MATHEMATICS 3 TEXTBOOK – 2 Name: ___________________________________ Section: ________________ Roll No.: _________ School: __________________________________
Preface ClassKlap partners with schools, supporting them with learning materials and processes that are all crafted to work together as an interconnected system to drive learning. Our books strive to ensure inclusiveness in terms of gender and diversity in representation, catering to the heterogeneous Indian classroom. ClassKlap presents the Traveller series, designed specifically to meet the requirements of the new curriculum released in November 2016 by the Council for the Indian School Certificate Examinations (CISCE). Guiding principles: The 2016 CISCE curriculum states the following as a few of its guiding principles for Mathematics teaching: D evelop mathematical thinking and problem-solving skills and apply these skills to formulate and solve problems. A cquire the necessary mathematical concepts and skills for everyday life and for continuous learning in Mathematics and related disciplines. R ecognise and use connections among mathematical ideas and between Mathematics and other disciplines. R eason logically, communicate mathematically and learn cooperatively and independently. Each of these principles resonates with the spirit in which the ClassKlap textbooks, workbooks and teacher companion books have been designed. The ClassKlap team of pedagogy experts has carried out an intensive mapping exercise to create a framework based on the CISCE curriculum document. Key features of ClassKlap Traveller series: Theme-based content that holistically addresses all the learning outcomes specified by the CISCE curriculum. T he textbooks and workbooks are structured as per Bloom’s taxonomy to help organise the learning process according to the different levels involved. Student engagement through simple, age-appropriate content with detailed explanation of steps. Learning is supported through visually appealing images, especially for Grades 1 and 2. Increasing difficulty level in sub-questions for every question. Multiplication tables provided as per CISCE requirement. All in all, the Traveller Mathematics books aim to develop problem-solving and reasoning skills in the learners’ everyday lives while becoming adept at mathematical skills as appropriate to the primary level. – The Authors
Textbook Features I Will Learn About I Think Contains the list of concepts to be covered Arouses the student’s in the chapter along with the learning curiosity before objectives introducing the concept I Recall I RUenmdeermsbtearndand Pin-Up-Note Recapitulates the Elucidates the basic Highlights the key points or prerequisite knowledge for elements that form the definitions the concept learnt previously basis of the concept ? Train My Brain I Apply I Explore(H.O.T.S.) C hecks for learning to gauge Connects the concept E ncourages the student to the understanding level of the to real-life situations by extend the concept learnt student providing an opportunity to more complex scenarios to apply what the student has learnt Maths Munchies Connect the Dots Drill Time Aims at improving speed of Aims at integrating Revises the concepts with calculation and problem Mathematical concepts practice questions at the solving with interesting facts, with other subjects end of the chapter tips or tricks A Note to Parent E ngages the parent in the out-of- classroom learning of their child
Contents 6 Multiplication 6.1 Multiply 2-digit Numbers���������������������������������������������������������������������������������� 1 6.2 Mental Maths Techniques: Multiplication������������������������������������������������������� 8 7 Time 7.1 Read a Calendar�������������������������������������������������������������������������������������������� 11 7.2 Read Time Correctly to the Hour������������������������������������������������������������������� 16 8 Division 8.1 Division as Equal Grouping���������������������������������������������������������������������������� 27 9 Money 9.1 Number Operations on Money��������������������������������������������������������������������� 33 10 Measurement 10.1 Conversion of Standard Units of Length������������������������������������������������������ 41 10.2 Weigh Mass Using Non-Standard Units������������������������������������������������������� 48 10.3 M easure Volume Using Non-Standard Units����������������������������������������������� 53 11 Data Handling 11.1 R ecord Data Using Tally Marks�������������������������������������������������������������������� 61
6Chapter Multiplication I Will Learn About • using repeated addition to construct multiplication tables from 2 to 10. • estimation of two products. • mental multiplication of two numbers. 6.1 Multiply 2-digit Numbers I Think Neena bought 2 boxes of toffees for her birthday. Each box has 25 toffees inside it. If there are 54 students in her class, do you think she has enough toffees? I Recall In Class 2, we have learnt that multiplication is repeated addition. In Fig. (a), the number ‘3‘ is repeated 4 times. So, using repeated addition we can write, 3 + 3 + 3 + 3 = 12. Fig. (a) Fig. (b) 1
Similarly, in Fig. (b), the number ‘2‘ is repeated twice. Hence, using repeated addition we can write 2 + 2 = 4. Thus, we can say that repeated addition is adding the same number repeatedly (again and again). Let us answer a few questions on repeated addition. a) 5 + 5 + 5 + 5 + 5 = __________ b) 3 + 3 + 3 = __________ c) 10 + 10 + 10 + 10 = __________ d) 6 + 6 = __________ I Remember and Understand The symbol ‘×’ indicates multiplication. Multiplication means having a certain number of groups of the same size. In the multiplication of two numbers: The number written to the left of the ‘×’ sign is called the multiplicand. The number written to the right of the ‘×’ sign is called the multiplier. The number written to the right of the ‘=’ sign is called the product. Multiplication Fact ↓ ↓ ↓ Multiplicand Multiplier Product Note: (a) Representation of the multiplicand, multiplier and the product using the symbols ‘×’ and ‘=’ is called a multiplication fact. (b) The multiplicand and the multiplier are also called the factors of the product. For example, in the multiplication fact 2 × 7 = 14 = 7 × 2, 2 and 7 are the factors of 14. Similarly, in the multiplication fact 4 × 5 = 20 = 5 × 4, 5 and 4 are the factors of 20. 2
