UNIT 6 MEASUREMENT AND MENSURATION Structure 6.1 Introduction 6.2 Objectives 6.3 Informal Measurement of Length, Weight, Area, Capacity, Money and Time 6.4 Standard Units of their Interrelationship 6.5 Conversion of Units in Metric System 6.6 Four Fundamental Operations using the Units of Measurement 6.7 Let Us Sum Up 6.8 Unit-end Exercises 6.9 Answers to Check Your Progress 6.1 INTRODUCTION Measurement is the most important area which children can apply their knowledge of numbers. In primary classes children are made aware of units of measurement and about the devices by which measurements are made. They are also provided with first hand experiences in which measures are applied. Through activities of daily life children are led to describe many of the quantitative aspects of the environment. Our system of measurement is metric. As such the rules and procedures followed for operations on whole numbers apply to measures of length, weight and capacity. 6.2 OBJECTIVES After going through this unit, you will be able to teach your students to: appreciate the need for standard units of measurement of length, weight, area, capacity, money and time; recall different units of measurement of length, weight, area, capacity, money and time; identify the relationship between different units of measurement; convert one unit of measurement to another; estimate and compare the lengths of familiar objects and short distances, weights of given objects, floor areas or areas of fields, capacity of containers, and time; and use proper units of measurement in different situation. 6.3 INFORMAL MEASUREMENT OF LENGTH, WEIGHT, AREA, CAPACITY, MONEY AND TIME The first stage in the teaching of measurement consists of informal experiences in which they compare two (like) things such as two distances, two masses, 5
Teaching of Basic Mathematical Concepts II two capacities etc. The purpose is to train the senses and give a feeling of more, less, larger, shorter, heavier, lighter, bigger, smaller, etc. To measure means to compare using multiplication i.e. the word times. The second stage in the teaching of measurement is to introduce defined but varying units, Body parts have been used as units. The following activities illustrate these two stages. Activity 1 Take any two rods of different sizes as shown in fig. A B P Q (Note that rods represent segments AB and PQ.) Stage 1: Ask pupils to put PQ against AB and use phrases such as AB is longer than PQ; PQ is shorter than AB. A little later they may say AB is two times PQ, PQ is one half of AB. This would give informally an idea of unit of measure and measure. AB = 2PQ; PQ is the unit; 2 is the measure. The measure of a segment is called its length; length is a number. Stage 2: Ask pupils to use body parts such as span or palm of the hand or any other rod selected at random and measure the two rods. Let them observe that AB = span and PQ = span and compare the measures of AB and PQ in terms of span which is the unit of measure. Sizes of span/palm will differ with different pupils. Hence the need for using a standard unit of measure. Activity 2 Take any two different objects (masses) say banana and a feather; a book and an empty lunch box; an apple and a lemon. Stage 1: Ask pupils to hold the objects in two hands and feel which of the two is heavier and which is lighter. Stage 2: Next take a balance and put the objects in two pans and find out which is heavier. Now take marbles or pebbles and find out how many marbles or pebbles balance object 1 and how may balance object 2. A pebble is a unit and the number of pebbles which balance the object give the weight of the objects. The weight of an object is a number. Activity 3 Stage 1 : Ask pupils to observe that the size of some rooms is small and of some is big. Small rooms accommodate less number of children and larger rooms accommodate more children. Hence, we are actually comparing the floor of the rooms. The floor of a room represents rectangular region. Fig. 6.1 Similarly, pupils may compare the table tops. The teacher s table has a bigger top than the top of the desk of the pupil. 6 Stage 2: Next ask pupils to take sheets of paper and spread them on the table top. How many sheets of paper cover the table top completely? Say 9; ask them
to sense that the area of the table top is 9 sheets of paper. Here a sheet of paper is the unit and 9 is the area. 7 8 9 6 5 4 1 2 3 Similar, activity may be performed with post cards and children may observe that when the unit is large the area is a smaller number. When the unit is smaller, the area is larger number. The sheet of paper or the post card are informal units. Activity 4 Comparison of capacities. Stage 1: Ask pupils to take a mug and a bucket. Let them observe that a mug holds less water than a bucket. Let pupils observe that many things like milk, oil, medicines etc. are sold in containers of different sizes. Stage 2: Next let pupils pour water in a bucket with the mug and observe that a bucket holds say 20 mugs filled with water. Let them say; capacity of bucket = 20 mugs. Here a mug is a unit of capacity and 20 is the measure of the capacity of the bucket. Mugs may be of different sizes and hence a mug is an informal unit. Time Measure Time was originally measured with natural units such as the day, month or year. Latter, mechanical devices such as water clocks, sand clocks and shadow clocks