Synergy for Success in Mathematics 7 is designed for Grade 7 students. The textbook contains all the required learning competencies and is supplemented with some additional topics for enrichment. Lessons are presented using effective Singapore Math strategies that are intended for easy understanding and grasp of ideas for its target readers. Various exercises are also provided to help the learners acquire the necessary skills needed. The book is organized with the following recurring features in every chapter: Learning Goals Introduction Historical Note This gives the specific objectives that are intended to be achieved in the end. The reader is given a bird's eye view of the contents. A brief historical account of a related topic is included giving the reader an awareness of some important contributions of some great mathematicians or even stories of great achievements related to mathematics. Method/Exam Notes These additional tools help students recall important information, formulas, and shortcuts, needed in working out solutions. Examples Enhancing Skills Linking Together Chapter Test Chapter Project Making Connection PREFACE Step-by-step and detailed demonstrations of how a specific concept or technique is applied in solving problems. These are practice exercises found after every lesson, that will consolidate and reinforce what the students have learned. This visual tool can help the students realize the connection of all the ideas presented in the chapter. This is a summative test given at the end in preparation for the expected actual classroom examination containing the topics included in the chapter. A challenging task is designed for the learner giving him an opportunity to use what he/she has learned in the chapter. This may be a manipulative type of activity that is specifically chosen to enhance understanding of the concepts learned in the chapter. The students are exposed to facts and information that connect mathematics and culture. This is for the purpose of letting the learners appreciate the subject because of tangible or true-tolife stories that show how mathematics is useful and relevant. Every effort has been made in order for all the discussions in this book to be clear, simple, and straightforward. This book also gives opportunities for the readers to see the beauty of mathematics as an essential tool in understanding the world we live in. With this in mind, appreciation of mathematics goes beyond seeing; realizing its critical application to decision making in life completes the purpose of knowing and understanding mathematics.
Table of C ntents CHAPTER 1 BASIC IDEA OF SETS Introduction... 1 Historical Note... 2 1.1 Introduction to Sets... 3 1.2 Operations on Sets... 14 Linking Together... 24 Chapter Test... 25 Chapter Project... 29 Making Connection... 30 CHAPTER 2 REAL NUMBERS Introduction... 31 Historical Note... 32 2.1 The Real Number System... 33 2.2 Properties of Real Numbers... 39 2.3 Integers... 46 2.4 Absolute Value of a Real Numbers... 64 2.5 Fractions... 68 2.6 Decimals... 82 2.7 Approximation on Square Roots... 91 2.8 Scientific Notation... 96 2.9 Significant Digits...101 Linking Together...105 Chapter Test...106 Chapter Project...109 Making Connection...110
CHAPTER 3 MEASUREMENT Introduction...111 Historical Note...112 3.1 Measuring Length, Perimeter, Mass, and Volume...113 3.2 Measuring Area, Temperature, and Time...131 Linking Together...145 Chapter Test...146 Chapter Project...149 Making Connection...150 CHAPTER 4 ALGEBRAIC EXPRESSIONS Introduction...151 Historical Note...152 4.1 Number Sequence and Pattern Finding...153 4.2 Algebraic Expressions...159 4.3 Integral Exponents...170 4.4 Evaluation of Algebraic Expressions...186 4.5 Translation of Mathematical Phrases into Symbols...194 4.6 Operations of Polynomials...202 4.7 Special Products...215 Linking Together...227 Chapter Test...228 Chapter Project...231 Making Connection...232 CHAPTER 5 SIMILARITY AND RIGHT TRIANGLES Introduction...233 Historical Note...234 5.1 Linear Equations in One Variable...235 5.2 Solving Absolute Value Equations...254 5.3 Linear Inequalities in One Variable...259 5.4 Solving Absolute Value Inequalities...278 5.5 Solving Word Problems Involving Linear Equations and Inequalities...286 Linking Together...310 Chapter Test...311 Chapter Project...315 Making Connection...316
