Table of C ntents - SAP Education › wp-content › uploads › wcdp...Chapter Test This is a summative test given at the end in preparation for the expected actual classroom examination

Synergy for Success in Mathematics 7 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 Thisgivesthespecificobjectivesthatareintendedtobeachieved in the end.

Introduction The reader is given a bird's eye view of the contents.Historical Note 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 Step-by-stepanddetaileddemonstrationsofhowaspecificconcept or technique is applied in solving problems.

Enhancing Skills These are practice exercises found after every lesson, that will consolidate and reinforce what the students have learned.

Linking Together This visual tool can help the students realize the connection of all the ideas presented in the chapter.

Chapter Test 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.

Chapter Project Thismaybeamanipulativetypeofactivitythatisspecificallychosen to enhance understanding of the concepts learned in the chapter.

Making Connection The students are exposed to facts and information that connect mathematics and culture. This is for the purpose of letting the learnersappreciatethesubjectbecauseoftangibleortrue-to-life 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.

PREFACE

Table of C ntents

Introduction ............................................................................................................................1Historical Note .......................................................................................................................21.1 IntroductiontoSets ...............................................................................................31.2 OperationsonSets ............................................................................................... 14Linking Together ................................................................................................................ 24Chapter Test ........................................................................................................................ 25ChapterProject ................................................................................................................... 29Making Connection ........................................................................................................... 30

Introduction ......................................................................................................................... 31Historical Note .................................................................................................................... 322.1 TheRealNumberSystem .................................................................................. 332.2 PropertiesofRealNumbers ............................................................................. 392.3 Integers ..................................................................................................................... 462.4 AbsoluteValueofaRealNumbers ............................................................... 642.5 Fractions ................................................................................................................... 682.6 Decimals ................................................................................................................... 822.7 ApproximationonSquareRoots .................................................................... 912.8 ScientificNotation ................................................................................................ 962.9 SignificantDigits .................................................................................................101Linking Together ..............................................................................................................105Chapter Test ......................................................................................................................106ChapterProject .................................................................................................................109Making Connection .........................................................................................................110

CHAPTER 1 BASIC IDEA OF SETS

CHAPTER 2 REAL NUMBERS

Introduction .......................................................................................................................111Historical Note ..................................................................................................................1123.1 MeasuringLength,Perimeter,Mass,andVolume .................................1133.2 MeasuringArea,Temperature,andTime.................................................131Linking Together ..............................................................................................................145Chapter Test ......................................................................................................................146ChapterProject .................................................................................................................149Making Connection .........................................................................................................150

Introduction .......................................................................................................................151Historical Note ..................................................................................................................1524.1 NumberSequenceandPatternFinding ....................................................1534.2 AlgebraicExpressions .....................................................................................1594.3 IntegralExponents .............................................................................................1704.4 EvaluationofAlgebraicExpressions .........................................................1874.5 TranslationofMathematicalPhrasesintoSymbols ............................1954.6 OperationsofPolynomials .............................................................................2034.7 SpecialProducts ..................................................................................................216Linking Together ..............................................................................................................229Chapter Test ......................................................................................................................230ChapterProject .................................................................................................................233Making Connection .........................................................................................................234

Introduction .......................................................................................................................235Historical Note ...................................................................................................................2365.1 LinearEquationsinOneVariable ................................................................2375.2 SolvingAbsoluteValueEquations ...............................................................2565.3 LinearInequalitiesinOneVariable ............................................................2615.4 SolvingAbsoluteValueInequalities ...........................................................2805.5 SolvingWordProblemsInvolving Linear Equations and Inequalities ..............................................................288Linking Together ..............................................................................................................312Chapter Test ......................................................................................................................313ChapterProject .................................................................................................................317Making Connection .........................................................................................................318

CHAPTER 3 MEASUREMENT

CHAPTER 4 ALGEBRAIC EXPRESSIONS

CHAPTER 5 LINEAR EQUATIONS AND LINEAR INEQUALITIES IN ONE VARIABLE

Introduction .......................................................................................................................319Historical Note ...................................................................................................................3206.1 ObjectsofGeometry ..........................................................................................3216.2 AnglesandAngleMeasures ...........................................................................3396.3 ReasoningandProving ....................................................................................3556.4 MoreObjectsofGeometry ..............................................................................3686.5 GeometricConstructions ................................................................................380Linking Together ..............................................................................................................387Chapter Test ......................................................................................................................388ChapterProject .................................................................................................................391Making Connection .........................................................................................................392

Introduction .......................................................................................................................393Historical Note ...................................................................................................................3947.1 PerpendicularandParallelLines .................................................................3957.2 ApplyingConceptsofPerpendicularandParallelLines ....................407Linking Together ..............................................................................................................416Chapter Test ......................................................................................................................417ChapterProject .................................................................................................................421Making Connection .........................................................................................................422

Introduction .......................................................................................................................423Historical Note ...................................................................................................................4248.1 IntroductiontoStatistics .................................................................................4258.2 TheFrequencyTable .........................................................................................4318.3 UseofGraphstoRepresentandAnalyzeData .......................................4408.4 MeasuresofCentralTendency(UngroupedData) ...............................449Linking Together ..............................................................................................................460Chapter Test ......................................................................................................................461ChapterProject .................................................................................................................465Making Connection .........................................................................................................466

Glossary .....................................................................................................................................467 Index ...........................................................................................................................................477 Bibliography ............................................................................................................................484

CHAPTER 6 TOOLS OF GEOMETRY

CHAPTER 7 PERPENDICULAR AND PARALLEL LINES

CHAPTER 8 STATISTICS

BASIC IDEA OF SETS1

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.

