SECONDARY MATHEMATICS WORKSHOP. ENGLISH LANGUAGE LEARNERS IN THE MATHEMATICS CLASSROOM ENGLISH LANGUAGE

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• SECONDARYMATHEMATICSWORKSHOP

• ENGLISH LANGUAGE

ENGLISH LANGUAGELEARNERSIN THEMATHEMATICSCLASSROOMENGLISH LANGUAGE

• SAMPLE QUESTION 1Questions to help students rely on their own understanding,ask the following :DO YOU THINK THAT IS TRUE? WHY?DOES THAT MAKE SENSE TO YOU?HOW DID YOU GET YOUR ANSWER?DO YOU AGREE WITH THE EXPLANATION?

• SAMPLE QUESTION : 2To promote problem solving, ask the following :

WHAT DO YOU NEED TO FIND OUT?WHAT INFORMATION DO YOU HAVE?WILL A DIAGRAM OR NUMBER LINE HELP YOU?WHAT TECHNIQUE COULD YOU USE?WHAT DO YOU THINK THE ANSWER WILL BE

• SAMPLE QUESTION : 3Questions to encourage students to speak out, ask the followingWhat do you think about what said?Do you agree what I have said?Why?Or why not?Does anyone have the same answer but a different way to explain it?Do you understand what ?Are you confuse?

• SAMPLE QUESTION : 4Question to check the students progress, ask the following:What have you found out so far?What do you notice about?What other things that you need to do?What other information you need to find out?Have you though of another way to solve the questions?

• SAMPLE QUESTION : 5Question to help students when they get stuck,ask the followingWhat have you done so far?What do you need to figure out next?How would you say the questions in your own words?Could you try it the other way round?Have you compared your work with anyone else?

• SAMPLE QUESTION : 6Question to make connection among ideas and application,Ask the following:What other problem does this remind you of?Can you give me an example of ?Can you write down the objective or aim?Can you write down the formulae?

• EXAMPLE TO COMMUNICATECAN YOU REPEAT THAT PLEASE?HOW DO YOU SPELL________?WHAT DOES ____MEAN?CAN YOU GIVE ME AN EXAMPLE?Teacher : I am reading a book about amphibiansStudents : Can you repeat that please?Teacher : I said : Im reading a book on amphibiansStudents : How do you spell amphibians?Teacher : A-M-P-H-I-B-I-A-N-SStudents : What does amphibians mean?Teacher : It is an animal that is born in water but can live on landStudent : Can you give me an example?Teacher : A frog

• KNOW YOUR KEY WORDSMORE THANLESS THANALTOGETHERAT FIRSTSUMDIFFERENTCOMPAREDIGITSFIND THE LENGTH /MASSPLACE VALUEWHOLE NUMBER

• KNOW YOUR KEY WORDSORDINAL NUMBERSUBTRACTSUBTRACT 2 FROM 5GREATER THANLESS THANSHORT/SHORTER/SHORTESTTALL/TALLER/TALLESTARRANGE THE NUMBER FROM THE GREATEST TO THE SMALLESTARRANGE THE STRINGS FROM THE SHORTEST TO THE LONGERSTREAD THE QUESTIONS CAREFULLY

• KNOW YOUR KEY WORDSLABEL THE FOLLOWINGEVALUATEHEAVY/HEAVIER/HEAVIESTNUMBER SEQUENCEHOW MUCH MONEY I LEFT?1 MORE THAN 103 LESS THAN 10HOW MANY MARBLE HAD SHE LEFT?HOW MUCH MORE MONEY JOHN HAVE THAN MARY?PRODUCT

• KNOW YOUR KEY WORDSFACTORSMULTIPLES OF 2, 3NUMBER LINESPOSITIVE NUMBERNEGATIVE NUMBERINTEGERS3 TO THE POWER OF 2PRIME NUMBERVENN DIAGRAMINEQUALITIESMULTIPLY

• KNOW YOUR KEY WORDSDIVIDEADD TWO NUMBER UP TO THREE DIGITSFACTIONMIXED NUMBERIMPROPER FRACTIONCONVERT THE FOLLOWING FRACTION TO DECIMALSEQUILATERALISOSCELESRIGHT ANGLE TRIANGLENUMBERATORDENOMINATOR

• FACTORS AND MULTIPLESWe can write a whole number greater than 1 as a product of two whole numbers.E.g.18=1x1818=2x918=3x6Tip : Note that 18 is divisible by each of its factors.Therefore, 1, 2, 3, 6, 9 and 18 are called factors of 18.The common factors of two numbers are the factors that the numbers have in common. Factors of a number are whole numbers which multiply to give that number.E.g.Factors of 12: 1, 2, 3, 4, 6, 12Factors of 21: 1, 3, 7, 21The common factors of 12 and 21 are 1 and 3.

