Mathematics Form 2

MATHEMATICS - FORM 2

DIRECTED NUMBER

MULTIPLICATION AND DIVISION OF INTEGERS

(A) Multiplying integers

1. The product of multiplication between two integers with the same sign is positive. If two integers have different signs, the product of the multiplication is negative.

2. If zero is multiplied by any integer, the product is always zero. For example,

50 x 0 = 0

(B) Solving problems involving multiplication of integers

(C) Dividing integer

1. The quotient of division between two integers with the same sign is positive. If two integers have different signs, then the quotient of the division is negative.

2. If zero divided by any integer (except zero), the quotient is zero. For example :

0 ÷ 10 = 0

3. Any integer divided by zero is undefined. For example:

6 ÷ 0 = undefined

(D) Solving problems involving division of integers

COMBINED OPERATIONS ON INTEGERS

(A) Performing computations involving combined operations on integers

1. Combined operations involve more than one operation on integers are also known as mixed operations.

2. For combined operations involving' + ' and ' - ' or ' x ' and ' ÷ ', work the calculations from left to right following the order given.

(B) Solving problems involving combined operations on integers

POSITIVE AND NEGATIVE FRACTIONS

(A) Comparing and ordering fractions

(B) Computations Involving Addition, Subtraction, Multiplication and

Division of Two Fractions

The same rule used for adding and subtracting integers is used in the addition and subtraction of positive and negative fractions.

Multiplication of positive and negative fractions is similar to the multiplication of integers.

POSITIVE AND NEGATIVE DECIMALS

(A) Comparing and ordering decimals

(B) Adding and Subtracting Decimals

The same rule used in adding and subtracting integers is used in the addition and subtraction of positive decimal.

(C) Multiplying and Dividing Decimals

The same rule used in multiplying and dividing integers is used in the multiplication and division of positive and negative decimals.

COMPUTATIONS INVOLVING DIRECTED NUMBERS(INTEGERS, FRACTIONS AND DECIMALS)

(A) Addition, Subtraction, Multiplication and Division Involving Two

Directed Numbers

Calculations involving two directed numbers with any operations are the same as that for integers, fractions and decimals.

(B) Combined Operations of Addition, Subtration, Multiplication and Division of Positive and Negative Integers.

For computations involving the use of brackets and combined operations of addition, subtraction, multiplication and division on positive and negative numbers, calculate according to the following order.

1. The brackets ( )

2. 'of' (which means x)

3. ÷ and - (from left to right)

4. + and - (from left to right)

(C) Combined Operations of Addition, Subtraction, Multiplication and

Division of Positive and Negative Fractions

(d) Combined Operations on Positive and Negative Decimals

(e) Combined Operations on Directed Numbers

(f) Solving Problems Involving Directed Numbers


SQUARES, SQUAREROOTS, CUBES AND CUBEROOTS

SQUARES OF NUMBERS

(A) Stating the square of a number

1. The square of number is a product of the number multiplied by itself.

2. The square of a number can be written either in the expanded form or the index form (with the square notation ( 2 )).

For example :

3. The square of any number is always positive.For example:

4. The square of a number can be read in several ways.For example : 242 is read as (i) twenty-four to the power of two (ii) twenty-four squared (iii) the square of twenty-four

5. a² is an index form for a x a and is not equal to a x 2.For example :

(B) Determining the squares of numbers without using a calculator

1. The value of the square of a number can be obtained by multiplying the number by itself.For example,

2. The number of decimal places (d.p) in the square of a decimals is twice the number of decimal places of the decimal being squared.For example :

3. A mixed number has to be changed to an improper fraction before squaring.

For example :

(C) Estimating the squares of numbers

There are two methods that can be used to estimate the squares of numbers.

Method 1: By approximation

Round off the number to the nearest whole number or to one decimal point that can be easily squared.For example:

Method 2: By determaining the range

The square of a number can be estimated by stating it between two values that can be easily squared.

