SECONDARY MATHEMATICS WORKSHOP

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SECONDARYMATHEMATICSWORKSHOP
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ENGLISH LANGUAGE
ENGLISH LANGUAGE

• ENGLISH LANGUAGE
• LEARNERS
• IN THE
• MATHEMATICS
• CLASSROOM

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SAMPLE QUESTION 1

• Questions 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?

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SAMPLE QUESTION 2

• To 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

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SAMPLE QUESTION 3

• Questions to encourage students to speak out, ask
  the following
• What 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?

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SAMPLE QUESTION 4

• Question 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?

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SAMPLE QUESTION 5

• Question to help students when they get stuck,ask
  the following
• What 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?

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SAMPLE QUESTION 6

• Question 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?

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EXAMPLE TO COMMUNICATE

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

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KNOW YOUR KEY WORDS

• MORE THAN
• LESS THAN
• ALTOGETHER
• AT FIRST
• SUM
• DIFFERENT
• COMPARE
• DIGITS
• FIND THE LENGTH /MASS
• PLACE VALUE
• WHOLE NUMBER

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KNOW YOUR KEY WORDS

• ORDINAL NUMBER
• SUBTRACT
• SUBTRACT 2 FROM 5
• GREATER THAN
• LESS THAN
• SHORT/SHORTER/SHORTEST
• TALL/TALLER/TALLEST
• ARRANGE THE NUMBER FROM THE GREATEST TO THE
  SMALLEST
• ARRANGE THE STRINGS FROM THE SHORTEST TO THE
  LONGERST
• READ THE QUESTIONS CAREFULLY

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KNOW YOUR KEY WORDS

• LABEL THE FOLLOWING
• EVALUATE
• HEAVY/HEAVIER/HEAVIEST
• NUMBER SEQUENCE
• HOW MUCH MONEY I LEFT?
• 1 MORE THAN 10
• 3 LESS THAN 10
• HOW MANY MARBLE HAD SHE LEFT?
• HOW MUCH MORE MONEY JOHN HAVE THAN MARY?
• PRODUCT

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KNOW YOUR KEY WORDS

• FACTORS
• MULTIPLES OF 2, 3
• NUMBER LINES
• POSITIVE NUMBER
• NEGATIVE NUMBER
• INTEGERS
• 3 TO THE POWER OF 2
• PRIME NUMBER
• VENN DIAGRAM
• INEQUALITIES
• MULTIPLY

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KNOW YOUR KEY WORDS

• DIVIDE
• ADD TWO NUMBER UP TO THREE DIGITS
• FACTION
• MIXED NUMBER
• IMPROPER FRACTION
• CONVERT THE FOLLOWING FRACTION TO DECIMALS
• EQUILATERAL
• ISOSCELES
• RIGHT ANGLE TRIANGLE
• NUMBERATOR
• DENOMINATOR

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FACTORS AND MULTIPLES

• We can write a whole number greater than 1 as a
  product of two whole numbers.

E.g. 18 1 x 18 18 2 x 9 18 3 x 6
Therefore, 1, 2, 3, 6, 9 and 18 are called
factors of 18.
Tip Note that 18 is divisible by each of its
factors.

• Factors of a number are whole numbers which
  multiply to give that number.

• The common factors of two numbers are the factors
  that the numbers have in common.

E.g. Factors of 12 1, 2, 3, 4, 6, 12 Factors
of 21 1, 3, 7, 21
The common factors of 12 and 21 are 1 and 3.
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FACTORS AND MULTIPLES

• When we multiply a number by a non-zero whole
  number, we get a multiple of the number.

E.g. 1 x 3 3 1 x 5 5 2 x 3
6 2 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
15 5 x 5 25
Therefore, the multiples of 3 are 3, 6, 9, 12,
15, and the multiples of 5 are 5, 10, 15, 20,
25,

• 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, ...
The first three common multiples of 4 and 6 are
12, 24 and 36.
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PRIME NUMBERS,PRIME FACTORISATION

• A 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.

• The numbers 2, 3, 5, 7, 11, 13, 17, are prime
  numbers.

• A composite number is a whole number greater than
  1 that has more than 2 different factors.

