More than two gears. A gear between two others (an idler) changes the direction but not the ratio; two gears on one axle make a compound train, whose stage ratios multiply. Students test an idler, work out compound trains, find one for a heavy brief, and see why big ratios come in stages.
The bucket is now 1500 g: more than any pair in the box can lift. You can use as many gears as you like. How?
In Gear up the winch, one pair of gears traded speed for strength. The strongest pair in the box, 8 → 40, makes the winch 5 times stronger: it lifts 1000 g.
The new brief: the same motor (120 turns a minute, lifts 200 g with no gears) and drum (0.1 m a turn). Lift a 1500 g bucket 1 m in 40 s or less. The box now has as many 8, 16, 24 and 40-tooth gears as you need.
Two new moves: a gear between two others, and two gears on one axle.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
A gear in the middle
TryTry it on the board and compare the result with your prediction.
Try: the idler
Was your prediction right? An idler changes the direction, not the ratio. Engineers use one to reverse a turn, or to bridge a gap between two gears.
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Press Turn the motor once. Watch the orange dots on the motor and winch gears: which way does each turn?
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Add 50 g at a time and lift, until the winch stalls.
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Change Gear 2 (the idler) to another size and lift 600 g again.
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With the idler in place, what is the heaviest bucket the winch can lift (in grams)?
InvestigateWork out why it happens, one step at a time.
Worked example: two gears on one axle
Worked exampleBefore each step, say what you think comes next.
stage 1: 24÷8=3The motor's 8 drives a 24: a ratio of 3, as before.
stage 2: 40÷8=5An 8 on the 24's axle turns with it, and drives a 40: the next stage starts small again.
3×5=15Stages multiply: the second stage slows down a gear that is already 3 times slower.
120÷15=8 turns a minuteSlower by the overall ratio.
200×15=3000 gStronger by the same ratio.
10÷8=1.25 min=75 s1 m takes 10 winch turns.
3000≥1500✓75≤40×Strong enough, but too slow. Too much ratio costs time.
Now 16 → 40 = 8 → 24. Stage 1 is 40÷16=2.5. Start with the overall ratio.
Kai uses the same train as the worked example.
Now 16 → 40, then 8 → 24 on the same axle as the 40. What is the overall ratio?
16 → 40 = 8 → 24: what is the heaviest bucket it can lift, in grams?
16 → 40 = 8 → 24: how many seconds to lift the bucket 1 m?
Kai thinks the train lifts the 1500 g bucket. Click the line where Kai's working first goes wrong.
ModifyChange the problem on purpose and see what happens.
Vary: choose your train
Need a hint?
The time limit sets the most ratio you can afford: 40 s is a ratio of 8 at most.
Another hint
The load sets the least: 1500 g ÷ 200 g = 7.5.
Another hint
Which two stage ratios multiply to between 7.5 and 8? Try 2.5 (16 → 40) and 3 (8 → 24).
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Choose four gears that lift 1500 g within 40 s, and lift with them.
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A 2500 g bucket, no time limit. Find the fastest train that can lift it, and lift.
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PractiseA few on your own, to check you have it.
Trains
10 → 30, then 10 → 20 on the same axle. What is the overall ratio?
A simple train 8 → 16 → 24: the 16 is an idler. What is the ratio?
Motor 120 turns a minute, train 8 → 40 = 8 → 24 (ratio 15). How many turns a minute does the winch make?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A brief only a train can meet
Need a hint?
The strongest single pair is 8 → 40: ratio 5, so 1000 g at most. Your bucket must be heavier.
Another hint
Each second of time limit allows a ratio of 0.2 here (40 s for a ratio of 8), so set a limit that leaves room.
Another hint
Find the least ratio (load ÷ 200), the most (time limit ÷ 5), then two stages that multiply to fit between.
Stretch: real meshes each lose about 5% of the turning force. Work out what your train can really lift, and whether it still meets your brief.
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Set a bucket heavier than 1000 g and a time limit, then choose a train that meets your brief and lift with it.
Use the board. This ticks when you get there.
For teachers
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. About 45 minutes.
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.
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
A student adds the stage ratios. Ask: Stage 1 turns the middle axle 3 times slower. What does stage 2 do to that? Fix: Each stage slows down what the stage before already slowed: multiply.
A student thinks a big idler makes the winch stronger. Ask: Write the two meshes as ratios. Which number appears on top and on the bottom? Fix: The idler's teeth cancel: the ratio is last ÷ first.
A student loses track of which gear drives which. Ask: Follow the turn from the motor: which gear does it push, and what turns with that gear? Fix: Meshing gears push each other; gears on one axle turn together.
A student picks the strongest train and ignores the time limit. Ask: It lifts it. Is it fast enough? Fix: Find the least ratio (load) and the most (time), then aim between.
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.
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.
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.
Now you can
Explain that an idler gear changes the direction of a turn but not the ratio.
Work out the overall ratio of a compound train by multiplying its stage ratios.
Choose a gear train that meets every criterion in a brief.