were used. Children begin to learn about time through the study of the seasons, the calendar and the clock in the classroom. They learn how the clock helps to regulate the activities of daily life. They should also familiarize themselves with time tables and how they make travel or any activity easy. Money Measure Money is the measure of the value of commodities. Since ancient times people exchanged commodities. Let pupils sense that trade and industry are not possible without money. Informal experiences of exchanging goods and using money in buying, selling activities should be encouraged. Before coming to primary classes children are already familiar with coins and currency notes. They know that our standard of money is the rupee and that 1 rupee = 100 paise. Check Your Progress 1. Ask any 5 students of a class to measure the length of the classroom with footsteps. Write the length measure by each students. 2. With the help of a balance, weight a stone with the help of pebbles and write the weight of the stone. 3. Take a sheet of paper and find its area in terms of a square sheet of width equal to length of a chalk piece. 4. Measure the capacity of a bottle in terms of a cup. 5. Write the Indian standard unit of measurement of money. 6. Write the names of different parts of a day. 7
Teaching of Basic Mathematical 6.4 STANDARD UNITS OF MEASUREMENT AND Concepts II THEIR INTERRELATIONSHIP Through informal activities mentioned earlier teacher may create in the mind of pupils need for a uniform standard system of measurement. Earlier a group of countries or even a country had its own system of measurement. If necessary teacher may give examples of use of earlier British measures used in our country. However, now the Metric System of weights and measures is recognized as the universally accepted system. Metric system is related to our number system as it used base-ten. The ratio between consecutive linear units in this system is the same as the decimal ratio of consecutive places in our number system. The units of measure in metric system are: Length metre, weight gram; capacity litre (m) (g) (l) The teaching of standard units should be organised in 3 stages. 1) Introduction of commonly used units metre, centimeter and kilometer for length; kilogram and gram for weight; litre and milliliter for capacity. 2) Introductions of multiple units and subunits and their relationships, conversion of units. 3) Four fundamental operations with measures (viz.). The students should be encouraged to learn with the help of suitable activities and examples. A few of these activities are given here. Activity 1 To introduce metre and centimetre Teacher may collect tools such as a metre rod, a 30 cm. ruler, tailor s tape, tape used for measuring the playground, etc. and elicit their use from pupils. Ask pupils to see the markings on them. Give pupils experiences in using these tools and find out their heights, sizes of ribbons, pencil, floor or table lengths etc. Through use let pupils understand that 1 metre (m) = 100 centimetres (cm) Ask pupils to record measures using these units. Ram s height = 1 metre, 20 centimetres Size of pencil = 12 centimetres Size of the eraser = 3 centimetres Ask pupils (i) express the measures given in metre and centimetre into centimetres. 1 m 20 cm = 120 cms. 3 m 5 cm = 305 cms. Ask pupils (ii) express the measures given in centimeters into metres and centimeters. 435 cm = 4 m 35 cms. 715 cm = 7 m 15 cms. 8
Activity 1 a) Let pupils recall their travel experiences. Far and near concepts should be informally elicited from their daily experiences of travelling within the city or between any two towns. Reference may be made to time-tables and sign stones/boards giving distances on the road sides. Long distances are measured in Kilometres. 1 Kilometre (Km.) = 1000 metres (m) Let pupils work out distances in Km. with the help of time-tables/reading meters of cars/buses/scooters etc. Let pupils measure distances around parks/tanks etc. in metres (using a tape) and convert into kilometers. Let pupils take rounds of football/hockey field land get a feel of kilometer distance. Activity 2 To introduce kilogram and gram Teacher may collect balances/weights and elicit their use. Let them recall what commodities they buy in grams and what they buy in kilograms. Let them make a record of daily purchases of vegetable, household materials etc. and note the use of different types of balances and weight measures. Let students note their weights on a weighing machine. Through use let pupils understand that Activity 3 1 Kilogram (Kg) = 1000 Gram (g) To introduce litre and milliliter Teacher may collect cartons/containers of milk, oil and water and let pupils recall that these commodities are measurd in litres. Teacher may also collect empty bottles of medicines, perfumes, etc. and let pupils notice that some commodities are used in smaller quantities and are measured in milliliters. Let pupils make a record of daily purchases of oils, milk, petrol etc. and note different sizes in which these commodities are available. Let pupils observe the containers used by milk vendors (these are cylindrical) and those used for oil (these are conical). Let pupils calibrate glasses/mugs and buckets in litres/milliliters. Through use let pupils understand that 1 litre = 1000 millilitres (ml) Activity 4 To introduce square measures Teacher may use square regions to find area of postcard; table top; floors etc. by counting the squares. This can be done by asking pupils to paste squares cut from paper or by drawing the figures on a paper with square grid drawn on it. Half and more than half squares are counted as one and squares less than half are ignored. Note: Both regular and irregular figures may be used. Let pupils cut squares with 1 cm side and 1 m side and use these to find out the area of sheets of notebook paper, book top, etc. and floors carpets, curtains etc. using cm 2 and m 2 respectively. 9
