matemáticas inglés

UNIT 1 DIVISIBILITY AND INTEGER NUMBERS

Things to remember: Divisions can be:

- Exact. - Integer.

Think:

When are negative numbers

necessary?

Types of numbers:

- Integer - Natural - Negative...

In mixed operations, calculate first the brackets, then multiply or divide and finally add or subtract.

Relation of divisibility

When we divide a number by another and the remainder is zero, there is a relation of divisibility between both numbers. For example, 30 and 6. Here, 30 is a multiple of 6; and 6 is a divisor of 30

Multiples

- Every time we multiply a number by any other natural number, we get a multiple of it: for example, to get multiples of 17: 17 · 2 = 34 17 · 3 = 51 17 · 4 = 68 ... - Any number is multiple of itself and of 1.

Activities

1. Calculate 10 multiples of 7:

2. Answer the questions:

- Is 7 a multiple of 7? ............................ Why? .............................................................. - How many multiples has 0 got? ......................... Why? .............................................. - How many multiples has 8 got? ......................... Why? ............................................ - How many multiples has 1 got? ......................... Why? ............................................... - Is there a multiple of ∞ ? ..................................Why? ...............................................

3. Check the property of adding two multiples of a number:

Get two multiples of 9: ______ and _______

Add them: ______ + _____ = ______

Is the sum a multiple of 9 ? Yes ? - No ?

Get another multiple of 9: ____________

Add it to 20 (no multiple): ____+ __20__ = ______

Is the sum a multiple of 9 ? Yes ? - No ?

Divisors

- All the numbers that can divide another number (and get zero as remainder) are called divisors of that number. Example: 5 is a divisor of 20, because 20 : 5 = 4 (and zero in the quotient) - Any number has got a finite quantity of divisors. - Any number has got, at least, two divisors: 1 and itself. - 1 is divisor of any number; the quotient is always that number.

Activities

1. Have these numbers got a relation of divisibility?

12 and 5 33 and 11

8 and 1 100 and 9

2. Write all the divisors of...:

18 : 25 :

43 : 21 :

3. Read the following list of numbers and then answer the questions:

2 3 5 9 14 27 35 50 68 72 84 99 111

- Which ones are divisors of 100?: ..................................................... - Which ones are divisors of 27?: ..................................................... - Which ones are multiples of 2? ..................................................... - Which ones are multiples of 9? .....................................................

4. Write three multiples of...:

6 : 11 :

21 : 200 :

5. Classify among the following numbers:

2 3 5 8 18 40 54 65 77 88 95 100 150

Multiples of 3 Multiples of 5 Multiples of 25 Divisors of 40 Divisors of 54 Divisors of 300

Criteria of divisibility

How to find out if a number is a multiple of 2, 3, 5, 9, 10? Remember the rules:

Multiples of 2 Multiples of 5 Multiples of 10

They are even numbers; their last digit is 0 , 2 , 4 , 6 or 8.

They always finish in 0 or 5.

They always finish in 0.

Multiples of 3 Multiples of 9

When a number is a multiple of 3, the sum of all its digits is a multiple of 3.

When the sum of the digits of a number is a multiple of 9, that number is consequently a multiple of 9, too.

Activities

1. Are these numbers multiples of 2, 3, 5, 9 or 10? Check the right squares:

Numbers Multiple of 2?

Multiple of 3?

Multiple of 5?

Multiple of 9?

Multiple of 10?

10

16

27

153

270

900

2. Think and answer:

- Are all the multiples of 3 also multiples of 9? ……………………………………………… - Are all the even numbers multiples of 10? …………………………………………………. - Can an odd number be a multiple of 2? ………………………………………..…………… - Can an odd number be a multiple of 3? …………………………………………………….. - Can an even number be a multiple of 9? …………………………………………………….

Prime numbers and composite numbers

A prime number can be divided only by itself and 1.

A composite number can be divided by other numbers than 1 and itself.

Composite numbers can be factored (written as a multiplication). Example: 15 = 3 · 5

Activities

1. Factor these composite numbers (with only prime factors in the answer):

16: 40:

85: 108:

2. Put these numbers in the correct columns:

7 15 17 24 35 53 69 75 92 113 131 150

Prime numbers

Composite numbers

Even numbers

Odd numbers

Multiples of 3

Divisors of 150

4. Factor these numbers vertically:

65

104 126 144 190 198

Lowest common multiple

In general, a common multiple is the multiple of more than one number. Example: 30 is a common multiple of 2, 3, 5, 6, 10 and 15.

The lowest common multiple is the smallest among all the common multiples of several numbers. 18 is the lowest common multiple of 6 and 9, but not of 2 and 3 (which is 6).

In order to find out the lowest common multiple of two (or more) numbers:

- First, get the prime factors of each number. - Then, multiply all those factors by each other, taking their highest power.

Example: the lowest common multiple of 30 and 40:

- Prime factors of both : 30 = 2 · 3 · 5 ; 40 = 23 · 5 - And multiplying them: 23 · 3 · 5 = 120

Activity

Find the lowest common multiple of these numbers:

14 and 16:

25 and 45:

33 and 52:

80 and 120:

Highest common factor

In general, a common factor can divide several numbers. 3 is a common factor of 6 and 9.

The highest common factor is the biggest among all the common factors we can find. Example: 3 is a common factor of 30 and 45, but not their highest common factor (it is 15).

In order to find out the highest common factor of two (or more) numbers:

- First, get the prime factors of each number. - Then, multiply only the factors they have in common, taking their lowest power (if there are several).

Example: the highest common factor of 30 and 40:

- Prime factors of both : 30 = 2 · 3 · 5 ; 40 = 23 · 5 - And multiplying them: 2 · 5 = 10

Activity

Find the highest common factor of these numbers:

26 and 104:

35 and 105:

14 and 60:

27 and 180:

About integer numbers

Things you should remember! Integer numbers (Z) include all whole positive (1, 2, 3, 4…) and negative (-1, -2, -3, -4…) numbers plus zero (0). Decimals and fractions are not integers. Integers are infinite.

The absolute value of an integer is the positive form of the number: |3| = 3 |-3| = 3

The opposite of an integer is the same number, but with different sign: Opposite of 5 is -5.

Comparing positive integers, the greater is the one with a bigger absolute value (6 > 4); but in negative integers the greater is that with a smaller absolute value (-6 < -4). Positive integers are greater than negative ones. Zero is greater than negatives.

Activities 1. What's the absolute value of...? | 7 | : ................. |-4| : ........................ |11| : ....................... |-22| : ...................... 2. What's the opposite of...? -4 : ................. 25 : ..................... -65 : ..................... 19 : .................. -12 : .............. 3. Write these numbers from the lowest to the greatest:

12 -12 |-12| 0 6 -4 |5| -19 -1 3 -7 .......................................................................................................................................

4. Write < or > to compare the numbers:

5 ..................... -3

|-6| ....................-6

-25 .....................-5

18 .................. 19

2 ..................... |-12|

-13 ...................... 12

7 …................. -1

- 1 .....................0

Operations with integer numbers

Additions and subtractions

With the same sign:

Examples: 7 + 9 = +16 -5 – 8 = -13

With different signs:

Examples:-8 + 3 = -5 -5 + 9 = 4 7 – 2 = 5

Additions and subtractions with brackets

Adding a positive number:

Example: + (+7) = +7

Adding a negative number:

Example: + (-7) = -7

Subtracting a positive number:

Example: - (+7) = -7

Subtracting a negative number:

Example: - (-7) = +7

Adding and subtracting inside the brackets

( GENERAL RULE: add all the positive numbers; add all the negative numbers; subtract both results; and get the sign of the result with greater absolute value )

+(-7 + 2 -3) = +(-8) = -8 +(7 – 2 + 3) = +(+8) = +8

-(-7 + 2 -3) = -(-8) = +8 - (7 – 2 + 3 ) = -(+8) = -8

Activities

1. Solve the operations:

+ 3 -4 + 5 - 1 = - 3 - 6 – 4 + 9 =

7 + 3 – 8 + 1 = -5 + 6 +9 -8 =

- 5 + 2 +3 -8 = - 4 – 2 - 5 =

8 + 6 + 3 -5 = + 14 – 3 – 4 + 2 =

2. Write without the brackets:

+(-6) = - (+3) =

+(+7) = -(-5) =

-(+15) = +(+12) =

-(-18) = +(-30) =

3. Solve the operations with brackets:

+(+3) - (+6) =

+(-23) - (-7) =

-(+9) + (+12) =

-(-2) - (-4) =

+(+2) - (+8) =

+(+5) + (+20) =

+(-6 – 8 + 10) =

-(-9 + 4 + 1) =

+(+6 – 7 - 5) =

Multiplication of integer numbers

The rule of the signs: The product of two integer numbers is:

Positive, if the factors have the same sign: + · + = + - · - = + Negative, if the factors have different signs: + · - = - - · + = -

With more than two factors, the product is:

Positive, if there is an even number of negative factors: (+3) · (-2) · (-1) · (+4) = +24 Negative, if there is an odd number of negative factors: (+5) · (-2) · (-4) · (-1) = - 40

Division of integer numbers

It is exactly the same as in the multiplication, but there are only two “elements”: dividend and divisor. And the rules of the signs work just the same as in the other operations:

(-6) : (-2) = 3 (+6) : (+2) = 3 (-6) : (+2) = -3 (+6) : (-2) = -3

Mixed operations

When different operations are mixed together: - First solve the operation inside the brackets. - Then, do the multiplications and divisions. - Finally, do the additions and subtractions.

