Very often two or more numbers are grouped together to indicate that they are to be considered together as a single number. To avoid confusion, grouping, grouping symbols are used.
The grouping symbols are:
() parenthesis [] brackets
{} braces ─ vinculum
Rules:
1. Simplify expression inside parentheses and other grouping symbols first.
2. Do multiplication and division next, in the order they occur that is, from left to
right.
3. Do addition and subtraction last, in the order they occur, that is from left t
right.
In short, you can remember this by using the mnemonic “Please Mind Dear Aunt Sally’. The first letters will remind you of “Parentheses, Multiplication and Division, Addition and Subtraction.’ (PMDAS)
Example:
Simplify 16 x 2 ÷ 8 + 12 x 3 – 48 ÷ 6 x 3 = 32
EXPONENTS AND POWERS
An exponent is the number that indicates how many times a whole number is used as a factor.
Example:
25 = 2 x 2 x 2 x 2
x 2
FRACTIONS
The symbol a/b, where a,b are the elements of whole nos. and b ≠ 0, is called a fraction.
The number above the bar is the numerator and the number below the bar is the denominator. The word fraction if derived from the latin word fraction, meaning’ to break into parts.
Kind of Fractions:
1. Proper Fractions – is a fraction in which the numerator is less tan the denominator.
Example: 1/3, 2/3, 15/28, 105/201
2. Improper fraction – A fraction in which the numerator is greater than the denominator.
Example: 5/2, 25/8, 128/121
3. Mixed Fractions – is composed of a whole number and a fraction.
Example: 2 1/3, 3 2/5, 15 16/17, 100 25/31
4. Equivalent Fraction – fractions that show the same value.
Example: 2/5 = 8/20, 16/20 = 32/40, 101/120 = 303/360
Addition and Subtraction
Adding and Subtracting Fractions with similar or Like Denominators
To add or subtract fractions with similar or like denominators, add or subtract the numerators. Write the sum or difference over the common denominator.
Examples:
1. 14/27 + 11/27 = 25/27 2. 7/18 , 4/18 = 3/18 = 1/6
Adding and Subtracting Fractions with Dissimilar or Unlike Denominators
1. Find the least common denominator (LCD)
2. Change each fraction to equivalent fraction using LCD.
3. Add or subtract the numerators and write the result over the LCD
4. If necessary, reduce the answer to lowest terms.
Example: Calculate 4 ¼ + 2 4/5 – 1 1/3
4 ¼ = 4 15/60
+ 2 4/5 = +2 48/60
- 1 1/3 + -1 20/60
_______________________
5 43/60
Multiplication and
Division
Rules for Multiplying Fractions
To multiply two fractions, multiply the numerators and
multiply the denominators. Write the product of the numerators over the product
of the denominators. If necessary, reduce the answer to lowest terms.
Example: Find the product
a. 2/9 x 5/8 = 5/36
b. 3/5 of 5 = ¾ x 5 = 15/14 = 3 ¾
c. (2/3) 3 = 2/3 x 2/3 x 2/3 = 8/27
d. 49/10 x 2/3 x 15/77 = 49/10
x 2/3 x 15/17 = 7/11
Rules for Multiplying Mixed Numbers
1. Change each mixed number to an improper fraction
2. Multiply the numerators.
3. Place the result over the product of the denominators.
4. Express the answer as a mixed number or as a proper fraction
reduced to lowest terms.
Example: Simplify 1 ½ x 1/ 7/9= 3/2 x 16/9
3/2 x 1 6/9 = 8/3 = 2 2/3
Rules for Dividing Fractions
If a, b, c and d are all real numbers and if b≠ 0, c≠0, d≠-, then a/b ÷ c/d = a/b x d/c
Example: Divide: 3 3/8 by 2 ¼
= 27/8 ÷ 9/4= 27/8 x 4/9 =
27/8 x 4/9= 3/2 = 1 ½
DECIMALS
A decimal fraction is a
fraction whose denominator can be expressed as a power
of ten
Examples
(1) 6/10 = 6/101 = .6 (2.) 16/100 = 16/102= .16 (3.) 425/1000 = 425/ 103 =
.425
Operations on Decimals To add Decimals
1. Write the numbers to be added vertically and line up the decimal points.
2. Add all digits with the same place value, beginning with the rightmost column.
3.
Be sure to place a decimal point in the sum in
the correct location Example: Add 6.47 + 340.8 + 73.523
To Subtract Decimals
1. Write the numbers to be subtracted vertically such that the decimal points are in a column.
2. If the number of decimal places in the subtrahend exceeds that in the minuend, insert the necessary number of zeros to the right of the last decimal place in the minuend.
3. Subtract all digits with the same place value, beginning with the rightmost column and regroup when necessary.
4.
Write the decimal point in the difference. This
should be in line with the other decimal points.
Example: Subtract 462 – 26.528
To Multiply Decimals
1. Calculate the product, in the same manner as whole numbers.
2.
Put a decimal point in the appropriate location
so that the number of decimals places in the product equals the sum of the
number of decimal places in all the factors.
Examples: Find the product of
43.7 and 0.00035
To Divide Decimals
1. Move the decimal point in the divisor up to the necessary number of places to the right to make it a whole number.
2. Move the decimal point in the dividend to the same number of places to the right.
3. Put a decimal point in the quotient immediately above the new decimal point in the dividend.
4.