(c) A ny number multiplied by 0 gives a Order Property: Changing product 0. For example, 6 × 0 = 0. the order in which numbers are multiplied does not (d) A ny number multiplied by 1 gives the change the product. This is number as the product. called the order property of multiplication. For example, 6 × 1 = 6. Using multiplication facts and order property, let us now construct the multiplication tables from numbers 2 to 10. 2 Multiplication Tables 4 2×1=2 4×1=4 2×2=4 3 4×2=8 2×3=6 3×1=3 4 × 3 = 12 2×4=8 3×2=6 4 × 4 = 16 2 × 5 = 10 3×3=9 4 × 5 = 20 2 × 6 = 12 3 × 4 = 12 4 × 6 = 24 2 × 7 = 14 3 × 5 = 15 4 × 7 = 28 2 × 8 = 16 3 × 6 = 18 4 × 8 = 32 2 × 9 = 18 3 × 7 = 21 4 × 9 = 36 2 × 10 = 20 3 × 8 = 24 4 × 10 = 40 3 × 9 = 27 3 × 10 = 30 5 6 7 5×1=5 6×1=6 7×1=7 5 × 2 = 10 6 × 2 = 12 7 × 2 = 14 5 × 3 = 15 6 × 3 = 18 7 × 3 = 21 5 × 4 = 20 6 × 4 = 24 7 × 4 = 28 5 × 5 = 25 6 × 5 = 30 7 × 5 = 35 5 × 6 = 30 6 × 6 = 36 7 × 6 = 42 5 × 7 = 35 6 × 7 = 42 7 × 7 = 49 5 × 8 = 40 6 × 8 = 48 7 × 8 = 56 5 × 9 = 45 6 × 9 = 54 7 × 9 = 63 5 × 10 = 50 6 × 10 = 60 7 × 10 = 70 Multiplication 3
8 9 10 10 × 1 = 10 8×1=8 9×1=9 10 × 2 = 20 8 × 2 = 16 9 × 2 = 18 10 × 3 = 30 8 × 3 = 24 9 × 3 = 27 10 × 4 = 40 8 × 4 = 32 9 × 4 = 36 10 × 5 = 50 8 × 5 = 40 9 × 5 = 45 10 × 6 = 60 8 × 6 = 48 9 × 6 = 54 10 × 7 = 70 8 × 7 = 56 9 × 7 = 63 10 × 8 = 80 8 × 8 = 64 9 × 8 = 72 10 × 9 = 90 8 × 9 = 72 9 × 9 = 81 10 × 10 = 100 8 × 10 = 80 9 × 10 = 90 Solve these Multiplying 2-digit numbers by 1-digit numbers H TO There are two ways to multiply numbers: 1) Standard algorithm 2) Lattice algorithm 17 Let us now learn both these methods. ×9 Standard Algorithm H TO Example 1: Find the product: 23 × 7 Solution: Follow these steps to find the product. 15 ×4 Steps Solved Step 1: Multiply the ones. 3 × 7 = 21 Step 2: Regroup the product. 21 ones = 2 tens and 1 ones Step 3: Write down the ones TO digit of the product from step 2. Carry forward its tens digit 2 to the tens place. 23 ×7 1 4
Steps Solved Solve these Step 4: Multiply the tens. Step 5: Add the carry over 2 × 7 = 14 H TO from step 3 to the product. 23 14 + 2 = 16 ×8 Step 6: Write the sum in the tens place. H TO 2 23 ×7 161 Lattice Algorithm Important features of the lattice algorithm: • Setting up the lattice before we begin multiplying • Doing all the multiplications first, followed by additions • There is no carry over in the multiplication phase of the algorithm Example 2: Multiply: 29 × 3 Solution: Follow these steps to find the product. Steps Solved Solve these Step 1: Construct a lattice as shown 3 2× such that: 5 (a) Number of rows = Number of digits in the multiplier. (b) N umber of columns = Number of digits in the multiplicand. Step 2: Write the multiplicand 2 9× 5 2× along the top of the lattice and 3 4 the multiplier along the right, one digit for each row or column. Draw diagonals to divide each box into parts as shown. Multiplication 5
Steps Solved Solve these 6 1× Step 3: Multiply each digit of the 2 9× 3 multiplicand by each digit of the multiplier. Write the products in the 2 3 5 7× cells where the corresponding rows 7 3 and columns meet. 2 9× Step 4: If the product is a single digit 0627 3 number, put 0 in the tens place. (2 × 3 = 6) = 06 Step 5: Add the numbers along the 2 9× 6 3× diagonals from the right to find the 02 3 product. Regroup if needed. Write 0 6 73 the sum from left to right. 87 Therefore, 29 × 3 = 087 = 87. ? Train My Brain c) 19 × 8 Find the product: a) 17 × 7 b) 28 × 9 I Apply Let us now see some real-life situations where we use multiplication of 2-digit numbers. Example 3: 42 people were sitting in a row of a stadium to enjoy a cricket match. How many people would be there in all if there were 6 such rows? Solution: Number of people sitting in one row = 42 Number of rows = 6 6
Total number of people in 6 rows = 42 × 6 4 2× = 252 21 Therefore, there are 252 people in the 2 4 26 stadium. 52 Example 4: A school had 54 students in each class. There were 8 such classes. How many students were there in the entire school? Solution: Number of students in one class = 54 H TO Number of classes in the school = 8 3 Number of students in the school = 54 × 8 54 ×8 Therefore, there were 432 students in the entire school. 4 3 2 I Explore (H.O.T.S.) Framing word problems: Using multiplication tables, we can frame word problems from the given clues. Let us now try to frame a few word problems using multiplication. Example 5: Frame a word problem using the following data: Number of chocolates in a box = 9 Number of such boxes = 5 Solution: Word problem: A box contains 9 chocolates. There are 5 such boxes. Find the total number of chocolates. Example 6: Frame a word problem using the multiplication fact: 8 × 2 = 16 Solution: Word problem: There are 2 rows with 8 students in each row. What is the total number of students? Multiplication 7
6.2 Mental Maths Techniques: Multiplication Mental Maths Now, let us learn to double a 2-digit number mentally. Steps Solved Solve this 53 41 Step1: Multiply the tens digit The tens digit is 5. The tens digit is ______. by 2. So, 5 × 2 = 10. So, _____ × _____. Step 2: Check if the ones digit The ones digit is 3. The ones digit is ______ is less than or equal to 4. If yes, 3 < 4 (True) _____ < ______ leave the product in step 1 as (True / False) it is. Else, add 1 to it. _____ × 2 = _____ Step 3: Multiply the ones digit 3×2=6 by 2 mentally. This is the ones _____ × 2 = _____ digit of the required product. Step 4: Combine the products from steps 1 and 3. This gives 53 × 2 = 106 the required number. ? Train My Brain e) 74 Double the given numbers mentally: a) 22 b) 36 c) 51 d) 23 8
