4 ESO ACADEMICS - UNIT 0 - NUMBERS

Unit 00 September

1. READING AND WRITING NUMBERS.

1.1. CARDINAL AND ORDINAL NUMBERS.

The Cardinal numbers (one, two, three, and so on) are adjectives referring to

quantity, and the Ordinal numbers (first, second, third, and so on) refer to distribution.

MATH VOCABULARY: Cardinal numbers, Ordinal Numbers, Quantity. Distribution.

NUMBER CARDINAL ORDINAL NUMBER CARDINAL ORDINAL

1 One First (1st) 16 Sixteen Sixteenth (16th)

2 Two Second (2nd) 17 Seventeen Seventeenth (17th)

3 Three Third (3rd) 18 Eighteen Eighteenth (18th)

4 Four Fourth (4th) 19 Nineteen Nineteenth (19th)

5 Five Fifth (5th) 20 Twenty Twentieth (20th)

6 Six Sixth (6th) 21 Twenty-One Twenty-First (21st)

7 Seven Seventh (7th) 22 Twenty-Two Twenty-Second (22nd)

8 Eight Eighth (8th) 23 Twenty-Three Twenty-Third (23rd)

9 Nine Ninth (9th) 24 Twenty-Four Twenty-Fourth (24th)

10 Ten Tenth (10th) 25 Twenty-Five Twenty-Fifth (25th)

11 Eleven Eleventh (11th) 26 Twenty-Six Twenty-Sixth (26th)

12 Twelve Twelfth (12th) 27 Twenty-Seven Twenty-Seventh (27th)

13 Thirteen Thirteenth (13th) 28 Twenty-Eight Twenty-Eighth (28th)

14 Fourteen Fourteenth (14th) 29 Twenty-Nine Twenty-Ninth (29th)

15 Fifteen Fifteenth (15th) 30 Thirty Thirtieth (30th)

MATH VOCABULARY: Table, Row, Column.

If a number is in the range 21 to 99, and the second digit is not zero, we should

write the number as two words separated by a hyphen:

Axel Cotón Gutiérrez Mathematics 4º ESO 0.1

Unit 00 September

25 twenty-five

57 fifty-seven

89 eighty-nine

MATH VOCABULARY: Range, Digit, Zero, Hyphen.

NUMBER CARDINAL ORDINAL NUMBER CARDINAL ORDINAL

40 Forty Fortieth (40th) 80 Eighty Eightieth (80th)

50 Fifty Fiftieth (50th) 90 Ninety Ninetieth (90th)

60 Sixty Sixtieth (60th) 100 A/One Hundred Hundredth (100th)

70 Seventy Seventieth (70th) 1,000 A/One Thousand Thousandth (1,000th)

Numbers over 100 are generally written in figures. If you want to say them

aloud or want to write them in words rather than figures you put ‘and’ in front of the

number expressed by the last two figures. For example:

203 two hundred and three (AmE: two hundred three)

622 six hundred and twenty-two (AmE: six hundred twenty-two)

MATH VOCABULARY: Figure.

NOTE: The British use ‘and’ before tens and ones but the Americans usually leave the

‘and’ out.

NUMBER CARDINAL ORDINAL

10,000 Ten Thousand Ten Thousandth

100,000 A/One Hundred Thousand Hundred Thousandth

1,000,000 A/One Million Millionth

Numbers between 1000 and 1,000,000 are usually said or written in words as:


Unit 00 September

1,803 one thousand, eight hundred and three

1,963 one thousand, nine hundred and sixty-three

2,840 two thousand, eight hundred and forty

NOTE: Notice that hundred, thousand, and so on is NOT followed by an ‘s’: two

hundred (NOT two hundreds ).

Four-figure numbers ending in 00 can also be said or written as a number of

hundreds. For example:

1800 can be said or written as ‘eighteen hundred’

If the number 1963 is being used to identify something, it is said as ‘one nine six

three’. We always say each figure separately like this with telephone numbers. If a

telephone number contains a double number, we used the word ‘double’:

561 6603 five six one (pause) double six ‘oh’ three

(AmE: five six one (pause) six six ‘oh’ three)

MATH VOCABULARY: Double, Ones, Tens.