CHAPTER 6 TOOLS OF GEOMETRY Introduction...317 Historical Note...318 6.1 Objects of Geometry...319 6.2 Angles and Angle Measures...337 6.3 Reasoning and Proving...353 6.4 More Objects of Geometry...366 6.5 Geometric Constructions...378 Linking Together...385 Chapter Test...386 Chapter Project...389 Making Connection...390 CHAPTER 7 PERPENDICULAR AND PARALLEL LINES Introduction...391 Historical Note...392 7.1 Perpendicular and Parallel Lines...393 7.2 Applying Concepts of Perpendicular and Parallel Lines...405 Linking Together...414 Chapter Test...415 Chapter Project...419 Making Connection...420 CHAPTER 8 STATISTICS Introduction...421 Historical Note...422 8.1 Introduction to Statistics...423 8.2 The Frequency Table...429 8.3 Use of Graphs to Represent and Analyze Data...438 8.4 Measures of Central Tendency (Ungrouped Data)...447 Linking Together...458 Chapter Test...459 Chapter Project...463 Making Connection...464 Glossary...465 Index...479 Bibliography...486
1 BASIC IDEA OF SETS Learning Goals At the end of the chapter, the students should be able to: 1.1 1.2 Our daily activities often involve groups or collection of objects, such as set of wardrobe, group of students, a collection of toys, a list of formulas, and many others. One of the important foundation for some topics in mathematics is the idea of sets. This chapter covers the fundamental concepts of a set, kinds of sets, union of sets, and intersection of sets. Define and describe a set and use a Venn diagram to illustrate a set and properties of set operations Describe and illustrate complement of a set, and union and intersection of sets
Historical Note George Ferdinand Ludwig Philipp Cantor (1845 1918) is a German mathematician known as the founder of set theory. Cantor set forth the modern theory on infinite sets that developed all the disciplines in mathematics. Cantor defined well-ordered and infinite sets. He established the importance of one-to-one correspondence between the members of two sets. He showed that not all infinite sets have the same size, therefore, infinite sets can be compared with one another. He then proved that the real numbers are numerous than the natural numbers. He defined what it means for two sets to have the same cardinal number. He proved that the set of real numbers and the set of points in ndimensional Euclidean space have the same exponent. Cantor s early interests were in number theory, indeterminate equations, and trigonometric series. In 1874, he started his radical work on set theory and the theory of the infinite. Cantor created a whole new field of mathematical research. 2
Synergy for Success in Mathematics Chapter 1 1.1 Introduction to Sets A set is a collection of objects which are clearly defined as belonging to a well-defined group. Each object in a set is called an element of a set. Each element is separated by a comma. The set is enclosed by braces { }. Normally, a capital letter is used to name or label a set. For example, set A consists of all subjects offered in secondary school. A = {set of subjects in secondary school} A = {English, Math, Science, CLE, Filipino, Social Studies, MAPEH} A set must be well defined so that we can determine whether an object is an element of the set. A set may be described using a set notation. The two main methods of set notation are the rule method or set builder notation and the roster or listing method. Rule Method A={ x x } Roster or Listing Method : is a counting number from 1 to 5 A ={ 12345,,,, } B={ x x } C : is a month that starts with letter A B ={ April, August} ={ x: x is a prime factor of 15} C ={ 35, } In roster or listing method, the elements are separated by commas and are enclosed within a pair of brace { }. 3
