by classklapTM MATHEMATICS 1 TEXTBOOK - PART ALPINE SERIES Enhanced Edition 3 Name: ___________________________________ Section: ________________ Roll No.: _________ School: __________________________________ Alpine_Maths_G1_TB_ToC.indd 1 12/12/2018 2:46:42 PM
Preface IMAX Program partners with schools, supporting them with learning materials and processes that are all crafted to work together as an interconnected system to drive learning. IMAX Program presents the latest version of this series – updated and revised after considering the perceptive feedback and comments shared by our experienced reviewers and users. This series endeavours to be faithful to the spirit of the prescribed board curriculum. Our books strive to ensure inclusiveness in terms of gender and diversity in representation, catering to the heterogeneous Indian classroom. The books are split into two parts to manage the bag weight. The larger aim of the curriculum regarding Mathematics teaching is to develop the abilities of a student to think and reason mathematically, pursue assumptions to their logical conclusion and handle abstraction. The Mathematics textbooks and workbooks offer the following features: S tructured as per Bloom’s taxonomy to help organise the learning process according to the different levels involved S tudent engagement through simple, age-appropriate language S upported learning through visually appealing images, especially for grades 1 and 2 Increasing rigour in sub-questions for every question in order to scaffold learning for students W ord problems based on real-life scenarios, which help students to relate Mathematics to their everyday experiences Mental Maths to inculcate level-appropriate mental calculation skills S tepwise breakdown of solutions to provide an easier premise for learning of problem-solving skills Overall, the IMAX Mathematics textbooks, workbooks and teacher companion books aim to enhance logical reasoning and critical thinking skills that are at the heart of Mathematics teaching and learning. – The Authors Alpine_Maths_G1_TB_ToC.indd 2 12/14/2018 12:10:03 PM
Textbook Features Let Us Learn About Think Contains the list of learning objectives Introduces the concept and to be covered in the chapter arouses curiosity among students Recall Discusses the prerequisite knowledge for the concept from the previous academic year/chapter/ concept/term Remembering and Understanding Explains the elements in detail that form the Application basis of the concept Ensures that students are engaged in learning throughout Connects the concept to real-life situations by enabling students to apply what has been learnt through the practice questions Higher Order Thinking Skills (H.O.T.S.) Encourages students to extend the concept learnt to advanced scenarios Drill Time Additional practice questions at the end of every chapter
Contents 3Class 1 Shapes 1 1.1 Vertices and Diagonals of Two-dimensional Shapes 9 2 Patterns 18 24 2.1 Patterns in Shapes and Numbers 31 3 Numbers 35 40 3.1 Count by Thousands 3.2 Compare 4-digit Numbers 45 48 4 Addition 53 4.1 Estimate the Sum of Two Numbers 59 4.2 Add 3-digit and 4-digit Numbers 63 4.3 Add 2-digit Numbers Mentally 69 5 Subtraction 5.1 Estimate the Difference between Two Numbers 5.2 Subtract 3-digit and 4-digit Numbers 5.3 Subtract 2-digit Numbers Mentally 6 Multiplication 6.1 Multiply 2-digit Numbers 6.2 Multiply 3-digit Numbers by 1-digit and 2-digit Numbers 6.3 Double 2-digit and 3-digit Numbers Mentally
Chapter Shapes 1 Let Us Learn About • identifying 2D shapes with straight and curved lines. • identifying sides, corners and diagonals. • making a tangram. • recognising 3D shapes and their faces and edges. Concept 1.1: Vertices and Diagonals of Two-dimensional Shapes Think There is a paper folding activity in Farida’s class. Her teacher asked the students to fold the paper across the vertices or the diagonals. How will Farida fold the paper? Recall We have learnt various shapes formed by straight lines or curved lines. Let us recall them. AB A BA B line line segment ray 1
horizontal lines vertical lines slant lines curved lines The straight and curved lines help us make closed and open figures. Figures which end at the point from where they start are called closed figures. Figures which do not end at the point from where they start are called open figures. closed figures open figures Try this! Write ‘open figure’ or ‘closed figure’ in the given blanks. ____________ ____________ ____________ ____________ Shapes such as rectangle, triangle, square and circle that can be drawn flat on a piece of paper are called two-dimensional shapes. Their outlines are called two-dimensional figures. In short, they are called 2D figures. Identify the following shapes and separate them as 1D or 2D shapes. One has been done for you. Object Shape Triangle 2D Name of the shape 1D or 2D 2
& Remembering and Understanding As we have already learnt various shapes, let us now name their parts. Consider a rectangle ABCD as shown. DC In the given rectangle, AB, BC, CD and DA are called A B its sides. There are lines joining A to C and B to D. These lines named AC and BD are called the diagonals of the rectangle. Points A, B, C and D where two sides of the rectangle meet are called the vertices. Vertex: The point where at least two sides of a figure meet is called a vertex. The plural of vertex is vertices. Diagonal: A straight line inside a shape that joins the opposite vertices is called a diagonal. A square also has sides, diagonals and vertices. Note: A triangle and a circle do not have any diagonals. Try this! Complete the table with vertices, sides and diagonals of the given different shapes. One has been done for you. CS R Y D Shape ZX W A B Q P Vertices A, B, C, D Sides AB, BC, CD, DA Diagonals AC, BD Application A We know that a 2D shape has length and breadth. Let us now learn to find the number of sides of a 2D shape. Consider a triangle as shown. B C Shapes 3