Learning GoalsAt the end of the chapter, the students should be able to:

1.1 Defineanddescribeasetand use a Venn diagram to illustrate a set and properties of set operations

1.2 Describe and illustrate complement of a set, and union and intersection of sets

2

George Ferdinand Ludwig Philipp Cantor (1845–1918) is a German mathematician known as the founder of set theory. Cantorsetforththemoderntheoryoninfinitesetsthatdevelopedallthedisciplinesinmathematics.Cantordefinedwell-orderedand infinite sets. He established the importance of one-to-one correspondence between the members of two sets. He showedthatnotallinfinitesetshavethesamesize,therefore,infinitesetscanbecomparedwithoneanother.Hethenprovedthatthereal numbers are “numerous” than the natural numbers.He definedwhat it means for two sets to have the same cardinal number. Heproved that the set of real numbers and the set of points in n-dimensionalEuclideanspacehavethesameexponent.

Cantor’s early interests were in number theory, indeterminate equations, and trigonometric series. In 1874, he started his radical workonsettheoryandthetheoryoftheinfinite.Cantorcreatedawholenewfieldofmathematicalresearch.

Historical Note

3

Synergy for Success in Mathematics Chapter 1

A set is a collection of objectswhich are clearly defined as belongingtoawell-definedgroup.Eachobjectinasetiscalledanelementofaset.Eachelementisseparatedbyacomma.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}

Asetmustbewelldefinedsothatwecandeterminewhetheran 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 Roster or Listing Method

A x x={ }: is a counting number from 1 to 5 A ={ }1 2 3 4 5, , , ,

B x x={ }: is a month that starts with letter A B ={ }April, August

C x x={ }: is a prime factor of 15 C ={ }3 5,

In roster or listing method, the elements are separated by commas and are enclosed within a pair of brace { }.

1.1 Introduction to Sets

4

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 thesetofelementx,suchthatxisacountingnumberfrom1 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 infiniteortoomanytoencloseinsidebraces.Suchsetsareratherdefinedusingtherulemethod.

Forexample:

A x x={ }: is an even number between 1 and 100

A ={ }2 4 6 8 96 98, , , , , ,…

The three dots (...) are called ellipsis which means "continue on." It represents the other elements which are no longer practical to include in the list.

Listalltheelementsofthefollowingsets.

(a) A = {x:xisaletterinthewordSUBTRACT}(b) B = {x:xisacountingnumbergreaterthan8}

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, ...} Theellipsisisusedtoacknowledgetheexistenceof

other elements. It indicates that there are infinitecounting numbers greater than 8, which is impossible to list them all down.

Example 1

5


The table below indicates the common symbols used to show the relationship between sets and elements.

Symbol 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 ∈and ∉ .

Forexample:If A = {a, e, i, o, u}, then u and b∈ ∉A A.This implies that “u belongs to A” and “b is not an element of A.”

Given: B ={ }all the factors of 24Fill in the blanks with ∈ ∉ or .

(a) 1 B(b) 15 B(c) 8 B(d) 4 B(e) 12 B(f) 16 B

SOLUTION

(a) 1∈ B(b) 15∉ B(c) 8 ∈ B(d) 4 ∈ B(e) 12∈ B(f) 16 ∉ B

Example 2

6

The factors 1, 2, 3, 4, 6, 8, and 12 are numbers which can exactly divide 24. Thus, these numbers are consideredfactors of 24.

Thenumbers1,2,3,4,6,8,12,and24canexactlydivide24. Thus, these numbers are considered factors of 24.

The numbers 15 and 16 are not factors of 24 because of theexistenceofaremainderwhen24isdividedbyeitherof 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.

Forexample:U = {set of whole numbers less than 10}U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

1 2

5

3

6

8

4

7

9

U

7


Empty or Null Sets

A set with no elements in it is known as an empty set or null set. It is represented by ∅ { } or by .However,itisnev-er represented by ∅{ }.

Forexample: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 a kind of triangle having sides of

different lengths} (b) Q = {x : x is a factor of 16 and 20 30< < }x(c) R = {x : x is a prime number and 8 10< < }x

SOLUTION

(a) A scalene triangle has sides of different lengths. Hence,P ≠ ∅.

(b) The factors of 16 are 1, 2, 4, 8, and 16. There are no factorsof16between20and30.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 = ∅.

Example 3

8

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.

SetQ 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 x Q x P⊂ ∈ ∈, if , then .

The following generalizations are consequences of thedefinition.

(1) Everysetisasubsetofitself. Notation: A A⊂

(2) An empty set is a subset of every set. Notation: ∅⊂ A

Fill in each of the following blanks with the symbol ⊂ or ⊄ .