Activities 1. Solve these multiplications:

(+3) · (+5) =

(+7) · (-2) =

(+5) · (-9) · (+4) =

(-6) · (-5) · (+8) =

(-1) · (-7) · (-5) =

(+4) · (-6) · (+9) · (-3) = 2. Divide:

(+45) : (-3) =

(-72) : (-9) =

(+35) : (+7) =

(+88) : (-11) =

(-144) : (+12) =

(+18) : (-3) = 3. Solve the mixed operations in the right order:

15 – 3 · [(-8) + (-4) : (+2)] = 8 + 2 · [(-2) - (+5) · (-2)] = (-5) + (-11) : [(-2) + (-3) · (-2)] =

20 : 5 + [(+6) · (-4) + 5] = [(-3) · (+8) + (-5)] · 2 - (-2) = 4 · [(-25) : (-5) + (+4)] – 4 · [(-8) : (-2) -1] =

Powers of integer numbers

Remember that... : A power is a short form to write the factors in a multiplication if the factors are equal:

3 · 3 · 3 · 3 · 3 = 35 Here, 3 is called the base and 5 is the index or power (or exponent in American English).

BASIC INFORMATION ABOUT THE POWERS: – A positive index corresponds to a multiplication. Example: xxxxxx ⋅⋅⋅⋅=5 – When the base is negative:

• the product will be positive if the index is an even number: (-4)2 = +16 • but the product will be negative if the index has got an odd number: (-4)3 = -64

– If the index is zero the result is always equal to 1. Example: x0 = 1

– If the base is one, the answer will always be one. Example: 112 = 1

Activities

1. Calculate the base ten powers:

103 = (-10)5 =

(-10)4 = (-10)0 =

2. Simplify and calculate:

4 · 4 · 4 · 4 · 4 = 6 · 6 · 6 =

5 · 5 · 5 = 10 · 10 · 10 · 10 =

3. Search the value of x:

2x = 512 x6 = 1

3x = 27 x4 = 625

4. Calculate the powers with a negative base:

(-7)2 =

(-4)4 =

(-3)6 =

(- 8)3 =

(-2)5 =

(-1)7 =

Operations with powers

a) When the powers are equal:

To multiply two powers together, you should multiply the bases and keep the same power. Example: 45 · 55 = (4 · 5)5 = 205

To divide two powers, you should divide the bases and keep the same power. Example: 106 : 26 = (10 : 2)6 = 56

b) When the bases are equal:

To multiply two powers together you should add the indices. This works for both positive and negative index numbers. Example: 32 · 33 = 32+3 = 35

To divide one power by another you should subtract the indices. This works for both positive and negative index numbers. Example: 46 : 42 = 46-2 = 44

To evaluate the power of a power, we multiply their indices. Ex: (24)2 = 24 · 2 = 28

Activities

1. Resolve the operations:

24 · 34 = 63 - 42 =

52 + 2-3 = 54 : 52 =

32 – 22 = 54 · 34 =

103 : 53 = 95 · 23 =

2. Find out the value of x :

5 · 5 · 5 · 5 = x4 =

(42)5 = 4x =

43 · 45 = 4x =

x5 · 55 = 155

36 : 34 = 3x =

x4 : 44 = 64


362 : (62 · 22) =

524 : (53 · 57) =

(-10)3 · (-4)-4 =

[(-5)2 · (+3)2] : 82 =

[(-2)3 · (+4)2] · 44 =

-54 · [(-4)4 · (+4)4] =

(153 · 155) : 154 =

33 · (-3)4 · (-3)2 =

Square roots of integer numbers

General characteristics:

- Square roots and powers are opposite operations: 39 = and 32 = 9

- A positive number has two square roots, one positive and one negative. The square roots of 25 are 5 and -5 (because -5 · -5 = 25). A negative number hasn’t got any square roots.

- As well as square roots, there are cube roots3 , fourth roots4 , etc.

Activities

1. Get the opposite operation:

81 = 9 and __________ 25 = 5 and _________ 144 = 12 and _________

4. Calculate the roots, if they really exist:

( ) =+ 36 ( ) =−3 64 ( ) =+4 16

0 =

=6 1

( ) =− 88

( ) =+169

=3 3

( ) =− 2

UNIT 2 DECIMAL SYSTEM OF UNITS. SEXAGESIMAL SYSTEM

THINGS TO REMEMBER!

- In the decimal system the value of a digit depends on its position:

Hundreds Tens Units DOT ( . ) Tenths Hundredths Thousandths

For example, in the number 238.136 the 3 in the tens has a value of 30 units (because of its position) but the other 3 in the hundredths has a value of 0.03 units.

- How to multiply or divide decimal numbers, paying attention to the dot and the quantity of decimals involved in the operation. Example: 25.346 · 12.41 = 314.54386 (the factors have altogether 5 decimals, then the product has 5 decimals, except if the last one(s) is/are zero).

- Rounding decimals to the nearest whole number, to the tenths, hundredths or thousandths:

Example:

762.8469

Whole number: Tenths: Hundredths: Thousandths:

763

762.8

762.85

762.847

- Do not mix quantities and operations made in the decimal system with others in the sexagesimal system. Example: 1 hour and ten minutes is not 1.10 hours!

Activities

1. What’s the value of each digit?

184.35: the value of 1 is ................; the value of 8 is ..............; the value of 4 is ...............; the value of 3 is ....................... and the value of 5 is ...........................

2. How many hundredths are there in ...?

7 units: 8 tens: 8 tenths: 1 hundred: 3 thousandths: 4 hundredths:

3. Calculate (remember in English there isn’t a comma to separate decimals, but a dot):

15.67 · 10 = 148.1 · 100 = 1.481 · 100 = 16.5 : 10 = 1.65 : 10 = 505.1 : 10 =

4. Round this number:

Example:

43.6491

Whole number: Tenths: Hundredths: Thousandths:

Decimal numbers (general concepts)

The decimal (base ten) numeral system has ten as its base. It is the most widely used numeral system, perhaps because humans have ten digits over both hands.

Decimal numbers have an integer part, a dot and a decimal part, where parts of a full unit are indicated. Example: 15.369 (and we read it: “fifteen point three six nine”).

Why are decimal numbers necessary? Because we can have more precise quantities of anything with decimal numbers; one unit is divided in ten tenths; one tenth is divided in ten

hundredths, one hundredth is divided in ten thousandths, and so on with even smaller submultiples. We generally don’t use full units in our lives, but parts of a unit. I buy things that cost 3.15 € or I drink half a litre of water (0.5), for example.

There are different types of decimal numbers:

- Terminating decimals (also called exact): in the operation 15 : 4 = 3.75 there is a decimal number with a finite quantity of digits.

- Repeating decimals (also called recurring decimals) have an infinite quantity of digits. We can get two possibilities among recurring decimals:

• 20 : 3 = 6.6666… (just repeating the same digit). • 35 : 6 = 5.83333 (in the decimal part, some digits aren’t

repeating until it becomes periodic). - Non terminating non repeating decimals: there are infinite digits, without

being repeated: 3 = 1.7320508075688772935274463415059…

Activities

1. Are these decimals exact, repeating or non terminating non repeating?

2.236067977499…

6.6161616161…

8.265

9.748888888…

Working with decimals (Review):

- How to order decimal numbers:

• Look at the integer part; the decimal with a higher integer part is the highest: 5.75 < 6.75 • If the integer parts are equal, the decimal with the highest tenths is the highest: 2.5 < 2.7 • If the integer and the tenths are equal, look at the hundredths. Example: 2.55 < 2.56 • And then to the thousandths: 2.552 < 2.558

- Tip : you can add extra zeros at the end of each decimal so that all the decimals have the same number of digits: remember that 2.2 is equal to 2.200, for example.

- Between two decimal numbers we can always find other decimals, in fact, there is an infinity of decimals. Example: between 6.4 and 6.5, you can find 6.41, 6.42, 6.421 ...

- A number with many decimals is not often used in fast calculations, so we round it. How to round decimals? Delete the digits you don't need beginning with the lowest in value on the right; if the first deleted digit is equal or higher than 5, add 1 to the previous digit. The most common rounding is called to one decimal place (there is only one digit in the decimal part).