Beyond a million, the names of the numbers differ depending where you live.

The places are grouped by thousands in America and France, by the millions in Great

Britain, Germany and Spain.

NAME AMERICA UK

Billion 1,000,000,000 1,000,000,000,000

Trillion 1 with 12 zeros 1 with 18 zeros

Quadrillion 1 with 15 zeros 1 with 24 zeros

1,412,605 one million four hundred twelve thousand six hundred and five

2,760,300 two million seven hundred sixty thousand three hundred


Unit 00 September

1.2. SAYING YEARS AND DATES.

YEARS

We normally say a year in two parts. In the case of years ending in ‘00’, we say

the second part in ‘hundred’:

1058 ten fifty-eight

1706 seventeen hundred and six (or ‘seventeen oh six’)

1865 eighteen sixty-five

1900 nineteen hundred

There are two ways of saying years ending in ‘01’ to ‘09’ before 2000. For

example:

1901 can be said as ‘nineteen oh one’ or ‘nineteen hundred and one’

The year 2000 is read ‘two thousand’, 2006 ‘two thousand and six’ (AmE: two

thousand six).

Post-2010 dates are often said as normal (2010 would be ‘twenty ten’).

DATES

We can say dates either with the day before the month, or the month before

the day.

The first of January / January the first

NOTE: Be careful! The second of December 2007 is written in British English like this:

2/12/07 and in American English like this: 12/2/07


Unit 00 September

1.3. FLIGHT NUMBERS AND TITLES.

FLIGHT NUMBERS

We pronounce a flight number in two parts or digit-by-digit. For example:

110 one ten (or ‘one one oh’)

1248 twelve forty-eight

2503 twenty-five oh three

3050 three oh five oh (or ‘three zero five zero’, ‘thirty fifty)

TITLES

In names for kings and queens, ordinal numbers are written in Roman numbers.

In spoken English, the definite article is used before the ordinal number:

Charles II - Charles the Second

Edward VI - Eduard the Sixth

Henry VIII - Henry the Eighth

1.4. NUMBER 0.

The figure 0 is normally called nought in UK and zero in USA. When numbers

are said figure by figure, 0 is often called like the letter O Examples:

My telephone number is nine six seven double two oh four six oh (967 220460)

My telephone number is nine six seven double two oh treble/triple six (967 220666)

In measurements (for instance, of temperature), 0 is called zero

Water freezes at zero degrees Celsius


Unit 00 September

Zero scores in team-games are usually called nil in UK and zero in USA. In

tennis, table-tennis and similar games the word love is used (this is derived from the

French l’oeuf, meaning the egg, presumably because zero can be egg-shaped)

Examples:

Ribera del Fresno three Real Madrid nil (nothing)

Nadal is winning forty-love

MATH VOCABULARY: Measurement, Love, Nought, Nil.

1.5. DECIMALS, FRACTIONS AND PERCENTAGES.

DECIMALS

Decimal Fractions are said with each figure separate. We use a full stop (called

“point”), not a comma, before the fraction. Each place value has a value that is one

tenth the value to the immediate left of it.

3.375 three point three seven five

2.36 two point three six

0.5 nought point five (AmE: zero point five)

0.75 nought point seven five (AmE: zero point seven five)


Unit 00 September

MATH VOCABULARY: Decimal, Fraction, Value.

FRACTIONS

Simple fractions are expressed by using “ordinal numbers” (third, fourth,

fifth...) with some exceptions:

1/2 One half / a half

1/3 One third / a third

2/3 Two thirds

3/4 Three quarters

5/8 Five eighths

4/33 Four over thirty-three

MATH VOCABULARY: Over. Quarter, Half.

PERCENTAGES.

We don't use the article in front of the numeral.

10% of the people – Ten per cent of the people

37% thirty-seven per cent

MATH VOCABULARY: Percentage, Per Cent, Numeral.

REMEMBER: Use (,) and (.) in the opposite way –the points for the decimals and the

commas for thousands, millions, and so on.