(D) Determining the squares of numbers using a calculator

1. To determine the square of a number by using a calculator, press the number keys followed by the X2 key on the calculator.

2. Different models of calculators have different buttons for squaring a number.For example:

3. By pressing a number, then Xfollowed by pressing the number again, we can also determine the square of the number.For example:

4. There are other function keys that will be used to determine the squares of the directed numbers.

5. For square values that exceed 10 digits, the reading on a calculator is an approximation.

(E) Listing perfect squares

Perfect square is the product of a whole number multiplied by itself.

(F) Determining if a number is a perfect square

1. A number is a perfect square if it is a product of the same two whole numbers.

2. To determine whether a number is a perfect square, find all its prime factors and then pair up all the same factors.

3. Perfect squares are whole numbers. Thus, decimals and fractions are not perfect squares.

(G) Solving problems involving squares of numbers

SQUARE ROOTS OF NUMBERS

(A) Writing the Square Roots of a Positive Number

(B) Determining the Square Roots of Perfect Squares Without Using a Calculator

Finding the square root of a number is the inverse process (opposite) of squaring.

(C) Determining the Square Roots of Positive Numbers Without Using a Calculator

The square root of a fraction is computed by finding the square root of the numerator and the denominator.Example :

Some fractions are reduced to lowest terms in order to find their square roots.Example :

To find the square root of a mixed number, change it to an improper fraction first.

Example :

The square root of a decimal can be found by changing the decimal to a fraction and then finding the square root of its numerator and denominator.Example :

(D) Multiplying Two Square Roots

The product of two square roots of the same number will produce the number itself.

Example :

The product of two square roots of two different numbers can be found by multiplying the two numbers first and then finding its square root.

Example :

(E) Estimating the Square Roots of Numbers

Method 1 : Estimating to the Nearest Value

Round off the given number to the nearest whole number from which the square root can be found easily.

Method 2 : Stating the Square Root of a Number Between Two Values

Determining the range of the square root of a number

The square root of a number can be estimated between two values (the range), from which the square root can be found easily.

(F) Finding the Square Roots of Numbers Using a Scientific Calculator

(G) Solving Problems Involving Squares and Square Roots

CUBES OF NUMBERS

(A) Stating the cube of a number

1. The cube of a number is the product of the number multiplied by itself twice.

2. The symbol of cube is ' ³ '. The cube can be represented in two ways, that are the expanded form and the index form.For example :

3. The cube of numbers can be read in several ways.For example :

5 ³ is read as (i) five cubed (ii) five to the power of three (iii) the cube of five

(B) Determining cubes of numbers without using a calculator

1. The value of a cubed number can be obtained by multiplying the given number by itself twice. For example :

(- 6)³ = - 6 x ( - 6 ) x ( - 6 ) = 36 x ( - 6 ) = - 216

2. The cube value of a positve number always positive.

3. The cube value of a negative number always negative.

(C) Estimating the cubes of numbers

Like the square and square root, we can estimate the cube of a number by approximation or by determining the range where the value of the cube lies.

(D) Determining the cubes of numbers using a calculator

To obtain the cube of a given number using a calculator, press the given number followed by :

(E) Solving problems involving cubes of numbers

CUBE ROOTS OF NUMBERS

(A) Writing the Cube Root of a Number

The cube root of any given number is the number that, when multiplied by itself twice, is equal to the given number.

(B) Estimating the Cube Roots of Numbers

Method 1: Estimating to the Nearest Value

Method 2: Stating the Cube Root of a Number Between Two Values Determining the range of the cube root of a number.

(C) Determining the Cube Roots of Integers Without Using a Calculator

(D) Determining the Cube Roots of Numbers Without Using a Calculator

(E) Determining the Cube Roots of Numbers Using a Calculator

(F) Solving Problems Involving Cubes and Cube Roots

(G) Computations Involving Addition, Subtraction, Multiplication, Division and Mixed Operations on Squares, Square Roots, Cubes and Cube Roots

When performing mixed operations on squares, square roots, cubes and cube roots, follow the order of operations as below.