Maths Munchies Multiplying by 10 and 100 When numbers are multiplied by 10, the products are the numbers followed by '0'. That is, the ones digit in the product is 0. Similarly, when numbers are multiplied by 100, the products are the numbers followed by '00'. That is, the ones and the tens digits in the product are 0. For example: a) 5 × 10 = 50 b) 9 × 10 = 90 5 × 100 = 500 9 × 100 = 900 c) 6 × 10 = 60 d) 4 × 10 = 40 6 × 100 = 600 4 × 100 = 400 Connect the Dots Social Studies Fun All the arrangements of Charbagh Garden of Taj Mahal, are based on four or its multiples. The entire garden is divided into four parts. There are 16 flower beds. It is said that each of the flower beds is planted with 400 plants. English Fun Compose a poem on multiplication as: Six times six. Magic tricks. Abracadabra. Thirty-six. Multiplication 9
Drill Time 6.1 Multiply 2-digit Numbers 1) Multiply the numbers using standard and lattice algorithms. a) 56 × 3 b) 33 × 7 c) 71 × 8 d) 50 × 5 e) 62 × 4 2) Word problems a) There were 23 boys in one group. The school had 4 such groups. How many boys were there in all the groups? b) V iraj travelled for 30 km in one day. He travelled the same distance for 7 days. How many kilometres did he travel in all? A Note to Parent Multiplication is used in many situations in our day-to-day activities. Calculating time, distance, money to be paid in a departmental store, the area of a room and so on are a few examples. Encourage your child to actively engage in these scenarios and help you with the calculations. 10
Time7Chapter I Will Learn About • identifying a day and a date on a calendar. • read the time correctly to the hour using a watch/clock. 7.1 Read a Calendar I Think Neena and her friends are playing a game using a calendar. They split into two groups. Each group says a date or day of a particular month. The other group answers with the corresponding day or date of another month. Can you also play such a game? I Recall Let us recall the days in a week and the months in a year. There are 7 days in a week. They are: 1) Sunday 2) Monday 3) Tuesday 4) Wednesday 5) Thursday 6) Friday 7) Saturday 11
There are 12 months in a year. They are: 1) January 2) February 3) March 4) April 5) May 6) June 7) July 8) August 9) September 10) October 11) November 12) December I Remember and Understand Using a calender we can find the day on which a given date falls. We can also find the dates in a given week and month. Let us see how we can find the day on which 11th July 2019 falls. Step 1: In the calendar of 2019, identify the month of July. Step 2: Find where the date 11 lies. Step 3: Match the day with the date identified 10 years = 1 decade in step 2. 100 years = 1 century Therefore, 11th July 2019 1000 years = 1 millennium falls on a Thursday. 12
Example 1: Observe the given calendar and answer the questions: a) How many days are there in the month of January? b) H ow many Sundays are there in this month? c) Which day appears 5 times? JANUARY 2019 SAT SUN MON TUE WED THU FRI Solution: d) O n which day is the Republic day? 12345 6 7 8 9 10 11 12 e) O n which date is the second 13 14 15 16 14 18 19 Saturday? 20 21 22 23 24 25 26 a) There are 31 days in this month. 27 28 29 30 31 b) There are four Sundays in this month. c) Tuesday, Wednesday and Thursday appear five times. d) The Republic day is on Saturday. e) Second Saturday is on 12th. Example 2: From the calendar for the year 2019, write the days of the following events. a) Independence Day - _____________ b) Republic Day - _____________ c) Christmas - _____________ d) Teachers’ Day - _____________ e) Children’s Day - _____________ f) Gandhi Jayanti - _____________ Solution: a) Independence Day - Thursday b) Republic Day - Saturday c) Christmas - Wednesday d) Teachers’ Day - Thursday e) Children’s Day - Thursday f) Gandhi Jayanti - Wednesday ? Train My Brain Answer the following: a) When is your father’s birthday? b) On which day is your birthday this year? c) When do you have Dussehra vacation in school? Time 13
I Apply We use the calendar on a daily basis. Events like planning holidays, conducting sports and examinations in school need the use of a calendar. Example 3: Renu wants to plan her holiday in October 2019 SAT October from Saturday to Thursday. 5 On the calendar, mark the days when SUN MON TUE WED THU FRI 12 Renu can plan her holiday. 1234 19 26 Solution: Renu’s trip will start on a Saturday and 6 7 8 9 10 11 end on a Thursday. 13 14 15 16 17 18 20 21 22 23 24 25 27 28 29 30 31 Saturdays in this month: 5, 12, 19, 26 October 2019 SUN MON TUE WED THU FRI SAT Thursdays in this month: 3, 10, 17, 24, 31 12345 6 7 8 9 10 11 12 Renu’s trip could be planned for 5th to 13 14 15 16 17 18 19 10th, 12th to 17th or 19th to 24th or 26th to 20 21 22 23 24 25 26 31st as marked on the calendar. 27 28 29 30 31 Example 4: Use the calendar of January 2019 shown below to answer the question. JANUARY 2019 Rupali is a clerk in a bank. She has SUN MON TUE WED THU FRI SAT 5 holidays on Sundays and on the first 1234 12 and the third Saturdays of the month. 6 7 8 9 10 11 19 She also has holidays on the New 13 14 15 16 17 18 26 20 21 22 23 24 25 Year’s Day and Republic Day. How 27 28 29 30 31 many holidays does she have in the month of January? Solution: Republic day is on 26th January. New Year’s day is on 1st January. The first and the third Saturdays fall on 5th and 19th January respectively. Sundays fall on 6th, 13th, 20th and 27thJanuary. Rupali has holidays on 1st, 5th, 6th, 13th, 19th, 20th, 26th and 27th January. Therefore, she has 8 holidays in January. 14
I Explore (H.O.T.S.) Observe the calendar for the month of February of different years. February 2012 February 2013 February 2014 SUN MON TUE WED THU FRI SAT SUN MON TUE WED THU FRI SAT SUN MON TUE WED THU FRI SAT 1234 12 1 5 6 7 8 9 10 11 3456789 12 13 14 15 16 17 18 10 11 12 13 14 15 16 2345678 19 20 21 22 23 24 25 17 18 19 20 21 22 23 26 27 28 29 24 25 26 27 28 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 February 2015 February 2016 February 2017 SUN MON TUE WED THU FRI SAT SUN MON TUE WED THU FRI SAT SUN MON TUE WED THU FRI SAT 1234567 123456 1234 8 9 10 11 12 13 14 7 8 9 10 11 12 13 5 6 7 8 9 10 11 15 16 17 18 19 20 21 14 15 16 17 18 19 20 12 13 14 15 16 17 18 22 23 24 25 26 27 28 21 22 23 24 25 26 27 19 20 21 22 23 24 25 28 29 26 27 28 February 2018 February 2019 February 2020 SUN MON TUE WED THU FRI SAT SUN MON TUE WED THU FRI SAT SUN MON TUE WED THU FRI SAT 123 12 1 4 5 6 7 8 9 10 3456789 11 12 13 14 15 16 17 10 11 12 13 14 15 16 2345678 18 19 20 21 22 23 24 17 18 19 20 21 22 23 25 26 27 28 24 25 26 27 28 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 We observe that February has 29 days in the years 2012, 2016 and 2020. In the other years, February has 28 days. Every four years, an extra day is added to the month of February. This is due to the revolution of the Earth around the Sun. The Earth takes 365 days and 6 hours to go around the Sun. An ordinary year is taken as 365 days only. 6 hours put together four times make an extra day for every four years. This is added on to get the leap year. So, there are 365 + 1 = 366 days in a leap year. Time 15
