WELCOMEWELCOME

•The laws of nature The laws of nature are but the are but the mathematical mathematical thoughts of God.thoughts of God. (Euclid)(Euclid)

Just think doing Just think doing things for others things for others could really end up could really end up benefit ing you in the benefit ing you in the long run.long run.

I. Contents:I. Contents: TEACHING-LEARNING REQUIREMENTS TEACHING-LEARNING REQUIREMENTS

OF GRADE 3 TO 6.OF GRADE 3 TO 6. CLASS-ROOM ENVIRONMENT OF THIS CLASS-ROOM ENVIRONMENT OF THIS

GRADE LEVEL.GRADE LEVEL. MATHEMATICAL FOUNDATIONS.MATHEMATICAL FOUNDATIONS. OBSTACLES TO QUALITY MATH-OBSTACLES TO QUALITY MATH-

LEARNING.LEARNING. HOW TO ERADICATE THE OBSTACLES.HOW TO ERADICATE THE OBSTACLES. TYPES OF INTELLIGENCES.TYPES OF INTELLIGENCES. HABITS OF HIGHLY EFFECTIVE MATH-HABITS OF HIGHLY EFFECTIVE MATH-

TEACHING.TEACHING.

1.TEACHING-LEARNING 1.TEACHING-LEARNING REQUIREMENTSREQUIREMENTS

Students in the grade 3-6 are intrigued with Students in the grade 3-6 are intrigued with mathematics. To nurture this interest , students mathematics. To nurture this interest , students at this grade level need to be involved in an at this grade level need to be involved in an active learning process rather than one that only active learning process rather than one that only builds memorization of concepts and procedures.builds memorization of concepts and procedures.

Concrete experiences are also important at this Concrete experiences are also important at this stage of development . Such experiences allow stage of development . Such experiences allow students to develop and strengthen the skills students to develop and strengthen the skills needed. needed.

The skills are :The skills are : To communicate , reason, solve mathematical To communicate , reason, solve mathematical

problems and reach to higher level of cognitive problems and reach to higher level of cognitive reasoning.reasoning.

2. CLASS-ROOM ENVIRONMENT 2. CLASS-ROOM ENVIRONMENT FOR THIS GRADE LEVEL.FOR THIS GRADE LEVEL.

An effectiveAn effective class-room environment provides class-room environment provides intellectually stimulating instructions and intellectually stimulating instructions and developmentally appropriate opportunities for developmentally appropriate opportunities for the students to learn mathematical concepts.the students to learn mathematical concepts.

This classroom environment fosters an This classroom environment fosters an atmosphere in which students are encouraged to atmosphere in which students are encouraged to find solutions through a variety of methods and find solutions through a variety of methods and feel less threatened about making and correcting feel less threatened about making and correcting mistakes.mistakes.

Class-room instruction includes opportunities for Class-room instruction includes opportunities for students to communicate them mathematical students to communicate them mathematical thinking by talking , writing and sharing with thinking by talking , writing and sharing with each other.each other.

3. MATHEMATICAL 3. MATHEMATICAL FOUNDATION.FOUNDATION.

Grade contents build a foundation of basic Grade contents build a foundation of basic number-sense , operations , quantitative number-sense , operations , quantitative reasoning , number-patterns , relationships , reasoning , number-patterns , relationships , geometric and spatial reasoning , geometric and spatial reasoning , measurements , probability and statistics. measurements , probability and statistics. This content builds on and expand This content builds on and expand conceptual understanding of math.conceptual understanding of math.

Through interweaving of mathematical Through interweaving of mathematical concepts and process students learn to value concepts and process students learn to value math , display confidence in the math , display confidence in the mathematically solved problems, and make mathematically solved problems, and make connections between math and other connections between math and other subjects .subjects .

4.OBSTACLES TO QUALITY 4.OBSTACLES TO QUALITY MATH-LEARNING.MATH-LEARNING.