Teaching of Basic Mathematical Concepts II Money Measures Activity 5 To introduce currency notes/coins Children are already familiar with currency notes/coins. The teacher should encourage them to recognise the shapes/sizes of coins and also understand the concept of their value. Encourage identification and selection of coins with the help of playway shops and display of prices on tags. Let pupils match values of coins. For example 50 paise coin = 2 coins of 25 paise = 2 coins of 20 paise and a coin of 5 paise = 5 coins of 10 paise etc. Let pupils note the cost/price of everyday things, they buy in rupees and paise. Postal articles, stamps etc. may also be used. Situations like how can I pay for an article costing 65 paise if I have 1 50 paise coin 2 10 paise coins 2 25 paise coins 4 5 paise coins, etc., may be discussed. Time Measure Children are already familiar with broad distribution of time in a day e.g. Night, Morning. Afternoon, Evening and Night again. But for regulating activities, we need a device called clock or watch. Teacher may display a model of clock. A clock tells us the time the numerals give us the hours. The long hand is the minutes hand, the short hand is the hour hand. Activity 6 Reading time from a Clock Teacher may elicit from pupils that we use clocks to tell the time. Time is measured in minutes and hours. There are two hands which move on the dial. The short hand is called the hour hand and shows the hour. The long hand is called the minutes hand and shows the minutes. In one full rotation of a minute hand, the hour hand moves one plan i.e. One hour is equal to sixty minutes. The minute hand takes five minutes to move from one numeral to another. The teacher may present figures as following and ask pupils to read time. Minutes hand is on 6 and hours hand in between 2 and 3. 10 Minutes is 6 5 = 30 minutes So time = 2 hours and 30 minutes.
Days, Weeks, Months and Year The teacher may display the calendar and develop the idea of week, month and year. Seven days make a week. They are Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday. Four weeks make a month. Twelve months make a year. The months are January, February, March, April, May, June, July, August, September, October, November and December. Every month does not have same number of days. Some months have 30 days, other have 31 days while February month has 28 or 29 days. To remember number of days in a particular month, we use our left hand figures base. For this start counting the months, associating the first month with the base first finger, second month between first and second figure, third month at the base of second figure and so on. Thus the eighth month will be associated with base of first finger, the ninth month between first and second fingers. Months which fall on the base of the fingers have 31 days other have 30 days except February which has 28 or 29 days (in leap years). This can also be learnt easily from the following Rhyme. Thirty days has September, April, June and November, All the rest have thirtyone, except February alone, which has twentyeight days clear, but twentynine each leap year. Let us sum all this in table from. 60 second = 1 minute 60 minutes = 1 hour 24 hours = 1 day 7 days = 1 week 30 or 31 days = 1 month 12 months = 1 year 365 days = 1 year 366 days = 1 leap year Note that a leap year falls every four years. Check Your Progress 7. Name the units of length in a sequence. 11
Teaching of Basic Mathematical Concepts II 8. Write the relationship between a kilogram and a hectogram. 9. 1 sq. m =... sq. cm. 10. What is the name used for dm 3 11. Write the smallest currency being used in India. 12. What is the number of days in a leap year. 6.5 CONVERSION OF UNITS IN METRIC SYSTEM As already mentioned the metric system follows the same place-value principles as our number system. Therefore, conversion of units does not require much computation. The teacher should introduce the complete table of metric measures and held the student to develop skills in reading and writing numbers on that table. Thousands Hundreds Tens Ones One One One tenth hundredth thousandth 1000 100 10 1 1/10 1/100 1/1000 Kilo Hecta Deca Unit Deci Centi Milli m/g/l 5 3 4 8 5 9 2 1 6 Meaning of the prefixes should be explained from the table. Deca means ten times Hecta means one hundred times Kilo means one thousand times Deci means one tenth times Centi mean one hundredths times Milli means one thousandths times. In actual every day usage some prefixes are not spoken. For example 5 Km 3 hectametre, 4 decametre, 8 metre will be read as 5 Km 348 m. 5 Decagram 9 gram 2 decigram 1 centigram 6 milligram will be read as 59 g 216 milligram. 12 To separate the multiple units from subunits a vertical bar is used. This can be later on replaced by a decimal point (.).