• FACTORS AND MULTIPLESWhen we multiply a number by a non-zero whole number, we get a multiple of the number.E.g.1 x 3 = 31 x 5 = 5 2 x 3 = 62 x 5 = 10 3 x 3 = 9 Multiples 3 x 5 = 15 Multiples 4 x 3 = 12 of 3 4 x 5 = 20 of 5 5 x 3 = 155 x 5 = 25Therefore, the multiples of 3 are 3, 6, 9, 12, 15, andthe multiples of 5 are 5, 10, 15, 20, 25, The first three common multiples of 4 and 6 are 12, 24 and 36.The common multiple of two numbers is a number that is a multiple of both numbers.E.g.Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, ...Multiples of 6 are 6, 12, 18, 24, 30, 36, ...

• PRIME NUMBERS,PRIME FACTORISATIONA prime number is a whole number greater than 1 that has exactly two different factors, 1 and itself. E.g. 5 = 1 x 5

Since 5 has no other whole number factors other than 1 and itself, it is a prime number.A composite number is a whole number greater than 1 that has more than 2 different factors. The numbers 2, 3, 5, 7, 11, 13, 17, are prime numbers.E.g.6 = 1 x 66 = 2 x 3

Therefore, 6 is a composite number.4 Factors

• PRIME NUMBERS,PRIME FACTORISATIONThe numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, are composite numbers. In other words, all whole numbers greater than 1 that are not prime numbers are composite numbers.Tip: 0 and 1 are neither prime nor composite numbers.E.g. The factors of 18 are 1, 2, 3, 6, 9, and 18. The prime factors of 18 are 2 and 3.Prime factors are factors of a number that are also prime.We can use either the factor tree or repeated division to express a composite number as a product of its prime factors. The process of expressing a composite number as the product of prime factors is called prime factorisation.

• PRIME NUMBERS, PRIME FACTORISATIONWORKED EXAMPLE 1:Express 180 as a product of prime factors. 180 2 x 90

2 x 2 x 45

2 x 2 x 3 x 15

2 x 2 x 3 x 3 x 5

Therefore, 180= 2 x 2 x 3 x 3 x 5= 22 x 32 x 5SOLUTION:Method I (Using the Factor Tree)Steps:Write the number to be factorised at the top of the tree.Express the number as a product of two numbers.Continue to factorise if any of the factors is not prime.Continue to factorise until the last row of the tree shows only prime factors.A quicker and more concise way to write the product is using index notation.

• PRIME NUMBERS, PRIME FACTORISATIONWORKED EXAMPLE 1:Express 180 as a product of prime factors.

2 1802 903 453 155 5 1

Therefore, 180= 2 x 2 x 3 x 3 x 5= 22 x 32 x 5SOLUTION:Method II (Using Repeated Division)Steps:Start by dividing the number by the smallest prime number. Here, we begin with 2.Continue to divide using the same or other prime numbers until you get a quotient of 1.The product of the divisors gives the prime factorisation of 180.

• INDEX NOTATIONIf the factors appear more than once, we can use the index notation to represent the product.E.g. 3 x 3 x 3 x 3 x 3 = 35In index notation, 3 is called the base and the number at the top, 5 is called the index.35 is read as 3 to the power of 5E.g. 2 x 2 x 2 x 5 x 5 = 23 x 5235indexbaseThe answer is read as 2 to the power of 3 times 5 to the power of 2.

• HIGHEST COMMON FACTOR (HCF)The largest common factor among the common factors of two or more numbers is called the highest common factor (HCF) of the given numbers.E.g.Factors of 12 are 1, 2, 3, 4, 6, and 12.Factors of 18 are 1, 2, 3, 6, 9, and 18.Another method to find the HCF of two or more numbers is by using prime factorisation which is the more efficient way. The common factors of 12 and 18 are 1, 2, 3 and 6. The highest common factor (HCF) of 12 and 18 is 6.We can also repeatedly divide the numbers by prime factors to find the HCF.

• HIGHEST COMMON FACTOR (HCF)WORKED EXAMPLE 1:Find the highest common factor of 225 and 270.SOLUTION:Therefore, the HCF of 225 and 270 is 45.225 =32x 52270 = 2 x33 x 5HCF =32 x 5 = 45Find the prime factorisation of each number first.To get the HCF, multiple the lowest power of each common prime factor of the given numbers.