Family Family disaster-stories about math disaster-stories about math difficulty.difficulty.

Terror producing teaching.Terror producing teaching. Fuzzier meaning less explanation.Fuzzier meaning less explanation. Lack of link between math and real life.Lack of link between math and real life. A rush into “formal abstract” math without A rush into “formal abstract” math without

any concrete physical or pictured any concrete physical or pictured experience without any concrete, experience without any concrete, physical or pictured experience.physical or pictured experience.

A robotic, imitate, memorized style with little A robotic, imitate, memorized style with little conceptual grounding.conceptual grounding.

A habitual disconnect between a child’s natural style A habitual disconnect between a child’s natural style and a superimposed alien math style.and a superimposed alien math style.

Sometimes the material is meaningless, monotone , Sometimes the material is meaningless, monotone , black and white, that leads to boredom.black and white, that leads to boredom.

Poor self-image causes constant forgetting, spacing Poor self-image causes constant forgetting, spacing out, blowing tests and in the end develops hatred for out, blowing tests and in the end develops hatred for math.math.

COMENTCOMENT :: How would you react with a daily How would you react with a daily portion of sawdust for your meal?portion of sawdust for your meal?

How to help them?How to help them? Youth can leap over any of these obstacles to math Youth can leap over any of these obstacles to math

success.success. Right approach can make up for a multitude of Right approach can make up for a multitude of

assumed impossibilities and disabilities.assumed impossibilities and disabilities. Young people or even the adults just need to get the Young people or even the adults just need to get the

real scoop behind the method in a clear and fun real scoop behind the method in a clear and fun way.way.

They are thirsty for concepts, the number sense, the They are thirsty for concepts, the number sense, the number patterns, the calculations , the formulae etc.number patterns, the calculations , the formulae etc.

At his stage we can lead them to the right direction. At his stage we can lead them to the right direction. Nine-tenth of education is encouragement.Nine-tenth of education is encouragement. As soon as we discover the problem we will be able As soon as we discover the problem we will be able

to resolve it.to resolve it.

TYPES OF INTELLEGENCETYPES OF INTELLEGENCE There are nine There are nine types of intelligences by research, each types of intelligences by research, each has been proved to be equivalent to any other by any has been proved to be equivalent to any other by any scientific criterion you think of, when we speak they are scientific criterion you think of, when we speak they are intelligent we can actually and accurately mean they are intelligent we can actually and accurately mean they are intelligent nine other ways;intelligent nine other ways;

1.1. IntrapersonallyIntrapersonally (self smart)(self smart)2.2. KinestheticallyKinesthetically (mind-body union)(mind-body union)3.3. SpatiallySpatially (Picture smart)(Picture smart)4.4. InterpersonallyInterpersonally (people smart)(people smart)5.5. MusicallyMusically (music smart)(music smart)6.6. NaturalisticallyNaturalistically (nature smart)(nature smart)7.7. Logically/mathematically (number/reason smart)Logically/mathematically (number/reason smart)8.8. LinguisticallyLinguistically (word smart)(word smart)9.9. Existentially Existentially (deep thinking ,questions)(deep thinking ,questions)

If that is so it stands to reason that if you teach math through If that is so it stands to reason that if you teach math through all the intelligence channels, you have a vastly all the intelligence channels, you have a vastly increased increased chance of reaching every chance of reaching every brain you are working with.brain you are working with.

FOUR HABITS OF HIGHLY FOUR HABITS OF HIGHLY EFFECTIVE MATH TEACHINGEFFECTIVE MATH TEACHING

1.LETS MAKE 1.LETS MAKE SENSE:SENSE: Dilemma between conceptual vs. procedural Dilemma between conceptual vs. procedural

understanding.understanding. 2.REMEMBER THE GOALS:2.REMEMBER THE GOALS: connect your ultimate goals to sub goals .connect your ultimate goals to sub goals . 3.KNOW YOUR TOOLS:3.KNOW YOUR TOOLS: Learn how to use tools and add to your toolbox.Learn how to use tools and add to your toolbox. 4.LIVING AND LOVING MATH:4.LIVING AND LOVING MATH: You are a teacher. Show the way, with your You are a teacher. Show the way, with your

attitudes.attitudes.