Proceed with the technique for dividing whole
numbers, take note of the location of the decimal point in the quotient,
Example: Find the quotient of
0.02904 ÷ 0.04
INTEGERS
The set of integers consists of the positive whole numbers, negative whole numbers, and zero. { . . . . , -5, -4 , -3, -2 , -1 ,0 ,1 ,2 ,3 , 4 ,5 . . . }
Examples: +45 or 45 ‘Positive forty-five’
-57 ‘negative fifty-seven’
0 ‘Zero’
THE NUMBER LINE
Negative Integers Positive Integers
For any numbers a, the
opposite of a is denoted by –a.
Examples: The opposite of 100
is -100
The opposite of -99 is 99
6 stands for a gain of P6,
while -3 stands for a loss of P3
OPERATIONS ON INTEGERS
Addition
1. if the addends have the same signs, add the numbers disregarding the signs. The sign of the sum will be the common sign of the addends.
Example: +138 + + 200 = 338
If the addends are of different signs, subtract the smaller number from the larger number and write the sign of the larger number.
Example: - 285 + 100 = -185
Subtraction
1. In subtracting signed numbers, change the sign of the subtrahend then proceed to addition of integers.
Example
-10,348
-
- 9,753
___________________
-595
Multiplication
1. When two numbers of the same signs are multiplied, the product is positive.
2.
When two numbers of opposite signs are
multiplied, the product is negative.
Example:
Multiply + 12 by + 4 = 48
Multiply -16 by 5 = -80
Division
1.
In dividing numbers of the same signs, their quotient
is a positive number.
2.
In dividing numbers of different signs, their
quotient is a negative number.
Examples: Find the quotient
-96 -84
a. _______= -32 b. _______= 12
3 -7
Cognitive Reasoning and Problem Solving
- Finding Patterns Objectives
- Recognize simple non-mathematical patterns
- Recognize number patterns
- Recognize patterns that will help gain mathematical maturity
Many discoveries in mathematics are based on patterns. A mathematician often uses the strategy of generalizing to form a rule based on a few examples.
The process of finding patterns is not quick or direct as following a recipe or list of instructions. It requires the concept of sequence.
A sequence is a set of numbers in a particular order. The numbers in a sequence are called terms of the sequence. If we have the sequence 1,3,5,7,9 . . . The first term in 1, the second term is 3, the third is 5 and so on.
Example
Given the following sequence of letters, A, C, D, F, G, I What are the next two letters?
Solution:
To discover the pattern for this problem, let us look at the first nine letters of the English alphabet, A B C D E F G H I, Let us cross out those letters that are not included in the sequence, A B C D E F G H I. Following this pattern, J should come after I. Then we should skip K and write L. A B C D E F G H I J K L. Hence, the two succeeding letters are J and I
From the examples shown, it is seen that patterns involve not only numbers, but also non-mathematical objects like letters, words, and figures.
Reasoning based on patterns is sometimes called cognitive reasoning. Students often use cognitive reasoning when they answer tests. Employees also often use this type of reasoning in the performance of their jobs. All of us, in one way or another, use cognitive reasoning in our daily lives.
A collection of numbers arranged in order from left to right,
such that there is a first term, second term, etc, and are separated by commas,
is called a number sequence.
Example:
What are the next three terms of this sequence?
1, 1, 2, 4, 8, 16, 32, 64
Solution:
To describe a sequence, we
often try to find a pattern that relates the number of a term to the term
itself. Now for the sequence 1, 1, 2, 4,8 , 16, 32 ,64, the pattern is not
obvious. First, let us look at the relationship between the terms. The first
term, I and the second term, 1, are equal but the third term is 2, which may be
treated as the sum of the forst two terms.
Considering the sums of the terms, we have
1 + 1 = 2
And if we add the first, the
second, and the third terms, we have 1 + 1 + 2 = 4, which is fourth term.
Continuing the pattern
1 + 1 + 2 + 4= 8, fifth term
1 + 1 + 2 + 4 + 8 = 16, sixth term
1 + 1 + 2 + 4 + 8 + 16 = 32, seventh term
1 + 1 + 2+ 4 + 8 + 16 + 32 =
64,eight term Hence for the next three terms,
64 + 64 = 128 ninth term
128 + 128 =256 10th term
256 + 256 = 512, eleventh term
READING FOR UNDERSTANDING AND ANALYSIS
Objectives
Analyze the problem to be
solved step by step with the aid of a picture or a diagram. Sometimes a math
problem looks more difficult than it really is. You must dirst understand the
problem before you attempt to solve it. It often helps to rewrite the problem
using only the important information. This makes it easier to figure out what
you must do.
Guideline for comprehension
1. Jot down the key words or phrases.
2. Write down what is being asked for.
3. Restate what you have read in your own words.
4. Prepare a visual representation
Example Given the numbers 6, 2
,3 ,1 ,4 and 5. If the second number is less then the fourth circle the number
that is the difference between the first numbers, circle the second number.
Solution:
To do this, we have the following:
1. The key word is less than.
2. We are to circle a number in 6 2 3 1 4 5.
3. If the second number is less than the fourth number, then circle
the number that is the difference between the first and the sixth numbers.
Circle the second number if the third is less than the first. 5th no.