Example 1: The distance of school from house is given as 3 Km 252 m. We have to convert it into metres. This can be done in two ways. 1. By computing 1 Km. = 1000 m. Therefore, 3 Km = 3 1000 m = 3000 m. So the distance of the school from house is 3 Km 252 m = 3000 m + 252 m = 3252 m. 2. By using the table We write unit s digits on the table. Digit 3 will go in Km. Column. The unit s digit 2 will go in the metre column in the table, we read 3252 m. Example 2: Change 5857 cm into metre. 1000 100 10 1 Km Hm Dm m 3 2 5 2 1. By computation We know 1 m = 100 cm. So we divide by 100 5857 cm = 5857/100 m = 58.57 m. 2. By using tables Deca unit 1/10 1/100 1/1000 deci centi milli 5 8 5 7 We write 5857 cm on the table. The first digit 7 will go in centimetre column. We read 58 m 57 cm or 58.57 m. Example 3: Change 1856 m into Kilometres. 1. By computing We know that 1 Km = 1000 m so we divide by 1000 1856 = 1856/1000 Km. = 1.856 Km. 2. By using tables Kilo Hecta Decca Unit 1 8 5 6 We write on the table. The units digits will go in metre column. It will be followed by writing 581 on the left of 6. Thus in Kilometre column, we get integer 1. We read 1 Km 356 m or 1.856 Km. Example 4: Change 5 gram 7 decigram 8 centigram into centigram. 1. By computing 5 gram = 5 100 cg. = 500 cg. 7 dg. = 7 10 cg. = 70 cg. 8 cg. = 8 cg. Total = 578 cg. 13
Teaching of Basic Mathematical Concepts II We convert 5 grams in terms of cg. by multiplying by 100, 7 decigram to cg. by multiplying by 10 and 8 cg. Thus we find the sum of all the figures to get 578 cg. 2. By using the table unit deci centi milli g 5 7 8 We write on the table and read 578 cg. Area You are aware that the unit of measurement of area is the square of a unit. If the side of the square is 1 cm then the unit of the area will be 1 sq. cm or cm 2 and if the side of the square is 1 m. then the area will be in terms of sq. m. or m 2 and so on. For conversion of one unit into another, let us use this relationship between the units of length and area. Units of length Units of area 1 km = 10 hm 1 sq. Km. = 10 10 sq. hm. hm 2 1 hm = 10 dam 1sq. Hm = 10 10 sq. dam dam 2 1 dam = 10 m 1sq. dam = 10 10 sq. m m 2 1 m = 10 dm 1sq.m = 10 10 sq. dm dm 2 1 dm = 10 cm 1sq.dm = 10 10 sq. cm cm 2 1 cm = 10 mm 1sq.cm = 10 10 sq. mm mm 2 From this we get 1 sq. km = 100 100 100 sq. m. = 1000000 sq. m 1 sq. m = 100 100 sq. cm = 10000 sq. cm The above relationship is to be used for conversion of one unit of area into another. Let us consider the following examples. Example 5: Convert 5 sq. cm 50 sq. cm into sq. cm First we convert 5 sq. m. into sq. cm We know that 1 sq. m = 5 10000 sq. cm = 50000 sq. cm 5 sq. m 50 sq. cm = (50000 + 50) sq. cm = 50050 sq. m. 14 Example 6: Convert 80000 sq. cm into sq. m Here, we have to take 100 100 sq. cm = 1 sq. m or 1000 sq. cm = 1 sq. m Therefore 80000 sq. cm = 80000/10000 sq. m = 8 sq. m. Thus 80000 sq. cm is the same as 8 sq. m.