II.CONTENTSII.CONTENTS

1. 1. Number patternsNumber patterns 2.Concept of multiplication2.Concept of multiplication 3.multimlication table.3.multimlication table. 4.division4.division 5.Factors5.Factors 6.prime numbers6.prime numbers 7. integers7. integers 8.fractions8.fractions 9.problem solving.9.problem solving.

NUMBER PATTERNSNUMBER PATTERNS

PREVIOUS KNOWLEDGE:PREVIOUS KNOWLEDGE: Whole numbersWhole numbers Natural numbersNatural numbers

1.EVEN AND ODD NUMBERS1.EVEN AND ODD NUMBERS

Objectives: to identify even and odd Objectives: to identify even and odd numbers.numbers.

Use a hundred chart.Use a hundred chart. Skip-count by two’s, start from 2 ,the numbers Skip-count by two’s, start from 2 ,the numbers

you will land on are even, shade the numbers , you will land on are even, shade the numbers , by the end of the page we get the shaded by the end of the page we get the shaded numbers which are even.numbers which are even.

The remaining numbers are odd numbers.The remaining numbers are odd numbers. Activity:Activity: use hundred chart and colors use hundred chart and colors

A HUNDRED CHARTA HUNDRED CHART11 22 33 44 55 66 77 88 99 10101111 1212 1313 1414 1515 1616 1717 1818 1919 20202121 2222 2323 2424 2525 2626 2727 2828 2929 30303131 3232 3333 3434 3535 3636 3737 3838 3939 40404141 4242 4343 4444 4545 4646 4747 4848 4949 50505151 5252 5353 5454 5555 5656 5757 5858 5959 60606161 6262 6363 6464 6565 6666 6767 6868 6969 70707171 7272 7373 7474 7575 7676 7777 7878 7979 80808181 8282 8383 8484 8585 8686 8787 8888 8989 90909191 9292 9393 9494 9595 9696 9797 9898 9999 100100

ACTIVITYACTIVITY :: Material: HUNDRED CHART TABLEMaterial: HUNDRED CHART TABLE SKIP-COUNT by 3’s, 4’s, 5’s,….10’s.SKIP-COUNT by 3’s, 4’s, 5’s,….10’s. Which helps to learn and understand table Which helps to learn and understand table

formation.formation.

Result;Result; The colouredThe coloured boxes have even numbers, where 0, 2, 4, 6 boxes have even numbers, where 0, 2, 4, 6

and 8 are in ones place.and 8 are in ones place. The boxes not coloured are odd numbers have 1, 3, 5, The boxes not coloured are odd numbers have 1, 3, 5,

7and 9 in ones place.7and 9 in ones place. Practice/exercisePractice/exercise;; use hundred chart and ask the use hundred chart and ask the

students whether the number is even or odd.students whether the number is even or odd.a) 7. b) 4. c)8. d) 9. e) 21 f) 54.a) 7. b) 4. c)8. d) 9. e) 21 f) 54.

Activity:Activity: Students can form different patterns e.g, start at Students can form different patterns e.g, start at 5, skip- count by 5, where do you land after 4 skips? 5, skip- count by 5, where do you land after 4 skips? skip count by 4 ,move 6 skips, what do you land on ?skip count by 4 ,move 6 skips, what do you land on ?

2, 4, 6, _, _, 12, 14,……..2, 4, 6, _, _, 12, 14,……..5, 10, _, 20, _, …………. 5, 10, _, 20, _, ………….

