Teaching and Learning about Metric measurement

1. Teaching and Learning about Metric measurement Key words: Imperial, Metric, metric prefixes, centi-, milli-, kilo- Purpose: This unit is designed to help tutors who teach courses that require metric measurements e.g. recognizing units, matching to quantities, and straightforward conversions between units Tutor Outcomes: By the end of the unit tutors should be able to: Develop lessons that help learners recognize which metric units are best to use in measuring situations Develop lessons in their teaching context that help learners to work out how to solve problems involving metric units and conversions

2. Section 1: Mathematical Background • Metric Measurement – conceptions • Stevin (1585, Flanders) suggests a decimal system for weights and measures’ when he writes “The Tenth” • Mouton (1670, France) proposes a ‘metric’ system based around multiples of 10. • Jefferson (1790, USA) suggests a similar decimal based measurement system for the United States but the motion is lost by just one vote. • During the French Revolution, Talleyrand (1790s) and the National Assembly direct their Academy of Sciences to . . . • “deduce an invariable standard for all the measures and all the weights”

3. Section 1: Mathematical Background • By 1795, this system of metric measure becomes the officially adopted system of measurement in France. • 1798-1799 officials from France, Denmark, Switzerland, Spain, Italy, and the Netherlands meet to agree on weights and measures according to this new system • 1840, France – the Metric system becomes compulsory during this year

4. Section 1: Mathematical Background • By 1900, thirty-nine nations have subscribed to the metric system, and by World War II that number has trebled • In 1960 a more modern system of metric measurements was established, and became known as the International System of Units, called SI. • In 1965, New Zealand’s major trading partner, the United Kingdom is (again) considering a shift to the metric system • 1967, in New Zealand, a National Metric Advisory Committee suggests that such a change here is ‘inevitable’ so work begins with many advisory boards contributing to a gradual changeover

5. Section 1: Mathematical Background • On 10 July 1967, New Zealand adopts Decimal Currency (changing from pounds, shillings, and pence to dollars and cents) • To give metrication a human face, a baby girl is presented as “Miss Metric”. Her progress (news, pictures ) is intermingled with the progress of metrication. • In 1971, a Metric logo is presented: • By 1972, metric road speed signs are up, and milk is being sold in millilitres. • In 1976, Parliament states that by 1977, everything would be sold and advertised with metric units, and that new non-metric weights and measures would no longer be verified. • The major convenience for users of the Metric System is that the sizes of the units for the same quantity are all related by factors of 10 or 100 or 1000 . . .

6. Section 2: Discussion One Q. Ask your learners: What were the Imperial measures we used to use? Do you have learners who could recognise any of these? How many are still in use today? Where? Length . . . inches, feet, yards, chains, furlongs, miles, . . . Mass(or how much an object weighs) . . . ounces, pounds, stones, hundredweights, tons, . . . Area . . . sq inches, sq feet, sq yards, roods, acres, sq miles, . . . Volume (or cubic measure) . . . cu inches, cu feet, bushels, cu yards, cords (e.g. firewood), , . . . Capacity (or liquid measure) . . . drachms, fluid ounces, pints, quarts, gallons, barrels, . . . Time . . . seconds, minutes, hours, days, weeks, fortnights, months, years, decades, ...

7. Section 2: Activity One Matching Metrics Give each pair of learners a set of metric matching cards (Copy Master 1) Each set of cards has a list of attributes of common items, and a metric measure that matches. Invite each pair to match each of their situations to the most appropriate metric measure.

8. Section 2: Activity Two • Q. How long is a metre? • Invite learners to stand up and model what one metre might be. • Ask several what they used to do this (i.e. what is their benchmark?) • Discuss the key measures: • the metre is one of the foundation or base units of the metric system • originally . . . . NP • if you could draw a ‘line’ across the surface • of the Earth, starting from the North Pole (NP) • and ending at the Equator (E) passing near • France . . . • and you divided this ‘line’ into ten million E • (10,000,000) equal parts . . . • then one of those parts is one metre

9. Section 2: Activity Three • Estimating lengths • Write 39 cm on the board. What does this look like? • Write 243 mm on the board. What does this look like? • Give an A3 piece of paper to pairs of learners and ask them to draw what each length looks like, on their paper. • Invite each pair to stick their A3 ‘posters’ up. Ask each pair how they arrived at their lengths. • Invite each person to show a one centimetre length . • Show something which could be one millimetre thick. • (e.g. coins, fingernails, wire, cardboard ?) • Discuss some of the more common vocabulary like: • centi- means “one hundredth () of the base unit” • one centimetre must be  of a metre • milli- means “one thousandth or of the base unit” • one millimetre must be of a metre

10. Section 2: Activity Four Where do we use lengths (examples)? Provide each group of learners with the cards fromCOPY MASTER 2. This provides authentic measures which need to be matched to lengths. Ask the learners for examples of lengths from their own contexts. Hand out sheets of A3 for them to record their ideas. (Invite them to draw diagrams or provide sketches of contexts they might be working on, or of projects they have completed in the past) Share the ideas from each group.