Example 7: Let us use the table for conversion of 6578 litre into hectoliter. Kl hl decl l dl cl ml 1000 100 10 1 1/10 1/100 1/1000 6 5 7 8 To write 6578 l, start writing from litre column and put 8 in litre column 7 in decl, 5 in hl. and 6 in Kl. column. Thus 6578 litre reads as 6 Kl, 5 hl, 7 decl and 8 l. For conversion 6578 into h1 put decimal point just after h1. column i.e. Money 6578 l = 65.78 hl. For conversion of money let us consider following examples. Example 8: Change Rupees into paise. We know 1 Rupee = 100 paise Rs. 2 = Re. 1 + Re. 1= 100 P + 100 P = 200 P Rs. 3 = Re. 1 + Re. + Re. 1 = 100 P + 100 P + 100 P = 300 P So 5 rupees = 5 100 = 500 paise Similarly Rs. 6 = 6 100 = 600 P Rs. 7 = 7 100 = 700 P Rs. 4.65 = 4.65 100 = 465 P 1 Re. = 100 paise To change the Rupee into paise we have to multiple by 100 or you can say remove the point. Example 9: Change paise into rupees and paise. 115 P = 100 P + 15 P = 1 Re + 15 P = Rs. 1.15 225 P = 200 P + 25 P = 2 Rs. + 25 P = Rs. 2.25 To change paise in rupees, you put decimal point after two digits from the end. Thus, Time 565 P = Rs. 5.65 407 P = Rs. 4.07 350 P = Rs. 3.50 You know that 1 day has 24 hours, 1 hour has 60 minutes and 1 minute has 60 seconds. But if we consider the days and months or year, we face the problem. A month may have 28 to 31 days, and a year may have 365 or 366 days. So before starting the conversion of units of time, one must be fully aware of the relationship between different units. For conversion of hours, minutes and seconds, we directly use the relationship but for conversion of months into days, we have to take care of the name of the month. Let us consider the following examples. 15
Teaching of Basic Mathematical Concepts II Example 10: Convert 2 hours 35 minutes into minutes. Here 1 hour = 60 minutes So 2 hours = 2 60 mts. = 120 mts. Therefore, 2 hrs. 35 mts. = 120 mts. + 35 mts. = 155 mts. Example 11: Find the number of days during the last 6 months of the year. Here we know that the last six months of the year are July, August, September, October, November, December. Let us consider the number of days in each month. Month No. of days July 31 August 31 September 30 October 31 November 30 December 31 184 So, there are 184 days in the last six month of the year. Here you find that we cannot multiply the number of months by 30 or 31 days. You may take other examples also e.g. number of days from 25 March to 2 nd April, and so on. Check Your Progress 13. Convert: Illustrate using table. i) 2.5 m into cm... ii) 13 Kg into g... iii) 5.43 sq. cm. into sq. mm... iv) 135 litres into kilo litres... v) Rs. 5.75 into paise and... vi) 300 second into hours... 16
14. Write the years between 1950 and 2000 which are leap years. 15. Write the names of the months which have exactly 30 dyas. 16. If you have to pay 45 rupees 60 paise, write at least 5 sets of denominations of money for making this payment. 6.6 FOUR FUNDAMENTAL OPERATIONS USING THE UNITS OF MEASUREMENT The four fundamental operations on metre measures are exactly like four fundamental operations on whole numbers since the same place-value principle holds in both the cases. The operations are done firstly by writing quantities in appropriate columns and then by using decimal notation. The following examples illustrate the process. A) Length Measures Example 12 : Rita purchases a cloth 3m 65 cm in length and Sita purchases the cloth of 6 metre 25 cm length. What is the total length of the cloth they have purchased? To find the total of length of the cloth we i) write measures in appropriate columns ii) add each column separately. m cm Cloth purchased by Rita 3 65 Cloth purchased by Sita + 6 25 9 90 Add 65 cm + 25 cm = 90 cm and write it in cm column Add 3 m + 6 m = 9 m and write it in m column or We write 3m 65 cm = 3.65 m 6 m 25 cm = 6.25 m and add in metres = 9.90 m 17