Notice that set A in the rule method is properly described so that it could be easier to list down all the possible elements. ={ } A x: x is a counting number from 1 to 5 is read as A is the set of elements x, such that x is a counting number from 1 to 5. There are cases when it is too tedious or impossible to list all the elements of a set. There are sets whose elements are infinite or too many to enclose inside braces. Such sets are rather defined using the rule method. For example: A={ x: x is an even number between 1 and 100} A ={ 2468,,,,, 9698, } The three dots (...) are called ellipsis which means "continue on." The ellipsis represents the other elements which are no longer practical to include in the list. List all the elements of the following sets. (a) Example 1 A = {x : x is a letter in the word SUBTRACT} (b) B = {x : x is a counting number greater than 8} SOLUTION (a) A = {S, U, B, T, R, A, C} Although there are two T's, this letter must be written only once within the brace. (b) B = {9, 10, 11, 12,...} The ellipsis is used to acknowledge the existence of other elements. It indicates that there are infinite counting numbers greater than 8, which is impossible to list them all down. 4
The table below indicates the common symbols used to show the relationship between sets and elements. Synergy for Success in Mathematics Chapter 1 Symbol Î Ï Ì Ë Æ È Ç Words element not an element subset; part of not a subset; not a part of empty; no element; null set union; combine elements intersection; common element(s) To relate or describe the relationship between an element and a set, we use Îand Ï. For example: If A = {a, e, i, o, u}, then u A and b A. This implies that u belongs to A and b is not an element of A. Given: B ={ all the factors of 24} Fill in the blanks with (a) 1 B (b) 15 B (c) 8 B (d) 4 B (e) 12 B (f) 16 B SOLUTION (a) (b) (c) (d) (e) (f) Example 2 1Î B 15Ï B 8Î B 4Î B 12Î B 16Î B or. 5
The factors 1, 2, 3, 4, 6, 8, and 12 are numbers which can exactly divide 24. Thus, these numbers are considered factors of 24. The numbers 1, 2, 3, 4, 6, 8, 12, and 24 can exactly divide 24. Thus, these numbers are considered factors of 24. The numbers 15 and 16 are not factors of 24 because of the existence of a remainder when 24 is divided by either of these two numbers. Universal Set A set that contains everything or all elements under consideration and are relevant to the problem is called a universal set, denoted as U. A universal set could be drawn (usually as a rectangle) to contain all the members which are considered. For example: U = {set of whole numbers less than 10} U = {1, 2, 3, 4, 5, 6, 7, 8, 9} U 1 2 3 4 5 6 7 8 9 6
Empty or Null Sets Synergy for Success in Mathematics Chapter 1 A set with no elements in it is known as an empty set or null set. It is represented by or by {}(a set with no elements). However, it is never represented by { }. For example: E = {the month of the year with more than 31 days} E = { } or E = since there are no months with more than 31 days. Determine whether each of the following sets is empty or not. (a) P = { x : x is kind of triangle having sides of different lengths} (b) Q = {x : x is a factor of 16 and 20< x < 30} (c) R = {x : x is a prime number and 8< x < 10} SOLUTION (a) (b) Example 3 A scalene triangle has sides of different lengths. Hence, P. The factors of 16 are 1, 2, 4, 8, and 16. There are no factors of 16 between 20 and 30. Hence, Q. (c) A prime number has only two factors, itself and 1. 9 is between 8 and 10. 9 has three factors: 1, 3, and 9. Hence, R. 7