Activities

1. Order these decimals from greatest to lowest:

0.901 0.91 0.9 0.091 0.1 9.01 9.101 9.11 9.1

> > > > > > > >

2. Round these decimal numbers:

Number: To the nearest unit: To one decimal place: To the nearest hundredth:

7.4893 186.5067 12.999 29.081

3. Which decimal is the nearest to 5.15 ? (underline it):

5.05 5.50 15.15 5.14 5.149 5.1415 5.155 5.152

4. Are these roundings correct? If not, make them correct:

49.056 → 49.05 10.936 → 10.94 25.7 → 25.8

17.564 → 17.56 88.9 → 89 12.78469 → 12.785

Operations with decimal numbers

Additions and subtractions

- Write the whole numbers in columns. Make sure the decimals are in the same place. - Add or subtract units with units, tenths with tenths... - Negative decimals behave exactly like negative integers (see the rules). Examples: 2.5 + 3.64 = 6.14 5.76 – 3.426 = 2.334

Multiplications

- Multiply the factors like integers. - In the product, be careful with the decimal: count the total number of decimals in both factors and add that quantity in the product (example: if there are two decimals in one factor and three in the other, the product should have five decimals). Example: 5.2 · 2.42 = 12.584

Divisions with decimal numbers

When only the dividend is a decimal number: When both dividend and divisor have decimals:

- Divide like integer numbers. - When the decimal (dot) appears in the dividend, write it in the quotient. - Continue dividing the rest of the digits. - Add zeros in the divident for a more precise decimal quotient.

- Multiply dividend and divisor by the unit (1) followed by as many zeros as the number of decimals in the divisor. -Then, the divisor becomes integer and the quotient won't change (as it is equivalent). -Divide following the rules.

Activities

1. Solve the operations vertically:

7.04 + 10.203 =

105.88 - 7.99 = 12.67 · 8.3 = 25.83 : 9.2 =

2. Calculate without using your calculator (round the products to the hundredths):

8.4 + 6.51 – 5.02 =

7.8 · (6.3 + 1.9 – 2.6) = 23.54 : (3.55 · 6.21) =

3. Solve the problem:

John works as a shop-assistant. He receives 4.70 euros an hour; he works for 4.5 hours every day, six days a week. Solve:

How much does he earn in one day?

His weekly salary? How many hours does he work in 4 weeks?

Square roots and decimal numbers

Solve square roots with decimal numbers in the same way you solved square roots of integers. Just be careful when you get to the decimal point: keep the decimal point in the same spot and continue to solve. If necessary, you can round large decimals.

Activities about square roots

1. Search these square roots; round large decimals to the thousandths:

4.6 =

54.32 = 16.140 =

86.0 =

55.524 = 302.254 =


- Can negative decimals have square roots?................. Why? ……………………………. - Can decimals have two square roots, or just one? ……………………………………... - Can we subtract one decimal square root from another? ………………………………. - What’s the difference between square roots of integers and those of decimal numbers?....... - Which internal operations must we do when calculating square roots? ………………..

SEXAGESIMAL SYSTEM OF UNITS

At the beginning of the unit we said that the decimal (base ten) numeral system has ten as its base. Now we are going to learn about the sexagesimal system, where number sixty is the base for operations. With it we can measure:

- Time: there are sixty seconds in one minute and sixty minutes in one hour; consequently, half an hour is 30 minutes. Be careful not to mix it up with decimal system operations:

Example: 1 minute is 60

1 in the sexagesimal system, but

10

1or even

100

1in the decimal one.

The quantity expressed can be rather different, so careful not to become confused.

- Angles: as a complete circle has got 360º, we can have four right angles measuring 90º each. One degree has got sixty minutes (60’) and one minute has got sixty seconds (60”). Geographic coordinates of latitude and longitude are also expressed in sexagesimal degrees.

Finally, we can express the same sexagesimal quantity in two ways:

- “Complex form”: the measure is expressed with different units: “2 hours and 20 minutes”. - “Non-complex form”: the measure is expressed with a same unit: “140 minutes”.

We have to know how to change a quantity from one form to another. It is necessary to remember that a degree, or an hour, is made up of 60 minutes and each minute is 60 seconds.

Activities 1. Are these quantities complex or non-complex?

4.33 hours

37º 15’ 30 minutes 240’

95º

3 hours, 20 minutes 10’ 10” 70 seconds

2. Change the complex quantities in the previous activity into non-complex ones, and vice versa:

3. Change these complex quantities to non-complex ones:

3º 60’: .............. min. 3º 60’: .............. sec.

24 hours, 10 minutes: ................... seconds.

90º 14’ 60”: .................... seconds.

3598 min., 120 sec.: ...................... hours.

4. Change these non-complex quantities to complex:

6500 minutes:

24.5 hours:

3000º:

2000’:

SEXAGESIMAL SYSTEM OPERATIONS:

ADDITION SUBTRACTION

When time or the measure of two angles are expressed in a complex form, we have to sum separately the hours (or degrees) and then their minutes. Examples:

Time: (2 h. 5 m.) + (4 h. 15 m.) = 6 h. 20 m. Degrees: (23º 35’) + (17º 14’) = 40º 49’

But if the minutes sum to more than 60, know that every 60’ forms a new hour or degree:

Time: (2 h. 55 m.) + (4 h. 7 m.) = 7 h. 2 m. Degrees: (23º 55’) + (17º 14’) = 41º 9’

First, the minuend should be bigger than the subtrahend; we have to subtract the hours or degrees and then the minutes. Examples:

Time: (5 h. 35 m.) – (3 h. 30 m.) = 2 h. 5 m. Degrees: (55º 30’) – (25º 25’) = 30º 5’

If the minutes in the minuend are less than those in the subtrahend, add 60’ to the minuend; subtract 1 hour or degree on its left:

Time: (5 h. 35 m.) – (3 h. 36 m.) = 1 h. 59 m. Degrees: (55º 30’) – (25º 31’) = 29º 59’

MULTIPLICATION BY A NUMBER DIVISION BY A NUMBER

First get the product of the hours or degrees and then the product of the minutes.

Example: (12º 14’) · 4 = 48º 56’

If the product of the minutes is greater than 60, remember that every 60 minutes completes another full hour or degree. So, divide that quantity by 60 to get new degrees. The remainder in that division is the seconds.

Example: (40º 15’) · 5 = 200º 75’ = 201º 15’

First, divide the hours or degrees by the given number; its remainder becomes minutes to add to the previous minutes; then continue dividing all the minutes by the given number.

Example:

(50º 18’) : 4 = 12º (120’ + 18’) : 4 =

= 12º (138’: 4) = 12º 34’ and 2’ left.

Activities

1. Operating with angles:

63º 15’ + 29º 48’

56º 28’ - 22º 57’

(123º 25’) · 3 = (225º 15’) : 5 =

2. Operating with time:

( 3 h. 54 m.) + ( 4 h. 18 m.)

( 8 h. 15 m.) - ( 5 h. 19 m.)

(12 h. 50 m.) · 6 = (15 h. 40 m.) : 7 =

3. Problems:

a) Yesterday, Tom started reading a book. He read for 2 hours and 30 minutes. Today he read for another hour and 45 minutes until he finished the book. How long did he read for in total? b) John’s sister, Paula, read the same book last month. She read it in 7 hours and 12 minutes. Who read the book faster? What’s the difference in minutes between the two?

UNIT 3 FRACTIONS

INTRODUCTION: THINGS TO REMEMBER!

In Mathematics, a fraction expresses a part of a whole. There are full objects, but sometimes

we need to speak about just a part of it. Each fraction consists of a numerator (at the top,

showing the part of a whole) and a denominator (at the bottom, expressing the whole). When

we say 2/3 (read “two thirds”) of a cake, we are talking about two parts of the complete cake

(which was previously divided in three parts).

A fraction can also be:

A division An operator

It is the quotient of the numerator divided by the

denominator. Then, we can express it with a decimal number:

6.05:35

3 ==

If the numerator is a multiple of the denominator, the quotient

is a natural number without decimals:

24:84

8 ==

But, as a division the quotient can also be a repeating decimal

with infinite digits. Then, a fraction is more accurate than a

decimal:

1.0...11111.09

1 ⌢

==

It is a number that changes

another quantity.

To calculate a fraction of a

full number, the quantity is

divided by the denominator

and then multiplied by the

numerator. Example:

4

3 of a five litres bottle:

5 : 4 = 1.25 1.25 · 3 = 3.75 l.

Activities

1. Express these fractions as decimal numbers (dividing numerators by denominators). Then,

say if the decimals are exact, repeating or non terminating non repeating

4 6 7 2 5 9 23 12 1 8

5 2 3 7 5 4 16 13 4 8

0.8

3

Exact

decimal

Integer

2. Now, classify the previous fractions from the greatest to the lowest:

> > > > > > > > >

3. Fractions as operators. Calculate:

5 / 8 of 30

2 / 7 of 80

4 / 10 of 35

1 / 6 of 60

4 / 11 of 150

Equivalent fractions

Definition: If two fractions have the same quotient when the numerator is divided by the

denominator, they are equivalent.

Example: 6 / 8 and 12 / 16 are equivalent because the quotient in both divisions is 0.75.

How to get equivalent fractions: multiply or divide both numerator and denominator

by the same number. Then, the quotient won’t change. Consequently, there are infinite

equivalent fractions. This is the main property of fractions.

Example: 2 / 3 equivalents are 6 / 9 (multiplying it by 3) and 10/15 (by 5).

How to simplify fractions: divide both numerator and denominator by the same

number. This process is called cancelling. When nothing more can be cancelled, we say that

the fraction is simplified or reduced to its lowest terms (“irreducible”).

Example: 8 / 10 is simplified dividing both terms by 2: 4 / 5 and that fraction is irreducible.