Teaching of Basic Mathematical Concepts II Example 13: Find the sum of two lengths. 16 m 38 cm and 25 m 78 cm m cm 16 38 + 25 78 42 16 Add 38 cm + 78 cm = 166, 116 cm = 10 cm + 16 cm Write 16 cm in cm and 1 m under m column Add 16 m + 25 m = 41 m. So we get 41 m + 1 m = 42 m or 16 m 38 cm = 16.38 m 25 m 78 cm = 25.78 m 1 1 carry 42.16 Example 14: I have a rope of length 18 m 60 cm. Out of it 12 m 20 cm length is cut and given to Gita. How much rope is left? m cm 18 60 12 20 6 40 Subtract 60 cm 20 cm and put it under cm column Subtract 18 m 12 m = 6 m and put it under m column We get 6 m 40 cm or 18 m 60 cm = 18.60 m 18.60 12 m 20 cm = 12.20 12.20 6.40 Example 15: Subtract 3 m 70 cm from 8 m 20 cm. It can be explained that 29 cm is less than 70 cm and so cannot be subtracted. Thus we borrow 1 metre and add to the cm 20 cm + 100 cm = 120 cm m cm 7 120 3 70 4 50 Now subtract 120 cm 70 cm = 50 cm write it in cm Now in metre column we have 7 m left. On subtracting 3 m from 7 m we get 4 m. Write it in the m column. 18
So, we get 4 m and 50 cm or 8 m 20 cm = 8.20 m 7.12 3 m 70 cm = 3.70 m 3.70 4.50 Example 16: The length of a rope is 3 dm 5 cm What will be the length of 8 such ropes? To find the length of 8 ropes we have to multiply 3 dm 5 cm by 8 as follows. dm cm 3 5 8 24 40 4 28 00 First, we multiply 5 cm by 8 to get 40 cm. We convert 40 cm to dm and get 4 dm. Then we multiply 3 dm by 8 to get 24 dm and add 4 dm to it. Thus the length of 8 such ropes is 28 dm or 3 dm 5 cm = 3.5 dm = 35 cm. So we multiply 3.5 dm by 8 or 35 cm by 8 35 cm 8 = 280 cm = 28 dm = 2.8 m Example 17: There is a piece of rope of length 6 m 9 dm 6 cm. We want to divide it in four equal parts. How can we find the length of each part? To find the length of each part, we have to divide 6 m 9 dm 6 cm by 4. We do it in two ways. We write quantities in columns and do long division or we change the quantities to centimeters and do as in the case of whole numbers. Divide 6 m by 4 once and get 2 m as remainder. 2 m is changed in dm. 2m = 2 10 dm. Add 9 dm in it. We get 29 dm. Again we divide 29 dm by 4 by 7 times and get one dm as remainder. 1 dm is changed into cm as 1 dm = 10 cm. Now add 6 cm to 10 cm and get 16 cm. Divide 16 by 4. 4 times and have zero as remainder. 1 7 4 m dm cm 4 6 9 6 4 2 m 10 20 dm + 9 dm 4 29 dm 28 1 dm 10 19
Teaching of Basic Mathematical Concepts II 10 cm + 6 4) 16 cm 16 or 4)696(174 cm 4 29 28 16 16 We get 174 cm We conclude that the length of each part is 1 mt. 7 dm and 4 cm or 174 cm. B) Weight Measures Example 18: Two students weight 19 Kg 200 g and 200 Kg 500 g respectively. Find their combined weight. We write quanties in columns. Kg g 19 200 + 20 500 39 700 Add 200 g + 500 g = 700 g, and write it under gram column. Add 19 Kg + 20 Kg = 39 kg, and write it under Kg column. So, we can conclude that their weight will be 39 Kg 700 g. We change quantities to Kg or gram and add as in whole numbers. kg gram 19.200 19200 + 20.500 + 20500 39.700 39700 or Here, you must have observed that the process of addition is the same as for the unit of length. So, you can yourself have an example for subtraction and find that the process of subtraction is also the same. Example 19: If the weight of a rice bag is 5 Kg, what will be the weight of 8 such bags? Let us find out the weight of 8 bags by adding them 8 times. 5 Kg+5Kg+5Kg+5Kg+5Kg+5Kg+5Kg+5Kg = 40 Kg But this is a inconvenient and time consuming. So, we multiple 5 kg by 8 and get 40 kg. 20
Example 20: If weight of a bag of apples is 8 kg 500g. Find the weight of 7 such bags. kg g 8 500 7 56 3500 + 3 59 500 Multiplying by 7 in 500 g, we get = 3500 g. Here, the next higher unit after g is kg. So write 3500 g = 3 kg + 500 g. Write 500 g under g and 3 kg along with kg. Now multiply, 8 kg 7, we get 56 kg and add 3 kg we get 56 + 3 = 59 kg or kg kg 8.500 8500 7 kg or 7 59.500 59500 So we conclude that weight of 7 such bags is 59 kg 500 g. C) Capacity Measures Activity 1 Let us take two empty bottles. Fill each of them with water using a measuring cylinder. We find that the respective capacities of these bottles are 850 ml and 780 ml. Now pour the water of these bottles into a large measuring cylinder. Note the amount of water in it. We get the measurement as 1 litre and 630 ml. We can also find the combined capacity of these bottle in the following way i.e. by adding. ml 850 +780 1630 Now 1630 ml = 1000 ml + 630 ml = 11 + 630 ml This activity can be repeated by using larger vessels. Example 21: If a barrel contains 851850 ml oil and another has 201680 ml, how much oil do both the barrels have? l ml 85 850 +20 680 105 1530 + 1 106 530 21
