments would visually misrepresent the true curved relationship.