Example 5: Find the leap years in the following years: 2020, 2021, 2022, 2024, 2025 Solution: In a leap year, the number formed by the last two digits is an exact multiple of 4. In 2020, the number formed by the last two digits is 20, which is a multiple of 4. In 2021, the number formed by the last two digits is 21, which is not a multiple of 4. In 2022, 22 is not a multiple of 4. In 2024, 24 is a multiple of 4. In 2025, 25 is not a multiple of 4. Thus, 2020 and 2024 are the leap years. Example 6: How many days were there from Christmas 2014 to Christmas 2015? Solution: 2015 was not a leap year. So, there were 365 days from Christmas 2014 to Christmas 2015. 7.2 Read Time Correctly to the Hour I Think Neena’s teacher taught her to read time. She now knows the units of time. Neena is able to read the time when her dad points to the clock as shown. Can you also read the time in this clock? I Recall We have learnt that the long hand of the clock shows minutes and the short hand shows hours. In some clocks, we see another hand, thinner than the hour and the minute hands. This is the seconds hand. Let us recall reading time from a clock by answering the following. a) 7 o’clock is _______________ hours more than 4 o’clock. 16
b) The _______________ hand takes one hour to go around the clock. c) The _______________ hand is the short hand of the clock. d) T he time is _______________ when both the hour hand and the minute hand are on 12. e) 2 hours before 10 o’clock is _______________. I Remember and Understand We see numbers 1 to 12 on the clock. These numbers are for counting hours. There are 60 parts or small lines between these numbers. They stand for minutes. The minute hand takes 1 hour to 1 hour = 60 minutes go around the clock face once. 1 day = 24 hours The minute hand takes 5 minutes to go from one number to the next number. We multiply the number to which the minute hand points by 5 to get the minutes. For example, the minute hand in the figure is at 6. So, it denotes 6 × 5 = 30 minutes past the hour (here, after 3). Therefore, the time is read as 3:30. The hour hand takes one hour to move from one number to the other. Let us now read the time shown by these clocks. Fig. (a) Fig. (b) Fig. (c) Fig. (d) In figure (a), the minute hand is at 9. The hour hand is in between 5 and 6 . The number of minutes is 9 × 5 = 45. Thus, the time shown is 5:45. Time 17
In figure (b), the minute hand is at 6. The number of minutes is 6 × 5 = 30. The hour hand is between 7 and 8. Therefore, the time shown is 7:30. In figure (c), the minute hand is at 3. The number of minutes is 3 × 5 = 15. The hour hand is between 9 and 10. Therefore, the time shown is 9:15. In figure (d), the minute hand is at 4. So, the number of minutes is 4 × 5 = 20. The hour hand is between 2 and 3. Therefore, the time shown is 2:20. Example 7: On which number is the minute hand if the minutes are as given below? a) 30 minutes b) 15 minutes c) 45 minutes Solution: To find the minutes when the minute hand is at a number, we multiply by 5. So, to get the number from the given minutes, we must divide it by 5. a) 30 ÷ 5 = 6. So, the minute hand is at 6. b) 15 ÷ 5 = 3. So, the minute hand is at 3. c) 45 ÷ 5 = 9. So, the minute hand is at 9. Quarter past, half past and quarter to the hour. We know that, ‘quarter’ of an hour is a part of the four equal parts into which the hour is divided. In Fig (a), the minute hand of the clock has travelled a quarter of an hour. So, we call it quarter past the hour. Train My Brain The time shown is 2:15 or 15 minutes past 2 or quarter past 2. Fig. (a) ‘Half’ of an hour is a part of the two equal parts into which the hour is divided. In Fig (b), the minute hand of the clock has travelled a half of an hour. So, we call it half past the hour. The time shown is 2:30 or 30 minutes past 2 or half past 2. Fig. (b) If the minute hand has to travel a quarter of the clock before it completes one hour, we call it quarter to the hour. The time shown in Fig (c) is 7:45 or 45 minutes past 7 or quarter to 8. Fig. (c) 18
Example 8: Read the time in each of the given clocks and write it in two different ways. Fig. (a) Fig. (b) Fig. (c) Fig. (d) Solution: To read the time, observe the positions of the hour and the minute hands. Solved Solve these Fig. (a) Fig. (b) Fig. (c) Fig. (d) The hour hand is The hour hand is The hour hand is The hour hand is between 3 and 4. between _____ between _____ between _____ and _____. The and _____. The and _____. The So, the minutes minutes are after minutes are after minutes are after are after 3 hours. ____hours. The ____hours. The ____hours. The The minute hand minute hand is at minute hand is at minute hand is at is at 6. So, the _____. So, the time _____. So, the time _____. So, the time time is 30 minutes is _____ minutes is _____ minutes is _____ minutes after 3. We write after _____. We after _____. We after _____. We it as 3:30 or half- write it as _____ or write it as _____ or write it as _____ past 3. _______________. _____________. or _____________. Example 9: Draw the hands of a clock to show the given time. a) 1:15 b) 6:15 c) 7:30 d) 9:45 Time 19