11. Important points are: • Ability to benchmark so there is a quantity in our mind or on our person which can help us compare what we know with an as yet unknown length • Recognising the words which identify where a quantity is such as length, width, height, depth, diameter, thickness, size (clothing), distance, lineal, stroke • Becoming familiar with the abbreviations for each unit and prefix • Recognising the various prefixes (centi-milli-kilo-), and knowing which represent smaller lengths and which are larger • How precise our length measurements might be depends on the context; for instance in an automotive or engineering context, fractions of millimetres might be critical • What might be the best unit of measure; a plumbing job might need pipes with a certain diameter in millimetres but they would be ordered in pipe lengths in metres * Realising that there are no plurals with metric measure – so even though we might say “25 kilograms”, or “25 kilos” we write 25 kg

12. Section 2: Activity Five Measuring lengths with a ruler We use rulers and measuring tapes to find the lengths we wish to know. These have repeated units and intervals which our learners need to recognise. • Give out pieces of paper each with an unmarked 22 cm length on it. Invite learners to guess the length; then using a ruler ask them to measure this length and compare their estimated length with the measured length. • Suggest that they write their answer with the correct unit beside the line, and look for some who write 22 cm and those who give 220 mm. Ask if these are the same? • Give out a second piece of paper with a line which is 118 mm long. Distribute ‘broken rulers / tapes’ and invite learners to measure this length; then ask them how they did this.

13. Give learners a piece of paper with lines of various lengths to estimate. Call out various lengths and ask learners to draw a line of that length, without using a ruler. Points that may arise: *Recognising how many millimetres fit into a centimetre; for instance recognising that 11.8cm is the same length as 118mm * Seeing how “zeroing” a ruler / tape around 0, 10, 20, …makes it easier to measure

14. Section 2: Discussion Two Mass We usually say the word “weight” when we mean mass. In science, weight is considered a force exerted on a body by gravity. The main point is that adult learners know what we are talking about so we would more likely use the word “weight”. Words used in questions about mass / weight which could be asked: • How heavy is that . . .? • How light is that . . . ? • What does that weigh . . . ? Though the base unit for mass / weight is the gram, since this is comparatively small (compared with the imperial pounds and ounces it replaced) the standard unit of mass is the kilogram (1000 g); the kilogram (or kg) is a ‘decent’ sized unit.

15. Section 2: Activity Six Prepare a set of random weights for people to pick up and compare. Items could be heavy books like phone or text books, plastic bags with salt or sand, timber, bricks, and so on. Ensure that a few of the items have similar weights. Also have two known weights (e.g. bags of sand) – one of 250g and another of 1kg. Invite people to put the unknown weights in order from lightest to heaviest. Get them to come to a consensus about the order of the weights – and justify any decisions on the order which were debated. Leave the weights in position. Introduce the two known weights - 250g and 1kg and invite people to place these in the line up. Once the positions are agreed on, ask people to estimate the weights on either side of the 250g (or the 1kg). Record on the board and agree on a median. Have some scales ready to measure the weights; compare these to their estimates.

16. Either using the same objects or a set of common items like cell phones and books, get people to measure these on a set of scales. Observe what they do in order to accurately read the display. If the scales are digital, discuss what the numbers might mean. If the scales are traditional, depending on the size of the display, discuss what the intervals might represent. For instance, the intervals could be . . . . . . in 10g, in 20g, in 25g, in 50g, in 100g, etc. Important points are: • Benchmarks for mass, unlike those for length, are not carried with us on our person so we rely on familiar objects (sugar, butter, etc) to help us compare what we know with an as yet unknown mass • Digital scales rely on users having some knowledge of decimal place values. • Traditional scales rely on users recognising the units and the intervals and scaling on the display. Learners may need assistance to see that a reading of 1kg and an extra 230g could also be given as 1.23kg or 1230g • Grams are abbreviated to just ‘g’ but not gr nor gms

17. Section 2: Discussion Three Volume and Capacity These two measures have virtually the same units and might seem interchangeable, though there are subtle differences in emphasis. Volume usually means what we call “cubic volume” where the amount of space inside an object is measured in units such as cubic centimetres (cu. cm or c.c. or cm³) or in cubic metres (cu. m or m³). On the other hand Capacity usually describes how much liquid (or gas) could fit into or be held by a container, usually measured in litres (l or L or 𝓵) or millilitres (mL). There are familiar situations like fridges whose (cubic) size or volume is usually given in litres. Again the main point should be that adult learners recognise what is being disussed.