Subset A subset is a portion of a set. A set is a subset of another set if and only if all the elements of a set are contained in another set. Set Q is a subset of set P if every element of set Q is also an element of set P. If set Q is a subset of P, but not equal to set P, then Q is a proper subset of P. Notation: Q P, if x Q, then x P. The following generalizations are consequences of the definition. (1) Every set is a subset of itself. Notation: AÌ A (2) An empty set is a subset of every set. Notation: A Fill in each of the following blanks with the symbol Ì or Ë. (a) {6, 7, 8} {0, 1, 4, 5, 6, 7, 8} (b) { j, l, q} {vowels} (c) Example 4 {blue, red} {rainbow colors} (d) {8, 16} {multiples of 16} SOLUTION (a) {6, 7, 8} Ì {0, 1, 4, 5, 6, 7, 8} 6, 7, and 8 can be found in {0, 1, 4, 5, 6, 7, 8}. (b) { j, l, q} Ë {vowels} j, l, and q are not vowels. (c) {blue, red} Ì {rainbow colors} Blue and red are two of the rainbow colors. (d) {8, 16} Ë {multiples of 16} 8 is not a multiple of 16. 8
The number of subsets of a certain set is 2 n, where n is the number of elements in the set. If A ={ 123,, }, then A has 8 subsets. 3 Number of subsets = 2 = 8, where the exponent 3 is the number of elements of A. Here is a complete list of the 8 subsets. { } { } { } { 23, } Improper subset: 123,, Proper subset with two elements: 12,, 13,, {}{}{} Proper subset with one element: 1, 2, 3 Improper subset with no element: {} Synergy for Success in Mathematics Chapter 1 Example 5 Determine the number of subsets for each of the following sets. Then, list all the subsets. (a) D ={ 79, } (b) E ={ p} (c) F ={ a,e,i,o} SOLUTION 2 (a) Number of subsets = 2 = 4 Subsets of D: { }, {7}, {9}, {7, 9} 1 (b) Number of subsets = 2 = 2 Subsets of E: { }, {p} 4 (c) Number of subsets = 2 = 16 Subsets of F: { }, {a, e, i, o}, {a}, {e}, {i}, {o}, {a, e}, {a, i}, {a, o}, {e, i }, {e, o}, {i, o} {a, e, i }, {a, e, o}, {e, i, o}, {a, i, o} 9
Finite or Infinite Set Sets having finite or exact list of elements are called finite sets. For a long list, a definition or a set builder has to be used. If the list is short like the one shown below, it could be simply described by listing all its members. For example: P = {set of two-digit positive integers ending with the digit 9} P = {19, 29, 39, 49, 59, 69, 79, 89, 99} The number of elements in a finite set is denoted by P. The symbol P is read as cardinality of set P. There are situations where the list could be infinite. A set is classified as infinite when its elements cannot be counted. For example, the set of even numbers starting from 0 and continuing indefinitely has to be stated as N = {x : x is an even number} which means the list 0, 2, 4, 6, 8,... continues indefinitely. Given: Example 6 B = {x : x is a letter in the word MATHEMATICS} C = {x : x is a factor of 21} D = {x : x is an integer between 4 and 5} (a) List all the elements of sets B, C, and D. (b) Find B, C, and D. SOLUTION (a) B = {A, C, E, I, M, S, T, H} Although M, A, and T appear more than once in the word MATHEMATICS, these three letters must be written only once inside the brace. C ={ 13721,,, } These four elements of set C are numbers which can equally divide 21. ={ } = D or D, since all the numbers between 4 and 5 are non-integers or fractions. 10
Synergy for Success in Mathematics Chapter 1 (b) B = 8 C = 4 D = 0 Special Sets A subset is a set contained within another set, or it can be the entire set itself. The set {1, 2} is a subset of the set {1, 2, 3}, and the set {1, 2, 3} is a subset of the set {1, 2, 3}. When the subset is missing some elements that are in the set it is being compared to, it is a proper subset. When the subset is the set itself, it is an improper subset. The symbol used to indicate is a proper subset of is Ì. When there is the possibility of using an improper subset, the symbol used is Í. Therefore, { 12, } { 123,, } and { 123,, } { 1, 23, }. The universal set is the general category set, or the set of all those elements under consideration. The empty set, or null set, is the set with no elements or members. Both the universal set and the empty set are subsets of every set. 11