E.g. 6 1 x 6 6 2 x 3 Therefore, 6
is a composite number.
4 Factors
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PRIME NUMBERS,PRIME FACTORISATION

• The 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.

• Prime factors are factors of a number that are
  also prime.

E.g. The factors of 18 are 1, 2, 3, 6, 9, and
18. The prime factors of 18 are 2 and 3.

• The process of expressing a composite number as
  the product of prime factors is called prime
  factorisation.

• We can use either the factor tree or repeated
  division to express a composite number as a
  product of its prime factors.

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PRIME NUMBERS, PRIME FACTORISATION

• WORKED EXAMPLE 1
• Express 180 as a product of prime factors.

SOLUTION Method I (Using the Factor Tree)
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 5

• 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.
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PRIME NUMBERS, PRIME FACTORISATION

• WORKED EXAMPLE 1
• Express 180 as a product of prime factors.

SOLUTION Method II (Using Repeated Division)
2 180 2 90 3 45 3 15 5 5
1 Therefore, 180 2 x 2 x 3 x 3 x 5 22 x 32
x 5

• 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.

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INDEX NOTATION

• If 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 35
35 is read as 3 to the power of 5
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index
base
In index notation, 3 is called the base and the
number at the top, 5 is called the index.
E.g. 2 x 2 x 2 x 5 x 5 23 x 52
The answer is read as 2 to the power of 3 times 5
to the power of 2.
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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.
The common factors of 12 and 18 are 1, 2, 3 and
6.
The highest common factor (HCF) of 12 and 18 is 6.

• Another method to find the HCF of two or more
  numbers is by using prime factorisation which is
  the more efficient way.

• We can also repeatedly divide the numbers by
  prime factors to find the HCF.

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HIGHEST COMMON FACTOR (HCF)

• WORKED EXAMPLE 1
• Find the highest common factor of 225 and 270.

SOLUTION
225 32x 52 270 2 x 33 x 5
Find the prime factorisation of each number first.
To get the HCF, multiple the lowest power of each
common prime factor of the given numbers.
HCF 32 x 5 45
Therefore, the HCF of 225 and 270 is 45.
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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, ...
The common multiples of 8 and 12 are 24, 48, ...
The lowest common multiple (LCM) of 8 and 12 is
24.

• Another method to find the LCM of two or more
  numbers is by using prime factorisation which is
  the more efficient way.

• We can also repeatedly divide the numbers by
  prime factors to find the LCM.

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LOWEST COMMON MULTIPLE (LCM)

• WORKED EXAMPLE 1
• Find the lowest common multiple of 24 and 90.

SOLUTION
24 23 x 3 90 2 x 32 x 5
To get the LCM, multiple the highest power of
each set of common prime factors. Also include
any uncommon factors
LCM 23 x 32 x 5 360
Therefore, the LCM of 24 and 90 is 360.
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SQUARES AND SQUARE ROOTS

• When a number is multiplied by itself, the
  product is called the square of the number

E.g. 5 x 5 25 or 52 25

• 5 is the positive square root of 25.

E.g. v25 5

• The numbers whose square roots are whole numbers
  are called perfect squares.

E.g. 1, 4, 9, 16, 25, ... are perfect squares.
Tip 22 4 and v 4 2 32 9 and v 9
3 42 16 and v16 4
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SQUARES AND SQUARE ROOTS

• WORKED EXAMPLE 1
• Using prime factorisation, find the square root
  of 5184.

SOLUTION
5184 26 x 34 2 5184 v5184 v26 x 34
2 2592 23 x 32 2 1296 8 x
9 2 648 72 2 324 2
162 3 81 3 27 3
9 3 3 1
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CUBES AND CUBE ROOTS

• When a number is multiplied by itself thrice, the
  product is called the cube of the number

E.g. 5 x 5 x 5 125 or 53 125

• 125 is the cube of 5 and 5 is the cube root of
  125.

E.g. ?125 5

• The numbers whose cube roots are whole numbers
  are called perfect cubes.

