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SECONDARY MATHEMATICS WORKSHOP

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

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SECONDARY MATHEMATICS WORKSHOP

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  1. SECONDARYMATHEMATICSWORKSHOP

  2. ENGLISH LANGUAGE ENGLISH LANGUAGE ENGLISH LANGUAGE LEARNERS IN THE MATHEMATICS CLASSROOM

  3. 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?

  4. 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

  5. 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?

  6. 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?

  7. 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?

  8. 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?

  9. 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 : “I’m 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

  10. KNOW YOUR KEY WORDS • MORE THAN • LESS THAN • ALTOGETHER • AT FIRST • SUM • DIFFERENT • COMPARE • DIGITS • FIND THE LENGTH /MASS • PLACE VALUE • WHOLE NUMBER

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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.

  16. 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.

  17. 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

  18. 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.

  19. 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.

  20. 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.

  21. 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’ 35 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.

  22. 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.

  23. 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.

  24. 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.

  25. 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.

  26. 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. √25 = 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 √ 4 = 2 32 = 9 and √ 9 = 3 42 = 16 and √16 = 4

  27. SQUARES AND SQUARE ROOTS WORKED EXAMPLE 1: Using prime factorisation, find the square root of 5184. SOLUTION: 5184 = 26 x 34 2 5184 √5184 = √26 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

  28. 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

  29. 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

  30. 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.

  31. REAL NUMBERS -5 -4 -3 -2 -1 0 1 2 3 4 5 Negative Integers Positive Integers • A number line showing integers is shown below: 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 > -3 We can also write -3 is smaller than 2 and is denoted by -3< 2

  32. REAL NUMBERS • >, <, > and < are called inequality signs. > means ‘is greater than’ < means ‘is smaller than’ > means ‘is greater than or equal to’ < 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

  33. ADDITION OF INTEGERS Rules for adding two integers:

  34. 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

  35. MULTIPLICATION OF INTEGERS Rules for multiplying integers:

  36. DIVISION OF INTEGERS Rules for dividing two integers:

  37. 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

  38. 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 = 28 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.

  39. 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

  40. 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

  41. 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

  42. 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.

  43. 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.

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

  45. 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

  46. 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

  47. 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)

  48. 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 – (a+b)/ac, given that a = 4, b = 2, c = -3. SOLUTION: 3a + 2b – 4c = 3(4) + 2(2) – 4(-3) = 12 + 4 + 12 = 28

  49. EVALUATION OF ALGEBRAIC EXPRESSIONS AND FORMULA SOLUTION: a(2b – c) – 3b2= 4[2(2) – (-3)] – 3(2)2 = 4(4+3) – 3(4) = 4(7) – 12 = 28 – 12 = 16 a/b – (a+b)/ac = 4/2 – (4+2)/4(-3) = 2 – (6/-12) = 2 + ½ = 2½

  50. 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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