The given triangle has 3 sides named as AB, BC and CA. We can also name them as BA, CB and AC. The different number of markings on the sides of the triangle show that the lengths of all the 3 sides are different. If all the sides have the same number of markings, we can say that the lengths of all the 3 sides are the same. Let us now find the number of sides of a few 2D shapes and name them. Shape S RD C A P QA B BC Name of the shape Square Rectangle Triangle Number of sides 4 4 3 (All sides are equal.) (Opposite sides are (All sides are equal equal.) in this case.) Names of sides PQ, QR, RS, SP AB, BC, CD, DA AB, BC, CA We find objects of various shapes around us. Complete in the following table by writing the basic shapes, number of the vertices and diagonals of the given objects. Object Basic shape 36 Number of vertices 4 1 Number of diagonals 7 5 Tangram 2 A tangram is a Chinese geometrical puzzle. It consists of a square that is cut into pieces as shown in the given figure. To create different shapes, we arrange these tangram pieces with their sides or vertices touching one another. 4
Let us make our own tangram. Materials needed: a square sheet of paper a pair of scissors a ruler (optional) Procedure: Figure Steps Step 1: Fold the square sheet of paper as shown. Step 2: Cut the square into two triangles, A across the fold. B Step 3: Cut one of the triangles obtained A1 in step 2, into two equal parts. We get two 2 smaller triangles as shown. Step 4: Fold the bigger triangle as shown. B Step 5: Unfold this piece and cut it across the fold. We get one more triangle. 3 Shapes 5
Steps Figure Step 6: Fold the boat-shaped piece from one 4 end as shown. We get a triangle again on cutting at the fold. Step 7: Fold the remaining part of the paper 5 as shown. We get a square on cutting at the fold. Step 8: Fold the remaining paper again as 6 shown. We now get one more triangle on 7 cutting at the fold. We, thus, get the seven pieces of the tangram. Step 9: Colour these shapes using different colours. You can use these tangram pieces to make different shapes. Higher Order Thinking Skills (H.O.T.S.) Observe the given figure. It looks like a box. Each side of the box is a square. In the figure, AB is the length and BF is the breadth of the box. AD is E F B called the height of the box. So, this shape has three dimensions - A length, breadth and height. Such shapes are called three-dimensional shapes or 3D shapes or HG solid shapes. DC In the figure, cube • The points A, B, C, D, E, F, G and H are called vertices. • The lines AB, BC, CD, DA, BF, FE, EA, CG, GH, HD, HE and GF are called edges. • The squares ABCD, ABFE, BFGC, GCDH, EFGH and AEHD are called faces. Solid shapes with all flat square faces are called cubes. 6
Let us learn how to draw a cube in a few simple steps. Step 1: Draw a square Step 2: Draw another Step 3: Join DH, AE, BF Steps ABCD. square EFGH cutting and CG. square ABCD as shown. Figure D C H G A B HG D C DC E F EF A B AB A few other such three-dimensional shapes are cuboids and cones. Solid shapes with flat rectangular faces are called cuboids. A solid shape with a circular base, a vertex and a curved surface is called a cone. Cuboid Cone Try this! Draw a cuboid and a cone showing the formation of the figure in steps. Shape Step 1 Step 2 Step 3 Cuboid Cone Shapes 7
Drill Time Concept 1.1: Vertices and Diagonals of Two-dimensional Shapes 1) Find the number of vertices and diagonals of the following shapes: a) b) c) d) e) 8
Chapter Patterns 2 Let Us Learn About • tiling of the given shape. • identifying and creating patterns in shapes and numbers. Concept 2.1: Patterns in Shapes and Numbers Think Farida went to her father’s office on a Sunday. She saw that the floor of each hall in the office is of different designs. She found that the designs are made up of triangles, squares, circles and rectangles. She wanted to know if such repetition of a design has any special name. Do you also want to know? Recall There are many patterns around us. Patterns are similar to drawings. Let us see some of the patterns around us. Saree borders 9
Carpets Window grills Nature & Remembering and Understanding A pattern is an arrangement of shapes or numbers that follow a particular rule. Consider these examples: a) b) c) 150, 152, 154, 156 We see that in each example some shapes or numbers are repeated to form a pattern. Each shape or a group of shapes that repeats is called a basic shape. In example a), one and one make a pattern. In this pattern, the basic shape is . In example b), two and one make a pattern. In this pattern, the basic shape is . 10
In example c), the first number is 150. We get the next numbers are got by adding 2 to the previous number. Patterns in lines and shapes Observe the following patterns. These are made up of lines and shapes. a) b) c) d) Let us see a few examples of patterns. Example 1: Complete the following patterns. a) b) Solution: a) b) In the same way, we can use numbers to make different patterns. Patterns in numbers We have seen that patterns are formed by repeating shapes in a particular way. Similarly, we can repeat the numbers and create patterns. Each number pattern follows a rule. Patterns in odd and even numbers are the easiest patterns that we usually come across. Let us learn to form patterns of odd and even numbers. Pattern with even numbers: The numbers ending with 2, 4, 6, 8 or 0 are called even numbers. You can make a pattern with even numbers by adding 2 to the given even number. Patterns 11
For example, 2 + 2 = 4; 4 + 2 = 6; 6 + 2 = 8 and so on Therefore, the pattern is 2, 4, 6, 8… In this pattern, 2 is the first term, 4 is the second term, 6 is the third term, 8 is the fourth term and so on. Similarly, 18, 20, 22, 24, 26… and 246, 248, 250, 252…. are some more patterns of even numbers. Pattern with odd numbers: The numbers ending with 1, 3, 5, 7 or 9 are called odd numbers. You can make a pattern with odd numbers by adding 2 to the given odd number. For example, 1 + 2 = 3; 3 + 2 = 5; 5 + 2 = 7 and so on. Therefore, the pattern is 1, 3, 5, 7… In this pattern, 1 is the first term, 3 is the second term, 5 is the third term, 7 is the fourth term and so on. Similarly, 27, 29, 31, 33… and 137, 139, 141, 143… are some more patterns of odd numbers. Growing patterns Growing patterns can be found in shapes. Let us see a few examples. Example 2: Complete the following patterns. a) b) c) Solution: a) b) 12