(a) {6, 7, 8} ____ {0, 1, 4, 5, 6, 7, 8}(b) { j, l, q} ____ {vowels}(c) {blue, red} ____ {rainbow colors}(d) {8, 16} ____ {multiples of 16}

SOLUTION

(a) {6, 7, 8} ⊂ {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} ⊄ {vowels} j, l, and q are not vowels.(c) {blue, red} ⊂ {rainbow colors} Blueandredaretwooftherainbowcolors.(d) {8, 16} ⊄ {multiples of 16} 8 is not a multiple of 16.

Example 4

9


The number of subsets of a certain set is 2n, where n is the number of elements in the set.

If A ={ }1 2 3, , , then A has 8 subsets.

Number of subsets = =2 83 , where the exponent3isthenumberofelementsofA.

Hereisacompletelistofthe8subsets. Improper subset: 1 2 3, ,{ } Propersubsetwithtwoelements: 1 2 1 3

2 3

, , , ,

,

{ } { }{ }

Propersubsetwithoneelement: 1 2 3{ } { } { }, , Improper subset with no element: { }

Determine the number of subsets for each of the following sets. Then, list all the subsets.

(a) D ={ }7 9,(b) E ={ }p(c) F ={ }a,e,i,o

SOLUTION

(a) Number of subsets ==

2

4

2

SubsetsofD: { }, {7}, {9}, {7, 9} (b) Number of subsets =

=2

2

1

SubsetsofE: { }, {p}(c) Number of subsets =

=2

16

4

SubsetsofF: { }, {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}

Example 5

10

Finite and Infinite Sets

Setshavingfiniteorexact listofelementsarecalledfinite sets.Fora long list, adefinitionora setbuilderhas tobeused. If the list is short like the one shown below, it could be simply described by listing all its members.

Forexample: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 byP P. The symbol is read as “cardinality of set P.”

Therearesituationswherethelistcouldbeinfinite.Asetisclassifiedasinfinite when its elements cannot be counted. Forexample, the setof evennumbers starting from0andcontinuingindefinitelyhastobestatedas

N = {x : x is an even number}

which means the list 0, 2, 4, 6, 8, ... continues indefinitely.

Given: B = {x : xisaletterinthewordMATHEMATICS} C = {x : x is a factor of 21} D = {x : x is an integer between 4 and 5}(a) ListalltheelementsofsetsB, C, and D.(b) Find B C D, , . and

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 ={ }1 3 7 21, , , These four elements of set C are numbers which

can equally divide 21.

D D={ } = ∅ or , since all the numbers between 4 and 5 are non-integers or fractions.

Example 6

11


(b) B

C

D

=

=

=

8

4

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 ⊂ . When there is the possibility of using an improper subset, the symbol used is ⊆ . Therefore, 1 2 1 2 3, , ,{ }⊂{ } and 1 2 3 1 2 3, , , , .{ }⊆{ } 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.

Boththeuniversalsetandtheemptysetaresubsetsofevery set.

12

Equal Sets and Equivalent Sets

Twosetsmaycontainexactlythesameelementsorthesamenumber 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 B={ } ={ }2 5 1 1 2 5, , , , . and

SinceA and Bhaveexactlythesameelements,then A B= .They are also equivalent sets since they contain the same number of elements. This is stated as A B= .

Determine whether the following pairs of sets are equal or equivalent.

(a) X Y={ } ={ }4 5 6 7 5 7 4 6, , , ; , , ,(b) A ={common multiples of 5 and 7

which are less than 880} B ={ }35 75,(c) C D={ } ={ }0 ;

SOLUTION

(a) X Y

X Y X Y

={ } ={ }={ }= =

4 5 6 7 5 7 4 6 4 5 6 7, , , ; , , , , , ,

.So, and

(b) A B

A B A B

={ } ={ }= ≠

35 70 35 75, ; ,

,

So,A is equivalent to B but not equal.

(c) C DC D

≠

= =1 0 and

So,C and D are neither equal nor equivalent.

Example 7

Method NoteThe order of the elements is not important.Equalsetsareequivalentsets,but not all equivalent sets are equal sets.

13


ENHANCING SKILLS

A Write the following sets in rule method.(1) G={x:xisaletterinthewordALGEBRA}(2) B={x:xisapositiveintegerdivisibleby2or3}(3) C={x:xisaninteger}(4) D={x:xisamultipleof2and3between20and40}(5) E={x: x is a reciprocal of 0}

B Giventhefollowingsets,fillineachblankwith⊂ or ⊄ . F={x:xisapositiveintegerdivisibleby2or3} D={x:xisamultipleof2and3between20and40}

(6) 5 F(7) 15 F(8) 20 F(9) 36 D(10) 39 D

C Check ()theclassificationofthesetorfollowingsets.Refertothesetsinpart A.

Set(s) Finite Infinite Empty

B

C

D

E

Documents

Table of C ntents - SAP Education › wp-content › uploads › wcdp...Chapter Test This is a summative test given at the end in preparation for the expected actual classroom examination