(a) Carry out the operations in the brackets [( )] first.

(b) Next, multiply or divide; calculating from left to right.

(c) Finally, add or subtract; again calculating from left to right.


ALGEBRAIC EXPRESSIONS II

ALGEBRAIC TERMS IN TWO OR MORE UNKNOWNS

(A) Identifying unknowns in algebraic terms

1. An unknown is a quatity with a value which is not yet known.

2. Any symbol or letter of the alphabet can be used to represent an unknown quantity.

(B) Identifying algebraic terms as the product of a number with the unknowns

1. An algebraic term in two or more unknowns is the product of a number and the unknowns.

2. The term an means a multiplied by itself n times (power of n).

3. In general, the number in the algebraic terms is written first before the unknowns.

For example :An algebraic term is written as 5ab and not ab5.

4. We usually write the unknowns in alphabetical order.

For example :4acb² is written as 4 ab²c

(C) Identifying coefficients of given algebraic terms

The coefficient of the unknown(s) in an algebraic term is the other factors of the term.

(D) Identifying like and unlike algebraic terms

(E) Stating like terms for a given algebraic terms

The state like terms for a given algebraic term, substitute the numerical coefficient of the term with another number.

MULTIPLICATION AND DIVISION OF TWO OR MORE ALGERAIC TERMS

(A) Finding the Product of Two Algebraic Terms

To find the product of two algebraic terms, multiply the numbers with the numbers and the unknowns with the unknowns.

(B) Finding the Product of Two Algebraic Terms

To find the quotient of two algebraic terms, write the division process as a fraction. Then, simplify the numbers and unknowns in the

numerator and denominator using the cancellation method.

(C) Multiplication and division involving algebraic terms

CONCEPT OF ALGEBRAIC EXPRESSIONS

(A) Writing algebraic expressions for given situation using letter symbols

1. An algebraic expression contains one or more algebraic terms that are separated by the ' + ' and ' - ' signs.

2. An algebraic expression can be written based on given situations by using letters.

(B) Recognizing algebraic expressions in two or more unknowns

If two or more algebraic terms are combined by plus or minus sign, then an algebraic expression is obtained.

For example :

3b + 7xy is an algebraic expression with three unknown, b, x and y.

(C) Determining the number of terms in given algebraic expressions

(D) Simplifying algebraic expressions by grouping like term

1. For an algebraic expression with like and unlike terms, group the like terms and the unlike terms separately. The

operation signs must follow the terms during the grouping process. Then, add or subtract the coefficients of the like terms.

2. Terms with different unknowns cannot be simplified.

(E) Evaluating expressions by substituting numbers for letters

COMPUTATIONS INVOLVING ALGEBRAIC EXPRESSIONS

(A) Multiplying and dividing algebraic expressions by a number

Multiply or divide each term in an expression by the given number.

(B) Adding and subtracting two algebraic expressions

1. Simplify the algebraic expression to find the sum or difference of two algebraic expressions using the following

steps.

(a) Firstly, remove brackets and group the like terms.

(b) Then, determine the sum or difference of the like terms.

2. If there is a ' - ' sign in front of the bracket, change all the signs within the bracket

when removing the bracket.

(C) Simplying algebraic expressions


LINEAR EQUATIONS

EQUALITY

(A) Stating the relationship between two quantities by using the symbols ' = ' or ' ≠ '

1. Equality is the relationship between two quantities with the same value.

2. The symbol ' = ' read as 'is equal to', is used when the two quantities are equal. For example :

1 cm = 10 mm 4 - 2 = 5- 3

3. For two quantities with different values, the symbol used is ' ≠ ', which is read as 'is

not equal to'. For example :