c) In these patterns, we observe that each term has one more basic shape than the previous term. Some patterns have terms increasing by a certain number. We can find this number by subtracting any two consecutive terms. Consider the following patterns. a) 20, 30, 40, 50... b) 100, 200, 300... c) 11, 21, 31, 41... d) 145, 155, 165... e) 246, 346, 446... In pattern a), 40 – 30 = 10 and 30 – 20 = 10. So, the terms increase by 10. Similarly, the terms in c) and d) also increase by 10. In pattern b), 300 – 200 = 100 and 200 – 100 = 100. So, the terms increase by 100. Similarly, the terms in e) also increase by 100. Therefore, we can define the rule of the patterns in a), c) and d) as: increase by 10. The rule of the patterns in b) and e) as: increase by 100. Some patterns can be formed by decreasing the terms by a certain number. Consider the following patterns. a) 820, 720, 620, 520… b) 100, 90, 80, 70… c) 61, 56, 51, 46… d) 165, 155, 145… e) 846, 646, 446… In pattern a), 820 – 720 = 100 and 720 – 620 = 100. So, the terms decrease by 100. Similarly, the terms in e) decrease by 200. In pattern b), 100 – 90 = 10 and 90 – 80 = 10. So, the terms decrease by 10. Similarly, the terms in d) also decrease by 10. In pattern c), 61 – 56 = 5 and 56 – 51 = 5. So, the terms decrease by 5. Therefore, we can define the rule of the pattern in a) as decrease by 100; in pattern e) as decrease by 200; in patterns b) and d) as decrease by 10; in pattern c) as decrease by 5. Patterns 13
Application We see and use patterns in real life every day. We use ceramic tiles, marble, granite and other such stones for the floors of our houses. Covering a surface with flat shapes like tiles without any gaps or overlaps is called tiling. We see tiling of floors and roofs of buildings and houses. Parking areas have parking tiles laid. Some tiling patterns are as follows. Tiling can also be done using different shaped tiles as shown here. Example 3: Draw the basic shape in the given tiling patterns. a) b) Solution: a) b) Higher Order Thinking Skills (H.O.T.S.) We have seen that patterns in shapes and numbers follow certain rules. Using the rule, we can form the pattern with the given basic shapes. 14
Consider the following examples. 1) Rule: Turn the shape upside down. Basic shape: Pattern: 2) Rule: Turn the shape horizontally to the right and then back vertically. Basic shape: Pattern: 3) Rule: Rotate the shape quarter way to the right. Basic shape: Pattern: Number patterns also follow certain rules. Once the rule is identified, we can continue the given pattern. For example, the rule for a pattern is 'Begin with 1, add 3 and subtract 1 alternately'. The pattern is: 1, 4, 3, 6, 5, 8, 7... Example 4: Complete the given pattern: 8, 16, 24, ____, ____ , _____, ____. Solution: In the given pattern, the first term is 8, the second term is 16 and the third term is 24. This pattern has numbers increasing by 8. So, the next terms of the pattern are: 24 + 8 = 32; 32 + 8 = 40; 40 + 8 = 48; 48 + 8 = 56. So, the rule of this pattern is adding 8. Therefore, the pattern is 8, 16, 24, 32, 40, 48, 56. Patterns 15
Try these! Find the rule of the following patterns and write the next terms. a) 12, 24, 36, _____, _____, _____. b) 1+ 2 = 3, 2 + 3 = 5, 3 + 4 = 7, ____________, ____________, ____________. Example 5: Form a pattern using the rule, 'Begin with 5 and multiply by 2'. Solution: If the rule is 'Begin with 5 and multiply by 2', the terms in the pattern are: 5, 10, 20, 40... Drill Time Concept 2.1: Patterns in Shapes and Numbers 1) Complete the following patterns. a) ___________ ___________ ___________ ☺☺☻ ☺☺☻ b) ______________ ______________ c) _____________ ____________ d) ____________ ____________ e) ________________ ______________ 2) Fill the blanks with the next two terms of the given patterns. a) 122, 133, 144, ______, ______ b) 303, 304, 305, ______, ______ c) 40, 42, 44, _______, ________ d) 8, 24, 40, ________, ________ e) 35, 30, 25, ________, ________ f) 82, 72, 62, _______, ________ 16
3) Draw the basic shapes in the given tiling patterns. a) b) c) d) Patterns 17
Chapter Numbers 3 Let Us Learn About • writing 4-digit numbers with place value chart. • identifying and forming the greatest and the smallest number. • writing the standard and the expanded forms of the number. • comparing and ordering numbers. Concept 3.1: Count by Thousands Think Farida went to buy one of the toy cars shown. She could not read the price on one of the cars. Can you read the price on ` 1937.00 both the cars and understand what they mean? ` 657.00 Recall We know that 10 ones make a ten. Similarly, 10 tens make a hundred. Let us now count by tens and hundreds as: Counting by 10s: 10, 20, 30, 40, 50, 60, 70, 80 and 90 Counting by 100s: 100, 200, 300, 400, 500, 600, 700, 800 and 900 18
When we multiply a digit by the value of its place, we get its place value. Using place values, we can write a number in its expanded form. Let us answer these to revise the concept. a) The number for two hundred and thirty-four is _____________. b) In 857, there are _______ hundreds, _______ tens and _______ ones. c) The expanded form of 444 is _______________________. d) The place value of 9 in 493 is _____________. e) The number name of 255 is _______________________________________. & Remembering and Understanding To know about 4-digit numbers, we count by thousands using boxes. Suppose shows 1. Ten such boxes show a 10. So, = 10 ones = 1 ten Similarly, 10 such strips show 10 tens or 1 hundred. = 10 tens = 1 hundred Numbers 19
= 1 hundred = 100 = 2 hundreds = 200 = 3 hundreds = 300 = 4 hundreds = 400 In the same way, we get 5 hundreds = 500, 6 hundreds = 600, 7 hundreds = 700, 8 hundreds = 800 and 9 hundreds = 900. Using a spike abacus and beads of different colours, we represent 999 as shown. 9 blue, 9 green and 9 pink beads on the abacus represent 999. H TO Remove all the beads and Th H T O represent 999 put an orange bead on the represent 1000 next spike. This represents one thousand. We write it as 1000. 1000 is the smallest 4-digit number. Now, we know four places: ones, tens, hundreds and thousands. Let us represent 4732 in the place value chart. 20