E.g. 1, 8, 27, 64, 125, ... are perfect cubes.
Tip 23 8 and ? 8 2 33 27 and ?27
3 43 64 and ?64 4
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CUBES AND CUBE ROOTS

• WORKED EXAMPLE 1
• Using prime factorisation, find the cube root of
  1728.

SOLUTION
1728 26 x 33 2 1728 ?1728 ?26 x 33
2 864 22 x 3 2 432 4 x
3 2 216 12 2 108 2
54 3 27 3 9 3
3 1
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REAL NUMBERS

• Numbers with the negative sign ( - ) are
  called negative numbers.

E.g. -1, -2, -3, -4, -5, ...

• Integers refer to whole numbers and negative
  numbers.

E.g. ..., -3, -2, -1, 0, 1, 2, 3, 4, ... are
integers.

• Positive integers are whole numbers that are
  greater than zero.

E.g. 1, 2, 3, 4, 5, ...

• Negative integers are whole numbers that are
  smaller than zero.

E.g. -1, -2, -3, -4, -5, ...

• Zero is an integer that is neither positive nor
  negative.

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REAL NUMBERS

• A number line showing integers is shown below

-5 -4 -3 -2 -1 0 1 2 3 4
5 Negative Integers Positive Integers

• The arrows on both ends of the number line show
  that the line can be extended on both ends.

• Every number on the number line is greater than
  any number to its left.

-5 -4 -3 -2 -1 0 1 2 3 4 5
E.g. 2 is greater than -3 and is denoted by 2 gt
-3 We can also write -3 is smaller than 2 and is
denoted by -3lt 2
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REAL NUMBERS

• gt, lt, gt and lt are called inequality signs.

gt means is greater than lt means is smaller
than gt means is greater than or equal to lt
means is smaller than or equal to

• 1, 2, 3, 4, 5, 6, 7, ... are called natural
  numbers. The natural numbers are also called
  positive integers.

• The 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.

E.g. 2 2, 0 0, -2 2
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ADDITION OF INTEGERS

• Rules for adding two integers

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SUBTRACTION OF INTEGERS

• To 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
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MULTIPLICATION OF INTEGERS

• Rules for multiplying integers

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DIVISION OF INTEGERS

• Rules for dividing two integers

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RULES FOR OPERATING ON INTEGERS

• Addition and multiplication of integers obey the
  Commutative Law.

Commutative Law of Addition of Integers a b
b a Commutative Law of Multiplication of
Integers a x b b x a
E.g. 1 2 (-10) (-10) 2 -8 E.g. 2 2 x
(-10) (-10) x 2 -20

• Addition and multiplication of integers obey the
  Associative Law.

Associative 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
6 E.g.2 3 x (-5) x 8 3 x (-5) x 8 -120
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RULES FOR OPERATING ON INTEGERS

• Multiplication of integers is distributive over
• a) addition b) subtraction

Distributive 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)
E.g. 1 -2 x (-3 5) -2 x (-3) (-2) x 5
-4 E.g. 2 -2 x (-8 6) -2 x (-8) (-2) x 6
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• The order of operation on integers is the same as
  those for whole numbers

• Order of operations
• Simplify expressions within the brackets first.
• Working from left to right, perform
  multiplication or division before addition or
  subtraction.

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RULES FOR OPERATING ON INTEGERS

• WORKED 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)2

• SOLUTION
• 25 36 (-4) (-11)
• 25 (-9) (-11)
• 25 9 11
• 23

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RULES FOR OPERATING ON INTEGERS

• SOLUTION
• (-10) (-6) (-9) 3
• (-10) (-6) (-3)
• -10 6 3
• -7

• -15 15 (-9)2 (-3)
• -15 (15 9)2 (-3)
• (-15 62) (-3)
• (-15 36) (-3)
• (-51) (-3)
• 17

• (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

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INTRODUCTION TO ALGEBRA

• Using Letters to Represent Numbers

• In algebra, we use letters (e.g. x, y, z, a, b,
  P, Q, ) to represent numbers.

E.g. There are n apples in a bag. If there are
5 bags, then the total number of apples is 5 x
n. 5 x n can be any whole number value. It
can be 5, 10, 15, depending on the value of
n. i.e. n 1, 2, 3, Here, n is called the
variable and 5 x n is called the algebraic
expression.

