Pistons, Pulleys and Pencil Work: A Family Count Round the Pump House

Abbey Pumping Station on Corporation Road is one of those Leicester places that rewards a second look. The first visit is all smell and scale: polished metal, high ceilings, the wonderful heaviness of Victorian engineering. The second visit, pencil in hand, is a maths lesson hiding in plain sight. This trail is designed for families and curious classrooms, with four stopping points and a short puzzle at each. It works best with a clipboard, a pencil and a calculator on a phone for checking — though the estimating is the fun part.

Check opening times before you travel, stay behind the barriers, and ask staff if you are unsure about anything. The puzzles below use simplified models rather than real engine specifications, so they are safe to attempt anywhere.

Stop one: the beam engine as a giant lever

A beam engine is a wonderful first stop because the main idea is visible from across the room. A rocking beam transfers force from one side to the other. In maths language, it is a lever with a pivot, a load arm and an effort arm. The longer the effort arm compared with the load arm, the smaller the effort needed to lift a given load. That is why a small person can, in principle, move something surprisingly heavy with a long enough lever.

Classroom-ready puzzle: A simplified beam has an effort arm of 5 metres and a load arm of 1.5 metres. A load of 3 kilonewtons sits at the short end. Using moments, what effort is needed at the long end?

Answer: effort × 5 = 3 × 1.5, so effort = 4.5 ÷ 5 = 0.9 kilonewtons. The long arm gives a mechanical advantage.

Try it with a pencil: Draw a seesaw with a pivot. Mark a load twice as far from the pivot as the effort. What does that tell you about the force ratio? Now change the distances and write the ratio in simplest form. No engine required — just a ruler and a steady hand.

Stop two: gear teeth and the ratio you can count

Gears are a gift to a maths trail because they turn a ratio into something you can literally count. If a large gear with 60 teeth drives a smaller gear with 20 teeth, the smaller one turns three times for every single turn of the large one. The speed goes up, but the turning force goes down. That trade-off is the heart of gear trains, and it is exactly the kind of proportional reasoning that appears in KS3 mechanics and GCSE physics.

Classroom-ready puzzle: A 72-tooth gear drives an 18-tooth gear. What is the speed multiplier? If the large gear turns at 30 revolutions per minute, how fast does the small gear turn?

Answer: 72 ÷ 18 = 4, so the small gear turns four times faster: 120 revolutions per minute.

Try it with a pencil: Invent a gear pair with a ratio of 3:1. Write down two possible tooth counts. Then invent another pair with the same ratio but different numbers. What stays the same? What changes? This is a neat way to show that ratios are about relationship, not size.

Stop three: cylinders, volume and the invisible bucket

A pump house is full of cylinders, even if the cylinders are tucked inside machinery. A piston moves inside a cylinder and sweeps out a volume. The formula is the same one students meet in class: volume of a cylinder = π × radius² × height, or πr²h. Here the height is the stroke length. Once you start estimating volumes, the numbers become pleasingly physical: litres, baths, buckets.

Classroom-ready puzzle: A simplified cylinder has a bore of 0.6 metres and a stroke of 1.2 metres. Using π ≈ 3.14, estimate its swept volume in litres.

Answer: radius = 0.3 metres. Volume = 3.14 × 0.3² × 1.2 = 3.14 × 0.09 × 1.2 ≈ 0.339 cubic metres. Since 1 cubic metre is 1,000 litres, that is about 339 litres.

Try it with a pencil: A typical bath holds around 80 litres. How many baths is 339 litres? Now double the bore and keep the stroke the same. What happens to the volume? The answer is not double — it is four times bigger, because the radius is squared. That surprise is worth writing down.

Stop four: boilers, heat and sensible estimates

Boilers are where the maths gets gloriously messy. Energy goes in as fuel, useful work comes out, and a great deal disappears as heat, friction and noise. Efficiency is simply useful output divided by total input, usually written as a percentage. You do not need to know the exact coal or gas figures for the museum’s engines to practise the reasoning. A simplified model is enough.

Classroom-ready puzzle: A model boiler takes in 30 megajoules of energy from fuel and produces 9 megajoules of useful work. What is its efficiency?

Answer: 9 ÷ 30 = 0.3, or 30%.

Try it with a pencil: List three places energy might go in a real pumping station besides useful work. Then write a sentence using the words input, output and efficiency. If you want a stretch, try this: if efficiency is 25% and the input is 40 megajoules, what is the useful output? Answer: 10 megajoules.

Turning the trail into a lesson

The beauty of this route is that it moves between ratio, area, volume and proportion without feeling like a worksheet. For a family visit, award a point for each sensible estimate and a bonus point for each correct calculation. For a school group, split into pairs and give each pair one stop to explain to the others. The explaining is where the learning settles.

A few practical notes: keep pencils away from moving machinery, do not lean over barriers, and treat all figures as teaching models rather than engineering data. The real value is in the habit of asking “what is the ratio here?” and “what would change if this doubled?” Abbey Pumping Station gives you the scenery; the pencil does the rest.

If you try the trail, take a photo of your working-out sheet — not the machinery — and compare answers at the café. The best maths trails leave you with a few wrong estimates, one satisfying correction and a reason to come back.