• LOWEST COMMON MULTIPLE (LCM)The smallest common multiple among the common multiples of two or more numbers is called the lowest common multiple (LCM) of the given numbers.E.g.Multiples of 8: 8, 16, 24, 32, 40, 48, ...Multiples of 12: 12, 24, 36, 48, 60, ...Another method to find the LCM of two or more numbers is by using prime factorisation which is the more efficient way. The common multiples of 8 and 12 are 24, 48, ... The lowest common multiple (LCM) of 8 and 12 is 24.We can also repeatedly divide the numbers by prime factors to find the LCM.

• LOWEST COMMON MULTIPLE (LCM)WORKED EXAMPLE 1:Find the lowest common multiple of 24 and 90.SOLUTION:Therefore, the LCM of 24 and 90 is 360.24= 23x 390= 2x 32 x 5LCM= 23x 32 x 5 = 360To get the LCM, multiple the highest power of each set of common prime factors. Also include any uncommon factors

• SQUARES AND SQUARE ROOTSWhen a number is multiplied by itself, the product is called the square of the numberThe numbers whose square roots are whole numbers are called perfect squares.5 is the positive square root of 25.Tip :22 = 4and 4 = 232 = 9and 9 = 342 = 16and 16 = 4

E.g.5 x 5 = 25or52 = 25E.g. 25 = 5E.g.1, 4, 9, 16, 25, ... are perfect squares.

• SQUARES AND SQUARE ROOTSWORKED EXAMPLE 1:Using prime factorisation, find the square root of 5184.

5184= 26 x 342 51845184= 26 x 34 2 2592= 23 x 322 1296= 8 x 92 648= 722 3242 1623 813 273 93 3 1SOLUTION:

• CUBES AND CUBE ROOTSWhen a number is multiplied by itself thrice, the product is called the cube of the numberThe numbers whose cube roots are whole numbers are called perfect cubes.125 is the cube of 5 and 5 is the cube root of 125.Tip :23 = 8and 8 = 233 = 27and 27 = 343 = 64and 64 = 4

E.g.5 x 5 x 5 = 125or53 = 125E.g. 125 = 5E.g.1, 8, 27, 64, 125, ... are perfect cubes.

• CUBES AND CUBE ROOTSWORKED EXAMPLE 1:Using prime factorisation, find the cube root of 1728.

1728= 26 x 332 17281728= 26 x 33 2 864= 22 x 32 432= 4 x 32 216= 122 1082 543 273 93 3 1 SOLUTION:

• REAL NUMBERSNumbers with the negative sign ( - ) are called negative numbers.E.g.-1, -2, -3, -4, -5, ...Positive integers are whole numbers that are greater than zero.Zero is an integer that is neither positive nor negative.Integers refer to whole numbers and negative numbers.E.g...., -3, -2, -1, 0, 1, 2, 3, 4, ... are integers.E.g.1, 2, 3, 4, 5, ... Negative integers are whole numbers that are smaller than zero.E.g.-1, -2, -3, -4, -5, ...

• REAL NUMBERSA number line showing integers is shown below:Every number on the number line is greater than any number to its left.The arrows on both ends of the number line show that the line can be extended on both ends.E.g.2 is greater than -3 and is denoted by 2 > -3We can also write -3 is smaller than 2 and is denoted by -3< 2 -5 -4 -3 -2 -1 0 1 2 3 4 5Negative Integers Positive Integers-5 -4 -3 -2 -1 0 1 2 3 4 5

• REAL NUMBERS>, and < are called inequality signs.1, 2, 3, 4, 5, 6, 7, ... are called natural numbers. The natural numbers are also called positive integers.> means is greater than< means is smaller than> means is greater than or equal to< means is smaller than or equal toE.g.|2| = 2, |0| = 0, |-2| = 2The numerical or absolute value of a number x, denoted by |x|, is its distance from zero on the number line.Since distance can never be negative, the numerical or absolute value of a number is always positive.