Equal Sets and Equivalent Sets Two sets may contain exactly the same elements or the same number of elements. Two sets are considered equal only if each member of one set is also a member of the other, in which case it can be stated that A= B. Two sets are considered equivalent if they contain the same number of elements. Consider set A={ 251,, } and B={ 125,, }. Since A and B have exactly the same elements, then A= B. They are also equivalent sets since they contain the same number of elements. This is stated as A = B. Example 7 Determine whether the following pairs of sets are equal or equivalent. ={ } ={ } (a) X 4567,,, ; Y 5746,,, (b) A ={common multiples of 5 and 7 which are less than 80} B ={ 35, 75} ={ 0; } ={ } (c) C D SOLUTION ={ } ={ }={ } (a) X 4567,,, ; Y 5746,,, 4567,,, So, X = Y and X = Y. ={ } ={ } (b) A 35, 70 ; B 35, 75 A = B, A B So, A is equivalent to B but not equal. Method Note The order of the elements is not important. Equal sets are equivalent sets, but not all equivalent sets are equal sets. (c) C D C = 1 and D = 0 So, C and D are neither equal nor equivalent. 12
Synergy for Success in Mathematics Chapter 1 ENHANCING SKILLS A Write the following sets in rule method. (1) G = {x : x is a letter in the word ALGEBRA} (2) B = {x : x is a positive integer divisible by 2 or 3} (3) C = {x : x is an integer} (4) D = {x : x is a multiple of 2 and 3 between 20 and 40} (5) E = {x : x is a reciprocal of 0} B Given the following sets, fill in each blank with Î or Ï. F = {x : x is a positive integer divisible by 2 or 3} D = {x : x is a multiple of 2 and 3 between 20 and 40} (6) 5 F (7) 15 F (8) 20 F (9) 36 D (10) 39 D C Check ( ) the classification of the set or following sets. Refer to the sets in part A. Set(s) Finite Infinite Empty B C D E 13
1.2 Operations on Sets Venn Diagrams Venn diagrams are schematic diagrams used to depict collections of sets and represent their relationships. Any closed geometrical shapes (circles, ovals, rectangles,...) can be used to represent Venn diagram. Consider set A is specified after a universal set has been defined. Set A has to be located within the universal set as it must be a subset of the universal set. Set A could be drawn as any plane figure within the universal set. U A 14
Synergy for Success in Mathematics Chapter 1 Set Notation Venn Diagram A = {a, e, i, o, u} U A a e u i o B = {Albert, Alex, Anne} U B Albert Alex Anne C = {4, 5, 6, 7} D = {4, 6, 8} U C D 5 4 8 7 6 E = {2, 4} F = {1, 2, 3, 4} U 1 F E 2 4 3 15
Example 1 Construct a Venn diagram to represent the following sets. Given U as the universal set. (a) A = {multiples of 4 which are less than 25} (b) B = {x : x is an integer, 5 x < 9} (c) C = { two-digit numbers that end with the digit 1} (d) D = {letters in the word ARITHMETIC} (e) E = {x : x is an even number, 21< x < 30} SOLUTION (a) U A 4 8 20 24 12 16 (d) U D A E I R C H T M (b) U B (e) U E 5 7 6 8 22 26 24 28 (c) U C 11 31 21 41 51 81 61 91 71 16
Synergy for Success in Mathematics Chapter 1 Example 2 Draw a Venn diagram to illustrate the relationship between each pair of sets. ={ } (a) U x: x is a rational number Z={ x: x is an integer} ={ } { } ={ 510} (b) U x: x is a factor of 20 D= y: y is a factor of 10 and 0< y< 12 F, (c) U J ={ 23456,,,,, 78910,,,, 11} ={ 35911,,, } K = { y: y is an integer, 2< y < 9} SOLUTION (a) U Z integers (b) U D F 5 10 2 4 1 20 (c) U J K 9 11 3 5 4 7 6 8 10 2 17
Complement of a Set The complement of set A with respect to the universal set U, denoted by A c, is the set of all points that are in U but not in A. Consider universal set U ={ 123456,,,,, } and set A ={ 12} then the complement of A is given by A C ={ 3456,,, }.,, U A A c 5 6 1 2 4 3 Union of Sets and Intersection of Sets The operations on sets somewhat behave in a similar manner to the basic operations on numbers. The union of a set is the result of adding or combining the elements of two or more sets. The union of set A and set B, denoted by A È B and read as A union B, is the set of all elements belonging to either of the sets. Each element of the union is an element of either set A and/or set B. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, and B = {4, 5, 6, 7}, then A È B = {1, 2, 3, 4, 5, 6, 7}. A Ì (A È B) B Ì (A È B) Exam Note U A B 1 2 4 6 3 5 7 9 8 10 18