Solution: T o draw the hands of a clock, first note the minutes. If the minutes are between 1 and 30, draw the hour hand between the given hour and the next. But take care to draw it closer to the given hour. If the minutes are between 30 and 60, draw the hour hand closer to the next hour. Then, draw the minute hand on the number that shows the given minutes. Fig. (a) Fig. (b) Fig. (c) Fig. (d) ? Train My Brain Answer the following: a) Write the time: quarter past 7. b) How many numbers do you see on the clock? c) H ow much time does the hour hand take to move from one number to the next? I Apply We have learnt how to read the time. But in a day we notice that the same time is seen twice. When the hour hand goes around the clock once, it completes 12 hours. A day has 24 hours, so the hour hand goes around the clock twice in a day. To identify morning or evening, we write a.m. or p.m. along with the time. a.m. is from 12 midnight to 12 noon and p.m. is from 12 noon to 12 midnight. 20
Morning Evening 6:00 a.m. 6:00 p.m. Roshan brushes his teeth at 6:45 a.m. Amit studies at 4:30 p.m. Sejal eats her dinner at 8:00 p.m. Example 10: What time is it? a) One hour before 4:45 p.m. b) Three hours after 6:15 a.m. c) Two hours after 11:00 a.m. d) One hour before 12:15 p.m. Solution: To get the time after the given time, we add the number of hours. To get the time before the given time, we subtract the number of hours. Therefore, a) One hour before 4:45 p.m. is 3:45 p.m. b) Three hours after 6:15 a.m. is 9:15 a.m. c) Two hours after 11:00 a.m. is 1:00 p.m. d) One hour before 12:15 p.m. is 11:15 a.m. Time 21
I Explore (H.O.T.S.) We have learnt to read and show time, in a 12-hour clock. Let us now learn to read and show time in a 24-hour clock. In a 24-hour clock: • the time is expressed as a 4-digit number (hh:mm). • 12 o’clock midnight is written as 00:00. • time post noon is written by adding 12 to the number of hours. • when the number of hours is more than 12, then the time indicates post noon. For example, 17:30, 18:15, 22:45 and so on. 12-hour clock 24-hour clock Morning 5:00 a.m. 05:00 Evening 5:00 p.m. 17:00 After 12 noon Convert time from 24-hour Convert time from 12-hour clock to 12-hour clock clock to 24-hour clock Subtract 12 from the Add 12 to the number number of hours. of hours. Write p.m. after the Omit writing p.m. difference. 22
12-hour clock time 24-hour clock time Read as 4:45 a.m. 04:45 Four forty-five hours 11:00 a.m. 11:00 Eleven hundred hours 5:30 p.m. 17:30 Seventeen thirty hours 7:15 p.m. 19:15 Nineteen fifteen hours Example 11: Convert the given time to 12-hour clock time. a) 13:45 b) 05:30 c) 09:15 d) 21:15 e) 19:45 f) 23:00 Solution: a) (13 – 12):45 = 1:45 p.m. b) 5:30 a.m. c) 9: 15 a.m. d) (21 – 12):15 = 9:15 p.m. e) (19 – 12):45 = 7:45 p.m. f) (23 – 12):00 = 11 p.m. We can find out the number of hours in the given days by multiplying it by 24. We can also find the number of minutes in the given hours by multiplying it by 60. Example 12: Convert the following: a) 3 days into hours b) 6 hours into minutes Solution: a) 1 day = 24 h Therefore, 3 days = 3 × 24 h = 72 h b) 1 hour = 60 minutes Therefore, 6 hours = 6 × 60 min = 360 min Maths Munchies A year with last two digits as a multiple of 4 is a leap year. The rule is different for century years. Century years are the years which have 0 in the ones and tens places. Years such as 1300, 1400 and so on are century years. For century years to be leap years, the number formed by the digits in their thousands and hundreds places must be a multiple of 4. For example, the years 1600 and 2000 are leap years whereas the years 2100 and 2200 are not. Time 23
Connect the Dots English Fun A poem to remember what a calendar tells us: When we see the calendar we learn the month, the date, the year. Every week day has a name there are lots of numbers that look the same. So let’s begin to show you how we see the calendar right now. Science Fun Have you noticed that you start feeling hungry between noon and 2 o’clock? Why don’t you feel hungry before that? It is because our body gets used to a sequence of events. This sequence of events is called our ‘body cycle’. Another example of the body cycle is that if you sleep daily by 10:00 p.m. then you will feel sleepy at that time even when you are not in your bed. Drill Time 7.1 Read a Calendar 1) Observe the calendar and answer the following questions. a) How many weekends and weekdays are JANUARY 2019 shown in the calendar? (Consider Saturday SUN MON TUE WED THU FRI SAT and Sunday to be one weekend.) 12345 b) Write the day and date two days before 6 7 8 9 10 11 12 13 14 15 16 17 18 19 the fourth Saturday of January. 20 21 22 23 24 25 26 c) On which day does the month end? 27 28 29 30 31 24
2) Word Problems a) R aju bought a new dress on the 1st of June. JUNE 2019 He bought another new dress 10 days SUN MON TUE WED THU FRI SAT after the first day of the month. On which 1 date did he buy the other dress? 23 45678 b) S onu’s birthday was on the 2nd of June, 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2019. What is the date, if he celebrated it 23 24 25 26 27 28 29 on the same day of the third week? 30 c) R am solved problems from one chapter of his book on the 9th of September. He solved problems from the next chapter 5 days later. On which day did he solve problems from the next chapter? 7.2 Read Time Correctly to the Hour 3) Draw the hands of a clock to show the given time. a) Half past 2 b) 4:15 c) Quarter to 12 d) 4:30 e) 6:45 4) What is the time shown on each of these clocks? Time 25
5) Write the time along with a.m. or p.m. Shilpa goes to school at ____________. Manas goes to play at ____________. 6) What is the time? b) Three hours after 11:45 a.m. a) Two hours before 5:15 p.m. d) Two hours before 1:00 p.m. c) One hour after 7:30 a.m. A Note to Parent Whenever you visit a railway station with your child, make him or her observe the 24 hour clock of the various trains arriving at the station. 26
8Chapter Division I Will Learn About • equal grouping and sharing. • repeated subtraction and division facts. 8.1 Division as Equal Grouping I Think Neena and her brother Piyush got a chocolate bar with 14 pieces for Christmas. Piyush divided it and gave Neena 6 pieces. Do you think Neena got an equal share? How can we find out? I Recall Suppose, we have to divide a group of 12 chocolate bars among 3 students. We can subtract 3 from 12 repeatedly to find how many chocolate bars each student will get. 27
