TheMathBooklets: Primary Maths : Notes and Formulae

WHOLE NUMBERS

Place Values

Value is the positional value of the digit in the number or where the digit is in the number. Example: tens, thousands

Digits are symbols or characters to represent a number in writing.

Example: What is the value of the digit 4 in 854013?

854 013 Step1: Underline 4

4000 Step2: Replace number to right of 4 to 0

Odd number

Numbers ending with 1, 3, 5, 7, 9 Examples: 7, 15, 129

Even number

Numbers ending with 0, 2, 4, 6, 8 Examples: 2, 10, 1356

Rounding

Rule for rounding off

If number is from 0 and to 4, then DROP or Fall (4) off / let it go

Example: Round 193 to the nearest 10. 3 is less than 4, => 190

If number is from 5 to 9, then ADD 1 or high 5 --> go higher

Example: Round 195 to the nearest 10. 5 is more than 4, => (19+1=20), 200

Estimate

Rounding numbers is a way of estimating numbers

Dividing Whole Numbers By Tens, Hundreds, And Thousands

When any whole number is DIVIDE by 10, 100, or 1000, ‘cancel’ the corresponding 0s

Example: 500000/100 = 500000/100 = 5000

Quotient And Remainder

There are special names for each number in a division.

12 ˜ 4 = 3

dividend ˜ divisor = quotient

Factors

Factors are numbers that divide exactly into another number.

The factors of 12, for example, are 1, 2, 3, 4, 6 and 12.

Multiples

Multiples are extended times tables

The multiples of 2 are all the numbers in the 2 times table:
2, 4, 6, 8, 10, 12 ... The fifth multiple is 10

Order of Operations

FIRST Order : ( ) /P² Bracket u² = u x u

SECOND Order : ÷ x Division and Multiplication

THIRD Order : + - Addition and Subtraction

Measure and Compare

Smallest, Largest

Smallest: of the least value as compared to others

Largest: of the biggest value as compared to others

Vocabulary

Naming Position/Words

First (1^st), second (2^nd), third (3^rd) , Fourth (4^th), … tenth(10^th)

Between, odd number, even number, left, altogether, twice, thrice

Equal/Add/Subtract/Multiple/Divide

Equal: value of, same as, answer, represents, means, will be, whole lot

Add: more than, total, sum of, increase, plus, additional, and

Minus: take, decrease, subtract, remove, subtraction, take away, pull, difference, less than, take

Multiplication: multiply, times, lots of, by, product of, of, power of, bracket () = square

Division: the fraction line -, divide, /, split, group, out of, give

Compare and Order

(1) Great/greater/greatest; as great as, greater than

(2) Small/smaller/smallest; as small as, smaller than

(3) Many/More/Most; as many as, more than

(4) Little/Less/Least; as little as, less than

(5) Few/Fewer/Fewest; as few as, fewer than

(6) Long/Longer/Longest; as long as, longer than

(7) Short/Shorter/Shortest; as short as, shorter than

(8) Tall/Taller/Tallest, taller than, as tall as

(9) Heavy/Heavier/Heaviest; as heavy as, heavier than

(10) Light/Lighter/Lightest; as light as; lighter than

(11) Decreasing / Increasing

Additional Words

Exchange, Transfer, Age, Equal at the start, Finally, Increase by, Altogether

FRACTIONS

Numerator -> part

Denominator -> whole

Proper Fractions

1 , 3 , 6 numerator is less/smaller than

3 7 9 denominator

Improper Fractions

3 , 8 , 22 numerator is more/bigger than

2 7 9 denominator

Mixed Number

8 ½ A whole number with a proper fraction

Lowest Term

1/2, 3/5, 6/7 Cannot be reduced anymore

Equivalent Fractions

Representing the same number/fraction.

-> 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8

3 6 9 12 15 18 21 24

Add/Subtract Like Fractions (same denominators)

- Add or subtract the numerators

- Do not add/subtract the denominator (it indicates the total portion)

Example 3 + 4 = 7 5 - 2 = 3

9 9 9 7 7 7

Add/subtract Unlike Fractions (Different denominators)

Example

2 + 1 = 2 x 3 + 1 x 5

5 3 5 x 3 3 x 5 Step1: Convert to same value (equivalent)