Teaching of Basic Mathematical Concepts II Add 850ml + 680ml, we get 1530 ml which can be expressed in litres and millilitre as 1530 ml = 1 l + 530 ml. Write 530 ml under ml and 1 l along with litre. Add 85 l + 20 we get 105 l. Now 105+1 l = 106 litre So we get 106 l and 530 ml as the combined amount of oil in the two barrels. litres or ml 85.850 85850 or + 20.680 +20680 106.530 ml 106530 Example 22: Let us take 6 empty bottles of same capacity. If each bottles contains 850 ml of water, can we find total quantity of water that can be filled in 6 bottles. Yes, we can by adding. 850 ml + 850 ml + 850 ml + 850 ml + 850 ml = 5100 ml. or 5100 ml can be written in 5 l 100 ml. But this will be inconvenient if we have large numbers. This can be found by multiplication of 6 by 850 ml i.e. 850 ml 6 5100 ml or 5100 ml = 5 l + 100 ml Example 23: Let us take a bucket having 6 l 250 ml of water. Take out 2 l 150 ml water from the bucket. Can you find how much water is left in the bucket? Yes we can by measuring the remaining water in the bucket. This can be measured but this will take more time. But the remaining water can be calculated by subtraction as: L ml 6 250 2 150 4 100 Subtracting 150 ml from 250 ml we get 100 ml. Subtract 2 l from 6 l we get 4 l. So 4 l and 100 ml is the amount of remaining water in the bucket. litres or millilitre 6.250 6250 or 2.150 2150 4.100l 4100 ml 22
Money Let us consider some example pertaining to the Four Fundamental Operations involving money. Example 24: Find the sum of Rs. 48.64 and Rs. 19.28. Addition Rs. Ps. i) Add paise first 64 P + 28 P = 92 P 48 64 Write 92 under paise 19 28 ii) Add rupees Rs. 48 + Rs. 19 = Rs. 67 67 92 Write 67 under rupees Thus the sum is Rs. 67 and 92 P. Example 25: Find the sum of Rs. 25.62, Rs. 28.52 and Rs. 672.86. Rs. Ps. i) Add paise first 25 62 65 P + 52 P + 86 P = 200 P 28 52 200 P = Rs. 2.0 P 62 86 Write 0 under paise and Rs. 2 under Rs. 115 200 ii) Add rupees Hence the sum is Rs. 117.00 P Subtraction Example 26: Subtract. Rs. 25 + Rs. 28 + Rs. 62 = Rs. 115 Rs. 115 + Rs. 2 = Rs. 117 Rs. Ps. i) Subtract 6 85 85 P - 30 P = 55 P 2 30 Write 55 under paise 4 55 ii) Subtract Rs. 6 Rs. 2 = Rs. 4 Write Rs. 4 under rupees Hence the remainder is Rs. 4.55 paise. Example 27: Subtract. Rs. Ps. i) Here you cannot subtract 60 P from 25 P 53 25 So rewrite Rs. 53.25 as RS. 52 + 125 paise 26 60 ii) Subtract 26 65 125 P 60 P = 65 paise Hence we get Rs. 26. 65 P. Write it under paise iii) Rs. 52 Rs. 26 = Rs. 26 Write it under rupees 23
Teaching of Basic Mathematical Concepts II Division Example 28: Divide Rs. 25.20 by 5 5) 25.20 (5.40 25 20 20 Hence the answer is Rs. 5.04 Multiplication Example 29: Multiply. Rs. Ps. 25 65 8 i) Multiply 65 P 8 = 520 P 200 520 520 P = Rs. 5 + 20 P 5 ii) Multiply Rs. 25 8 = 200 rupees 205 20 Rs. 200 + Rs. 5 = Rs. 205 Hence, we get Rs. 205. 20 P. Write 205 under rupees Note: We can also change the amount given in Rs. And Paise into Paise and do the same operation upon it. The answer obtained in paise should be again changed into Rs. and Paise. Time Let us consider following examples for four fundamental operations involving the measurement of time. Addition Rahim took one hour and 40 minutes to walk form his home to school and he returned from school to home in just 1 hour 30 minutes. Can you tell the total time Rahim has taken? Yes, by adding, we can find it thus. Hour Minutes 1 40 1 30 2 70 + 1 3 10 Minute Adding the minute column, we get 40 minutes + 30 minutes = 70 minutes 70 minutes can be written as 60 mts + 10 mts = 1 hours + 10 minutes Write 1 hr under hour column we get 1 hr + 1 hr = 2 hrs 2 hr + 1 hr = 3 hrs 24 So, we get 3 hr 10 minutes.