• A variable is a letter that is used to represent
  some unknown numbers/quantity .

E.g. x, y, z, a, b, P, Q, are variables
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INTRODUCTION TO ALGEBRA

• Using Letters to Represent Numbers

• An algebraic expression is a collection of terms
  connected by the signs , -, x, .

E.g. 3x y, a2 ab, 2x2 3x 4.
Tip An algebraic expression does not have an
equal sign (). An algebraic expression is
different from an algebraic equation. An
equation is a mathematical statement that says
that two expressions are equal to each other
E.g. A lb is an equation. A and lb are
algebraic expressions.
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ALGEBRAIC NOTATIONS

• We use the signs , -, x, and in
  Algebra the same way as Arithmetic.

• The examples below show how we rewrite
  mathematical statements as algebraic expressions.

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ALGEBRAIC NOTATIONS
More examples below show how we rewrite
mathematical statements as algebraic expressions.
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ALGEBRAIC NOTATIONS

• In Algebra, we use the same index notation as in
  Arithmetic.

Index Notation Recall 5 x 5 x 5 53 53 is read
as 5 to the power of 3 In Algebra, a x a a2
(read as a squared) a x a x a a3 (read as a
cubed) a x a x a x a x a a5 (read as a to the
power of 5)
index
base
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ALGEBRAIC NOTATIONS

• WORKED EXAMPLE 1
• 3x x 4y 6z
• 2a x 3b x a
• 5p 10q 7s x 2

• SOLUTION
• 3x x 4y 6z
• 3 x x x 4 x y 6z
• 12xy 6z
• 12xy/6z
• 2xy/z

• 2a x 3b x a
• 6a2b
• 5p 10q 7s x 2
• 5p/10q 14s
• p/2q 14s

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ALGEBRAIC NOTATIONS

• WORKED EXAMPLE 2
• Subtract 3 from the sum of 5a and 4b.
• Add the product of c and d to the cube of e.
• Multiple 2 to the quotient of f divided by g.

• SOLUTION
• Sum of 5a and 4b
• 5a 4b
• Required expression
• 5a 4b 3 (ans)

• Product of c and d
• c x d cd
• Cube of e e x e x e e3
• Required expression
• cd e3 (ans)

• Quotient of f divided by g f/g
• Required expression 2 x f/g 2f/g (ans)

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EVALUATION OF ALGEBRAIC EXPRESSIONS AND FORMULA

• To evaluate an algebraic express, we substitute a
  number for the variable and carry out the
  computation.

• WORKED EXAMPLE 1
• 3a 2b 4c,
• a(2b c) 3b2,
• a/b (ab)/ac,
• given that a 4, b 2, c -3.

• SOLUTION
• 3a 2b 4c 3(4) 2(2) 4(-3)
• 12 4 12
• 28

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EVALUATION OF ALGEBRAIC EXPRESSIONS AND FORMULA

• SOLUTION
• a(2b c) 3b2 42(2) (-3) 3(2)2
• 4(43) 3(4)
• 4(7) 12
• 28 12
• 16

• a/b (ab)/ac 4/2 (42)/4(-3)
• 2 (6/-12)
• 2 ½
• 2½

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ALGEBRAIC EXPRESSIONS
a) Find the total cost of m cups and n plates if
each cup cost 3 and each plate costs 4.
1 cup 3 1 plate 4 m cups m x 3 n
plates n x 4 3m 4n Total Cost 3m
4n (3m 4n) (ans)
b) Find the total cost of 7 bars of wafers at p
cents each and q packets of sweets at 1 each.
1 bar p cents 1 packet 100 cents 7
bars 7 x p cents q packets q x 100 cents
7p cents 100q cents Total Cost 7p cents
100q cents (7p 100q) cents
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ALGEBRAIC EXPRESSIONS
c) John has 100, He bought n comic books at 9
each. How much money had he left?
1 book 9 n books n x 9 9n Amt left
100 - 9n (100 9n) (ans)
d) The cost of 3 caps is x. Find the cost of 5
caps. Each cap costs the same.
3 caps x 1 cap x 3 x/3 5
caps x/3 x 5 5x/3 (ans)
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RATIONAL NUMBERS

• A rational number is any number that can be
  written as a ratio of two integers. In other
  words, a number is a rational number if it can be
  written as a fraction where both the numerator
  and denominator are integers.