Unit 00 September

2. NATURAL NUMBERS.

The numbers we use for counting are called natural numbers or sometimes just

counting numbers. The possible combination of natural numbers is endless. There is

no largest natural number, so we say the set of all natural numbers is infinite. We use

the letter ℕ to refer to the set of all natural numbers.

ℕ = {1, 2, 3, 4, . . . , 10, 11, . . . }

If we include the number zero or 0, then our set now has a new name, which is

the set of whole numbers.

{0, 1, 2, 3, 4, . . . , 10, 11, . . . }

The Hindu-Arabic system is more useful and more efficient than the systems

used by the Egyptians, Romans, and Mayans.

• It uses only 10 digits to construct all the natural numbers.

• It uses the digit 0 or zero to show an empty place value.

• It has a place value system where digits represent different numbers when

placed in different place value columns.

Each digit in a number has a place value.

This system is called also Decimal system (its base is 10).


Unit 00 September

MATH VOCABULARY: Natural Numbers, Whole Numbers, Infinite, Counting Numbers,

Decimal System, Twice, Average.

3. REPRESENTATION AND ORDER NATURAL NUMBERS.

The Natural numbers can be represented by a line like that:

We can use symbols < and > to express which is the order relation between

two numbers: less than and more than.

Example: “Order the following pairs of numbers: 3 and 9; 4 and 5; 9 and 8”.

We can write:

3 < 9 “3 is smaller than 9”

4 < 5 “4 is smaller than 5”

9 > 8 “9 is greater than 8”

If you have any doubts then check the correspondent position in the number line:

To compare and order whole numbers:

• Align the digits by place value.

• Compare the digits in each place, starting with the greatest place.

39,630 19,578 130,434 36,415

There are no hundred thousands in the other numbers. 130,434 is greatest.

39,630 36,415

3 = 3, but the ones of thousands period 9 is bigger than 6, so 39,630 is greatest.

In order from greatest to least, the numbers are:

130,434 > 39,639 > 36,415 > 19,578


Unit 00 September

4. OPERATIONS WITH NATURAL NUMBERS.

Many situations in real life involve operations with numbers so it is useful to

practice them. We strongly recommend you practice them mentally.

4.1. ADDITION.

The addition is an operation that combines numbers to get a total. The symbol

is +.

Example: 3 + 5 = 8

We read 3 + 5 = 8 as: “Three plus five is equal to eight” or “Three plus five

equals eight” or “Three plus five is eight”. The terms in the addition are called addends

and the result is called the sum.

Example: The library lent 45 books last Monday, 50 books on Tuesday and 73

books on Wednesday. How many books have they lent?

45 + 50 + 73 = 168 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

Answer: They have lent 168 books

We can use the Addition table:


Unit 00 September

ADDITION PROPERTIES.

The properties of addition are: closure, commutative, associative, and additive

identity.

• Closure property: Addition of two natural numbers is always another natural

number.

6 + 7 = 13 (13 is another natural number)

• Commutative property: When we add two natural numbers, the sum is the same

no matter what the order of the addends is.

𝒂𝒂 + 𝒃𝒃 = 𝒃𝒃 + 𝒂𝒂

4 + 2 = 2 + 4

• Associative Property: When we add three or more natural numbers, the sum is

the same no matter how we group the addends.

𝒂𝒂 + 𝒃𝒃 + 𝒄𝒄 = (𝒂𝒂 + 𝒃𝒃) + 𝒄𝒄 = 𝒂𝒂 + (𝒃𝒃 + 𝒄𝒄)

(2 + 3) + 4 = 2 + (3 + 4)

• Additive Identity Property: The sum of any natural number and zero is the

original number.

𝒂𝒂 + 𝟎𝟎 = 𝒂𝒂

5 + 0 = 5

MATH VOCABULARY: Operation, Addition, Plus, Equal, Term, Addends, Sum,

Properties, Closure Property, Commutative Property, Associative Property, Additive

Identity Property, To Group.


Unit 00 September

4.2. SUBTRACTION.

Subtracting is removing or taking away some objects from a group. The symbol

is −.

