Students add a third gear between two others and find it changes only the direction (an idler), then put two gears on one axle to make a compound train, work out that stage ratios multiply, choose a train for a load no single pair can lift, and see why big ratios come in stages.
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Students will be able to
I can explain that an idler gear changes the direction of a turn but not the ratio.
I can work out the overall ratio of a compound train by multiplying its stage ratios.
I can choose a gear train that meets every criterion in a brief.
I can explain why big ratios are built in stages.
Timing
Predict and Try 8 min · Worked example 12 min · Vary 12 min · Practise and Make 13 min
Before class
Do Gear up the winch first: this lesson assumes ratio = driven ÷ driver, and that speed and strength change by the same factor.
Try every board yourself. With LEGO Technic or Meccano, build 8 → 24 = 8 → 40 on a frame: students can count 15 motor turns for one winch turn, and feel how much easier the output is to hold.
Simplifications: no friction or losses (each real mesh loses a few per cent, so a two-stage train loses a little more than one pair), the motor lifts exactly 200 g with no gears, and the gears on a shared axle are drawn on top of each other (in a real train they sit side by side along the axle).
Curriculum hooks: Design and Technologies, engineering principles and systems (primary); Science, forces and simple machines; Maths, multiplying ratios and rates (applied).
Running the lesson
Predict: take a show of hands for each option before anyone opens the board.
Try: ask what the idler is for, if not to change the ratio (reverse the direction; bridge a gap).
Worked example: point at the two gears on one axle. They turn together, so the second stage starts from a small gear again.
Find the mistake: ask a student to say what the first stage does to the second stage's input.
Vary: the hints give the least and the most ratio. Write 7.5 ≤ ratio ≤ 8 on the board.
Make: compare briefs. Who has the narrowest gap between least and most ratio?
The hard part (on purpose)
Students want to add stage ratios, and expect a big idler to do something. Keep asking 'what does the next stage start from?' and 'which teeth cancel?'
When students get stuck
What you will see
A question to ask
The fix
A student adds the stage ratios.
Stage 1 turns the middle axle 3 times slower. What does stage 2 do to that?
Each stage slows down what the stage before already slowed: multiply.
A student thinks a big idler makes the winch stronger.
Write the two meshes as ratios. Which number appears on top and on the bottom?
The idler's teeth cancel: the ratio is last ÷ first.
A student loses track of which gear drives which.
Follow the turn from the motor: which gear does it push, and what turns with that gear?
Meshing gears push each other; gears on one axle turn together.
A student picks the strongest train and ignores the time limit.
It lifts it. Is it fast enough?
Find the least ratio (load) and the most (time), then aim between.
Answers to the checks
a gear in the middle: Put a 40-tooth gear between the 8 and the 24: 8 → 40 → 24. Compared with 8 → 24 on its own, the winch now… Same speed, same strength, other direction. 40 ÷ 8 × 24 ÷ 40 = 24 ÷ 8: the idler's teeth cancel out, and each mesh reverses the turn.
Which way?: Press Turn the motor once. Watch the orange dots on the motor and winch gears: which way does each turn?
The motor and the winch turn the same way: two meshes, and each one reverses the turn. With 8 → 24 alone they turn opposite ways.
Find the stall: Add 50 g at a time and lift, until the winch stalls.
It lifts 600 g and stalls at 650 g: exactly what 8 → 24 does on its own.
Change the idler: Change Gear 2 (the idler) to another size and lift 600 g again.
Any idler gives the same 600 g. The ratio is still last ÷ first: 24 ÷ 8 = 3.
The heaviest lift: With the idler in place, what is the heaviest bucket the winch can lift (in grams)? 600 600 g, the same as 8 → 24: 40 ÷ 8 × 24 ÷ 40 = 3, and 200 × 3 = 600.
Check 8 → 24 = 8 → 40, then 16 → 40 = 8 → 24: Now 16 → 40, then 8 → 24 on the same axle as the 40. What is the overall ratio? 7.5 40 ÷ 16 = 2.5, 24 ÷ 8 = 3, and 2.5 × 3 = 7.5.
Strength: 16 → 40 = 8 → 24: what is the heaviest bucket it can lift, in grams? 1500 200 × 7.5 = 1500 g: just enough.
Time and criteria: 16 → 40 = 8 → 24: how many seconds to lift the bucket 1 m? 37.5 The winch turns 120 ÷ 7.5 = 16 times a minute. 10 turns ÷ 16 = 0.625 min = 37.5 s. It lifts 1500 g within 40 s: it passes.
Find the mistake: Kai thinks the train lifts the 1500 g bucket. Click the line where Kai's working first goes wrong. line 4 Line 4: stage ratios multiply, they don't add. Each stage makes the next one 3 times (then 5 times) slower and stronger again: 3 × 5 = 15, so it lifts 3000 g (but takes 75 s).