Thousands (Th) Hundreds (H) Tens (T) Ones (O) 4 7 32 We count by 1000s as 1000 (one thousand), 2000 (two thousand)... till 9000 (nine thousand). The greatest 4-digit number is 9999. Expanded form of 4-digit numbers The form in which a number is written as the sum of the place values of its digits is called its expanded form. Let us now learn to write the expanded form of 4-digit numbers. Example 1: Expand the following numbers. a) 3746 b) 6307 Solution: Write the digits of the given numbers in the place value chart, as shown. Expanded forms: Th H TO a) 3746 = 3000 + 700 + 40 + 6 a) 3 7 46 b) 6307 = 6000 + 300 + 0 + 7 b) 6 3 0 7 Writing number names of 4-digit numbers Observe the expanded form and place value chart for a 4-digit number, 8015. Th H TO Place values 80 15 5 ones = 5 1 tens = 10 0 hundreds = 0 8 thousands = 8000 We can call 8015 as the standard form of the number. Let us look at an example. Example 2: Write the expanded forms and number names of these numbers. a) 1623 b) 3590 Numbers 21
Solution: To expand the given numbers, write them in the correct places in the place value chart. Expanded forms: Th H T O a) 1623 = 1000 + 600 + 20 + 3 a) 1 6 2 3 b) 3590 = 3000 + 500 + 90 + 0 b) 3 5 9 0 Writing in words (Number names): a) 1623 = One thousand six hundred and twenty-three b) 3590 = Three thousand five hundred and ninety We can write the standard form of a number from its expanded form. Let us see an example. Example 3: Write the standard form of 3000 + 400 + 60 + 5. Solution: Write the numbers in the place value Th H T O chart in the correct places. Write the 3 46 5 digits one beside the other, starting from the thousands place. 3000 + 400 + 60 + 5 = 3465 So, the standard form of 3000 + 400 + 60 + 5 is written as 3465. Application We can solve a few real-life examples using the knowledge of 4-digit numbers. Example 4: Ram has some money with him as shown. Calculate the amount that Ram has and write it in figures and words. 22
Solution: 1 note of ` 2000 = ` 2000 1 note of ` 100 = ` 100 3 notes of ` 10 = ` 30 1 coin of ` 5 = ` 5 So, the amount that Ram has = ` 2000 + ` 100 + ` 30 + ` 5 = ` 2135 In words, ` 2135 is two thousand one hundred and thirty-five rupees. Example 5: The number of students in different schools is given in the table. Read the table and answer the questions that follow. Name of schools Number of students Unique High School 2352 Modern High School 4782 Ideal High School 7245 Talent High School 9423 Concept High School 1281 a) What is the number of students in Ideal High School? Write the number in words. b) H ow many students are there in Concept High School? Write the number in words. Solution: a) The number of students in Ideal High School is 7245. In words, it is seven thousand two hundred and forty-five. b) The number of students in Concept High School is 1281. In words, it is one thousand two hundred and eighty-one. A place value chart helps us to form numbers using given digits. Here is an example. Example 6: A number has 6 in the thousands place and 5 in the hundreds place. It has 1 in the tens place and 4 in the ones place. What is the number? Solution: Write the digits in the place value chart according to Th H T O their places as shown. So, the required number is 6514. 6 5 1 4 Higher Order Thinking Skills (H.O.T.S.) We have learnt the concepts of expanded form and place value chart. Now, we will solve a few examples to identify numbers from the abacus. Numbers 23
Example 7: Write the numbers represented by these abacuses. a) b) c) Th H T O Th H T O Th H T O Solution: Follow these steps to write the numbers. Step 1: Write the number of beads in each Th H T O Number Step 2: place in the place value chart. a) 1 3 3 2 1332 Put a 0 in the places where there are b) 5 0 3 0 5030 no beads. c) 4 0 3 4 4034 Example 8: Draw beads on the abacus to show the given numbers. a) 3178 b) 6005 c) 4130 Th H T O Solution: Step 1: Follow these steps to show the given numbers. a) 3 1 7 8 Write the digits of the given numbers in the place b) 6 0 0 5 value chart. c) 4 1 3 0 Step 2: Draw the number of beads on each spike of the abacus to show the digit in each place of the number. Th H T O Th H T O Th H T O a) 3178 b) 6005 c) 4130 Concept 3.2: Compare 4-digit Numbers Think Farida has 3506 paper clips and her brother has 3605 paper clips. Farida wants to know who has more paper clips. But the numbers appear to be the same, and she is confused. Can you tell who has more number of paper clips? 24
Recall In class 2, we have learnt to compare 3-digit numbers and 2-digit numbers. Let us quickly revise the concept. A 2-digit number is always greater than a 1-digit number. A 3-digit number is always greater than a 2-digit number and a 1-digit number. So, a number with more number of digits is always greater than a number with lesser digits. We use the symbols >, < or = to compare two numbers. & Remembering and Understanding Comparing two 4-digit numbers is similar to comparing two 3-digit numbers. Let us understand the steps to compare through an example. Example 9: Compare: 5382 and 5380 Solution: Follow these steps to compare the given numbers. Steps Solved Solve this Step 1: Compare the number of digits 5382 and 5380 7469 and 7478 Count the number of digits in the given numbers. The number having more number of digits is Both 5382 and greater. 5380 have 4 digits. Step 2: Compare thousands If two numbers have the same number of digits, 5=5 ____ = ____ compare the thousands digits. (If two numbers have an equal number of digits, start comparing 3=3 ____ = ____ from the leftmost digit.) The number with the greater digit in the thousands place is greater. Step 3: Compare hundreds If the digits in the thousands place are the same, compare the digits in the hundreds place. The number with the greater digit in the hundreds place is greater. Numbers 25