Example: 13 – 7 = 6

We read 13 – 7 = 6 as: “Thirteen minus seven equals six” or “Thirteen subtract

seven equals six” (sometimes you can see “thirteen take away seven equals six” but it

is better to use the other expressions). The terms of subtraction are called minuend

(M) and subtrahend (S), the result is called the difference (D).

𝑴𝑴− 𝑺𝑺 = 𝑫𝑫

The minuend is the first number, it is the number from which you take

something and it must be the larger number. The subtrahend is the number that is

subtracted and it must be the smaller number. The difference is the result of the

subtraction.

To check if the subtraction is correct we add up the subtrahend and the

difference. The result must be the minuend:

𝑺𝑺 + 𝑫𝑫 = 𝑴𝑴

Example: We have saved 3,520 euros but we have spent € 745 on a computer.

How much money do we have left?

3,520 – 745 = 2,775

Answer: We have 2,775 euros left.

SUBTRACTION PROPERTIES.

• The Subtraction is NOT Commutative.


Unit 00 September

𝒂𝒂 − 𝒃𝒃 ≠ 𝒃𝒃 − 𝒂𝒂

3 − 2 ≠ 2 − 3

• The Subtraction in Natural Numbers has NOT Closure Property when 𝑺𝑺 > 𝑴𝑴.

2 − 3 = −1 (−1 is not a Natural Number)

• Associative Property: Only when 𝒂𝒂 > 𝒃𝒃 > 𝒄𝒄 𝒂𝒂𝒏𝒏𝒅𝒅 (𝒂𝒂 − 𝒃𝒃) > 𝒄𝒄.

𝒂𝒂 − 𝒃𝒃 − 𝒄𝒄 = (𝒂𝒂 − 𝒃𝒃) − 𝒄𝒄 = 𝒂𝒂 − (𝒃𝒃 − 𝒄𝒄)

(5 − 4) − 2 = 5 − (4 − 3)

5 > 4 > 3

• Additive Identity Property: To subtract zero to any natural number is the original

number.

𝒂𝒂 − 𝟎𝟎 = 𝒂𝒂

5 − 0 = 5

MATH VOCABULARY: Subtraction, To Remove, To Take Away, Minus, To Subtract,

Minuend, Subtrahend, Difference, Result, To Be Correct, To Spend.

4.3. MULTIPLICATION.

Multiplying is doing an addition of equal addends. . The symbol is 𝑥𝑥 or ∙.

3 + 3 + 3 + 3 + 3 = 3 𝑥𝑥 5 = 15

We read 3 𝑥𝑥 5 = 15 as: “Three times five equals fifteen” or “Three times five is

fifteen”. The factors are the numbers that are multiplied together. The product is the

result of multiplying.


Unit 00 September

Example: In my living-room I have a bookcase with three shelves. If there are

five books on each shelf, how many books are there?

5 𝑥𝑥 3 = 15

Answer: There are 15 books in my bookcase.

We use the Multiplication Table:

To multiply a number by a unit followed by zeroes, you add the zeroes to the

right of the number.

28 ∙ 10 = 280

15 ∙ 1,000 = 15,000

PROPERTIES OF MULTIPLICATION.

The properties of multiplication are: closure, commutative, associative, and

additive identity:


Unit 00 September

• Closure property: Multiplication of two natural numbers is always another

natural number.

6 ∙ 7 = 42 (42 is another natural number)

• Commutative property: When two numbers are multiplied together, the product

is the same no matter what the order of the factors is.

𝒂𝒂 ∙ 𝒃𝒃 = 𝒃𝒃 ∙ 𝒂𝒂

4 ∙ 2 = 2 ∙ 4

• Associative Property: When three or more numbers are multiplied, the product is the

same no matter how we group the factors.

𝒂𝒂 ∙ 𝒃𝒃 ∙ 𝒄𝒄 = (𝒂𝒂 ∙ 𝒃𝒃) ∙ 𝒄𝒄 = 𝒂𝒂 ∙ (𝒃𝒃 ∙ 𝒄𝒄)

(2 ∙ 3) ∙ 4 = 2 ∙ (3 ∙ 4)

• Additive Identity Property: The product of any number and one is that number.