4 x 3 ≠ 4 x 2

RM1 ≠ 10 sen

4. An equality has two characteristics :

(a) Reflective characteristic If a = b, then b = a For example : 10 - 3 = 2 + 5 Thus, 2 + 5 = 10 - 3

(b) Transitive characteristic If a = b and b = c, then a = c For example : 2 x 8 = 16 and 16 = 4 x 4 Thus, 2 x 8 = 4 x 4

LINEAR EQUATIONS IN ONE UNKNOWN

(A) Recognizing linear algebraic terms

1. A linear algebraic term is a term with only one unknown raised to the power of 1.For example :

2. If a term has more than one unknown or/and the power of the unknown(s) is greater than one, then, the term is a non-linear algebraic term.For example :

(B) Recognizing linear algebraic expressions

A linear algebraic expression has one or more linear algebraic terms that are linked by plus or minus sign or both.

(C) Determining if a given equation is a linear equation or a linear

equation in one unknown

1. An equation must have the symbol ' = '.

2. A linear equation is an equation with one or more linear algebraic terms.For example :

(a) 3k + 4s = 10 is a linear equation.(b) 2ab = 5 and c2 - 6 = 1 are not linear equation.

3. A linear equation in one unknown is a linear equation that consists of only one unknown.For example :

(D) Writing linear equations in one unknown for given statements and vice versa

1. To solve a problem, we can write the given information as an equation and then solve it.

SOLUTIONS OF LINEAR EQUATIONS WITH ONE UNKNOWN

(A) Determining if a Numerical Value is a Solution of a Given Linear Equation with One Unknown

Solving a linear equation is the process of finding the value of the unknown itn the

equation that satisfies or makes the equation true.

If we add, subtract, multiply or divide one side of the equation by a number, we must do likewise on the other side of the of the equation.

The number that satisfies the eqution is called the solution or root of the equation.

A linear equation with one unknown has only one solution.

(B) Determining the Solution of a Linear Equation with One Unknown by Trial and Improvement Method

The trial and improvement method in determining the solution of a linear equation with one unknown is to quess at a solution and then test it in the equation. If the value satisfies the equation, then it is the solution for the equation.

(C) Solving Equations in the Form of x + a = b , where a and b are Integers and x is an Unknown

(D) Solving Equations in the Form of x - a = b, where a and b are Integers and x is an Unknown

(E) Solving Equations in the Form of ax = b, where a and b are Integers andx is an Unknown

(F) Solving Equations in the Form of x = b, where a and b are Integers and x is an Unknown. a

(G) Solving Equations in the Form of ax + b = c, where a, b and c are Integers and x is an Unknown

(H) Solving Linear Equations with One Unknown

(I) Solving Problems Involving Linear Equations with One Unknown

When solving a problem involving equation with one unknown,(a) form a linear equation from the statements and information given.(b) solve the equation formed.


RATIOS, RATES ANDPROPORTIONS

RATIOS OF TWO QUANTITIES

(A) Comparing two quatities in the form a : b or a b

1. A ratio of two quatities is a comparison between two quantities of the same units. It

is also a fraction showing the first quantity as part of the second quantity.

2. A ratio of two quantities can be written as a : b or a (b ≠ 0). a : b is read as 'a is to b'. b

3. The order in which the ration is written is of great important. Note that a ≠ b, therefore b a a : b ≠ b : a.

4. A ratio has no unit. It is only a number which indicates how many times one quatity is greater than the other.

5. The units need to be standardized for quantities of different units of measurement, before they are expressed as a ration.

(B) Determining whether given ratios are equivalent

1. By multiplying or dividing each part of a ratio with the same common factor, we can obtain the equivalent ratio.

2. A ratio of two quatities remains the same if the two quantities of the ratio are both multiplied of divided by the same number.

(C) Simplifying ratios to the lowest term

1. The ratio a : b is in the lowest term if a and b are whole numbers with no other common factor except 1.

2. A simplified ratio can be obtained by multiplying or dividing both quantities of the ratio with the same number.

(D) Stating ratios related to a given ratio

When a ratio is given, other ratios related to the given ratio can be written accordingly.