Activities

1. Find the two equivalent fractions in every group:

8 4 6

12 6 4

5 4 1

15 14 5

9 30 10

8 21 7

50 1 25

6 4 100

2. Find two equivalent fractions for those without any in the previous activity:

3. Simplify these fractions, if possible, and express them in their simplest form (irreducible):

15

12

2

7

20

5

75

125

20

36

24

72

3

2

108

27

4. These fractions are equivalent; calculate the value of “x”:

16 and 4

20 x

6 and 24

18 x

8 and 40

40 x

x and 9

15 45

2 and x

6 78

25 and 5

x 20

15 and x

9 45

9 and 1

x 10

5. Find the irreducible fraction in every group:

6 5 16

10 8 21

17 8 3

11 32 33

40 30 24

15 23 30

21 2 14

14 14 5

6. Choose the greatest fraction in every group; then, find one equivalent and its irreducible one:

8 1 10

15 8 12

12 18 30

30 4 14

100 15 16

15 100 2

2 2 3

8 10 18

Finding the common denominator

In many operations we can't use fractions because they have different denominators. So it is

necessary to find a common denominator. There are two steps to finding it:

a) Get the lowest common multiple of the denominators. Remember how to do it (getting

their prime factors and multiplying them using their highest index):

Example: 2

1and

3

1(the denominators are 2 and 3; 6 is their lowest common multiple).

b) Change every fraction into its equivalent with that common denominator. Consequently,

we have to divide their lowest common multiple by the denominators. Then, we get the

number we need to be multiplied by the numerators and get equivalent fractions:

Example:2

1 (multiplying by 3) is equivalent to

6

3and

3

1(multiplying by 2) is equivalent to

6

2.

Activities

1. Find the lowest common denominator:

2

3 and

5

2

8

5 and

7

6

7

4,

11

2 and

14

1

10

7,

5

3 and

25

6

20

11,

24

5 and

28

10

9

5,

5

6 and

15

8

2. Write the equivalent fractions from the previous activity:

3. Order the previous fractions from the greatest to the lowest:

>

>

> >

>

>

> >

4. Find the common denominator of these fractions; get their equivalent ones and order them

from the lowest to the greatest:

7

6

15

4

4

5

20

8

25

7

18

6 Common denominator: _ _ _ _

- Equivalent fractions:

- Order them: < < < < <

Addition and subtraction of fractions

With the same denominator With different denominators

- Just add or subtract the nominators

among themselves and keep the same

denominator.

Examples: 5

5

5

3

5

2 =+ 7

3

7

1

7

4 =−

- Find the lowest common multiple of the

denominators.

- Get the equivalent fractions with that

denominator.

- Add or subtract the nominators. Examples:

28

15

28

8

28

7

7

2

4

1 =+=+ // 88

23

88

32

88

55

11

4

8

5 =−=−

Some other things to remember:

When we have to add or subtract fractions with integer numbers:

- An integer number is a fraction with 1 as a denominator.

Example : 1

66 = →

4

25

4

24

4

1

1

6

4

16

4

1 =+=+=+

When fractions to add or subtract are inside brackets, do the same as with integers:

Positive signs before the brackets

(+) don’t change the inner signs.

Negative signs before the brackets (-) change the inner

signs from positive to negative and vice versa.

If we add two fractions and the sum is zero, they are called opposite fractions.

Activities

1. Solve these additions and subtractions with different denominators:

=+7

2

2

5 =−

8

3

13

12 =++

5

1

9

2

7

6

=+7

1

6

8 =−

3

4

4

5 =−−

9

4

15

13

21

20

2. Calculate (with fractions and integers involved):

=+5

26 =+−

3

240 =++ 2

4

8

7

6

=33

4 − =−−2

1515 =+−−

8

3

9

25

3. Operate with brackets:

=

−+

+4

1

5

2

3

9

4

5

=

+−

−8

1

12

10

15

6

12

10 =

+−−

+−6

2

7

4

6

7

21

8

Multiplication and division of fractions

How to multiply fractions: How to divide fractions:

- Multiply the nominators together.

- Multiply denominators together.

Examples: 10

3

20

6

5

2·

4

3 == // 5

1

60

12

2

3·

6

1·

5

4 ==

Inverse fractions are those whose product is

the unity (= 1). Exam.: 120·21

21·20

20

21·

21

20 ==

- Multiply the first fraction by the inverse of

the second fraction. In other words:

• The numerator of the result will be

the product of the numerator of the

first fraction and the denominator of

the second fraction.

• The denominator of the result will

be the product of the denominator of

the first fraction and the nominator of

the second fraction.

Example: 8

21

2·4

7·3

7

2:

4

3 ==

Activities

1. Multiply the fractions and simplify (if possible) the product:

=·3

5

9

8

=·4

1

5

6

=·4

3·2

12

5

=·2

5

13

5

=·6

11

11

6

=2

5·

8

3·4

2. Write the inverse fractions.

6

5and

7

2and

3

4and

17

6and

52

11and

3. Divide the fractions and simplify if possible:

=3

7:

5

8

=2

1:

5

12

=11

10:

9

4

=2

3:

19

14

=27

6:9

=2:18

3

4. Operate with brackets and simplify:

=

6

1·

9

4:

7

2·

3

5

=

9

4·

7

3·

3

8:

5

2 =

9

2:

5

8:

5

6·

4

3

Problems with fractions involved

1. Suppose 3

2 of our class (of 24) was absent one day; and

4

3 of the absent people went to to

visit the museum. Calculate:

What fraction of the class

went to the museum?

Fraction:

Exact number of pupils:

What fraction of the class

expresses the non absent

pupils?

Fraction:


What fraction expresses the

students who weren’t in class

or in the museum?

Fraction:


2. My bedroom measures 4 9

8 metres by 4

9

8 metres. If I wanted to carpet the bedroom, and

I was able to buy a carpet measuring 5 metres by 5 metres, how many square metres of

carpet would I be wasting (if I measured really carefully)?

3. My grandparents gave me 300 euros because I had passed all my exams.

I spent the money like this: What fractions express every quantity?

- The whole amount of money:

- The private lessons:

- My sister’s present:

- The ticket:

- My savings:

200 €: I paid for my private English lessons.

25 €: I bought a present for my little sister.

30 €: I bought a ticket for a concert.

I kept the rest.

4. Joseph’s mum made a big pizza for him and his three friends. She divided it in 18 portions.

Joseph ate 3

1 of it; Robert ate

4

1of the rest of the pizza; David and Paul ate

3

1 of the rest

each.

Total amount: 18 portions

Joseph: 1/3 of 18.

Robert: 1/4 of the rest.

David: 1/3 of the second rest.

Paul: the same as David.

Guess how many portions everyone ate.

- Joseph:

- Robert:

- David:

- Paul:

Did they eat the

whole pizza?

Is there anything

left for me?

5. Johnny is a good football player. Last season, he scored 17 goals. That was 5

1 of the total

goals of his team. How many goals did the team score?

Powers and fractions

Powers and fractions have the same

First index law Index Law for Multiplication

mnmn

b

a

b

a

b

a+

=

·

When multiplying powers with

the same base, add the indices.

Fourth index law Index Law for Powers

mnm

n

b

a

b

a·

=

When a power of a fraction is

raised to another power, multiply

the indices.

Activities

1. Calculate (follow the above index laws):

=

2

5

3

·

7

2


=3

33

16

4·2

2

6

·3

3. Calculate, if possible, the powers of zero:

=

0

5

3 =

0

2

0

Things to remember about powers:

10 is called the base.

2 is called the index or power (or exponent

indicates the power to which the base, 2, is raised.

100 is the basic numeral (or number).

102 is read as '10 to the power 2' or simply '10 squared'.

Source: www.mathsteacher.com.au/year7/ch02_power/roots.htm

actions have the same rules as powers and integers. Let’s remember them:

Index Law for Multiplication

When multiplying powers with

the same base, add the indices.

Second index law Index Law for Division

mnmn

b

a

b

a

b

a−

=

:

When dividing powers with the

same base, subtract indices.

Third index lawThe Power of Zero

b

a

Any number, except 0, raised to

the power zero has a value of 1.

When a power of a fraction is

raised to another power, multiply

Fifth index law Index Law for Powers of Products

nnn

d

c

b

a

d

c

b

a

=

··

When a product is raised to a

power, every factor of the product

is raised to that power.

Sixth index lawIndex Law for Powers of Quotients

b

a

When a quotient is raised

power, numerator and denomi

nator are both

1. Calculate (follow the above index laws):

=

3

3

4·

=

4

9

2:

4

1

=2

24·

( ) ( ) =−−3

33

30

5·3

3. Calculate, if possible, the powers of zero:

=

0

0

2 =

0

7

2

Things to remember about powers:

exponent) because it

indicates the power to which the base, 2, is raised.

(or number).

is read as '10 to the power 2' or simply '10 squared'.

www.mathsteacher.com.au/year7/ch02_power/roots.htm

as powers and integers. Let’s remember them:

Third index law The Power of Zero

1

0

=

(for b ≠ 0)

Any number, except 0, raised to

the power zero has a value of 1.

Sixth index law Index Law for Powers of Quotients

n

nn

b

a= (for b ≠ 0)

When a quotient is raised to a

power, numerator and denomi-

both raised to that power.