18. Section 2: Activity Seven Discuss with learners what volume is and the possible units to use. In pairs, ask learners what a cubic centimetre might look like or perhaps sketch one. (If possible have some cubic centimetres available afterwards for them to look at) Prepare a set of cubic objects (books, bricks, timber off-cuts, etc) to compare. Ask them to estimate how many cubic centimetres could fill each object. This will be an estimate of how much space or volume that object could displace. Distribute rulers or tape measures so they can check their estimates. Ask them how they might do this by measuring the object. Is there a rule they might be using? If they are multiplying length by width by height invite them to explain why they do this – though obvious to some, others may not have realised that it is the area of the cross-section (or end slice) moving through the object much like a slice of bread though a loaf.

19. Section 2: Activity Eight Prepare a set of objects which could be filled with water, for pairs of people to measure. If possible remove or block out any labelling on each object which has millilitres or litres. Ensure you are at a venue where there is water and some measuring jugs with clearly marked metric scales available. Ask each pair to agree on an estimate of how many litres or millilitres of water that could fill each object. Many will have a feel for the size of milk or soft drink containers and they could refer to these when justifying their estimates. Then invite the pairs of learners to fill each object with water then pour this into the measuring jugs to work out the capacity of each. Ask them to record their figures ensuring that the quantities have the correct units beside them. Ask each pair to pick an object and report their estimated capacity and then their measured capacity.

20. Section 3: Modelling metric conversions The metric measures are related like a stairway – the base units (m, L, g) are in the centre ‘landing’, the smaller units lie below, while the larger units are above.Each step along the stairway is another factor of 10 –To change the unit of expression (eg., to express a kg measure as g) : - each step you move down is another  10 (since smaller units means there are more)- each step you move up is another  by 10(since larger units means there are less)

21. Section 3: Examples A house plan shows 2540 millimetres – how many metres is this?Since 2540 mm  10  10  10 = 2.540 m (So converting from mm  m, or small  large, goes UP 3 steps or places ) Common sense check: 1000 mm is 1 m, so 2000 mm is 2 m and 3000 mm is 3 m. 2540 mm is between 2 m and 3 m. 2540 mm = 2.540 m is reasonable.

22. Section 3: Examples A horse eats 12 kilograms of feed each day – how many grams is this?Since 12 kg  10  10  10 = 12,000 g (So converting from kg  g, takes three steps or place values down) Common sense check: 1 kg is 1000 g so 12 kg is 12 lots of 1000 g. 12 kg = 12 000 g is reasonable.

23. Section 3: Examples Try the following conversion examples with your learners, to practise the ideas introduced so far. • One of the Police Fitness Assessments is a 2.4 kilometre run which candidates must complete in a certain time. How far is this distance in metres? • Vince tries on a pair of jeans at the local Y-Mart. He has an 88 cm waist. What is this in metres? • Judy checks the new three litre bottle of juice in her fridge and it is only half full! - how many litres of juice are left? - convert this quantity to millilitres.

24. Section 3: Examples • One of the cars in Marvellous Merv’s Mighty Motors has an odometer reading of 103,465 km. - How many kilometres over 100,000 km is this? - What is this extra distance in metres? • Before selling one of the vehicles, Merv notices that one of his cars will fail a warrant because the tread depth in the front tyres is just 3 mm. How much is this in centimetres?

25. Section 3: Activity Metre to Centimetre and Millimetre snap • Play a game of snap with cards made from COPYMASTER 3. • This game is designed to practise straightforward conversions between metres, centimetres , and millimetres. Points that may arise: • Decimal values in metres can become whole numbered values of cm and mm - for instance 1.5 m is the same as 1500 mm or 150 cm • Need to appreciate how place values underpin conversions within the metric system of measures. • Play a game of snap with cards made from COPYMASTER 4. • This game is designed to practise straightforward conversions between grams and kilograms, and litres and millimetres.

26. Section 4: Assessment 1. Estimate the length of this line in centimetres 2. Measure this line to the nearest millimetre • Convert your answer in 2. to metres • What would be the best unit to measure the capacity of a bath?