• ADDITION OF INTEGERSRules for adding two integers:

Sign of numbersMethodBoth numbers have the same signs

(+a) + (+b) = +(a + b)(-a) + (-b) = -(a + b)Add the numbers while ignoring their signs.Write the sum using their common sign.E.g. (+3) + (+8) = +11 = 11E.g. (-3) + (-8) = -(3 + 8) = -11Both numbers have different signs

(+a) + (-b) = +(a b) if a>b(+a) + (-b) = - (b a) if b>a(-a) + (+b) = - (a b) if a>b(-a) + (+b) = +(b a) if b>aSubtract the numbers while ignoring their signs.The answer has the same sign as the number having the larger numerical value.E.g. 12 + (-4) = 12 4 = 8E.g. 5 + (-11) = -(11 5) = -6E.g. -8 + 3 = -(8 3) = -5E.g. -9 + 15 = 15 9 = 6

• SUBTRACTION OF INTEGERSTo subtract integers, change the sign of the integer being subtracted and add using the addition rules for integers.a b = a + (-b)E.g. 8 15 = 8 + (-15) = -(15 8) = -7 -11 7 = -11 + (-7) = -(11 + 7) = -18-6 (-10) = -6 + 10 = 10 6 = 4 3 (-13) = 3 + 13 = 16

• MULTIPLICATION OF INTEGERSRules for multiplying integers:

MultiplicationExamples(+a) x (+b)= +(a x b)(- a) x (- b)= +(a x b)(+a) x (- b)= - (a x b)(- a) x (+b)= - (a x b)3 x 4 = 12(-5) x (-6) = +(5 x 6) = 308 x (-3) = -(8 x 3) = -24(-12) x 4 = -(12 x 4) = -48

Rules for signs:( + ) x ( + ) = ( + )The product of two positive integers is a positive integer( - ) x ( - ) = ( + )The product of two negative integers is a positive integer( + ) x ( - ) = ( - )( - ) x ( + ) = ( - )The product of a positive and a negative integer is a negative integer.

• DIVISION OF INTEGERSRules for dividing two integers:

DivisionExamples(+a) (+b)= +(a b)(- a) (- b)= +(a b)(+a) (- b)= - (a b)(- a) (+b)= - (a b)16 2 = 8(-20) (-5) = +(20 5) = 436 (-4) = -(36 4) = -9(-24) 8 = -(24 8) = -3

Rules for signs:( + ) ( + ) = ( + )The quotient of two positive integers is a positive integer( - ) ( - ) = ( + )The quotient of two negative integers is a positive integer( + ) ( - ) = ( - )( - ) ( + ) = ( - )The quotient of a positive and a negative integer is a negative integer.

• RULES FOR OPERATING ON INTEGERSAddition and multiplication of integers obey the Commutative Law.Addition and multiplication of integers obey the Associative Law.E.g. 1 2 + (-10) = (-10) + 2 = -8E.g. 2 2 x (-10) = (-10) x 2 = -20Commutative Law of Addition of Integers:a + b = b + aCommutative Law of Multiplication of Integers:a x b = b x aAssociative Law of Addition of Integers:(a + b) + c = a + (b + c)Associative Law of Multiplication of Integers:(a x b) x c = a x (b x c)E.g. 1 [3 + (-5)] + 8 = 3 + [(-5) + 8] = 6E.g.2 [3 x (-5)] x 8 = 3 x [(-5) x 8] = -120

• RULES FOR OPERATING ON INTEGERSMultiplication of integers is distributive overa) additionb) subtractionThe order of operation on integers is the same as those for whole numbersE.g. 1 -2 x (-3 + 5) = -2 x (-3) = (-2) x 5 = -4E.g. 2 -2 x (-8 + 6) = -2 x (-8) = (-2) x 6 = 28Distributive Law of Multiplication over Addition of integers:a x (b + c) = (a x b) + (a x c)Distributive Law of Multiplication over Subtraction of Integers:a x (b c) = (a x b) (a x c)Order of operationsSimplify expressions within the brackets first.Working from left to right, perform multiplication or division before addition or subtraction.

• RULES FOR OPERATING ON INTEGERSWORKED EXAMPLE 1:Evaluate each of the following.25 36 (-4) + (-11)(-10) (-6) + (-9) 3{-15 [15 + (-9)]2} (-3)(3 5)3 x 4 + [(-18) + (-2)] (-3)2SOLUTION:

25 36 (-4) + (-11)= 25 (-9) + (-11)= 25 + 9 11= 23

• RULES FOR OPERATING ON INTEGERSSOLUTION:

(-10) (-6) + (-9) 3= (-10) (-6) + (-3)= -10 + 6 3= -7(3 5)3 x 4 + [(-18) + (-2)] (-3)2= (-2)3 x 4 + (-18 2) (-2)2= (-2)3 x 4 + (-20) (-2)2= (-8) x 4 + (-20) 4= -32 + (-5)= -32 5= -37{-15 [15 + (-9)]2} (-3)= [-15 (15 9)2] (-3)= (-15 62) (-3)= (-15 36) (-3)= (-51) (-3)= 17

• INTRODUCTION TO ALGEBRAUsing Letters to Represent NumbersA variable is a letter...