=

25

10

3·

10

3

=

4

2

5

3

- =

0

9

1

Powers of base ten

The Powers of Base 10 help with

scientific notation. Imagine the

difficulty to write high integers or

low decimals with so many zeros.

Think, for example, about errors in

the writing of a number with one

zero more or one zero less. Have a

look at the table on the left.

Activities


- What’s the difference between

American billions and ours?

- How do we write our billions?

- Why are there three different

colours in the table?

- What’s the difference between

positive and negative powers?

2. Say some very big things we

measure with billions: _________

____________________________

___________________________.

3. Say some very small things we

measure with billionths: ________

____________________________

___________________________.

4. Write the whole numbers:

8 · 106: …………………………...

5 · 10-9

: …….…………………….

6 · 107: ………………...…………

American Billions

109 = 1,000,000,000

10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000,000

Millions

106= 1,000,000

10 x 10 x 10 x 10 x 10 x 10 = 1,000,000

Hundred Thousands

105 = 100,000

10 x 10 x 10 x 10 x 10 = 100,000

Ten Thousands

104 = 10,000

10 x 10 x 10 x 10 = 10,000

Thousands

103 = 1,000

10 x 10 x 10 = 1,000

Hundreds

102 = 100

10 x 10 = 100

Tens

101 = 10

Ones

100 = 1

Tenths

10-1 = 1/10

1/10 = 0.1

Hundredths

10-2 = 1/102

1/102 = 0.01

Thousandths

10-3 = 1/103

1/103 = 0.001

Ten Thousandths

10-4 = 1/104

1/104 = 0.0001

Hundred Thousandths

10-5 = 1/105

1/105 = 0.00001

Millionths

10-6 = 1/106

1/106 = 0.000001

American Billionths

10-9 = 1/109

1/109 = 0.000000001

Convert fractions to decimals

The simplest method is to use a calculator:

- Get your calculator and type in "5 / 8 =", the answer should be 0.625

- Convert these fractions to decimals:

15

4: …………………

21

6: ………………………

81

9: ………………………

Sometimes we have to do it manually, however. Then, follow these steps for exact decimals:

- Find a number you can multiply by the bottom (denominator) of the fraction to make it 10,

or 100, or 1000, or any 1 followed by zeros.

- Multiply both the top and bottom by that number.

- Then write down just the top number, putting the decimal place in the correct spot (one

space from the right for every zero in the bottom number).

Example: “Express 3/4 as a decimal”

- We can multiply 4 by 25 to become 100.

- Multiply top and bottom by 25.

- Write down 75 with the decimal place 2 spaces from the right (because 100 has 2 zeros).

Convert decimals to fractions

- Write down the exact (or terminating) decimal divided by 1.

- Multiply both top and bottom by 10 for every number after the decimal point. (For

example, if there are two numbers after the decimal, then use 100, if there are three then use

1000, etc.)

- Simplify (or reduce) the fraction. (Repeating decimals get their fractions differently).

RATIONAL NUMBERS are those that can’t be expressed as a fraction, as they are non

repeating non terminating ( 2 =1,414213562373095048801688724209…). Consequently,

it is impossible to get a fraction for such decimals.

Example: “Express 0.75 as a fraction”

- Write down: 1

75.0

- Multiply both top and bottom by 100 (because there were 2 digits after the decimal place in 75):100

75

- Simplify the fraction: 4

3

Activities

a) Convert these fractions to decimals: b) Convert these decimals to fractions:

20

5:

4

1:

25

4:

0.92: 0.845: 0. 6⌢

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UNIT 5 ALGEBRA

Things to remember! - Algebra is the branch of Mathematics concerning the study of operations with

unknown numbers; when we have to operate with unknown numbers, we use letters. So,

in algebraic operations we can work with numbers and letters (such as a, x, y) together.

- The main properties (commutative, distributive and associative) will be useful to

operate with numbers and letters.

- The basic rules about brackets and how to eliminate them (with positive or negative

signs).

- How to simplify fractions: getting equivalent ones, their lowest common multiples

and greatest common divisors.

- When operating with powers, remember their properties:

Examples: 24 · 2

5 = 2

4 + 5 = 2

9 2

5 : 2

2 = 2

5 – 2 = 2

3

Activities

1. What property is involved?

a + 5 = 5 + a

2xy = 2 · y · x

x · y · z = y · z · x

a · (4 + b) = a · 4 + a · b

2. Give the value of a:

6a = 12 a + 4 = 0 20 – 2a = 10

a + 8 = 15 85

a a

2 = 64

3. Simplify the fractions:

9

6

14

4

21

6

45

5

80

16

4. Operate with powers:

35 · 3

2 =

82 · 8

3 · 8

4 =

x3 : x

2 =

y6 : y

2 =

Algebraic expressions

Remember the concepts we learned last year:

Monomial Degrees Similar monomials

This is the name given to the

simplest algebraic expression

and it is made up by products

(multiplications) of letters

and numbers. So, a monomial

consists of a known number

(called “coefficient”) and

one or more letters (named

“literal part”).

Example: 8·x

(Coefficient: 8; literal part: x)

The degree of a letter is its

power. Example: the degree

in x2 is 2 and the whole is a

second degree monomial.

The degree of a monomial is

the sum of its degrees.

Examples: 3x is a first degree

monomial. 7ab is a second

degree (1 for a + 1 for b = 2).

7ab2 is a third degree

monomial (1 + 2).

Similar monomials are those

which contain the same letters

with the same exponents or

powers. Their coefficients can

be different, however.

Example:

2ax ; -3ax ; ax ; 5ax

(They are similar monomials)

How to add or subtract monomials:

- It is possible to add or subtract similar monomials (if they have got the same literal

part):

Example: 3xy + 5xy – 2xy = 6xy (we add or subtract their coefficients keeping the literal

part)

- When the monomials are not similar, they can’t be added or subtracted: 8xy + 5x2y

3

Activities

1. Which expressions are monomials?

x – y 5ab 3x2

1/a

2a 8 + a - b 5 + b a3

2. Find the degree of the monomials:

3x5y x

3y

2 3x

25y

3 x

32y

43z

2

6xyz2

2x4y

5 7x6y

42z 2xy

3z5

3. Complete the table:

Monomial Coefficient Literal part Degree

-3xy3

1/4x4y

x2y

4

4. Underline the similar monomials:

8ab2 5ab 5a

2b 8a

2b 6a

3b 6a

2b

2 6a

2b

5. Add or subtract monomials, if possible:

3x + 5x – 4x =

2y + 3y – y =

7y3 – 2y

3 + 4y

2 =

x2 + 3x – 4x

2 =

Multiplication and division of monomials

Multiplication of monomials: Division of monomials:

- The product of two monomials is always

another monomial (remember how to

multiply powers).

Examples: x4 · x

3 = x

4+3 = x

7

x2 · y

2 = (x · y)

2

- The multiplication of a monomial by a sum:

when one of the factors is a sum, use the

distributive property. Examples:

7 · (2a + 3b) = 7 · 2a + 7 · 3b = 14a + 21b

4x · (3x3 + 2y

3) = 4x · 3x

3 + 4x · 2y

3 =

= 12x1+3

+ 8xy3 = 12x

4 + 8xy

3

The quotient of two monomials can be a

number, another monomial or an algebraic

fraction (a fraction with letters in the

denominator).

To operate, follow the same rules as with the

integers. Examples:

A number: (3a) : (6a) = a

a

6

3=

2

1

·2·3

·3

a

a

A monomial: (4a3) : (2a

2) = a

aa

aaa2

··2

···4

An algebraic fraction: (6a3) : (3a

4) =

aaaaa

aaa 2

····3

···3·2

Activities

1. Multiply a number by a monomial:

5 · 4x = y15·5

2

x6

8·3

6 · (-2y) = - )20·(4

1b

5

2·

4

3 x

2. Multiply the monomials:

18x · x2 = 3b

2 · 4b

4 · 3b

5 = -5y

6 · (-7y

3) =

3. Reduce:

(3ab) · (4ab) =

(2b2) · (5a

4b) =

6a2 · (5ab) =

4. Divide the monomials; simplify them if possible:

9a : 3 =

2a : 6a3 =

15a3b

2 : 5ab

4 =

x2

: x4 =

4x3y

2 : 2xy =

25xy4 : 5x

5y

3 =

Polynomials

- In general, a polynomial is an expression constructed from variables (or

indeterminates) and constants, using the operations of addition, subtraction or

multiplication. Examples:

x2 − 4x + 7 is a polynomial, but x

2 − 4/x + 7x

3/2 is not (because its second term

involves division by the variable x and also because its third term contains an exponent

that is not a number).

- We can also talk about binomials or trinomials (when two or three terms are

involved).

- The degree of a polynomial is that of its greatest monomial. Examples:

3x2 + 5xy -8 (second degree) 5x

4 + 4y (fourth degree) 6x -1 (first degree)

- When the literal part takes a supposed numerical value, we can calculate the value of

the polynomial for that case. Example:

If x = 3 in the polynomial 3x + 4, then the value of the polynomial is 13.

But if x = 4, the value of the polynomial would change (16).