Steps Solved Solve this 5382 and 5380 7469 and 7478 Step 4: Compare tens 8=8 ____ > ____ If the digits in the hundreds place are also same, So, compare the digits in the tens place. The number with the greater digit in the tens place is greater. ____ > ____ Step 5: Compare ones If the digits in the tens place are also the same, 2>0 compare the digits in the ones place. The So, - number with the greater digit in the ones place is greater. When the ones place are the same, the 5382 > 5380 numbers are equal. Note: Once we can decide a greater/smaller number, the steps that follow need not be carried out. Application We can apply the knowledge of comparing numbers and place value to: 1) arrange numbers in the ascending and descending orders. 2) form the greatest and the smallest numbers using the given digits. Ascending and descending orders Ascending Order: The arrangement of numbers from theTrsmaainllesMt toythBeragrienatest Descending Order: The arrangement of numbers from the greatest to the smallest Example 10: Arrange 4305, 4906, 4005 and 4126 in the ascending and descending orders. Solution: Follow these steps to arrange the given numbers in the ascending and descending orders. Ascending Order Step 1: Compare the digits in the thousands place: All the numbers have 4 in their thousands place. Step 2: Compare the digits in the hundreds place: 4005 – 0 hundreds, 4126 –1 hundred, 4305 – 3 hundreds and 26
4906 – 9 hundreds So, 4005 < 4126 < 4305 < 4906 Step 3: Arranging the numbers in ascending order: 4005, 4126, 4305, 4906 Descending Order Step 1: Compare the digits in the thousands place: All the numbers have 4 in their thousands place. Step 2: Compare the digits in the hundreds place: 4005 – 0 hundreds, 4126 – 1 hundred, 4305 – 3 hundreds and 4906 – 9 hundreds So, 4906 > 4305 > 4126 > 4005 Step 3: Arranging the numbers in descending order: 4906, 4305, 4126, 4005 Simpler way! The descending order of numbers is just the reverse of their ascending order. Forming the greatest and the smallest 4-digits numbers Let us learn to form the greatest and the smallest 4-digit numbers. Look at the following examples. Example 11: Form the greatest and the smallest 4-digit number using 4, 3, 7 and 5 (without repeating the digits). Solution: The given digits are 4, 3, 7 and 5. The steps to find the greatest 4-digit number are given below. Step 1: Arrange the digits in descending order as 7 > 5 > 4 > 3. Step 2: Place the digits in the place value chart from left to right. So, the greatest 4-digit number formed is 7543. Th H T O T he steps to find the smallest 4-digit number are given 7543 below. Step 1: Arrange the digits in ascending order as Th H T O 3 < 4 < 5 < 7. 34 5 7 Step 2: Place the digits in the place value chart from left to right. So, the smallest 4-digit number formed is 3457. Numbers 27
Example 12: Form the smallest 4-digit number using 4, 1, 0 and 6 (without repeating the digits). Solution: The given digits are 4, 1, 0 and 6. Step 1: Arrange the digits in ascending order as 0 < 1 < 4 < 6. Step 2: Place the digits in the place value chart from left to Th H T O right. But the number formed is 0146 or 146. 01 4 6 It is a 3-digit number. In such cases, we interchange the first two digits in Th H T O the place value chart. 10 4 6 So, the smallest 4-digit number formed is 1046. Example 13: Form the smallest and the largest 4-digit numbers using 4, 0, 8 and 6 (with repeating the digits). Solution: The given digits are 4, 0, 8 and 6. Follow the steps to form the smallest 4-digit number. Step 1: Find the smallest digit. 0 is the smallest of the given digits. (But a number cannot begin with 0.) Step 2: If the smallest digit is ‘0’, find the next smallest digit, which is 4. Write ‘4’ in the thousands place. Write ‘0’ in the rest of the places. Therefore, the smallest 4-digit number is 4000. Note: If the smallest of the given digits is not ‘0’, repeat the smallest digit four times to form the smallest number. Now, let us form the largest 4-digit number from the given digits. Step 1: The largest of the given digits is 8. Step 2: Repeat the digit four times to form the largest 4-digit number. Therefore, the largest 4-digit number that can be formed is 8888. Higher Order Thinking Skills (H.O.T.S.) Let us see a few real-life examples where we use the comparison of 4-digit numbers. Example 14: 4538 people visited an exhibition on Saturday and 3980 people visited it on Sunday. On which day did fewer people visit the exhibition? 28
Solution: Number of people who visited the exhibition on Saturday = 4538 Number of people who visited the exhibition on Sunday = 3980 Comparing both the numbers using the place value chart, Th H T O Th H T O 4 53 8 3 98 0 4 > 3 or in other words, 3 < 4 So, 3980 < 4538. Therefore, fewer people visited the exhibition on Sunday. Example 15: Razia arranged the numbers 7123, 2789, 2876 and 4200 in the ascending order as 2876, 2789, 4200, 7123. Reena arranged them as 2789, 2876, 4200, 7123. Who arranged them correctly? Why? Solution: Reena’s arrangement is correct. Reason: Comparing the hundreds place of the smaller of the given numbers, 7 hundreds < 8 hundreds. So, 2789 is the smallest number. Drill Time Concept 3.1: Count by Thousands 1) Write the numbers in the place value chart. a) 1451 b) 8311 c) 9810 d) 1000 e) 7613 2) Write the numbers in their expanded forms. a) 8712 b) 6867 c) 1905 d) 4000 e) 9819 3) Write the number names of the following numbers: a) 9125 b) 5321 c) 3100 d) 1900 e) 7619 4) Form 4-digit numbers from the following: a) 4 in the thousands place, 3 in the hundreds place, 0 in the tens place and 2 in the ones place b) 9 in the thousands place, 1 in the hundreds place, 4 in the tens place and 0 in the ones place Numbers 29