12 chocolate bars in 1 group can be represented as: By repeated subtraction, we can have 12 – 3 = 9; 9 – 3 = 6; 6 – 3 = 3; 3 – 3 = 0 (1) (2) (3) (4) Thus, each student will receive 4 chocolate bars. Repeated subtraction is subtracting the same number over and over again. Fill in these blanks to revise the concept repeated subtraction. a) 16 – 4 = 12 ; 12 – 4 = ______ ; ______ – 4 = 4 ; ______ – ______ = ______ b) 21 – 7 = 14; ; 14 – 7 = ______ ; ______ – 7 = ______ c) ______ – 6 = 18 ; 18 – 6 = ______ ; 12 – ______ = 6 ; 6 – ______ = 0 I Remember and Understand In Grade 2, we have learnt about equal grouping and equal sharing. If 9 balloons are to be shared equally among 3 girls, each one of them gets 3 balloons. We can write it as 9 divided by 3 equals 3. It is represented as: Division Fact ↓ ↓ ↓ Dividend Divisor Quotient Note: Representation of the dividend, The symbol for ‘is divided by’ is ÷. divisor and quotient using the symbols ‘÷’ and ‘=’ is called a division fact. We can represent division as: In a division, the number that is divided [Total number] ÷ [Number in is called the dividend. The number that each group] = [Number of divides is called the divisor. The answer in groups] division is called the quotient. The number 28
(part of the dividend) that remains is called the remainder. We use multiplication tables to find the quotient in a division. We find the factor which when multiplied by the divisor gives the dividend. Example 1: 18 pens are to be shared equally by 3 children. How many pens does each child get? Solution: Total number of pens = 18 Number of children = 3 Number of pens each child gets = 18 ÷ 3 = 6 (since 6 × 3 = 18) Therefore, each child gets 6 pens. Example 2: 10 flowers are put in some vases. If each vase has 2 flowers, how many vases are used? Solution: Number of flowers = 10 Number of flowers in each vase = 2 Number of vases used = 10 ÷ 2 = 5 (since 2 × 5 = 10) Therefore, 5 vases are used to put 10 flowers. We get two division facts from a multiplication fact. The divisor and the quotient are the factors of the dividend. Observe the following: Dividend ÷ Divisor = Quotient Multiplicand × Multiplier = Product 18 ÷ 6 = 3 ↓↓ ↓ 6 × 3 = 18 Product Factor Factor ↓ ↓↓ (Multiplicand) (Multiplier) Divisor Quotient Dividend From the multiplication fact 6 × 3 = 18, we can write two division facts: a) 18 ÷ 3 = 6 and b) 18 ÷ 6 = 3 Multiplication and division are reverse operations. We can show a multiplication fact on the number line. For example, 3 × 5 = 15 means 5 times 3 is 15. Division 29
To show 5 times 3 on the number line, we go forward from 0 to 15. While doing so, we jump forward 5 times, covering 3 steps in every jump. Similarly, we can show the division fact 15 ÷ 3 = 5 on the number line. To show 15 divided by 3 on the number line, we go backward from 15 to 0. While doing so, we go back 5 times, covering 3 steps each time. ? Train My Brain Write two multiplication facts for each of the following: a) 20 ÷ 5 = 4 b) 48 ÷ 6 = 8 c) 36 ÷ 4 = 9 I Apply Equal sharing and equal grouping are used in some real-life situations. Consider the following situations. Example 3: 25 buttons are stitched on 5 shirts. Each shirt has the same number of buttons. How many buttons are there on each shirt? Solution: Total number of buttons = 25 Number of shirts = 5 The division fact for 25 buttons distributed to 5 shirts is 25 ÷ 5 = 5. Therefore, each shirt has 5 buttons on it. Example 4: 24 marbles are to be divided among 4 friends. How many marbles will each friend get? Solution: Total number of marbles = 24 Number of friends = 4 Number of marbles each friend will get = 24 ÷ 4 = 6 Therefore, each friend will get 6 marbles. 30
I Explore (H.O.T.S.) Division is commonly used in our day-to-day lives. Let us see some examples. Example 5: Aman spends 14 hours a week for tennis practice. He spends 21 hours a week for doing homework and 48 hours a week at school. How much time does he spend in a day for these activities? (Hint: 1 week = 7 days. The school works for 6 days a week.) Solution: Time spent for tennis practice per day = 14 hours ÷ 7 = 2 hours Time spent for doing homework per day = 21 hours ÷ 7 = 3 hours Time spent at school per day = 48 hours ÷ 6 = 8 hours (School works for 6 days a week) Therefore, the total time spent by Aman in a day for all the activities = (2 + 3 + 8) hours = 13 hours (except on Sunday) Example 6: Deepa shares 15 lollipops among her 5 friends. If she shares among only 3 of them, how many more lollipops does each of them get? Solution: Number of lollipops = 15 If Deepa shares the lollipops among her five friends, the number of lollipops each would get = 15 ÷ 5 = 3 If Deepa shares the lollipops among only three of them, the number of lollipops each would get = 15 ÷ 3 = 5 Difference in the number of lollipops = 5 – 3 = 2 Therefore, distributing the lollipops among 3 friends, each of them would get 2 more lollipops. Maths Munchies Why is division of a number by 0 not possible? We know that division and multiplication are related. If we have to find 6 ÷ 3, we get the answer 2, because 2 × 3 = 6. Similarly, if we have to find 6 ÷ 0, what would be the answer? We must get a number which when multiplied by 0 gives 6. But any number when multiplied by 0 results in 0. Therefore, 6 ÷ 0 is not possible. This is true for any number. So, division by zero is undefined. Division 31
Connect the Dots Science Fun Some fruits have one seed. Some have many seeds. Peas have many seeds. Take out all seeds from the four pods. Divide them equally among your family. What is the quotient? What is the remainder? Social Studies Fun Division mean equal sharing. It exists in our neighbourhood and families also. Apartment buildings are divided into equal parts to create flats or apartment houses. In this way, many families can live together in the same building. Drill Time 8.1 Division as Equal Grouping 1) Give two division facts for the given multiplication facts. a) 4 × 5 = 20 b) 9 × 2 = 18 c) 7 × 4 = 28 d) 8 × 1 = 8 e) 2 × 7 = 14 2) Word Problems a) 2 0 students are to be divided equally into 2 groups. How many students will be there in each group? b) 14 pencils must be distributed equally among 7 children. How many pencils will each child receive? A Note to Parent Help your child share the things he would enjoy having with friends or family members. Teach him how sharing is caring and spreads happiness. 32