Activities

1. Say if the following expressions are polynomials or not; if not, explain why:

4a + 3b – 2c - d

9 · 3a · 8b

15x : 8y

9x + 1 = 19

5x2 + 2y

3 – 8z + 1

4 · (5 + 3) - 1

2. What’s the degree of these polynomials?

5x6 – 2x

4 + x

2 +3

(x – 2) · (y + 1)

- x2 + y

2 - 2

x4 – y3 + z -2

50x19

+ x12

– x3 +10

5x7 -6x

5 + 9x

2

3. Calculate the numerical value:

For x = 2

4x2 + 5

2x4 -8

For x = -3

5x2 – 3x + 8

6x4 + 9x

2 - 3

For x = 6

x3 + 8x

2 – x

3x2 – x

2 + x

Operations with polynomials

ADDING POLYNOMIALS SUBTRACTING POLYNOMIALS

Adding polynomials is simply the adding

of their like (similar) terms. One method is

to place like terms in columns and to find

the algebraic sum of the like terms. For

example, to add 3a + b - 3c, 3b + c - d and

2a + 4d, we would arrange the

polynomials as follows:

3a + b - 3c

3b + c - d

2a + 4d

5a + 4b – 2c + 3d

Subtraction is performed by using the

same arrangement (by placing terms of the

subtrahend under the like terms of the

minuend). But remember in subtraction

the signs of all the terms of the subtrahend

must first be mentally changed and then

the process completed as in addition. For

example, subtract 10a + b from 8a - 2b:

8a – 2b

10a + b

-2a - 3b

MULTIPLICATION OF POLYNOMIALS

A POLYNOMIAL BY A MONOMIAL A POLYNOMIAL BY A POLYNOMIAL

We make use of the distributive property

of multiplication. This is illustrated in the

following examples:

4 (5 + a) = 20 + 4a

3 (a + b) = 3a + 3b

ab (x + y –z) = abx + aby - abz

Thus, to multiply a polynomial by a

monomial, multiply each term of the

polynomial by the monomial.

Multiply each term of the multiplicand

separately by each term of the multiplier

and combine the results with due regard to

signs.

It is often convenient to place the

polynomial with the fewer terms beneath

the other and multiply term by term:

3x2 - 7x - 9

2x - 3

-9x2 + 21x + 27

6x3 - 14x

2 - 18x

6x3 - 23x

2 + 3x + 27

Activities

1. Add as indicated:

6x3 - x – 6

2x2 + 5x + 8

9x5 – 2x

3 – 5x

2x6 + 5x

4 + 8x

3 - 1

2. Subtract:

5x3 - 4x

2 – 7x + 6

- 2x3 + 5x

2 + 8x - 9

7x4 – 5x

3 + 3x

2 - x

- 2x5 + 7x

3 + 8x

2 - 1

3. Multiply these polynomials:

- 4x · (2x2 +6x – 2)

(x2 -5x +2) · (-3x

2 + 6)

Special products of polynomials

The products of certain binomials occur frequently. It is convenient to remember the

form of these products so that they can be written immediately without performing the

complete multiplication process. We present four such special products as follows:

SQUARE THE SUM OF TWO NUMBERS:

(x + y)2 = x

2 +2xy + y

2

SQUARE THE DIFFERENCE OF TWO

NUMBERS:

(x - y)2 = x

2 -2xy + y

2

PRODUCT OF THE SUM AND DIFFERENCE

OF TWO NUMBERS:

(x - y) · (x + y) = x2 - y

2

PRODUCT OF TWO BINOMIALS HAVING A

COMMON TERM:

(x + a) · (x + b) = x2 + (a + b)x + ab

(If you learn these usual products, you can save time when operating with polynomials.)

Activities

1. Calculate these special products:

(x + 5)2 =

(8 + x)2 =

(x – 3)2 =

(4 – x)2 =

2. Multiply these binomials:

(5x – 2y) · (5x + 2y) =

(8x + 5y) · (8x – 5y) =

(x – 3) · (x + 3) =

3. Calculate the product of these binomials having a common term:

(x+ 8) · (x + 5) =

(6 + x) · (6 + y) =

(2x + 3) · (2x + 6) =

4. Get the common factor:

5x + 5y

6x + 3

3a2 + 9a

UNIT 6 EQUATIONS

Things to remember:

- Definition: an equation is a mathematical equality where two things are exactly the same (or equivalent), but only for one or some values of the letters involved. Equations are written with an equal sign: 2x + 3 = 5. An identity is an equality that remains true for any value of the variables that appear: 2x + 3x = 5x

- How to combine like terms: it is the process used to simplify an equation by adding and subtracting the coefficients of terms. Example:

12x + 7 + 5x = 41 → 17x + 7 = 41 and finally → 17x = 34

-How to expand the brackets:

8 · (5x -3) → 40x -24 (distrib. property)

- How to multiply fractions and integers (integers are fractions with 1 as its denominator).

Example: 2 · 25

410

4·15·2

45

===

- How to operate with algebraic fractions, by getting the lowest common multiple of denominators and then combining like terms, or by using the properties:

435

432

43

18

438

438

)1(4)1(38

4433

=+=+=+=+−−

=+−−

xx

xx

- First degree equations are properly called linear equations; second degree equations are called quadratic. Examples: 4x – 6 = 14 (linear) 4x2 -6x = 70 (quadratic)

- Finally, the names for the parts of an equation:

-------------------------------------------------------------------------------------------------------------------------------------

Activities

1. Tick on the equations. Give reasons for those that aren’t equations:

2x - 4 = 12 4x + 9x = 13x x + y = 7 (5x – 3) : 2 = 21 4x2 + 3x

2. Name the parts of the following equation:

3x2 – 8x + 6 = -x

The coefficients are ………. The variable …….... Two (2) is …….… The operators …………………. The constant ……………………… The expressions are …………… The terms …………………………

3. Combine like terms and order them:

6x – 8x2 + 2 – 3x + 5 + 3x2= 2x + 1

12 – 8x + 5x2 + x – 3 + 2x2 = 860

4. Remove brackets and operate with algebraic fractions:

=

−+ 4·

263

xx

=+

−− 4

8855

xx

( ) ( ) =−−−

72·3

x

Working with equations

A) An equation is made up of two algebraic expressions which include numbers and letters (unknown factors or variables). The expressions are equal to each other and are separated by an equals sign.

B) An equation with one variable only has one letter (usually represented by the x). if there is a second variable involved, it is called y.

C) In order to solve an equation we need to find a value for x, which when substituted back into the equation satisfies it (i.e. both sides are equal to each other). Thus, solving an equation means to find its solution. Sometimes there is more than one solution.

D) Equivalent equations are those with the same solution in their variables. Example:

8x + 3 = 19 and 5x – 4 = 6 (x = 2 in both)

E) If we try to solve an equation numerically, we first need to get x "on its own", isolating it on one of the expressions. This involves carrying out certain operations and bringing terms over from one side to the other until x = a number. Remember how the signs change before the terms. Example:

6x - 6 = 8 - 4x - 4 (be careful with signs) 6x + 4x = 6 + 8 - 4; 10x = 10; x = 10/10 ; x = 1

(combine like terms and isolate x)

F) We can solve equations numerically and also graphically (not all of them).

- Follow these basic rules to get x on its own on the LHS (Left Hand Side) of the equation:

3x + 1 = x - 2 2x=-3

a) Add or subtract the same number to each side of the equation.

If you subtract 1 + x from each side you get: 3x +1 -1 - x = x - x - 2 -1, which when simplified gives us: 2x = -3. We get the same answer if we "change the sign (+ to - or - to +) when we take terms over to the other side of the equation".

b) Multiply or divide both sides of the equation by the same number.

In this case by 2, so: 2x/2 = -3/2, which when simplified gives us x = -3/2 which is the solution. We get the same answer by doing the following: "when you take a factor over to the other side of the equation divide what the other side is multiplied by or multiply what the other side is divided by."

Activities

1. Solve these linear equations:

5x + 6 = 46

22

27=

+x

x 5x – 3 + 6x = 10x + 1

54

3=

−x

(x + 6) · 3x = 30x 2

1210 +x = 6

2. Give the English names for the parts of the previous equations.

3. Are these equations equivalent or not?

6x – 5 = 13 and 2x – 2 = 4 8 – 4x = 0 and 16 – 2x = 0 x = 8 and x2 = 64

Solving real-life problems using linear equations

1. Five times a number less 7 is equal to 7 times the number plus 13. What is the number? (Tip: it may be a negative number)

2. I bought 3 packets of rice and 2 packets of sugar. One packet of rice costs 1 euro. How much does one packet of sugar cost if I paid 6 euros in total for everything?

3. My grandmother gives me some money every time I visit her. I always go once a week. If I have received 260 € in one full year, how much do I get in one single visit?

4. My father is 3 times older than me. Our ages sum 56 altogether. How old are we?

5. Last week I went to the shopping centre with my family. I bought a computer game. It cost exactly double that of a book that my sister bought. Then, my parents bought some other things which cost 47 €. We paid 80 € cash for everything. Calculate how much my computer game cost.