𝒂𝒂 ∙ 𝟏𝟏 = 𝒂𝒂

5 ∙ 1 = 5

• Zero Property of Multiplication: The product of zero and a number is zero.

𝒂𝒂 ∙ 𝟎𝟎 = 𝟎𝟎

5 ∙ 0 = 0

MATH VOCABULARY: To Multiply, Times, Factor, Product, Identity.


Unit 00 September

4.4. DIVISION.

Dividing is to share a quantity into equal groups. It is the inverse of

multiplication. In Spanish we write 6 : 2 , but in English it is always 6 ÷ 2, we never use

the colon (:).

We read 15 ÷ 5 = 3 as: “Fifteen divided by five equals three”. There are four

terms in a division: dividend, divisor, quotient and remainder.

The dividend is the number that is divided. The divisor is the number that

divides the dividend. The quotient is the number of times the divisor goes into the

dividend. The remainder is a number that is too small to be divided by the divisor.

Example: There are 72 sweets in a bag. If we want to distribute them among 12

children, how many sweets are there for each child?

72 ∶ 12 = 6

Answer: There are six sweets for each child.


Unit 00 September

The division can be:

• Exact: the remainder is 0.

• Non exact: the remainder isn’t 0.

To check if the division is correct we do the Division Algorithm:

𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝒅𝒅𝑫𝑫𝒏𝒏𝒅𝒅 = 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 ∙ 𝑸𝑸𝑸𝑸𝑫𝑫𝑸𝑸𝑫𝑫𝑫𝑫𝒏𝒏𝑸𝑸 + 𝑹𝑹𝑫𝑫𝑹𝑹𝒂𝒂𝑫𝑫𝒏𝒏𝒅𝒅𝑫𝑫𝑫𝑫

Example: Find out the result of the division 237 ÷ 13 and then check the result

with the division algorithm:

237 ÷ 13 = 18; 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 3

Dividend = Divisor x Quotient + Remainder; 13 x 18 + 3 = 237, so it is correct.

NOTE: We can never divide by Zero.

DIVISION PROPERTIES.

• Closure property: The Division in Natural Numbers has NOT Closure Property .

3 ÷ 2 = 1.5 (1.5 is not a Natural Number)

• The Division is NOT Commutative.

𝒂𝒂 ÷ 𝒃𝒃 ≠ 𝒃𝒃 ÷ 𝒂𝒂

𝟑𝟑 ÷ 𝟐𝟐 ≠ 𝟐𝟐 ÷ 𝟑𝟑

• Associative Property: Only when 𝒂𝒂 > 𝒃𝒃 > 𝒄𝒄 𝒂𝒂𝒏𝒏𝒅𝒅 (𝒂𝒂 ÷ 𝒃𝒃) > 𝒄𝒄 and with exact

divisions.

𝒂𝒂 ÷ 𝒃𝒃 ÷ 𝒄𝒄 = (𝒂𝒂 ÷ 𝒃𝒃) ÷ 𝒄𝒄 = 𝒂𝒂 ÷ (𝒃𝒃 ÷ 𝒄𝒄)

(100 ÷ 4) ÷ 2 = 100 ÷ (4 ÷ 2)


Unit 00 September

100 > 4 > 2

• Additive Identity Property: The division of any number by one is that number.

𝒂𝒂 ÷ 𝟏𝟏 = 𝒂𝒂

5 ÷ 1 = 5

MATH VOCABULARY: To Divide, To Share, Inverse, Dividend, Divisor, Quotient,

Remainder, Algorithm.

4.5. COMBINED OPERATIONS.

Look at these expressions: 6 + 2 ∙ 4 and (6 + 2) ∙ 4; although they have the same

numbers and the same operations, their values are different:

6 + 2 ⋅ 4 = 6 + 8 = 14

(6 + 2) ⋅ 4 = 8 ⋅ 4 = 32

When expressions have more than one operation, we have to follow rules for

the order of operations. To remember these rules we will use the BODMAS or

BEDMAS rule:

• B: Brackets. ( ) before [ ].