In general, if X : Y = a : b, with a > v, then the ratio

(a) X is to the sum of X and Y

= a : a + b

(b) Y is to the sum of X and Y the ratio

= b : a + b

(c) X is to the difference between X and Y

= a : a - b

(d) Y is to the difference of X and Y

= b : a - b

PROPORTIONS

(A) Stating Whether Two Pairs of Quantities is a Proportion

A proportion is an equality of ratios between two pairs of quantities. It is a statement with a ratio on each side stating that the two ratios are equal.

(B) Determining if a Quantity is Proportional to Another Quantity Given Two Values of Each Quantity

We can determine whether a quantity is proportional to another quantity when we are given two values of each quantity, by finding whether the ratios of the given quantities are equivalent.

(C) Finding the Value of a Quantity Given the Ratio of the Two Quantities and the Value of Another Quantity

Method 1 : Unitary Method

The solving of problems involving proportions by working out the vaue of one unit is called the Unitary Method.

Method 2 : Cross-Multiplication

Problems involving proportions can be solved using the cross-multiplication method by following the steps given below.

(a) Form the ratios of the two quantities from the given information.(b) Write an equation of proportion according to the information given.(c) Solve the equation of proportion by cross-multiplication.

(D) Finding the Value of a Quantity Given the Ratio of the Sum of the Two Quantities

(F) Finding the Sum of Two Quantity Given the Ratio of the Quantities and the Difference Between the Quantities

(G) Solving Problems Involving Ratios and Proportions

RATIOS OF THREE QUANTITIES

(A) Comparing three quantities in the form a : b : c

1. A ratio of three quantities is a comparison between three quantities that have the same unit.

2. The ratio of three quantities is qritten as a : b : c and is read as ' a is to b is to c '. For example :

(a) The ratio of 5 cm to 15 cm to 17 cm

= 5 : 15 : 17

(b) The ratio of 7 mm to 9 mm to 1.5 cm

= 7 : 9 : 15

(B) Determining whether the given ratios are equivalent ratios

Multiply of divide the three quantities in the given ratio by the same factor to determine whether the two ratios of three quantities are equivalent ratios.

(C) Simplifying a ratio of three quantities to the lowest term

(D) Stating the ratio of any two quantities given the ratio of three quantities

1. The ratio of any two quantities can be identified if the ratio of three quantities is given. In general, if S : T : U = x : y : x, then

(a) S : T = x : y (b) S : U = x : z(c) U : T = z : y(d) S + T : T + U = x + y : y + z

(E) Finding the ratio of a : b : c given the ratio of a : b and b : c

To write a : b : c if a : b : c are given, meke sure that the value of b is the same by using the lowest common multiple (LCM).

(F) Finding the value of the other quantities, given the ratio of three quantities and the value of one of the quantities

(G) Finding the values of each of the three quantities, given the ratio and the sum of three quantities

(H) Finding the values of each of the three quantities, given the ratio and the difference between two of the three quantities

(I) Finding the sum of three quantities, given the ratio and the difference between two of the three quantities

(J) Solving problems involving the ratio of three quantities

The ratio of the number of blue pens to the number of red pens is 3 : 5. The total number of pens is 184. Calculate the difference between the number of red and blue pens.

Let the number of blue pens and red pens be b and r respectively.

Thus, the difference between the number of red and blue pens

= 184 - 69 = 115

Let the number of blue pens and red pens be b and r respectively. The difference between the number of red and blue pens is r - b.

If p : q = 4 : 5, find the value of p when q = 20.

The ratio of the number of oranges in basket A to the number of oranges in basket B to the number of oranges in basket C is 8 : 2 : 3. The total number of oranges in basket A and C is 27 more than the number of oranges in basket B. Find the total number of oranges in these three baskets.