c) 5 in the thousands place, 4 in the hundreds place, 9 in the tens place and 7 in the ones place d) 8 in the thousands place, 2 in the hundreds place, 6 in the tens place and 5 in the ones place e) 1 in the thousands place, 2 in the hundreds place, 3 in the tens place and 4 in the ones place 5) Word problems a) The number of people in different rows in a football stadium is as given: Row 1: 2345 Row 2: 6298 Row 3: 7918 Row 4: 8917 Row 5: 1118 (A) What is the number of people in Row 1? Write the number in words. (B) How many people are there in Row 4? Write the number in words. b) R am has a note of ` 2000, a note of ` 500, a note of ` 20 and a coin of ` 2. How much money does he have? Write the amount in figures and words. Concept 3.2: Compare 4-digit Numbers 6) Compare the following numbers using <, > or =. a) 8710, 9821 b) 1689, 1000 c) 4100, 4100 d) 2221, 2222 e) 6137, 6237 7) Arrange the numbers in ascending and descending orders. a) 4109, 5103, 1205, 5420 b) 7611, 7610, 7609, 7605 c) 9996, 8996, 1996, 4996 d) 5234, 6213, 1344, 5161 e) 4234, 6135, 4243, 6524 8) Form the greatest and the smallest numbers using: a) 3, 5, 9, 2 b) 1, 5, 9, 4 c) 7, 4, 1, 8 d) 9, 1, 3, 5 e) 8, 2, 3, 4 9) Word problems a) 5 426 people visited a museum on a Friday and 3825 people visited it on the following Sunday. On which day did fewer people visit the museum? b) A shopkeeper sold 1105 milk chocolates and 2671 white chocolates. Which type of chocolates did he sell more? 30
Chapter Addition 4 Let Us Learn About • adding numbers with and without regrouping. • rounding off numbers to the nearest tens. • estimating the sum by adding mentally. Concept 4.1: Estimate the Sum of Two Numbers Think Farida has ` 450 with her. She wants to buy a toy car for ` 285 and a toy train for ` 150. Do you think she has enough money to buy the toys? Recall We have learnt addition of 2-digit and 3-digit numbers. Here is a quick recap of the steps. Step 1: Place the numbers one below the other, according to their places. Step 2: Start adding from the ones place. Step 3: Regroup the sum and carry it forward to the next place, if necessary. Step 4: Write the answer. 31
& Remembering and Understanding Many a times, knowing the exact number may not be needed. When we say there are about 50 students in class, we mean that the number is close to 50. Numbers which are close to the exact number can be rounded off. Rounding off numbers is also known as estimation. If the digit in the ones place is equal to or greater than 5, we round off the number to the closest multiple of ten, greater than the given number. Let us now learn to round off or estimate the given numbers. Rounding to the nearest 10 Observe the number line given. The numbers on it are written in tens. 12 is between 10 and 20 and is closer (12) (28) (35) (49) to 10. So, we round off 12 down to 10. 0 10 20 30 40 50 35 is exactly in between 30 and 40. So, we round it off up to 40. Let us now learn a step-wise procedure to round off numbers to the nearest 10. Example 1: Round off the following numbers to the nearest 10. a) 86 b) 42 Solution: Let us round off the given numbers using a step-wise procedure. Steps Solved Solve these 86 42 57 25 63 Step 1: Observe the digit in the ones place 86 42 57 25 63 of the number. Step 2: If the digit in 6>5 2 < 5 ____ > 5 ____ = 5 ____ < 5 the ones place is 4 or less, round the number 86 is 42 is ____ is ____ is ____ is down to the previous rounded rounded rounded rounded rounded ten. up to 90 down to 40 up to ____ up to ____ down to If it is 5 or more, round the number up, to the ____ next tens. 32
Rounding off numbers is used to estimate the sum of two 2-digit and 3-digit numbers. Let us understand this through an example. Example 2: Estimate the sum of: a) 64 and 15 b) 83 and 18 Solution: a) 64 + 15 Rounding off 64 to the nearest tens gives 60 (as 4 < 5). R ounding off 15 to the nearest tens gives 20 (as 5 = 5). So, the required sum is 60 + 20 = 80. b) 83 + 18 Rounding off 83 to the nearest tens gives 80 (as 3 < 5). Rounding off 18 to the nearest tens gives 20 (as 8 > 5). So, the required sum is 80 + 20 = 100. Example 3: Estimate the sum in the following: a) 245 and 337 b) 483 and 165 Solution: a) 245 + 337 R ounding off 245 to the nearest tens gives 250 (as 5 = 5). Rounding off 337 to the nearest tens gives 340 (as 7 > 5). So, the required sum is 250 + 340 = 590. b) 483 + 165 R ounding off 483 to the nearest tens gives 480 (as 3 < 5). Rounding off 165 to the nearest tens gives 170 (as 5 = 5). So, the required sum is 480 + 170 = 650. Application Here are a few examples where the estimation of the sum can be useful. Example 4: Arun wants to distribute sweets among students in the two sections of his class. In Section A, there are 43 students and in Section B, there are 36 students. Estimate the number of sweets that Arun should take to the class. Addition 33
Solution: Number of students in Section A = 43 Rounding off 43 to the nearest tens, we get 40. Number of students in Section B = 36 Rounding off 36 to the nearest tens, we get 40. Their sum is 40 + 40 = 80. Therefore, Arun should take about 80 sweets to the class. Example 5: Raj buys vegetables for ` 63 and fruits for ` 25. Estimate the amount he should pay to the shopkeeper. Solution: Amount spent on vegetables = ` 63 63 rounded to the nearest tens is 60. Amount spent on fruits = ` 25 25 rounded to the nearest tens is 30. Total amount to be paid = ` 60 + ` 30 = ` 90 HigShoe, rROajrsdheour lTdhpinakyinagboSukti`lls90(Hto.Oth.Te.Ssh.)opkeeper. Observe a few more situations where estimation of sum is used. Example 6: There are 416 walnut trees in a park. The park workers plant 574 more walnut trees. Estimate the number of walnut trees in the park after the workers finish planting. Solution: Number of trees in the park = 416 Rounding off 416 to the nearest tens, we get 420. Number of more trees the workers plant = 574 Rounding off 574 to the nearest tens, we get 570. Their sum is 420 + 570 = 990. Therefore, the park will have about 990 walnut trees after the workers finish planting. Example 7: Ramya has 26 cookies and 34 toffees. Renu has 42 cookies and 13 toffees. Estimate the total number of cookies and toffees. 34
Solution: Number of cookies with Ramya = 26 Number of toffees with her = 34 Rounding off 26 and 34 to the nearest tens, we get 30 and 30 respectively. Number of cookies with Renu = 42 Number of toffees with her = 13 Rounding off 42 and 13 to the nearest tens, we get 40 and 10 respectively. So, the sum of cookies = 30 + 40 = 70 Sum of toffees = 30 + 10 = 40 Therefore, they have 70 cookies and 40 toffees altogether. Concept 4.2: Add 3-digit and 4-digit Numbers Think Farida’s father bought her a shirt for ` 335 and a skirt for ` 806. Farida wants to find how much her father had spent in all. How do you think she can find that? Recall We can add 2-digit or 3-digit numbers by writing them one below the other. This method of addition is called vertical addition. Let us revise the earlier concept and solve the following. a) 22 + 31 = _________ b) 42 + 52 = _________ c) 82 + 11 = _________ d) 101 + 111 = _________ e) 100 + 200 = _________ f) 122 + 132 = _________ Addition 35