• O: Orders (Powers). E: Exponents

• D M: Divisions and Multiplications (left to right).

• A S: Additions and Subtractions (left to right).

Example 1:

Solve 3 + 6 𝑥𝑥 (5 + 4) 3 − 7 using the rules for the correct order of operations.

𝑺𝑺𝑸𝑸𝑫𝑫𝑺𝑺 𝟏𝟏: 3 + 6 𝑥𝑥 (5 + 4) ÷ 3 − 7 = 3 + 6 𝑥𝑥 9 ÷ 3 − 7 𝐵𝐵𝑟𝑟𝑟𝑟𝐵𝐵𝑏𝑏𝑟𝑟𝐵𝐵𝑏𝑏


Unit 00 September

𝑺𝑺𝑸𝑸𝑫𝑫𝑺𝑺 𝟐𝟐: 3 + 6 𝑥𝑥 9 ÷ 3 − 7 = 3 + 54 ÷ 3 − 7 𝑀𝑀𝑀𝑀𝑀𝑀𝐵𝐵𝑟𝑟𝑀𝑀𝑀𝑀𝑟𝑟𝐵𝐵𝑟𝑟𝐵𝐵𝑟𝑟𝑏𝑏𝑟𝑟

𝑺𝑺𝑸𝑸𝑫𝑫𝑺𝑺 𝟑𝟑: 3 + 54 ÷ 3 − 7 = 3 + 18 − 7 𝐷𝐷𝑟𝑟𝐷𝐷𝑟𝑟𝑏𝑏𝑟𝑟𝑏𝑏𝑟𝑟

𝑺𝑺𝑸𝑸𝑫𝑫𝑺𝑺 𝟒𝟒: 3 + 18 − 7 = 21 − 7 𝐴𝐴𝑟𝑟𝑟𝑟𝑟𝑟𝐵𝐵𝑟𝑟𝑏𝑏𝑟𝑟

𝑺𝑺𝑸𝑸𝑫𝑫𝑺𝑺 𝟓𝟓: 21 − 7 = 14 𝑆𝑆𝑀𝑀𝑏𝑏𝐵𝐵𝑟𝑟𝑟𝑟𝐵𝐵𝐵𝐵𝑟𝑟𝑏𝑏𝑟𝑟

Example 2:

Calculate 9 − 5 ÷ (8 − 3) 𝑥𝑥 2 + 6 using the correct order of operations.

𝑺𝑺𝑸𝑸𝑫𝑫𝑺𝑺 𝟏𝟏: 9 − 5 ÷ (8 − 3) 𝑥𝑥 2 + 6 = 9 − 5 ÷ 5 𝑥𝑥 2 + 6 𝐵𝐵𝑟𝑟𝑟𝑟𝐵𝐵𝑏𝑏𝑟𝑟𝐵𝐵𝑏𝑏

𝑺𝑺𝑸𝑸𝑫𝑫𝑺𝑺 𝟐𝟐: 9 − 5 ÷ 5 𝑥𝑥 2 + 6 = 9 − 1 𝑥𝑥 2 + 6 𝐷𝐷𝑟𝑟𝐷𝐷𝑟𝑟𝑏𝑏𝑟𝑟𝑏𝑏𝑟𝑟

𝑺𝑺𝑸𝑸𝑫𝑫𝑺𝑺 𝟑𝟑: 9 − 1 𝑥𝑥 2 + 6 = 9 − 2 + 6 𝑀𝑀𝑀𝑀𝑀𝑀𝐵𝐵𝑟𝑟𝑀𝑀𝑀𝑀𝑟𝑟𝐵𝐵𝑟𝑟𝐵𝐵𝑟𝑟𝑏𝑏𝑟𝑟

𝑺𝑺𝑸𝑸𝑫𝑫𝑺𝑺 𝟒𝟒: 9 − 2 + 6 = 7 + 6 𝑆𝑆𝑀𝑀𝑏𝑏𝐵𝐵𝑟𝑟𝑟𝑟𝐵𝐵𝐵𝐵𝑟𝑟𝑏𝑏𝑟𝑟

𝑺𝑺𝑸𝑸𝑫𝑫𝑺𝑺 𝟓𝟓: 7 + 6 = 13 𝐴𝐴𝑟𝑟𝑟𝑟𝑟𝑟𝐵𝐵𝑟𝑟𝑏𝑏𝑟𝑟

PROPERTY.