2.CONCEPT OF 2.CONCEPT OF MULTIPLICATIONMULTIPLICATION

Objectives: Objectives: To develop the concept of multiplication To develop the concept of multiplication

through addition.through addition. To prepare the multiplication table by skip-To prepare the multiplication table by skip-

count.count. To stimulate them through different To stimulate them through different

experiences.experiences.

MULTIPLICATION THROUGH MULTIPLICATION THROUGH ADDITIONADDITION

Form the arrays of some objects ; how Form the arrays of some objects ; how many in all?many in all?

444444

1212 3 groups of 4 = 3 groups of 4 =

Connect addition to the Connect addition to the multiplicationmultiplication

Repeat addition of a number and find:Repeat addition of a number and find: 3 groups of 4 =4+4+4 = 12 3 groups of 4 =4+4+4 = 12 Similarly; 2 groups of 5Similarly; 2 groups of 5 =___=___ AndAnd 4 groups of 74 groups of 7 =___ =___ 3+3+3+3+3 3+3+3+3+3 =15=15 5 Threes 5 Threes =15=15 5 times 35 times 3 =15=15

FinallyFinally 5 x35 x3 =15=15 This concept can be related by This concept can be related by Number l ine.Number l ine.

Practice:Practice:

XX 11 22 33 44 55 6644

XX 11 22 33 44 55 6666

XX 00 11 22 33 4455

STRUCTURED DRILLSTRUCTURED DRILL IS IS DEMONSRTATED, WHICH IS DEMONSRTATED, WHICH IS DESIGNED FOR TABLE LEARNING.DESIGNED FOR TABLE LEARNING.

The multiplication tableThe multiplication tableXX 00 11 22 33 44 55 66 77 88 99 101000 00 00 00 00 00 00 00 00 00 00 0011 00 11 22 33 44 55 66 77 88 99 101022 00 22 44 66 88 1010 1212 1414 1616 1818 202033 00 33 66 99 1212 1515 1818 2121 2424 2727 303044 00 44 88 1212 1616 2020 2424 2828 3232 3636 404055 00 55 1010 1515 2020 2525 3030 3535 4040 4545 505066 00 66 1212 1818 2424 3030 3636 4242 4848 5454 606077 00 77 1414 2121 2828 3535 4242 4949 5656 6363 707088 00 88 1616 2424 3232 4040 4848 5656 6464 7272 808099 00 99 1818 2727 3636 4545 5454 6363 7272 8181 90901010 00 1010 2020 3030 4040 5050 6060 7070 8080 9090 100100

XX 00 11 22 33 44 55 66 77 88 99 1010

00 00

11

22 1414

33

44

55 4040

66 1212

77 3535

88 8080

99 00 3636

1010

3.CONCEPT OF DIVISION3.CONCEPT OF DIVISION Obj: to develop the concept the concept of division by Obj: to develop the concept the concept of division by

connecting it to subtraction.connecting it to subtraction. To develop the skills by using the To develop the skills by using the multiplication table.multiplication table. 2 x =162 x =16 16 / 2 =16 / 2 = ________ 1616 1414 1212 1010 88 66 44 22 -2-2 -2-2 -2-2 -2-2 -2-2 -2-2 -2-2 -2-2 1414 1212 1010 8 8 6 6 44 22 00 8 times subtraction of 2 from 16 ended at 08 times subtraction of 2 from 16 ended at 0 16 /2 =8 or 2x 8 =1616 /2 =8 or 2x 8 =16 This concept can be illustrated by This concept can be illustrated by number l ine.number l ine.

108108 22 216216 dividedivide 22 *multiply by 2*multiply by 2 oo11 *subtract 2-2=0*subtract 2-2=0 00 *R: 1<2 (compare the diff*R: 1<2 (compare the diff 1166 with divisor)with divisor) 1616 * bring down the new digit.* bring down the new digit. 00 * repeat the same process.* repeat the same process. This conclusion: R = 0 shows it is completely This conclusion: R = 0 shows it is completely divisible by 2.divisible by 2.

practice:practice:

DivideDivideby by

88 1212 2020 1616 2828 2424

44

4.FACTOR AND MULTIPLES4.FACTOR AND MULTIPLES

Objectives: to develop the concept of factors and Objectives: to develop the concept of factors and multiples by using the multiplication table.multiples by using the multiplication table.