A rational number can be written in the form a/b
where a and b are integers and b ? 0
E.g. -3/5, ½, 5/3, 12/3, are rational numbers.

• All integers are rational numbers since each
  integer, n can be written as n/1.

E.g. 3 and -6 are rational nos. since 3 3/1
and -6 -6/1

• Most decimals can be expressed as rational
  numbers too.

E.g. 0.5 and 3.2 are rational nos. since 0.5
5/10 and 3.2 32/10
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RATIONAL NUMBERS
Recall 5/10 ½ 5/10 and ½ are equivalent
fractions. ½ is said to be in its simplest form
or in its lowest terms. Here, the numerator and
denominator have no common factors.
32/10 16/5 31/5 16/5 is called an improper
fraction. The numerator is greater than or equal
to the denominator in an improper fraction.
31/5 is called a mixed number. It represents the
sum of whole number and a proper fraction.
A proper fraction has its numerator smaller than
its denominator.
E.g. ½ , 3/7, and 5/9 are proper fractions.
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ADDITION AND SUBTRACTION OF RATIONAL NUMBERS

• To add or subtract rational numbers, express the
  rational numbers as equivalent fractions in the
  same denominators first

• WORKED EXAMPLE 1
• 61/6 23/4, b) (-51/4) (-12/5) (-½)

• SOLUTION
• 61/6 23/4
• (5 2) (11/6 ¾)
• 3 (7/6 ¾)
• 3 (14/12 9/12)
• 3 5/12
• 35/12

• SOLUTION
• (-51/4) (-12/5) (-½)
• - (51/4 12/5 ½)
• - (55/20 48/20 10/20)
• - (563/20)
• - (5 33/20)
• -83/20

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MULTIPLICATION AND DIVISION OF RATIONAL NUMBERS

• To multiply two rational numbers
• a) Convert all mixed numbers to improper
  fractions first.
• b) Simplify the fractions first by crossing out
  the common factors of the numerators and
  denominators.
• c) Multiply the numerators, then the
  denominators.
• d) Reduce answer to its simplest form.

a/b x c/d a x c/b x d , where a, b, c, d are
integers and b ? 0, d ? 0

• To divide a rational number by another number
• a) Convert all mixed numbers to improper
  fractions first .
• b) Invert the second fraction by interchanging
  its numerator and denominator .
• c) Multiply the numerators, then the
  denominators .
• d) Reduce answer to its simplest form.

a/b c/d a/b x d/c a x d/b x c , where a,
b, c, d are integers and b ? 0, c ? 0, d ? 0
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MULTIPLICATION AND DIVISION OF RATIONAL NUMBERS

• WORKED EXAMPLE 1
• (-21/2) x 31/5, c) (1/5 1/3) (-1/4 x 2/9)
• b) (-2/11) (-10/33)

• SOLUTION
• (-21/2) x 31/5
• - 5/2 x 16/5
• -8

• SOLUTION
• (3/15 5/15) (-1/4 x 2/9)
• - 2/15 (- 1/18)
• - 2/15 x (- 18/1)
• 36/15
• 22/5

• (-2/11) (-10/33)
• - 2/11 x (-33/10)
• 3/5

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TECHNIQUE STRATEGIES IN SOLVING MATHEMATICS
WORD PROBLEM SUM 1/2 SHARING HOW TO GO ABOUT
TEACHING
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SYNOPSIS

• In solving math problem sum at secondary school
  level, it is widely acknowledged that heuristics
  strategies play a major role. By using suitable
  heuristics, it could greatly enhance pupils
  problem solving performance. Heuristics,
  referred to the method or strategies of achieving
  a solution to a given problem sum