Subtraction Example 30: Tanu gets up at 6 hours 30 minutes. She goes to school at 8 hours 40 minutes. How much time does she take to get ready? In this case we subtract Hr Mts 8 40 6 30 2 10 We get 2 hrs and 10 minutes. For carrying on multiplication and division with units of time, we first convert mixed unit into a single unit as discussed earlier and then carry out the multiplication or division by any number and write the same unit with the quotient. Check Your Progress 17. Add i) 4 Kg 82 g + 3 Kg 95 g ii) 14 Kg 673 g + 10 Kg 495 g 18. Subtract i) 3 Kg 75 g 1 Kg 287 g 19. A grocer had 15 kg 500 g of sugar. On one day he sold 8 kg 625 g and the next day 5 kg 450 g. How much sugar has he left? 20. Add i) Rs. 11 and 62 paise Rs. 10 and 10 paise ii) Rs. 94 and 59 P Rs. 77 and 94 P 25
Teaching of Basic Mathematical Concepts II 21. Subtract Rs. Ps. 16 75 12 85 22. A dealer bought an article for Rs. 275 and 50 paise and sold it for Rs. 300 and 25 paise. Find his profit. 23. Add 5 hours 39 min + 6 hours 48 min 24. Subtract 15 hours 48 min 10 hours 56 min 6.7 LET US SUM UP To get uniformity in measurement, we need Standard Units of Measurement. For each of the measurement of length, weight, area, capacity, money and time, there are sequences of units ranging from very small to very large. The use of a unit depends upon the situation. Sometime, more than one unit are used simultaneously. So it is essential to know their interrelationship. For conversion of a smaller unit to a larger unit, division is required and for conversion of larger unit to smaller unit, multiplication is required. To make your students learn faster, similarity between the units of measurement of length, weight and capacity and the number system should be explained. This will help in doing computation with speed and accuracy. The rates and procedures followed in doing fundamental operations for numbers can be effectively applied to metric measures. 6.8 UNIT-END EXERCISES 1. Name some non-standard units of measurement for each (a) length (b) weight (c) area (d) capacity (e) money and (f) time 2. Why do we need Standard Units of Measurement? Give reasons. 3. Name the commonly used standard unit of measurement for each 26 (a) length (b) weight (c) area (d) capacity (e) money and (f) time
4. Explain the relationship between the units of measurement of length, area and capacity. 5. How will you illustrate the similarity between different units of measurement of length, weight and capacity in the classroom? 6. Give one example each for the use of smaller and bigger units of measurement of (a) length (b) weight (c) area (d) capacity (e) money and (f) time 7. Write one question each for addition, subtraction, multiplication and division involving units of measurement of length, weight, area, capacity, money and time. 6.9 ANSWERS TO CHECK YOUR PROGRESS 1. Length of classroom =... footsteps 2. Weight of the stone =... pebbles 3. Area of sheet of paper =...sheet of square paper 4. Capacity of bottle =... cups 5. Standard Unit of Measurement of money is Rupee 6. Morning, afternoon, evening and night 7. Millimetre, centimetre, decimeter, metre, decameter, hectometer and kilometer. 8. 1 kilometre = 10 hectogram 9. 1 m = 100... 1 sq. m = 100 100 = 10,000 sq. cm. 10. dm 3 = litre 11. Paisa 12. 366 days 13. i) 250 cm, ii) 13000 g, iii) 543 sq. mm, iv) 0.135 kl, v) 575 paise, vi) 1/12 hr 14. 1952, 1956, 1960, 1964, 1968, 1972, 1976, 1980, 1984, 1988, 1992, 1996 are leap years. 15. April, June, September and November 16. Rs. 10+ Rs. 10 + Rs. 10 + Rs. 10 + Rs. 5 + 50 paise + 10 paise Rs. 20 + Rs. 20 + Rs. 5 + 50 paise + 10 paise and so on 17. i) 7 kg 177 g, ii) 25 kg 168 g 18. 1 Kg 788 g 19. 1 Kg 425 g 20) i) Rs. 21 and 72 P, ii) Rs. 172 and 53 P 21. Rs. 3 and 90 paise 22. Profit is Rs. 24 and 75 paise 23. 11 hrs 87 min. or 12 hours 27 min. 24. 4 hrs. 52 min. 27