Meet the brief: Choose four gears that lift 1500 g within 40 s, and lift with them.
A ratio of exactly 7.5 does it, in 37.5 s: 16 → 40 = 8 → 24, or 16 → 24 = 8 → 40 (1.5 × 5). Either order of stages works.
Heavier still: A 2500 g bucket, no time limit. Find the fastest train that can lift it, and lift.
It needs a ratio of 2500 ÷ 200 = 12.5 or more; 5 × 2.5 = 12.5 (8 → 40 = 16 → 40) is the smallest, so the fastest: 62.5 s.
Why stages?: A ratio of 15 from one pair would need an 8-tooth gear driving a 120-tooth gear. Why do engineers use two stages (3 × 5) instead? A 120-tooth gear is 15 times as wide as the small one; two stages give 15 with gears that fit Big ratios come in stages so the gears stay a sensible size. Clocks, car gearboxes and electric screwdrivers all do this.
Compound ratio: 10 → 30, then 10 → 20 on the same axle. What is the overall ratio? 6 30 ÷ 10 = 3, 20 ÷ 10 = 2, and 3 × 2 = 6.
Idler: A simple train 8 → 16 → 24: the 16 is an idler. What is the ratio? 3 Last ÷ first: 24 ÷ 8 = 3. The idler cancels out.
Direction: Three gears in a row, each meshing with the next. The motor gear turns clockwise. Which way does the last gear turn? Clockwise Two meshes: clockwise, then anticlockwise, then clockwise again.
Speed: Motor 120 turns a minute, train 8 → 40 = 8 → 24 (ratio 15). How many turns a minute does the winch make? 8 120 ÷ 15 = 8.
A clock: A clock's hour hand turns 12 times slower than its minute hand. Its gears do this in two stages, 3 × 4. Which pairs could do it? 8 → 24, then 10 → 40 on the same axle 24 ÷ 8 = 3 and 40 ÷ 10 = 4: 3 × 4 = 12.
A brief for a train: Set a bucket heavier than 1000 g and a time limit, then choose a train that meets your brief and lift with it.
No single pair lifts more than 1000 g, so only a compound train meets your brief.
Solutions
One possible solution for each task. Other correct programs are fine.
Vary: choose your train › Meet the brief
Open task: any program that does what the step asks.
Vary: choose your train › Heavier still
Open task: any program that does what the step asks.
Vary: choose your train › Why stages?
Open task: any program that does what the step asks.
a brief only a train can meet › A brief for a train
Open task: any program that does what the step asks.
a brief only a train can meet › Evaluate
Open task: any program that does what the step asks.
Stretch and support
Stretch: Each real mesh loses about 5% of the turning force. With two stages the train passes on about 0.95 × 0.95 ≈ 90%. Does 16 → 40 = 8 → 24 still lift 1500 g? What would you change?
Support: Give a table with columns stage 1, stage 2, overall ratio, winch speed, heaviest lift, time, and fill the 8 → 24 = 8 → 40 row together.
Exit ticket
A motor turning 240 times a minute drives 10 → 30, then 10 → 40 on the same axle. How fast does the output turn, and how many times stronger is it?
Ratio 3 × 4 = 12, so 240 ÷ 12 = 20 turns a minute, 12 times stronger.
What each step is for
a gear in the middle: A big gear in the middle looks as if it must change something. Committing first makes the result stick: an idler changes the direction only. Each wrong option is a real slip: the middle gear's own ratio, or multiplying both meshes without noticing they cancel.
Try: the idler: Watching the dots settles the direction; loading until it stalls settles the strength (the same 600 g as 8 → 24). Swapping the idler for another size and still lifting 600 g shows its size does not matter.
Worked example: two gears on one axle: The worked example models the compound move: put a small gear on the same axle as a big one, so the next stage starts again from a small driver. Stage ratios multiply. The first train is strong enough but too slow; the faded one passes. Kai's adding is planted after the method.
Vary: choose your train: Students choose all four gears to meet the brief, then to lift a heavier load as fast as possible: the same reasoning from criteria back to ratio, now with products. The question names why compound trains exist.
trains: Stage ratios multiply, idlers cancel, each mesh reverses the turn, and the overall ratio does speed and strength as before.
a brief only a train can meet: Writing a brief no single pair can meet, then meeting it, reverses the process again: students reason from the load to the least ratio, from the time to the most, then to two stages that multiply into the gap.
Where it leads
Next: levers and pulleys (planned), the same trade of distance for force.
Real examples: clocks (hour and minute hands), electric screwdrivers and drills (motor to chuck), car gearboxes, bike hub gears.
Code links: a short Python loop can try every train from the box against a brief; the board's Make counts them for students.