Model Drawing is just one of the methods that can
be used. Of course there are also various
strategies that can be used.
The reason why Model drawing is used is because
it is one of the most common heuristics used to
solve word problems in Mathematics. It is
recognized internationally as an effective way
for young children to solve word problems and to
be exposed early to algebraic concepts.
Through-out this seminar, I will share with you
the various types of strategies used to solve
word problems sum.
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TECHNIQUES AND OTHER HEURISTICS STRATEGIES

• Guess Check method

• Making a Table

• Make a List (Listing method)

• Draw a Picture

• Find a Pattern

• Working Backwards

• Model Drawing

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COMMON DIFFICULTIES IN MATHEMATICAL PROBLEM
SOLVING

• Inability to read the problem

• Lack of comprehension of the problem posed

• Lack of strategy knowledge

• Inappropriate strategy used

• Inability to translate the problem into a
  mathematical form

• Computational error

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4 - Step in solving problem sum

• I dentify the problem
• (what is the questions exactly asking for?)

• D evise a plan
• (model method)

• E xecute the plan
• (work it out)

• A nswer check
• (number sense)

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4 - Step in solving problem sum

• I dentify the problem

After reading the problem sum, what is the
questions exactly asking for?

• D evise a plan

By drawing models, pupils can represent the
mathematical relationships in a problem
pictorially. This helps them understand the
problem and plan the steps for the solution
63
4 - Step in solving problem sum

• E xecuting the plan

In your plan, you might required to use one or
more of the strategies (heuristics) listed below
to help you solve the words problem sum.

• Guess and Check
• Making A Table
• Make a List
• Draw a Picture
• Find a Pattern
• Working Backwards
• Model Drawing

• A nswer check

Answer must be check to be able to satisfy the
condition of the question.
64
Strategy 5 Find a Pattern
Q1. A few children had to share a plate of
chicken wings. If each of them took 5 chicken
wings, there would be 4 left. In the end, they
decided to take 6 chicken wings each, leaving 1
chicken wing on the plate. How many children
shared the chicken wings? What was the original
number of chicken wings on the plate?
STEP 1 Draw a Table and determine the
information that needs to be found.
STEP 2 I find the patterns present in the data
to complete the table.
1 2 3 4 5
9 14 19 24 29
7 13 19 25 31
From the table, I can see that number 19
satisfies both conditions of the question.
Checking ( 3 x 5 ) 4 19 ( 3 x 6 ) 1 19
------- Correct
Ans There were 3 children sharing the chicken
wings. There were 19 chicken wings.
65
Strategy 5 Find a Pattern
Q2. Mr. Tom wanted to distribute his stamps
equally among a few of his students. If he were
to give each of them 5 stamps, he would have 4
left. If he gave each of them 6 stamps, he would
have 1 left. How many students and stamps did
he have?
STEP 1 Draw a Table and determine the
information that needs to be found.
STEP 2 I find the patterns present in the data
to complete the table.
1 2 3 4 5
9 14 19 24 29
7 13 19 25 31
From the table, I can see that number 19
satisfies both conditions of the question.
Checking ( 3 x 5 ) 4 19 ( 3 x 6 ) 1 19
------- Correct
Ans There were 3 students sharing the stamps.
There were 19 stamps.
66
Strategy 5 Find a Pattern
Q3. A Christmas tree was decorated with flashing
light bulbs. The red bulbs flashed every 2
minutes. The yellow bulbs flashed every 3
minutes and the blue bulbs flashed every 4
minutes. At 8pm, all the light flashed
simultaneously. Figure out the next time when
all the bulbs will flash together?
STEP 1 Draw a Table and determine the
information that needs to be found.
STEP 2 I find the patterns present in the data
to complete the table.
8.00 8.02 8.04 8.06 8.08 8.10 8.12
8.00 8.03 8.06 8.09 8.12 8.15 8.18
8.00 8.04 8.08 8.12 8.16 8.20 8.24
Looking at the pattern above, the starting time
is 8.00pm.
Next time the bulbs will flash together 8pm
12mins
Ans The next time all the bulbs will flash
together is at 8.12pm.
67
Strategy 6 Working Backwards
Q1. On my way to the shopping centre, I found
that I did not bring enough money for my
shopping. I then went to the bank to withdraw
100. Next , I bought a pair of shoes for 40.
Later, I paid for a T-shirt with half of the
money I had left. I was left with 65. How much
money did I have before I visited the bank to
withdraw the money?
To find out how much money I had at first, I have
to work backward by starting at the end and
undoing each step in reverse order.
You can draw a flow chart or an arrow to show
what happened.
Step 1 Step 2 Step 3 Amount I Withdrew
100 Spent 40 on a Spent half of
the Amount started with from bank pair of shoes
money on T-Shirt left
?
100
- 40
2
65
68
Strategy 6 Working Backwards
Q1. cont
Step 1 Step 2 Step 3 Amount I Withdrew
100 Spent 40 on a Spent half of
the Amount started with from bank pair of shoes
money on T-Shirt left
?
100
- 40
2
65
Amount I started with
Amount before Step 2
Amount before Step 3
Amount left
Ans I had 70 before I went to the bank to
withdraw the money.
69
Strategy 6 Working Backwards
Q2. Alice, Billy and John each bought some drink.
John poured his drink in a jug. Alice then
added 0.7l of drink into the jug. After that,
Billy added enough drink to double the amount in
the jug. All of them drank 1.2l of it, leaving
1.3l in the jug. How much drink did John bring ?
1.2l
2
- 0.7l
1.3l
2.5l
1.25l
0.55l
Amount left in the jug
Ans John brought 0.55 l of drink.
70
Strategy 6 Working Backwards
Q3. One evening, Lily baked some chocolate
cookies. She put 44 chocolate cookies in a bag
and packed another 24 in a tin. Then she divided
the remainder equally among herself and 2 of her
friends. She kept her share of 14 chocolate
cookies in a jar. How many chocolate cookies did
she bake that evening?
x 3
24
44
14
42
66
110
Amount left in the jar
Ans She baked 110 chocolate cookies.
71
UNDERSTANDING THE 8 DIFFERENTS MODEL. (MODEL
DRAWING)