& Remembering and Understanding Let us now understand the addition of two 3-digit numbers with regrouping. We will also learn to add two 4-digit numbers. Add 3-digit numbers with regrouping Sometimes, the sum of the digits in a place is more than 9. In such cases, we need to regroup the sum. We then carry forward the digit to the next place. Example 8: Add 245 and 578. Solution: Arrange the numbers one below the other. Regroup if the sum of the digits is more than 9. Step 1: Add the ones. Solved Step 3: Add the hundreds. H TO Step 2: Add the tens. H TO 1 11 245 H TO 245 11 +578 245 +578 3 +578 823 23 H TO Solve these H TO H TO 823 171 +197 390 +219 +121 Add 4-digit numbers without regrouping Adding two 4-digit numbers is similar to adding two 3-digit numbers. Let us understand this through an example. Example 9: Add 1352 and 3603. Solution: Arrange the numbers one below the other. 36
Solved Step 1: Add the ones. Step 2: Add the tens. Th H T O Th H T O 135 2 135 2 +3 6 0 3 +3 6 0 3 5 55 Step 3: Add the hundreds. Step 4: Add the thousands. Th H T O Th H T O 135 2 13 5 2 +3 6 0 3 +3 6 0 3 955 49 5 5 Solve these Th H T O Th H T O Th H T O 41 9 0 20 0 2 11 1 1 +2 0 0 0 +3 0 0 3 +2 2 2 2 Add 4-digit numbers with regrouping We regroup the sum when it is equal to or more than 10. Example 10: Add 1456 and 1546. Solution: Arrange the numbers one below the other. Add and regroup, if necessary. Solved Step 1: Add the ones. Step 2: Add the tens. Th H T O Th H T O 1 11 1456 1456 +1 5 4 6 +1 5 4 6 2 02 Addition 37
Solved Step 3: Add the hundreds. Step 4: Add the thousands. Th H T O Th H T O 111 111 1456 1456 +1 5 4 6 +1 5 4 6 002 3002 Th H T O Solve these O Th H T O Th H T 17 5 8 45 9 2 +5 6 6 2 26 7 8 +1 4 5 6 +1 3 3 2 Application Look at a few examples where we use addition of 3-digit and 4-digit numbers. Example 11: Vinod had some stamps out of which he gave 278 stamps to his brother. Vinod now has 536 stamps left with him. How many stamps did he have in the beginning? H TO Solution: Number of stamps Vinod has now = 536 11 Number of stamps he gave his brother = 278 5 36 Number of stamps Vinod had in the +2 78 beginning = 536 + 278 = 814 8 14 Therefore, Vinod had 814 stamps in the beginning. 38
Example 12: Ajit collected ` 2683 and Radhika collected ` 3790 for donating to a nursing home. What is the total money Th H T O collected? Solution: Amount collected by Ajit = ` 2683 11 Amount collected by Radhika = ` 3790 2 6 83 Total amount collected for the donation +3 7 9 0 =` 2683 + ` 3790 = ` 6473 6 4 73 Example 13: The number of Class 3 students in Heena’s school is 236. The number of Class 3 students in Veena’s school is 289. How many total number of students were present in Class 3 of both the school? Solution: Number of students in Heena’s school = 236 Number of students present in Veena’s school = 289 Total number of students present in Class 3 of both the schools = 236 + 289 = 525 Higher Order Thinking Skills (H.O.T.S.) Let us see a few more examples on the addition of 4-digit numbers. Example 14: Three pieces of ribbon of lengths 2134 cm, 1185 cm and 3207 cm are cut from a long ribbon. What was the total length of the ribbon before the pieces were cut? Solution: The pieces of ribbon are 2134 cm, 1185 cm Th H T O and 3207 cm long. 11 Length of the ribbon before the pieces were 2134 cut = 2134 cm + 1185 cm + 3207 cm +1 1 8 5 Therefore, the ribbon was 6526 cm long before + 3 2 0 7 6526 the pieces were cut. Example 15: Payal, Eesha and Suma have 1284, 7523 and 5215 stamps respectively. Frame an addition problem. Solution: An addition problem contains words such as - in all, total, altogether and so on. So, the question can be ‘‘Payal, Eesha and Suma have 1284, 7523 and 5215 stamps respectively. How many stamps do they have altogether?” Addition 39
Concept 4.3: Add 2-digit Numbers Mentally Think Farida had 18 colour pencils. Her sister gave her 71 more. Farida wanted to calculate the total number of pencils mentally. Do you know how Farida could do? Recall We have already learnt to add two 1-digit numbers mentally. To do so, we keep the larger number in mind and add the smaller one to it. Let us answer the questions to revise the concept. a) 5 + 4 = ________ [ ] (A) 5 (B) 4 (C) 1 (D) 9 b) 3 + 3 = ________ [ ] (A) 3 (B) 6 (C) 0 (D) 5 c) 1 + 4 = ________ [ ] (A) 3 (B) 4 (C) 6 (D) 5 d) 5 + 0 = ________ Train My Brain [ ] (A) 4 (B) 5 (C) 0 (D) 6 e) 6 + 3 = ________ [ ] (A) 4 (B) 6 (C) 3 (D) 9 & Remembering and Understanding Let us now learn to add two 2-digit numbers mentally, through these examples. Add 2-digit numbers mentally without regrouping Example 16: Add mentally: 53 and 65 Solution: To add the given numbers mentally, follow these steps: 40
Steps Solved Solve this 53 and 65 38 and 41 Step 1: Add the digits in the ones place of the two 3+5=8 _____ + _____ = _____ numbers mentally. Step 2: Add the digits in the The digits in the tens The digits in the tens place of tens place of the two numbers place of the two the two numbers are ___ and mentally. To mentally add numbers are 5 and 6. ____. Keep ____ in your mind, two 1-digit numbers, keep the Keep 6 in your mind, count ___ forward as ____, larger number in mind and the count 5 forward as 7, ____and ____. smaller on the fingers. 8, 9, 10 and 11. ____ + ____ = ___ 5 + 6 = 11 Step 3: Write sum of the digits So, 53 + 65 = 118. So, 38 + 41 = ___. obtained in step 1 and sum of the digits obtained in step 2 together. This is the sum of the given numbers. Add 2-digit numbers mentally with regrouping Example 17: Add mentally: 29 and 56 Solution: To add the given numbers mentally follow these steps. Steps Solved 29 and 56 Solve this 83 and 47 Step 1: Regroup the two 29 = 20 + 9 83 = ___ + ____ given numbers as tens and 56 = 50 + 6 47 = ___ + ____ ones mentally. ____ + ____ = ____ Step 2: Add the ones of the 9 + 6 = 15 ____ + ____ = ____ two numbers mentally. ____ + ___ = ____ Step 3: Add the tens of the 20 + 50 = 70 two numbers mentally. So, 83 + 47 = ___. Step 4: Add the sums from 70 + 15 steps 2 and 3 mentally = 70 + 10 + 5 (regroup if needed). = 85 So, 29 + 56 = 85. Step 5: Write the sum of the given numbers. Addition 41