= 6 + 5 = 11 Step2: Add or subtract the NUMERATOR

15 15 15

Multiplying fractions

What is ⅖ of ¾? <of  Multiply>

2 x 3

5 8 4 (1)2 is common for 2 and 8

= 1 x 3 = 3

5 4 20

Dividing Fractions

a ÷ c 1 ÷ 9 Step1: Invert the fraction after ÷ sign

b d 2 4

= a x d 1 x 4 2 = 2 Step2 : reduce to the lowest term

b c 2 9 9

<<Convert mixed number to improper fraction before division / multiplication>>

Compare Fractions

By Same Denominators

4/7 , 1/7 , 3/7 => 1/7 , 3/7, 4/7 Smallest numerator (smallest), biggest numerator (largest)

By Same numerators

9/4 , 9/7 ,9/13 => 9/4 , 9/7, 9/13 Smallest denominator (largest), biggest numerator (smallest)

Different numerators/denominators - convert to either same numerator or denominator and compare

Vocabulary
halves, quarters, fifths, tenths, whole, part

DECIMALS

In the number 17.3, the point separates 17 from the 3.

3 is in the tenth position.

=> 3 of tenths = 3 of 1/10 = 3/10

As a fraction, 17.3 = 17 3/10

Decimal Numbers Multiply/divide By 10s, 100s, 1000s

- MULTIPLY : the decimal value shifts by number of 0s to the RIGHT.

- DIVIDE : the decimal value shifts by number of 0s to the LEFT.

Vocabulary

Money

(1) Expensive/More Expensive/Most Expensive; as expensive as

(2) Cheap/Cheaper/Cheapest; as cheap as, cheaper than

(3) Mary receives a Change of $20.

(4) Judy’s saving is 80cents.

PERCENTAGE

"Percent %" means 'out of 100' or ‘per 100 of the total value’.

Converting Fraction and Decimal to Percent (Multiply by 100)

Example: Express 1/2 as percentage

½ x 100 = 50 Step 1: Multiply by 100

1/2 as percent is 50% Step 2: Express the fraction as percent

Example: Convert 0.345 to percent

0.345 = 0.0345 x 100% = 34.5%. Step: Move Decimal point 2 places to the right

Converting % To Fraction and Decimal (Divide by 100)

Example: Percent can be expressed as #/100

40% is 40/100 = 4/10 = 2/5

Formula

Percent = (Required) Value x 100%

Total Value

GST = Amount x GST rate

100

Increase/Discount = Original Price x Percent Discount/increase

RATIO
It is represented by TWO or more numbers with a colon:
Number1: Number2
The ratio’s position of the numbers is important -> 1:2 is not 2:1
Ratio is comparison of numbers, there is no unit of measure.
Comparing more than 1 ratios
1 : 2 : 5
Equivalent Ratio
1 : 2 , 2 : 4, 4 : 8
Ratio in its simplest form
2 : 4 = 1 : 2 <= simplest form (similar to lowest term in fractions)

Ratio and Fractions

A fraction is a part of a whole.

SPEED

Distance(D) = Speed (S) x Time (T)

Speed = Distance / Time

Time = Distance / Speed

ALGEBRA

Simplifying Algebraic Expression

Example

Simplify 4f + 3g -2f +4g

4f +3g -2f +4g. Step 1 Underline with +/- on left

= 4f - 2f + 3g + 4g Step 2 GROUP the different shapes together

= 2f + 7g. Step 3 SIMPLIFY expression

Solving Equation

Example

Solve 12 = U + 5

L = R Step 1: WRITE L=R putting variable to left side

12 = U + 5

U + 5 = 12

U + 5 – 5 = 12 – 5 Step 2: Move all numbers to right

U = 12 - 5

U = 7 Step 3: Solve

Finding The Value Of An Equation

When the variable is given a value, replace the variables with the value.

Example

Find the value of the equation when A = 4

(1) 4 + A

= 4 + 4 (replace A with A = 4)

= 8

Number Patterns

Usually in the form of structured series of numbers in uniform changes

Position(Term) 1 2 3 4 5 .... N

2 , 5 , 8 , 11 , 14 .... nth term <Common pattern>

(2 = a(1st) +3 +3 +3 +3 (d= difference between 2 numbers =3)

Formula/Equation: Nth term = a + (n - 1)d

Example: What is the value of the 11th term ?

15th term = 2 + (11 - 1) x 3 = 32

MEASUREMENT

Mass - Kilogram(kg) and Gram(g)
1 kg = 1000 g
kg ( x 1000) g Eg: 2kg = 2 x 1000 = 2000g
-------->
Gram (g) to Kg
kg (/1000) g 800g = 800/1000 = 0.8kg
<------------
Converter= (divide) / <- kg (1000) g -> x (multiply)