• The model drawing/diagram is a very important
  strategy in secondary school mathematics. Using
  it correctly, a child will be able to solve many
  types of challenging problems sum easily.

The 8 different types of model drawing is very
useful as it can be used a Diagnostic Tool.
The trainers or the teachers can straight away
identified what kinds or types of problem sum a
child has instead of spending time figuring out
where is the weakness of the child.
Therefore with constant practice on the model
drawing it not only reinforce the understanding
of the questions it also develop skill and the
process thinking skill in solving word problems.
72
The 8 Different Models can be used in the
following types of problem sum

• ADDITION

• SUBTRACTION

• COMPARISION

• MULTIPLICATION

• DIVISION

• 1 STEP PROBLEM SUM

• 2 STEP PROBLEM SUM

• 3 STEP PROBLEM SUM

• CHALLENGING PROBLEM SUM INVOLVING BEFORE AND
  AFTER MODEL CONCEPT

73
Model Drawing 2
Q1. Jim and his brother share a sum of 150. If
Jim gets 50 more than his brother. How much
money do Jim and his brother get?
Step 1 Look out for KEY PERSON (REFERENCE
POINT) Key person or reference point has only 1
UNIT ( )
Step 2 In this case, the key person is his
brother.
Step 3 Draw the MODEL drawing of his brother
first as 1 unit.
1 Unit
Brother
150
Jim
50
74
Model Drawing 2
Q1. Cont
1 Unit
Brother
50
150
Jim
50
150 - 50 100
100 2 50 1 Unit ( ) /
Brother
Step 4 After finding your 1 unit, look back at
the MODEL DRAWING. The one that has 1 unit is
Brother.
Therefore, the brother get 50
Step 5 As for Jim, look at the model drawing it
consist of 1 unit 50
50 50 100
Ans Therefore, Jim gets 100 and his brother
gets 50.
75
Model Drawing 2
Q2. Peter and David share a total of 300
sweets. If David gets 60 more than Peter. How
many sweets do Peter and David get?
Step 1 Identified KEY PERSON that has only 1
UNIT ( ) CAN YOU FIGURE IT OUT?
Step 2 DRAW THE MODEL
1 Unit
Peter
120
300
David
60
300 60 240
240 2 120 1 unit ( ) / Peter
120 60 180 David
Ans Peter gets 120 sweets and David gets 180
sweets.
76
Hand-On Exercise 2
Q1. 400 is to be shared between Susan and
Mary. If Susan gets 20 more than Mary. How
much money do Susan and Mary get?
1 Unit
Mary
190
400
Susan
20
400 20 380
380 2 190 1 unit ( ) / Mary
Step 4 After finding your 1 unit, look back at
the MODEL DRAWING. The one that has 1 unit is
Mary.
Therefore, Mary gets 190
Step 5 As for Susan, look at the model drawing
it consist of 1 unit 20
190 20 210 Susan
Ans Therefore, Mary gets 190 and Susan gets
210.
77
Hand-On Exercise 2