Flash cards for multiples.Flash cards for multiples. Flash card for different factors.Flash card for different factors. Factor Factor X X factorfactor = = multiplemultiple 66 XX 44 = 24= 24 find other factors of 24 by using M.table.find other factors of 24 by using M.table. List the factors of :18, 63, 36 and 54.List the factors of :18, 63, 36 and 54. List the multiples of: 6, 2, 5 and 7.List the multiples of: 6, 2, 5 and 7. Dril lDri l l : flash cards and number patterns .: flash cards and number patterns .

5.THE CONCEPT OF PRIME5.THE CONCEPT OF PRIME NUMBERSNUMBERS

The number has only two factors 1 and The number has only two factors 1 and itselfitself ..

As factors of two are; 1and 2.As factors of two are; 1and 2. Factors of 3; 1 and 3Factors of 3; 1 and 3 Factors of 4;1, 2 and 4Factors of 4;1, 2 and 4 2 and 3 are prime numbers but 4 is not.2 and 3 are prime numbers but 4 is not. Is 1 a prime number? Give reason.Is 1 a prime number? Give reason. The remaining numbers are Composite The remaining numbers are Composite

numbersnumbers Activity:Activity: use the hundred chart and colors. use the hundred chart and colors.

THETHE CHART OF PRIMECHART OF PRIME NUMBERSNUMBERS

11 22 33 44 55 66 77 88 99 10101111 1212 1313 1414 1515 1616 1717 1818 1919 20202121 2222 2323 2424 2525 2626 2727 2828 2929 30303131 3232 3333 3434 3535 3636 3737 3838 3939 40404141 4242 4343 4444 4545 4646 4747 4848 4949 50505151 5252 5353 5454 5555 5656 5757 5858 5959 60606161 6262 6363 6464 6565 6666 6767 6868 6969 70707171 7272 7373 7474 7575 7676 7777 7878 7979 80808181 8282 8383 8484 8585 8686 8787 8888 8989 90909191 9292 9393 9494 9595 9696 9797 9898 9999 100100

6.INTEGERS6.INTEGERS Obj: relate the Obj: relate the

integers in practical integers in practical situations.situations.

The numbers which The numbers which involve direction in involve direction in addition to the addition to the magnitudemagnitude..

Temperature, sea Temperature, sea level, profit and loss level, profit and loss etc.etc.

INTEGERS: The directed numbers.INTEGERS: The directed numbers. Z={….,-4,-3,-2,-1,0,+1,+2,+3,+4,….}Z={….,-4,-3,-2,-1,0,+1,+2,+3,+4,….} 00 negative numbersnegative numbers positive numberspositive numbers Order of integers:Order of integers: -4<-3<-2<-1<0<+1<+2<+3<+4-4<-3<-2<-1<0<+1<+2<+3<+4 The numerical value of -2 and +2 is same.The numerical value of -2 and +2 is same. But their direction is opposite.But their direction is opposite. Explanation: Explanation: Addition and subtraction on numberAddition and subtraction on number

lines. Multiplication on number lines.lines. Multiplication on number lines.

Activity:Activity: Algebra tilesAlgebra tiles are introduced in order to are introduced in order to

make, addition and subtraction of Integers, make, addition and subtraction of Integers, easy and interesting.easy and interesting.

Multiplication is illustrated on the charts.Multiplication is illustrated on the charts.

6.FRACTIONS• Equal parts of whole.

• • 3• 5

I am

denominator, I tell the name of parts into which the

unit is divided

Call me numerator, I am the number of parts that

are in fraction

EQUIVALENT FRACTIONS

Fractions that name the same amount are called equivalent fractions.