Money9Chapter I Will Learn About • add and subtract money. • make rate charts and bills. 9.1 Number Operations on Money I Think ` 56 ` 43 Neena’s father bought a toy aeroplane for ` 56 and a toy truck for ` 43. How much did he spend altogether? How much change does he get if he gives ` 100 to the shopkeeper? I Recall Recall that two or more numbers are added by writing them one below the other. This method of addition is called the column method. We know that rupees and paise are separated using a dot or a point. In the column method, we write money in such a way that the dots or points are placed exactly one below the other. The rupees are placed under rupees and the paise are placed under paise. 33
Let us recall a few concepts about money through these questions. a) 50 paise + 50 paise = ____________ b) ` 20 + ` 10 = _______________ c) ` 20 + ` 5 + 50 paise = ____________ d) ` 50 – ` 10 = _______________ I Remember and Understand Addition and subtraction of money is similar to Paise is always addition and subtraction of numbers. In the column written in two digits method, we write numbers one below the other and after the point. add or subtract as needed. Let us understand this through some examples. Example 1: Add: ` 14.65 and ` 23.80 Solution: We can add two amounts using the following steps: Steps Solved Solve these Step 1: Write the given numbers `p `p with the points exactly one below 1 4. 6 5 the other, as shown. + 2 3. 8 0 3 8. 4 5 + 3 5. 6 0 Step 2: First add the paise. `p Regroup the sum if needed. Write 1 the sum under paise. Place the dot just below the dot. 1 4. 6 5 + 2 3. 8 0 .4 5 Step 3: Add the rupees. Add `p `p the carry forward (if any) from 1 the previous step. Write the sum 4 1. 5 0 under rupees. 1 4. 6 5 + 4 5. 7 5 + 2 3. 8 0 3 8. 4 5 Step 4: Write the sum of the given ` 14.65 + ` 23.80 = ` 38.45 amounts. 34
Example 2: Write in columns and subtract: ` 56.50 from ` 73.50 Solution: We can subtract the amounts using the following steps: Steps Solved Solve these `p Step 1: Write the given numbers `p with the dots exactly one below 7 3. 5 0 the other, as shown. − 5 6. 5 0 Step 2: First subtract the paise. `p 8 0. 7 5 Regroup if needed. Write the 7 3. 5 0 − 4 1. 5 0 difference under paise. Place − 5 6. 5 0 the dot just below the dot. 00 Step 3: Subtract the rupees. `p `p Write the difference under 6 13 rupees. 7 3. 5 0 6 0. 7 5 − 5 6. 5 0 − 3 2. 5 0 Step 4: Write the difference of 1 7. 0 0 the given amounts. ` 73. 50 – ` 56. 50 = ` 17.00 ? Train My Brain c) ` 70.75 – ` 62.45 Solve the following: a) ` 28.65 + ` 62.35 b) ` 32.35 + ` 65.65 I Apply Making rate charts When rates of different items are written on them, it is difficult to compare them. So, we need to make a rate chart. In this chart, we write the rate of each item against its name. Money 35
Example 3: Anil and his friends are playing with play money. Anil runs a supermarket. Some items in his supermarket are given below along with their rates. ` 40 per kg ` 147 ` 50 ` 34 ` 240 per kg ` 149.50 ` 44 per litre ` 48 per kg ` 80 per kg ` 150 per kg ` 50 ` 20 per dozen He makes a rate chart to display the price of each item. How will the rate chart look? Solution: 1. Draw a table. 2. Complete the table with each item and its rate. Item Rate (in `) Item Rate (in `) 1 kg sugar 40 1 litre milk 44 Tomato Ketchup 147 1 kg wheat 48 Chocolate bar 50 1 kg oranges 80 Soap bar 34 1 kg apples 150 1 kg tea 240 1 kg pineapple 50 1 dozen bananas 20 Honey 149.50 36
Making bills To make a bill of items, we write the rate of the items and the quantities in the bill. We write the product of the rate and the quantity. We then add the products to find the total bill amount. Let us understand how to make bills through an example. Example 4: Look at the rates of the items from a stationery shop in the box below. Colour pencils Geometry Set Notebooks Sharpener ` 5 ` 140 ` 140 ` 40 Pencils ` 3 Pens ` 10 Water colours ` 100 Scissors ` 25 Sunil buys a few items as given in the list. Make a bill for the items he bought. Item Pencil Water colour Sharpener Pen Notebook Quantity 21 4 42 Solution: Follow the steps to make the bill. Step 1: Write the items and their quantities in the bill. Step 2: Then write the cost per item. Step 3: Find the total cost of each item by multiplying the number of items by their rates. Step 4: Find the total bill amount by adding the amount of each item. Money 37
S.No Item Bill Rate per Amount item `p 1 Pencil Quantity ` 3.00 6 00 2 Water colour 3 Sharpener 2 ` 100.00 100 00 4 Pen 1 ` 5.00 5 Notebook 4 ` 10.00 20 00 4 ` 40.00 40 00 2 Total 80 00 246 00 I Explore (H.O.T.S.) In some situations, we may need to add and subtract amounts together. In such cases, we need to identify which operation is to be carried out first. Let us see a few examples. Example 5: Surya went to a water park with his parents. The ticket price for each ride is given in the box. Surya went on two rides. He gave ` 60 and got Roller coaster: ` 35 ` 5 as change. Which two rides did he go on? Break dance: ` 32 Solution: Water ride: ` 20 Surya gave ` 60. The change he got is ` 5. The money spent on two rides = ` 60 – ` 5 = ` 55 So, we must add and check which two tickets cost ` 55. ` 35 + ` 32 = ` 67 which is not ` 55. ` 32 + ` 20 = ` 52 which is not ` 55. ` 35 + ` 20 = ` 55 Therefore, the two rides that Surya went on are the roller coaster and the water ride. Example 6: Add ` 20 and ` 10.50. Subtract the sum from ` 40. Solution: First add ` 20 and ` 10.50. `p ` 20 + ` 10.50 = ` 30.50 2 0. 0 0 Now, let us find the difference between ` 30.50 + 1 0. 5 0 3 0. 5 0 38
