Pariksha SaathiLesson 5 of 8 · Mathematics, Classes I-V (Paper I)
Measurement: Length, Weight, Capacity, Time and Money in Standard and Non-standard Units
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From comparing to standard units
Measurement develops in stages. Children first compare directly (which pencil is longer, which bag is heavier), then measure with non-standard units such as hand spans, cubits, foot lengths, steps, pencils or glasses, then see the need for a standard unit and finally convert between units. Math-Magic deliberately lets children get different answers with their own spans and steps (the Class 2 chapter The Longest Step), because the disagreement shows why a shop or a tailor needs the metre. On a ruler, measure from 0 or subtract the starting mark: a pencil lying from the 2 cm mark to the 9.5 cm mark is 7.5 cm long. Piaget found that young children do not yet conserve length or quantity: they may think a stick pushed forward has become longer, or that water poured into a taller, thinner glass has increased.
Length
10 mm = 1 cm, 100 cm = 1 m and 1,000 m = 1 km. To change a larger unit into a smaller one, multiply; to change a smaller unit into a larger one, divide. So 3 km 45 m = 3,045 m, 2 m 5 cm = 205 cm and 4,250 m = 4 km 250 m. The unit should suit the object: millimetres for the thickness of a coin, centimetres for a pencil, metres for a room and kilometres for the distance between towns. A typical problem: from a 10 m rope, 3 pieces of 2 m 75 cm each are cut; together they measure 825 cm, so 175 cm, that is 1 m 75 cm, is left.
Weight (mass)
A simple balance comes first: the heavier side goes down, and bigger is not always heavier (a pillow and a stone), as the Class 3 chapter Who is Heavier? and the Class 4 chapter How Heavy? How Light? show. 1 kg = 1,000 g, 1 g = 1,000 mg, 1 quintal = 100 kg and 1 tonne = 1,000 kg. Shop weights are usually 50 g, 100 g, 200 g, 500 g, 1 kg, 2 kg and 5 kg, so 700 g can be weighed with 500 g and 200 g, and 850 g with 500 g, 200 g, 100 g and 50 g. Adding with grams: 2 kg 350 g + 1 kg 875 g = 4 kg 225 g, since 350 g + 875 g = 1,225 g = 1 kg 225 g.
Capacity
Capacity is how much a container can hold. Children pour water, sand or rice from one container into another and count how many glasses fill a bottle or a bucket (the Jugs and Mugs chapters of Math-Magic), and then use the litre: 1 L = 1,000 mL, half a litre is 500 mL and a quarter litre 250 mL. A 2 L bottle fills 8 glasses of 250 mL. 3 L 400 mL - 1 L 750 mL = 1 L 650 mL, since 3,400 mL - 1,750 mL = 1,650 mL. A tall, thin bottle may hold less than a short, wide one, which children should check by pouring.
Time
Time is learnt first through daily routines and then through the clock and the calendar. The minute hand takes 1 minute for each small division and 5 minutes from one number to the next; the hour hand moves from one number to the next in 1 hour. 60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day, 7 days = 1 week and 12 months = 1 year of 365 days, or 366 days in a leap year, when February has 29 days. A year divisible by 4 is a leap year, but a century year is one only if it is divisible by 400 (2000 was, 1900 was not). January, March, May, July, August, October and December have 31 days; April, June, September and November have 30. 7:45 p.m. is 19:45 on the 24-hour clock. Intervals are found by counting on: a train leaving at 10:40 a.m. and arriving at 2:15 p.m. takes 20 minutes to reach 11:00 and 3 hours 15 minutes more, 3 hours 35 minutes in all. On a calendar, dates 7 apart fall on the same day: if 5 March is a Monday, so are 12, 19 and 26 March, and 1 April is a Sunday.
Money as a measure
Money measures value, and rupees and paise work like any other pair of units: ₹ 1 = 100 paise, so ₹ 12.50 = 1,250 paise. Children make the same amount in different ways (₹ 50 as two ₹ 20 notes and a ₹ 10 note, or five ₹ 10 notes) and give change by counting up from the price. Shopping problems use rate × quantity and the unitary method: if 12 pencils cost ₹ 60, 1 pencil costs ₹ 5 and 7 pencils cost ₹ 35. A bill for 2 kg of sugar at ₹ 46 per kg and 3 packets of biscuits at ₹ 15 each comes to ₹ 92 + ₹ 45 = ₹ 137, leaving ₹ 63 change from ₹ 200.
Key facts
- Stages of measuring: direct comparison, non-standard units (span, cubit, step), standard units, then conversion; different spans give different answers, which is why a standard unit is needed.
- Ruler: an object from the 2 cm mark to the 9.5 cm mark is 7.5 cm long; measure from 0 or subtract the starting mark.
- Length: 3 km 45 m = 3,045 m; 2 m 5 cm = 205 cm; 4,250 m = 4 km 250 m; larger to smaller unit multiply, smaller to larger divide.
- Weight: 1 kg = 1,000 g, 1 quintal = 100 kg, 1 tonne = 1,000 kg; 2 kg 350 g + 1 kg 875 g = 4 kg 225 g.
- Capacity: 1 L = 1,000 mL; a 2 L bottle fills 8 glasses of 250 mL; 3 L 400 mL - 1 L 750 mL = 1 L 650 mL.
- Leap year: divisible by 4, but a century year only if divisible by 400 (2000 yes, 1900 no); February then has 29 days.
- Time interval: 10:40 a.m. to 2:15 p.m. is 3 hours 35 minutes; 7:45 p.m. is 19:45 on the 24-hour clock.
- Calendar: dates 7 apart fall on the same weekday; if 5 March is a Monday, 26 March is a Monday and 1 April a Sunday.
- Unitary method: 12 pencils for ₹ 60 means 1 pencil for ₹ 5, so 7 pencils cost ₹ 35.
- Piaget: young children do not yet conserve length or liquid quantity; pouring and re-measuring builds the idea.
Quick revision
Compare, then non-standard units, then standard units, then convert; 10 mm = 1 cm, 100 cm = 1 m, 1,000 m = 1 km; 1 kg = 1,000 g, 1 quintal = 100 kg; 1 L = 1,000 mL; larger to smaller unit multiply, smaller to larger divide; 60 s = 1 min, 60 min = 1 h, 24 h = 1 day; leap year 366 days, century years only if divisible by 400; 10:40 a.m. to 2:15 p.m. = 3 h 35 min; ₹ 1 = 100 paise; bill = rate × quantity.