Units, symbols and metric prefixes

Picture this. You download a file for a phone stand and send it to the school 3D printer. Two hours later it hands you a perfect little phone stand, small enough to sit on your fingernail. Nobody touched the file, and every number in it is exactly right. So what shrank it?

Every measurement on a product has three parts. The quantity is what you are measuring, such as mass or length. The value is the number. The unit is what that number counts in, such as the kilogram or the metre. Each unit has a symbol, the short form you write down. Mass is measured in kilograms, symbol k g. Length is measured in metres, symbol m. Energy is in joules, symbol J, and time is in seconds, symbol s. The case of each letter matters. Kilo is always a small k, and milli is always a small m. The joule takes a capital J, because it is named after a person, the scientist James Joule. Capital K capital J, small k small j, or small k capital J: which is right for kilojoules? Small k, capital J. The first two each get one letter's case wrong.

A prefix is the part in front of a unit that scales it. Kilo means a thousand, so a kilogram is a thousand grams. Prefixes sit on a ladder, biggest at the top, and each rung is a power of ten. Ten to the three means three tens multiplied together, which is a thousand. Tera is ten to the twelve, the T on a one terabyte hard drive. Giga is ten to the nine, the G on a sixty-four gigabyte memory stick. Mega is ten to the six, a million. Below the plain unit the powers turn negative, and a minus power means divide. Centi is a hundredth, ten to the minus two. Milli is a thousandth, ten to the minus three. Micro is a millionth, and nano is ten to the minus nine, the nano in nanomaterials. This video works with three rungs: kilo, centi and milli. From kilo down to the plain unit is a step of a thousand, and from the plain unit down to milli is another thousand. Centi turns up in centimetres, and one centimetre is ten millimetres. Here is how to tell which way to go. Weigh a bag of sugar in grams and you need a thousand of them. Weigh it in kilograms and you need one. The bigger unit always needs the smaller number. Three kilograms written as three hundred grams. What has gone wrong? The factor. Kilo means a thousand, not a hundred, so three kilograms is three thousand grams. One examiner's report says: When answered incorrectly it was because there was a misconception that three kilograms was three hundred grams rather than three thousand grams. The same report notes that many candidates were able to convert kilograms into grams. Both are true of one question: many got it right, and the slips came from one wrong factor. The fix is the ladder. Write the factor on the step, times a thousand from kilo down, before you multiply anything, then check the bigger unit has the smaller number.

A parcel puts the ladder to work. A small business posts twenty-five laser-cut acrylic phone stands, each with a mass of eighty-four grams, in a box that weighs three hundred and fifty grams. The courier charges by the kilogram. Start with a units line: every quantity with its unit, all in grams for now. The stands first. Twenty-five times eighty-four grams is two thousand one hundred grams. Next, add the box. Two thousand one hundred grams plus three hundred and fifty grams is two thousand four hundred and fifty grams. Into kilograms now: multiply by a thousand, or divide by a thousand? Divide. The kilogram is the bigger unit, so it needs the smaller number. Two thousand four hundred and fifty divided by a thousand is two point four five. The parcel is two point four five kilograms. Divide by a hundred by mistake and you would post a twenty-four point five kilogram parcel. Another examiner's report, on a calculation with several steps, records that most missed the opportunity to convert the units. The numbers were there, but they were never turned into the unit the next step needed. The habit that prevents it is the units line, written before any multiplying, with every quantity in one unit.

Next, a label you can find on a bike light. Its battery pack says two point six amp-hours, and its LED draws five hundred and twenty milliamps. Amp-hours tell you how much the battery holds, and milliamps tell you how fast the light uses it. Battery life in hours is the capacity in milliamp-hours divided by the current in milliamps. The two numbers must carry the same prefix before you divide one by the other. So what is two point six amp-hours in milliamp-hours? Two thousand six hundred. Milli means a thousandth, so one amp-hour is a thousand milliamp-hours, and two point six times a thousand is two thousand six hundred. Now both numbers are in milli units, so they can be divided. Two thousand six hundred divided by five hundred and twenty is five. The light runs for five hours. The question wants that time in minutes. An hour is sixty minutes, so five hours is five times sixty, which is three hundred minutes. One report, on a question that wanted a time in minutes, puts it this way: A well answered question where very many candidates scored well. In the few instances where candidates only scored one mark, they did so because they did not convert the two hours into a hundred and twenty minutes. So the slip was the last line, not the method. Read which unit the question asks for, and make converting into it the final step you write. Times need the same care. A student times a laser cut on a stopwatch and writes the time down as a decimal. Two minutes forty-five seconds as two point four five minutes: what is wrong? A minute is sixty seconds, not a hundred. Forty-five seconds is three quarters of a minute, so the time is two point seven five minutes. Or put it all in seconds. Two minutes is a hundred and twenty seconds, plus forty-five makes a hundred and sixty-five seconds. Pick one unit for every time before you multiply.

Units belong in a design specification too. A specification point is one thing the finished product must do, written so it can be checked. A student's step stool specification says it must be light. A better version says it must have a mass of no more than two point five kilograms. Which of those two points could you test, and why? The second. It names the quantity, gives a value and carries a unit, so a set of scales gives a clear yes or no. The first has nothing to measure against. The other two points get the same treatment. Height: the standing surface is two hundred and thirty millimetres above the floor. Load: it holds a person with a mass of up to one hundred kilograms without breaking. The plywood for that stool comes from a supplier who lists its sheets as eight foot by four foot. Feet and inches are imperial units, an older system that some materials and components are still sold in. One number is worth knowing here: one inch is twenty-five point four millimetres. A foot is twelve inches, about three hundred and five millimetres, so the sheet is about two thousand four hundred and forty by one thousand two hundred and twenty millimetres. Now the phone stand. It was drawn three inches tall. Read as millimetres, how tall does it print? Three millimetres, when it should be about seventy-six. Each number was right, but it was counting in the wrong unit. That is what shrank the phone stand. The file was drawn in inches, and the printer software read the very same numbers as millimetres, a unit twenty-five point four times smaller.

Time to pull it together. Four quick questions, most of them using a rule from this video on a measurement it has not shown you. A sheet of acrylic is three millimetres thick. What is that in metres? Nought point nought nought three metres. The metre is the bigger unit, so you divide by a thousand. Next, how many seconds are there in one and a half hours? Five thousand four hundred. One and a half hours is ninety minutes, and ninety times sixty is five thousand four hundred seconds. Now the ladder: what power of ten does the prefix mega stand for? Ten to the six, which is a million. A megajoule is a million joules. One more: how would you turn 'the lamp must be small' into a point you could test? Name the quantity, give a value and add the unit, such as a height of no more than three hundred millimetres. Any sensible value works, as long as it can be measured. A number on its own never tells you a size. The phone stand had every number right and the wrong unit, and that is why the unit rides with every number you write.

Each video in this chapter stands on its own, so the thumbs up becomes your own record of where you are. If you feel you could teach this one, thumb it up and move on. If units still feel slippery, save it to a playlist, because three questions in, it starts to feel normal.

Next in the chapter: Decimals and standard form.

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