and ` 40. `p Therefore, ` 40 – ` 30.50 = ` 9.50 4 0. 0 0 − 3 0. 5 0 Maths Munchies 0 9. 5 0 It is easier to find the total amount without actually carrying out addition. If you want to know the exact amount in rupees, just group the coins equally. Suppose you have eight 50 paise coins, you group them to find the total amount. Put two 50 paise coins together, as two 50 paise = 1 rupee. So, 1 group = 1 rupee. Eight coins of 50 paise make 4 groups as each group has 2 coins. Therefore, the total amount is ` 4. Connect the Dots English Fun Here is a poem about the Indian rupee. Very odd are the things A rupee coin can make, A pleasure to give and take. Toss it up for head or tail, Buy a stamp for your mail, Offer it to god and pray, It can buy you toys of clay, Use it for a call you make, Or to check your body weight. Money 39
Social Studies Fun Different countries have different types of currency. Like, in India we have rupees and paise, the United States of America have dollars and cents. 1 rupee = 100 paise and 1 dollar = 100 cents. Drill Time 9.1 Number Operations on Money 1) The rates of some vegetables and fruits per kg are given in the box. ` 60 ` 40 ` 60 Item Quantity ` 50 ` 40 ` 19 (in kg) Tomato Carrot 2 Pumpkin 3 Cabbage 1 1 Raj buys a few items as given in the list. Make a bill for the items he bought. 2) Add: a) ` 23.24 + ` 10.80 b) ` 31.20 + ` 19.16 c) ` 61.21 + ` 29.20 d) ` 11.10 + ` 12.90 e) ` 60.90 + ` 24.23 3) Subtract: a ) ` 87.10 – ` 23.20 b) ` 20.12 – ` 10.13 c) ` 31.55 – ` 22.44 d) ` 99.99 – ` 22.22 e) ` 56.13 – ` 12.03 A Note to Parent While shopping with your child, show him or her what a bill looks like. Make your child calculate the total bill amount using addition, subtraction and multiplication. 40
10Chapter Measurement I Will Learn About • estimating and measuring length and distance. • weighing objects using simple balance. • comparing capacities using different containers. 10.1 Conversion of Standard Units of Length I Think Neena’s mother bought 3 cubits of garland. Neena observed that the same garland measured 5 cubits with her cubit. She wondered how she got more cubits than her mother. Do you know the reason for the difference? I Recall Recall that the length of an object is the distance between its two ends. We can measure the lengths of longer objects using some shorter objects. Also, we can measure objects using our hands, palm, foot and so on. 41
Hand span Cubit Foot Pace For example, consider the following: a) 9 paper clips long b) 5 erasers long c) 12 hand spans long d) 4 crayons long I Remember and Understand 3m When different people measure an object by using their body parts, they all get different lengths. The measures are different because the length of the same body part is different for different people. So, measures such as hand span, cubit, foot and leg span are called non-standard units. To get the same and accurate measurement of objects, we use a standard unit of measurement. The standard unit of measuring length is called metre. It is used to measure the length and width of a room, the height of a building and so on. We also use metre to measure the length of cloth needed to make a dress. The unit ‘metre’ is written in short as ‘m’. 42
To measure lengths smaller than a metre, we use another unit called the centimetre. Centimetre is used to measure a line, length of a ribbon and so on. We write ‘centimetre’ in short as “cm”. To measure much longer objects we use a larger unit called the “kilometre”. It is written in short as “km”. It is used to measure the length of a road and the distance between two places. The lengths of bridges and tunnels are also measured in km. A few of the standard instruments to measure length are the ruler and the measuring tape. A ruler is used to measure length in centimetres and inches. A measuring tape is used to measure longer lengths. Measuring objects using a ruler Relation between units of length. A ruler is made of plastic, wood or metal. It has two 1 m = 100 cm scales on both sides as shown in the figure. On one 1 km = 1000 m side, there is a centimetre scale and on the other side is the inch scale. We measure lengths of small objects such as a chalk, a duster and a sketch pen using these scales. We can also measure the length of a pencil, pencil box using a ruler. To measure the length of an object using a ruler, follow these steps: Step 1: Keep one end of the object at the zero of the ruler. Step 2: Note the number on the ruler which is at the other end of the object. Step 3: Write the units beside the number noted in step 2. The number along with the unit denotes the length of the object. Observe the following: a) The distance between the two ends of the given pencil is 6 cm. So, the pencil is 6 cm long. Measurement 43
Similarly, b) The water bottle is 12 cm long. c) The cell phone is 9 cm long. Let us consider a few examples. Example 1: Measure these objects and write their lengths with the correct unit. One has been done for you. Example 2: T ick the unit used to measure the lengths of the following. One has been done for you. 44
Object cm Units km Blue whale m Book Toothbrush Table Road ? Train My Brain T ick the unit we use to measure the following distances. Distance to be measured cm Units km m a) Distance between your house and school b) Distance between your desk and the blackboard c) Distance between your wrist and the tip of your middle finger Measurement 45
I Apply We can convert one unit of length to another using the relationship between them. Let us understand the conversion between metre and centimetre through a few examples. Example 3: Convert: a) 4 m into cm b) 8 m 6 cm into cm c) 300 cm into m d) 615 cm into m and cm Solution: To convert metres into centimetres, we multiply by 100. Solved Solve these a) 7 m = __________ cm a) Conversion of m into cm b) 4 m 5 cm = __________ cm c) 500 cm = __________ m 4 m = ___________ cm d) 130 cm = _______ m ______ cm 1 m = 100 cm So, 4 m = 4 × 100 cm = 400 cm b) Conversion of m and cm into cm 8 m 6 cm = ____________ cm 1 m = 100 cm So, 8 m = 8 × 100 cm = 800 cm 8 m 6 cm = (800 + 6) cm = 806 cm c) Conversion of cm into m 300 cm = ___________ m 100 cm = 1 m So, 300 cm = 3 × 100 cm =3m d) Conversion of cm into m and cm 615 cm = _________ m ________ cm 100 cm = 1 m So, 600 cm = 6 × 100 cm =6m 615 cm = (600 + 15) cm = 6 m 15 cm 46
Search