3/4 and 6/8 give the same amount.

1/3 =2/6 Understanding: use

Number line & fraction bars.

FRACTION BARS

ADD THE FRACTIONSADD THE FRACTIONSMake

the name same1/3+1/6

=2/6+1/6

Add the numbers of fraction(2+1)/6

Write the answer3/6=1/2

7.PROBLEM SOLVING.7.PROBLEM SOLVING.Key wordsKey words

++ Add, sum, total, more, increase, all. Add, sum, total, more, increase, all. Altogether.Altogether.

-- Subtract, difference, decrease, less Subtract, difference, decrease, less than, more than (comparison), than, more than (comparison), remaining, rest.remaining, rest.

XX Product, of, into, times, twice(x2), Product, of, into, times, twice(x2), thrice(x3), etc.thrice(x3), etc.

÷÷ Divided, parts, fractions, half(Divided, parts, fractions, half(÷2), ÷2), fourth(÷4), etc.fourth(÷4), etc.

== Is, equals, gives, results.Is, equals, gives, results.

STEPS TO SOLVE THESTEPS TO SOLVE THE PROBLEMPROBLEM

1.UNDERSTAND1.UNDERSTAND the the problem.problem.

What is given info?What is given info? What is to f ind?What is to f ind?2.PLAN 2.PLAN a strategy.a strategy. Decide the method.Decide the method.3.SOLVE3.SOLVE the problem. the problem. Apply the method.Apply the method.4.CHECK4.CHECK your answer. your answer. Look back, does the Look back, does the

answer make sense.answer make sense. I f not, what other If not, what other

strategy can be strategy can be used.used.

1.1.List theList the given information.given information.state the required info,state the required info,whether you need all of whether you need all of the info.the info.

2.2.Think about the problem Think about the problem solving strategies you can solving strategies you can use.use.

3.3.Follow your plan, show Follow your plan, show your solution.your solution.

4.4.Be sure that you Be sure that you answered the question answered the question asked.asked.or use other method to or use other method to check your work.check your work.

EXAMPLEEXAMPLE

Q.Q. If two angles of the triangle are 94 If two angles of the triangle are 94° and ° and 34° then what is the measure of the third 34° then what is the measure of the third angle?angle?

Let’s apply the four steps of problem Let’s apply the four steps of problem solving. solving.

Problem: class 3Problem: class 3

One kilometer equals 1000meters.How One kilometer equals 1000meters.How many meters are there in 24 kilometer.many meters are there in 24 kilometer.

Problem: class 4Problem: class 4

A shopkeeper bought 2175 ice-pops. If 15 A shopkeeper bought 2175 ice-pops. If 15 ice pops were packed in each box. How ice pops were packed in each box. How many boxes did he buy?many boxes did he buy?

Problem: class 5Problem: class 5

Sadaf bought 36 chickens.17 of them are Sadaf bought 36 chickens.17 of them are white. What fraction of the chicken are not white. What fraction of the chicken are not white?white?

Problem: class 6Problem: class 6

The length of a rectangle is The length of a rectangle is x x cm and its cm and its width is 4 Cm .If the rectangle is 72 cmwidth is 4 Cm .If the rectangle is 72 cm²² in in area, find area, find x.x.

Problem :class 3Problem :class 3

Order the fraction from least to greaterOrder the fraction from least to greater a. 1/2, 1/3, 1/4a. 1/2, 1/3, 1/4 b. 3/4, 3/6, 3/5b. 3/4, 3/6, 3/5

Research and presented byResearch and presented by Mrs. UMBER TARIQMrs. UMBER TARIQ Assisted byAssisted by Miss. UNIBAMiss. UNIBA Mrs. FOUSIA SHAHEENMrs. FOUSIA SHAHEEN

THANK THANK YOUYOU