6. There were 98 sandwiches at my birthday party. Every child ate 3. There were 20 adults, too, who had two sandwiches each. If at the end there were 13 sandwiches left, calculate how many children were at the party.

7. The perimeter of a rectangle is 80 cm. If the longest sides are 4 cm longer than the shortest sides, calculate the measure of all the sides.

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Quadratic equations

A quadratic equation is a polynomial equation of the second degree. They are called quadratic because quadratus is Latin for square; the leading term in the variable is squared.

The general form is:

ax2 + bx + c = 0 Example:

3x2 + 5x – 8 = 0

Some things to make note of about the formula:

a ≠ 0. (If a = 0, the equation becomes a linear equation). The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term or constant term. The signs may be positive or negative.

Most of the quadratic equations have got two different solutions or “roots”, but others may have no solution or just one solution.

How to solve quadratic equations: “By completing the square” “By using the formula”

If the equation has the form x2 = k : (no linear term and no constant involved)

Take the square root of each side. Then: kx ±= (If k is a negative number, there is

no solution, as its root is impossible).

In the full form of a quadratic equation (with all its terms present), we have to use the quadratic formula to solve it:

aacbbx

242 −±−

=

Comments about it:

- You should learn it by heart, and then change every letter in the formula by the real numbers in any given quadratic equation.

- If b2 – 4ac is negative, there are no solutions, because this would mean taking the square root of a negative number.

- If b2 = 4ac, there is just one solution as no square root is involved ( 0 ).

With the form ax2 + c = 0 (no linear term):

Bringing its terms over we get this solution:

acx −

±= (and there is only one solution,

the one of its positive root). In the form ax2 + bx = 0 (no constant):

If we factorise it: x · (ax + b) = 0 : Then, x = 0 or ax + b = 0. Two solutions:

1) x = 0

2) ax + b = 0 → x = ab−

Activities

1. Solve these quadratic equations, if possible, by completing the square:

x2 = 64 x2 = 144

6x2 + 6 = 300 6x2 + 6 = -300

2x2 – 10x = 0 3x2 – 36x = 0

2. Solve these quadratic equations by using the quadratic formula:

2x2 + 3x – 8 = 0

x2 – 5x + 6 = 0 3x2 -5x -2 = 0

UNIT 8 PYTHAGORAS THEOREM. SIMILARITY.

First of all, remember some basic things about triangles we learned last year:

How to name the parts of a triangle (or other polygons):

- The vertices are named with a capital letter (A, B and C). - The sides can be named with a small letter (a, b, and c) or with the two capital letters (AB, BC, AC) of its end points. - The angles are named with a capital letter and a ^ (Â).

How to measure angles:

- We use a protractor to measure the angles of any polygon. The unity is the degree. One degree: 1/90 part of a right angle. - About triangles, the sum of the measurement of angles equals 180 degrees (watch the yellow triangle on the right). →

Types of triangles:

A) According to the measurement of the angles: at least two of the three angles in any triangle must always be acute; looking at the third angle, we can classify triangles as:

- Acute: if the third angle is < 90º. - Right: if the third angle measures 90º. - Obtuse: if the third angle is > 90º.

B) According to the sides:

-Equilateral: all the sides measure the same. - Isosceles: two equal sides and one different. - Scalene: the three sides are different.

Activities

1. Draw three triangles using your ruler and protractor:

Acute and equilateral Right and isosceles Obtuse and scalene

2. Name their vertices, sides and angles. 3. Measure all their angles. Do they sum up to 180º in total? 4. Calculate the measurement of the third angle if we know the degrees of the other two angles:

32º, 59º and _____

45º, 45º and _____

90º, 24º and _____

120º, 41º and ____

5. According to their measurement, label the four triangles of the previous activity.

About areas and perimeters of polygons

Remember the definitions, names in English and areas of the most important polygons:

Area is the size of a figure on a two-dimensional surface. Perimeter is the distance around a given two-dimensional object.

(Area is the inside of a figure and perimeter is the measure of the line that sorrounds it).

In rectangles and squares: Area Perimeter Multiply its base by its height. It’s the sum of its four sides.

In the rest of parallelograms: Area Perimeter Multiply the measure of the longest side (called base) by its

height (which is not the measure of the smaller side). It’s the sum of its four sides.

In a rhombus: Area Perimeter Multiply the length of its two diagonals (D and d) and then divide the product by two.

It’s the sum of its four sides.

In a trapezium: Area Perimeter Sum the measure of its two parallel sides (called bases: B and b); then, multiply the result by its height (distance between both bases); finally, divide the product by two.

It’s the sum of its four sides.

In a triangle: Area Perimeter First, multiply the base (one of its sides) by its height or altitude; then, divide it by two. In right triangles, the height is equal to one of the sides (or catheti).

It’s the sum of its three sides.

In the rest of polygons: Area Perimeter First, divide the polygon in triangles; then calculate their

areas (as if they were isolated); finally sum their products. It’s the sum of all its sides. In regular ones, multiply sides.

In a circle: Area Perimeter If r is its radius and π is 3.14, then we’ll obtain the area with this operation: π · r2

Multiply its diameter by 3.14 (called π); or also: 2π · r

Activities

1. Calculate the areas of the following polygons:

A square: side 7 cm:

A parallelogram: base 7 cm and height 5 cm:

A rhombus: diagonals 30 and 40 cm:

A trapezium: bases, 15 and 20 cm; height, 12 cm:

A right triangle: base, 9 cm; height, 15 cm:

A circle: radius, 20 cm:

2. With the data given in the previous activity, can you calculate the perimeter of those

polygons, or do you need some other measurement of the sides? Try to calculate perimeters.

Tip: Pythagoras can sometimes help.

Pythagoras’ Theorem

Pythagoras’ theorem shows the relationship among three sides of a right triangle: the catheti or legs (a and b). The theorem is as

“In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs or catheti.” In other words:

“The square of the hypotenu

squares of the other two sides.”

Consequently, if we know the measure of two sides in a right triangle, we can find out the length of the third side. It doesn’t work with acute or obtuse triangles.

Many real life problems catheorem.

Activities

1. Find out using Pythagoras’ theorem:

a) If the sides of a right triangle measuwhat’s the approximate length of the hypotenuse? b) If the hypotenuse measures 12 cm and one side is 5 cm, what’s the approximate measure of the other side?

2. The sides of this rectangle measure 14 and 8 cm.

Calculate the approximate length of its diagonal:

3. My TV set has a screen whose base measures 60 cm. and its

diagonal is 80 cm. What’s the length of its height?

4. Think of other objects dimensions we can calculate by using

…………...., ……………….., ………………, ……………., ………………, ………............

Pythagoras’ Theorem

shows the relationship among three triangle: the hypotenuse (c) and the two

or legs (a and b). The theorem is as follows: “In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs or catheti.” In other words:

“The square of the hypotenuse is equal to the sum of the

squares of the other two sides.” Consequently, if we know the measure of two sides in a right triangle, we can find out the length of the third side. It doesn’t work with acute or obtuse triangles. Many real life problems can be solved using Pythagoras’s

1. Find out using Pythagoras’ theorem:

a) If the sides of a right triangle measure 9 and 6 cm, what’s the approximate length of the hypotenuse?

b) If the hypotenuse measures 12 cm and one side is 5 cm, what’s the approximate measure of the other side?

2. The sides of this rectangle measure 14 and 8 cm.

Calculate the approximate length of its diagonal:

3. My TV set has a screen whose base measures 60 cm. and its

diagonal is 80 cm. What’s the length of its height?

bjects dimensions we can calculate by using Pythagoras’ theorem:

…………...., ……………….., ………………, ……………., ………………, ………............

Pythagoras’ theorem:

…………...., ……………….., ………………, ……………., ………………, ………............

Practical problems using Pythagoras’ Theorem

1. I haven’t got the keys!

You're locked out of your house and the only open window is on the second floor, 6 metres above the ground. You need to borrow a ladder from one of your neighbours. There's a bush along the edge of the house, so you'll have to place the ladder 2 metres from the house wall. What length ladder do you need to reach the window? In this “real life triangle”, what are the hypotenuse and the catheti?

2. A baseball problem:

You've just picked up a ground ball at first base, and you see the other team’s player running towards third base. How far do you have to throw the ball to get it from first base to third base, and tag the runner out?

3. The height of this tree:

There are 8 metres from this tree to where I’m standing now. I am pulling a rope measuring 10 metres and coming from the top of the tree to my eyes. I am 1.70 m tall. Guess the height of the tree.

4. Dimensions of the football pitch:

I usually play football in this pitch. It measures 100 m from goal to goal and its width is 50 metres. But I’d like to know the measurement from this corner flag to the opposite flag. Can you help me?

Similarity in triangles

Similarity expresses the relationship between tmore objects, in this case triangles. Consequently, similarity studies the exact or approximate repetitions of patterns in the compared triangles. Two triangles (or any other geometrical objects) are called if they both have the same shape.

Equivalently and more precisely, one triangle is congruent to the result of a uniform (enlarging or shrinking) of the other. Corresponding sides of similar triangles are in proportion, and corresponding angles have the same measure.