Q2. Amy and John have 240 stickers altogether.
If Amy has 80 stickers more than John. How many
stickers does John have?
1 Unit
John
80
240
Amy
80
240 80 160
160 2 80 1 unit ( ) /
John
Step 4 After finding your 1 unit, look back at
the MODEL DRAWING. The one that has 1 unit is
John.
Therefore, John gets 80
Step 5 As for Amy, look at the model drawing
it consist of 1 unit 80
80 80 160 Amy
Ans Therefore, John gets 80 and Amy gets 160.
78
Model Drawing 3
Q1. Sharon and Janet share a total of 180
beads. If Janet gets 30 bead less than
Sharon. How many beads do Sharon and Janet get?
Step 1 Look out for KEY PERSON (REFERENCE
POINT) Key person or reference point has only 1
UNIT ( )
Step 2 In this case, the key person is Sharon.
Step 3 Draw the MODEL drawing of Sharon as 1
unit.
1 Unit
Sharon
105
180
Janet
30
180 30 210
210 2 105 1 unit ( ) /
Sharon
105 - 30 75 Janet
Ans Therefore, Sharon gets 105 and Janet gets 75.
79
Hand-On Exercise 3
Q1. 150 is shared between Judy and Susan. If
Susan gets 30 less than Judy. How much money do
Judy and Susan get?
1 Unit
Judy
90
150
Susan
30
150 30 180
180 2 90 1 unit ( ) /
Judy
90 - 30 60 Susan
Ans Therefore, Judy gets 90 and Susan gets 60.
80
Hand-On Exercise 3
Q2. David and John share a total of 360
stamps. If John gets 40 stamps less than
David. How many stamps do David and John get?
1 Unit
David
200
360
John
40
360 40 400
400 2 200 1 unit ( ) /
David
200 - 40 160 John
Ans Therefore, David gets 200 and John gets 160.
81
Model Drawing 4
Q1. 300 is to be shared between Jason and Kevin.
If Kevin gets twice as much as Jason. How much
money do Jason and Kevin get?
1 Unit
Jason
100
300
Kevin
300 3 100 1 unit ( ) / Jason
100 x 2 200 Kevin
Ans Jason and Kevin each get 100 200
respectively.
82
Model Drawing 4
Q2. Ken and Joseph share a sum of 250. If Ken
gets 4 times as much as Joseph. How much money
do Ken and Joseph get?
1 Unit
Joseph
50
250
Ken
250 5 50 1 unit ( ) / Joseph
50 x 4 200 Ken
Ans Joseph and Ken each get 50 200
respectively.
83
Model Drawing 4
Q3. Aaron has 4 times as many stamps as Jimmy.
If he has 24 stamps more than Jimmy. How many
stamps does Aaron have?
Step 1 Who is the KEY PERSON?
Step 2 Draw the model drawing of that person
first.
1 Unit
Jimmy
8
24
Aaron
24 3 8 1 unit ( ) / Jimmy
8 x 4 32 Aaron
Ans Jimmy and Aaron each get 8 32 stamps
respectively.
84
Model Drawing 4
Q4. Sarah has 50 more stickers than Jenny. If
Sarah has thrice as many stickers as Jenny. How
many stickers does Sarah have?
1 Unit
Jenny
25
50
Sarah
50 2 25 1 unit ( ) / Jenny
25 x 3 75 Sarah
Ans Jenny and Sarah each get 25 75 stickers
respectively.