Synergy for Success in Mathematics Chapter 1 Example 3 Determine the union of the following sets and represent it using a Venn diagram. (a) M = {-5, -4, -3, -2} N = {-2, -1, 0, 1, 2} (b) P = {x : x is a factor of 20} Q = {x : x is a factor of 36} R = {x : x is a factor of 9} SOLUTION (a) M È N = {-5, -4, -3, -2, -1, 0, 1, 2} Exam Note U M -3-4 -5-2 -1 0 1 2 N -2 is a common element. It must be written once only in the union of M and N. shaded region = M È N (b) P = {1, 2, 4, 5, 10, 20} Q = {1, 2, 3, 4, 6, 9, 12, 18, 36} R = {1, 3, 9} P È Q È R = {1, 2, 3, 4, 5, 6, 9, 10, 12, 18, 20, 36} Method Note Arrange the elements in increasing order. U P 5 10 20 36 2 3 1 4 9 15 12 Q 6 R shaded region = P È Q È R 19
The intersection of sets P and Q, denoted by P Ç Q, is the set of elements which are common to both sets P and Q. If P = {1, 2, 3, 4, 5} and Q = {2, 4, 6, 8}, then P Ç Q = {2, 4}. U P Q 1 2 5 3 4 8 6 Notice that (P Ç Q) Ì P and (P Ç Q) Ì Q. Sets with no common elements are disjoint sets. (a) (b) Example 4 A B ={ 25711,,,, 13, 15, 19} ={ 147811,,,,, 14, 17, 19} Find A B. P Q ={ a, c, d, f, h, j} ={ e, g, h, i, j, k} { c, f, h, i, l, m} R = Find P Q R. SOLUTION (a) A B ={ 25711131519,,,,,, } ={ 147811,,,,, 14, 17, 19} A B={ 71119,, } Exam Note The intersection of sets A and B are the common elements in both sets. (b) ={ a, c, d, f, h, j} ={ e, g, h, i, j, k} { c, f, h, i, l, m} P Q R = P Q R ={ h} Exam Note The intersection of sets P, Q, and R is the common elements in the three sets. 20
Synergy for Success in Mathematics Chapter 1 Example 5 Write down the intersection of the following sets and represent the intersection using a Venn diagram. SOLUTION C = 1 2 3 7 9,,,, 4 3 5 9 11 D = 1 2 3 4 9 10,,,,, 3 3 8 7 11 13 C = 1 2 3 7 9,,,, 4 3 5 9 11 D = 1 2 3 4 9 10,,,,, 3 3 8 7 11 13 U C Ç D C D 3 5 1 4 7 9 2 3 9 11 4 7 1 3 10 13 3 8 Method Note The region where both sets overlap represents the intersection between the two sets. C D = 2 9, 3 11 21
Write down the intersection of the following sets and represent the intersection using a Venn diagram. { } F = z: z is a multiple of 5 and 10 < z< 50 G= { y: y is an even number and 18 < y < 42} H Example 6 ={ 25, 27, 30, 33, 34, 40, 41} SOLUTION F ={ 15, 20, 25, 30, 35, 40, 45} G = { 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40} H ={ 25, 27, 30, 33, 34, 40, 41} U F G 15 22 24 20 35 26 28 30 32 36 45 40 25 34 38 27 33 41 H F Ç G Ç H Method Note The region where all the three sets overlap represents the intersection of the three sets. F G H ={ 30, 40} 22
Synergy for Success in Mathematics Chapter 1 ENHANCING SKILLS A Answer the following using the given sets. Indicate each solution in a Venn diagram. { } Given: U = all positive integers < 30 A= { x: x is a positive integer where x + 2< 19} B= x: x is a positive integer where 2x 3> 17 C = (1) AÈ B (2) AÇ B (3) A C (4) BÈ B C (5) AÇC C (6) C C C È A (7) ( A B) C (8) ( C B) A C ( ) (9) A B C C (10) A C ÈU { } { x: x is an odd integer < 30} B Solve the following problems using the Venn diagram and answer the related questions. In a certain school, a group of students in a class were enrolled in three subjects. How many students were enrolled in: (11) exactly one subject? (12) exactly two subjects? (13) at most two subjects? (14) at most one subject? (15) Algebra or Geometry? (16) Algebra and Geometry? (17) Algebra and Geometry but not Trigonometry? (18) Geometry and Trigonometry but not Algebra? (19) Algebra only? (20) If there are 50 students in all, how many did not enroll in any of the three subjects? U Algebra Geometry 10 6 5 9 7 4 7 Trigonometry 2 23
LINKING TOGETHER Basic Idea of Sets Describing Sets Rule Method Roster or Listing Method Kinds of Sets Number of Subsets in a Set = 2 n Empty Sets Finite Sets Infinite Sets Disjoint Sets No common elements Operations on Sets Union of Set Intersection of Sets 24