• Distributive Property: says that multiplying a number by a group of numbers

added together is the same as doing each multiplication.

𝒂𝒂 ∙ (𝒃𝒃 + 𝒄𝒄) = 𝒂𝒂 ∙ 𝒃𝒃 + 𝒂𝒂 ∙ 𝒄𝒄

𝒂𝒂 ∙ (𝒃𝒃 − 𝒄𝒄) = 𝒂𝒂 ∙ 𝒃𝒃 − 𝒂𝒂 ∙ 𝒄𝒄

3 ∙ (2 + 4) = 3 ∙ 2 + 3 ∙ 4

3 ∙ (4 − 2) = 3 ∙ 4 − 3 ∙ 2

MATH VOCABULARY: Combined Operation, Expression, Power, To Solve, To Calculate,

Step, Distributive Property.


Unit 00 September

5. WHOLE NUMBER PROBLEMS.

In order to solve problems you must follow the rules below:

• Start with a first reading of the problem to know what it is about.

• Then you do a second reading more slowly, in order to understand the problem

and find out what data it provides.

• Write down the data of the problem clearly. If it is a geometric problem, then

you can make a drawing. You must also check that the units are all the same. If

they are not, then you will have to change them to the appropriate ones.

• Now you can do all the necessary operations to solve the problem.

• Finally, answering: Reread the question of the problem and answer it with a

sentence. Don't forget to mention the correct unit and check that the answer

makes sense.

Example John has 350 euros in his bank account. He has received 37 euros as a

birthday present and then, he has bought 4 DVDs which cost €15 each. How much

money does he have now?

DATA:

Saved: 350 euros

Gift: 37 euros

Spend: 4 x 15 euros

SOLVE:

350 + 37 = 387 𝑟𝑟𝑀𝑀𝑟𝑟𝑏𝑏𝑏𝑏

4 ∙ 15 = 60 𝑟𝑟𝑀𝑀𝑟𝑟𝑏𝑏𝑏𝑏

387− 60 = 327 𝑟𝑟𝑀𝑀𝑟𝑟𝑏𝑏𝑏𝑏

ANSWER:

John has now 327 euros

MATH VOCABULARY: Geometry, To Draw, Unit, Answer.


Unit 00 September

This traditional Japanese game just need two pairs of hands and some strategy

(and simple adding) skills! Good fun for getting kids thinking - and every game ends up

differently! You need at least 2 children, but you can play with more if your adding

skills are up to it!

The object of the game is to be the last player left with a hand in the game. A

hand is "dead" and has to be hidden behind the back when all 5 fingers (or chopsticks)

are extended. Play as follows:

• Players each put one finger out and hold their hands in front of them. One

person starts by using one of his hands to touch any other person's hand. The

person he touches adds up the number of fingers involved (on this first go it

would be two - one on each of the children's hands) and holds that number of

fingers out. Play then passes to the next child (move clockwise around the group,

if there are more than two children).

• When it is your turn, if you have more than 1 finger extended on each hand, you

can choose to touch your own hands together rather than touch someone else's

hand. If you do this, you add together the number of fingers on both hands, then

split that number between your two hands in a different way. So, for example, if

you had 4 fingers extended on your right hand and 2 on your left hand, you

might tap them together then split them so that you have 3 extended on each

hand. Obviously it is a good idea to do this if you have 4 fingers extended on one

hand, because with one more touch that hand could become "dead" and have to

go out of the game. You can't do this if you 4 fingers extended on each hand, or 4

on one hand and 3 on the other, as you would just be swapping your fingers

around indefinitely.


4 ESO ACADEMICS - UNIT 0 - NUMBERS

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