Let the number of oranges in baskets A, B and C be a, b and c respectively.

a : b : c

8 : 2 : 3

Let the total number of oranges in the three baskets be x.

x = a + b + c = 8 + 2 + 3 = 13

By proportion,

Therefore, the total number of oranges in the three baskets is 39.

Yana, Freda and Lau invested in a food business in the ratio 2 : 5 : 8. The total amount invested by Yana and Freda was RM4 200. Find the amount of money invested by Lau.

Let the amount of money invested by Yana, Freda and Lau be C, S and H, respectively.C : S : H = 2 : 5 : 8

Let the amount of money invested by Yana, Freda and Lau be C, S and H, respectively.C : S : H = 2 : 5 : 8

The total amount invested by Yana and Freda = RM4 200i.e. C + S = 4 200

By proportion,

Therefore, the amount of money invested by Lau is RM4 800.


PYTHAGORAS'

THEOREM

SIDES OF A RIGHT-ANGLED TRIANGLE

(A) Identifying the Hypotenuse of a Right-angled Triangle

In a right-angled triangle, the hypotenuse is the side that is (a) opposite the right angle.(b) the longest among the three sides in the triangle.

Example :

(B) Relationship Between the Lengths of the Sides of a Right-angled Triangle

Pythagoras' Theorem states that in a right-angle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The relationship between the length of sides of the right-angled triangle in the Diagram 1 are:(a) c² = a² + b²(b) a² = c²- b²(c) b² = c² - a²

(C) Finding the Length of the Missing Side of a Right-angled Triangle Using Pythagoras' Theorem

The length of a side of a right-angled triangle can be calculated by using Pythagoras' Theorem when the length of the other two sides are known or given.

Pythagorean Triples are three whole numbers that form the sides of a right-angled triangle. Here are some of the Pythagorean Triples :

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (6,8, 10), (10, 24, 26), (9, 40, 41), (20, 21, 29), (9, 12, 15).

(D) Finding the Length of the Sides of Combined Geometric Shapes Using Pythagoras' Theorem

In the Diagram 2, PQR and QRS are rright-angled triangles. Find the length of QS, correct to two decimal places.

(E) Solving Problems Using Pythagoras' Theorem

Diagram 3 shows an outdoor camp. The entrance of the tent is built in a shape of an isosceles triangle PQR. Calculate the length of the slant height, correct to two decimal places.

THE CONVERSE OF PYTHAGORAS' THEOREM

(A) Determining whether a triangle is a right-angled triangle

1. The converse of Pythagoras' theorem states that in a triangle, if the square of the longest side is equal to the sum of the squares of the other two sides, then the angle

which faces the longest side is a right angle.

2. The converse of Pythaforas' theorem can be used to determine whether a triangle is a right-angled triangle.

3. Basically, there are two other type of angles besides the right angle. They are obtuse angle and acute angle.

(B) Solving problems involving the converse of Pythagoras'

theorem

The Pythagoras' theorem is usually applied in topics on trigonometry, perimeters, areas and similar triangles.

A cardboard is cut into a triangle shape with sides measuring 9 cm, 12 cm and 15 cm. Is the cardboard a right-angled triangle?

The longest side = 15 cm

15² = 225

The sum of the squares of two other sides

= 9² + 12² = 81 + 144 = 225 15² = 9² + 12²

The cardboard is a right-angled triangle.

Diagram 1 shows a cross-section of a pipe with a radius of 13 cm. The level of water in the pipe increases from JK to LM so that JK = LM = 10 cm. Find the increase in water level in the pipe.

Therefore, the increase in water lavel = 13.93 cm + 13.93 cm = 27.86 cm

Let the increase in water level be POQ.

In the diagram above, KLM is a straight line. Calculate the length, in m, of KM.

In the diagram above, PQRS is a trapezium. Find the area of the shaded region in cm².

Area of the shaded region = 115 cm² - 30 cm² = 85 cm²

Mathematics Form 2

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Transcript of Mathematics Form 2