Application We have seen how easy it is to add two 2-digit numbers mentally. Let us see some real-life situations in which mental addition of 2-digit numbers is useful. Example 18: Suraj has 34 sheets and Kamal has 27 sheets of paper. How many sheets of paper do they have in all? Solve mentally. Solution: Number of sheets of paper Suraj has = 34 Number of sheets of paper Kamal has = 27 Total number of sheets they have together = 34 + 27 Regrouping the given numbers in tens and ones and adding, we get 30 + 4 + 20 + 7 To add two 1-digit numbers mentally, keep the larger number in mind and add the smaller one to it. Add tens and ones accordingly. = 50 + 11 = 50 + 10 + 1 (Regroup and add) = 60 + 1 = 61 Therefore, Suraj and Kamal have 61 sheets of paper. Example 19: Vivek has 49 bags and Shyam has 29 bags. How many bags do they have in total? Solve mentally. Solution: Number of bags Vivek has = 49 Number of bags Shyam has = 29 Total number of bags they have together = 49 + 29 Regrouping the given numbers in tens and ones and adding, we get 40 + 9 + 20 + 9 To add two 1-digit numbers, keep the larger number in mind and add the smaller one to it. Add tens and ones accordingly. = 40 + 20 + 18 = 60 + 10 + 8 (Regroup and add) = 70 + 8 = 78 So, they have 78 bags in total. 42
Higher Order Thinking Skills (H.O.T.S.) We have seen mental addition of two 2-digit numbers. Let us now see a few examples to add three 2-digit numbers mentally. Example 20: Add mentally: 25, 37 and 19 Solution: To add the given numbers mentally follow these steps. Steps Solved Solve this 25, 37 and 19 40, 29 and 54 Step 1: Regroup the three given 25 = 20 + 5 40 = ___ + ____ numbers as tens and ones mentally. 37 = 30 + 7 29 = ___ + ____ 19 = 10 + 9 54 = ____+____ Step 2: Add the tens mentally. 20 + 30 + 10 = 60 ____ + ____+ ____ = ____ Step 3: Add the ones mentally. 5 + 7 + 9 = 21 ____+___ + ____ = ____ Step 4: Add the sums from steps 2 60 + 21 ____ + ___ = ____ and 3 mentally, regroup again if = 60 + 20 + 1 = 81 needed. So, 25 + 37 + 19 = 81. So, 40 + 29 + 54 = ___. Step 5: Write the sum of the given numbers. Drill Time Concept 4.1: Add 3-digit and 4-digit Numbers c) 288 + 288 1) Add 3-digit numbers with regrouping. a) 481 + 129 b) 119 + 291 d) 346 + 260 e) 690 + 110 Addition 43
2) Add 4-digit numbers without regrouping. a) 1234 + 1234 b) 1000 + 2000 c) 4110 + 1332 d) 5281 + 1110 e) 7100 +1190 3) Add 4-digit numbers with regrouping. a) 5671 + 1430 b) 3478 + 2811 c) 4356 + 1753 d) 2765 + 1342 e) 4901 + 2222 4) Word problems a) There are 142 people riding in Train A and 469 people in Train B. How many people rode in both the trains altogether? b) Ali scored 272 points in one level of a computer game. His friend, Jenny, scored 538 points in the next level. What is their total score in both the levels? Concept 4.2: Estimate the Sum of Two Numbers 5) Estimate the sum of the following: a) 211 and 115 b) 549 and 120 c) 385 and 190 d) 222 and 524 e) 672 and 189 6) Word problems a) Susan has 46 red roses and Mukesh has 22 yellow roses. Estimate the total number of roses. b) Rakesh has 67 pencils and Mona has 43 pencils. Estimate the number of pencils both of them have in all. Concept 4.3: Add 2-digit Numbers Mentally 7) Add 2-digit numbers mentally without regrouping. a) 31 and 22 b) 22 and 42 c) 45 and 51 d) 11 and 34 e) 32 and 61 8) Add 2-digit numbers mentally with regrouping. a) 45 and 47 b) 25 and 56 c) 12 and 19 d) 27 and 35 e) 17 and 37 44
Chapter Subtraction 5 Let Us Learn About • Subtracting 3-digit numbers with regrouping. • Subtracting 4-digit numbers with and without regrouping. • rounding off numbers. • estimating the difference between numbers. • subtracting two numbers mentally. Concept 5.1: Estimate the Difference between Two Numbers Think Farida had ` 450 with her. She wanted to buy a toy car for ` 185 and a toy train for ` 150. How much money will remain with Farida after buying the toys? Recall We know that in some situations where we do not need the exact number, we use estimation. Estimation can be done by rounding off numbers to a given place. Let us answer these to revise the concept of rounding off to the nearest 10. a) 87 = ______ b) 53 = ______ c) 65 = ______ d) 42 = ______ e) 33 = ______ & Remembering and Understanding Estimation is finding a number that is close enough to the right answer. Rounding off numbers can be used to estimate the difference between two 2-digit numbers and between two 3-digit numbers. 45
Let us understand this through examples. Example 1: Estimate the difference: a) 69 – 15 b) 86 – 12 Solution: a) 69 – 15 Rounding off 69 to the nearest tens gives 70 (as 9 > 5) and rounding off 15 to the nearest tens, gives 20 (as 5 = 5). So, the required difference is 70 – 20 = 50. b) 86 – 12 R ounding off 86 to the nearest tens gives 90 (as 6 > 5) and rounding off 12 to the nearest tens, gives 10 (as 2 < 5). So, the required estimated difference is 90 – 10 = 80. Example 2: Estimate the difference: a) 593 – 217 b) 806 – 124 Solution: a) 593 – 217 R ounding off 593 to the nearest tens gives 590 (as 3 < 5) and rounding off 217 to the nearest tens, gives 220 (as 7 > 5). So, the required estimated difference is 590 – 220 = 370. b) 806 – 124 R ounding off 806 to the nearest tens gives 810 (as 6 > 5) and rounding off 124 to the nearest tens, gives 120 (as 4 < 5). So, the required estimated difference is 810 – 120 = 690. Application Estimation of differences can be used in real-life situations. Let us see a few examples. Example 3: Parul has 83 pencils. She gives 32 pencils to her sister. Estimate the number of pencils that remain with Parul. Solution: Number of pencils Parul has = 83 83 rounded off to the nearest tens is 80 (since 3 < 5). Number of pencils given to Parul’s sister = 32 32 rounded off to the nearest 10 is 30 (since 2 < 5). 46
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