Thales was known for his innovative use of Geometry. His understanding was theoretical and practical. For example, he said: "Place is the greatest thing, as it contains all things".And he deduced: “Parallel lines cutting two other lines provide proportional measu

AE

AD

Thales used the same method to measure the distances of ships at sea, for example. We can find out measurements by comparing proportionality of similar triangles or polygons.

Now, we can use Thales’ discoveries to work with triangles.

Find the value of x and y:

Solution: we have to write involving corresponding sides

use cross products to solve:

96

4 x= ; 6x = 36;

y

7

6

4 = ; 4y = 42; y

Activities

1. Are the triangles shown in

Tip: Find the ratios of the corresponding sides; if the sides

are proportional, the angles are congruent.

2. Can you calculate the measurement of sides UW and KM

with the given data? Are they right triangles? Can we use

Thales’ Theorem?

Similarity in triangles

expresses the relationship between two or more objects, in this case triangles. Consequently, similarity studies the exact or approximate repetitions of patterns in the compared triangles. Two triangles (or any other geometrical objects) are called similar if they both have the same shape.

Equivalently and more precisely, one triangle is to the result of a uniform scaling

(enlarging or shrinking) of the other. Corresponding of similar triangles are in proportion, and

have the same measure.

as known for his innovative use of Geometry. His understanding was theoretical and practical. For example, he said: "Place is the greatest thing, as it contains all things". And he deduced: “Parallel lines cutting two other lines provide proportional measurements:

EC

DB

AE

AD =

Thales used the same method to measure the distances of ships at sea, for example. We can find out measurements by comparing proportionality of similar triangles or polygons.

Now, we can use Thales’ discoveries to

: we have to write proportions corresponding sides. Then, we

use cross products to solve:

= 36; x = 6

y = 10.5

What’s the measure of angle P?

Solution: P and S are corresponding

By definition of similar polygons,

1. Are the triangles shown in the figure similar?

Tip: Find the ratios of the corresponding sides; if the sides

tional, the angles are congruent.

2. Can you calculate the measurement of sides UW and KM

with the given data? Are they right triangles? Can we use

s the measure of angle P?

corresponding angles. By definition of similar polygons, P = S = 86º

MORE PROBLEMS ABOUT SIMILAR TRIANGLES

1. Find the value of x (create a

proportion matching the

corresponding sides):

2. Make couples of angles that

measure the same:

Many problems involving similar triangles have one triangle ON TOP OF another triangle. Since DE is parallel to AC, we know that we have angle BDE equal to angle DAC (corresponding angles). And angle B is shared by both triangles.

3. Calculate the measurement of BE:

(Tip: Use the rule related tmultiply and solve).

4. Find EC:

5. At a certain time of the day, the shadow of a 5' boy is 8' long. The shadow of a tree at this same time is 28' long. How tall is the tree?

MORE PROBLEMS ABOUT SIMILAR TRIANGLES

1. Find the value of x (create a

ching the

2. Make couples of angles that

Many problems involving similar triangles have one another triangle. Since DE is

parallel to AC, we know that we have angle BDE equal to angle DAC (corresponding angles). And angle B is shared by both triangles.

3. Calculate the measurement of BE:

(Tip: Use the rule related to parallel lines, cross

At a certain time of the day, the shadow of a 5' boy is 8' long. The shadow of a tree at this same time is 28'

Scale of a map

The scale of a map is the ratio of a single unit of distance on the map to the equivalent distance on the ground. The scale can be expressed: in words, as a fraction, a ratio, a fraction and as a graphical (bar) scale:

- The statement “one millimetre represents 25 metres” is an expression of scale in words.

- Scale expressed as a fraction, 1/25,000, means that any distance on the map is 1/25,000th the distance on the ground. It expresses the amount of reduction of distances used to represent detail on the map. The 25,000 value is called the scale denominator.

- Due to the curved surface of the earth on a flat map surface, the scale varies from place to place. Thus a representative fraction is correct at the centre of the map but varies elsewhere. So, a representative fraction is really a representative ratio.

- A graphical (bar) scale is a ruler with ground distances added, included in the margin of most maps. The graphical scale is used to measure distances on the map. The distance on the map is marked on the edge of a sheet of paper, which is then placed over the graphical bar scale and the distance read.

Activities

1. Label the four ways to express scales:

0 500 1.000 1.500 2.000 km

Scale: 1 : 41.500.000

“One centimetre represents 100 kilometres” 000,30

1

2. Order the previous scales and the maps they represent (from the highest to the lowest):

The highest:

After that:

Next:

The lowest:

3. Remember what you learned last year in Social Studies and match the columns:

A small scale map... A large scale map...

o 1

A * o 2

B * o 3

o 4

... shows a large portion of land.

... shows a more restricted area.

... is more useful to represent continents.

... is more useful to represent a city.

4. Calculate measurement using the different scales:

a) I measure 120 cm between two cities on a map. How many kilometres do they represent if the scale is 1: 50,000? b) According to the following scale, what’s the distance in kilometres between two points, if the measurement is 7 cm on the map?

Scale: 1 : 15.000.000

c) The fraction 000,5

1expresses the scale on a map. How many km do 6 cm represent?

d) If “one centimetre represents 5 kilometres”, how many km do 20 cm represent?

Unit 9 GEOMETRIC OBJECTS: POLYHEDRA

A polyhedron (plural polyhedra) is often defined as a geometric object with flat faces and

straight edges. We can say that a polyhedron contains different dimensions:

- 3 dimensions: The body is bounded by the faces, and is usually the volume inside them.

- 2 dimensions: A face is a polygon bounded by a circuit of edges, and usually including the

flat (plane) region inside the boundary. These polygonal faces together make up the

polyhedral surface.

- 1 dimension: An edge joins one vertex to another and one face to another, and is usually a

line of some kind. The edges together make up the polyhedral skeleton.

- 0 dimensions: A vertex (plural vertices) is a corner point.

Polyhedra are often named according to the number of faces, for example tetrahedron (4),

pentahedron (5), hexahedron (6), heptahedron (7), triacontahedron (30), and so on.

Often this is qualified by a description of the kinds of faces present, for example the rhombic

dodecahedron vs. the pentagonal dodecahedron.

Prisms, cubes and pyramids are the most common examples of polyhedra.

Activities

1. Describe the polyhedra shown above: number of faces, edges and vertices.

2. Are the above polyhedra regular or irregular? Why? Which ones are symmetrical?

3. According to the number of faces, what type of polyhedron is a pyramid?

Prisms, cubes and pyramids

A prism is made of two n

Thus these joining faces are parallelograms. All cross

the same. A right prism is the one in which the joining edges and faces are perpendicular to

the base faces. This applies if the joining faces are rectangular. If the joining edges and faces

are not perpendicular to the base faces, it is called an

A cube is a three-dimensional solid object bounded by

each vertex. The cube can also be called a

prism.

A pyramid is a polyhedron formed by connecting a

apex. Each base edge and apex form a

Pyramids can have from three to a virtually unlimited amount of sides. When unspecifi

base is usually assumed to be square.

Extra activities

1. Can you think of any other types of

2. Surf the Net for information about the names in English of other polyedra.

3. Name the polyhedron built with the lowest quantity of sides, edges and vertices

4. Draw different polyhedra (overleaf). The teacher will tell you their measurem

pyramids

n-sided polygonal bases and n faces joining corresponding sides.

Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are

is the one in which the joining edges and faces are perpendicular to


are not perpendicular to the base faces, it is called an oblique prism. Describe

ensional solid object bounded by six square faces

each vertex. The cube can also be called a regular hexahedron. It is a special kind of square

Activities

1. Describe this cube.

2. What are the differences

between a prism and a

cube?

3. How many edges and

vertices does a cube have?

is a polyhedron formed by connecting a polygonal base and a point, called the

. Each base edge and apex form a triangle. It is a conic solid with polygonal base.

Pyramids can have from three to a virtually unlimited amount of sides. When unspecifi

base is usually assumed to be square.

Activities

1. Describe these pyramids.

2. What’s the difference between

them?

3. What’s the meaning of

on the second pyramid?

1. Can you think of any other types of polyhedra?



4. Draw different polyhedra (overleaf). The teacher will tell you their measurem

joining corresponding sides.

sections parallel to the base faces are

is the one in which the joining edges and faces are perpendicular to


Describe these prisms:

six square faces, with three meeting at

. It is a special kind of square

Activities

1. Describe this cube.

2. What are the differences

between a prism and a

cube?

3. How many edges and

vertices does a cube have?

and a point, called the

. It is a conic solid with polygonal base.

Pyramids can have from three to a virtually unlimited amount of sides. When unspecified, the

Activities

1. Describe these pyramids.

2. What’s the difference between

3. What’s the meaning of h and s

on the second pyramid?



4. Draw different polyhedra (overleaf). The teacher will tell you their measurements.

Polyhedra: their dimensions and areas

Length is a measurement in one dimension (sides of a polyhedron).

Area is a measurement in two dimensions (faces and bases).

Volume is a measurement in three dimensions (the full object, “its inside”).

Surface area is the sum of all the areas of all the shapes that cover the surface of an object.

Activities

1. Calculate the surface areas of these polyhedra:

2. What’s the surface area of this prism? (Round decimals

to the nearest tenth).

3. Calculate the surface area of a cube whose edge measures 15 cm.

4. A right pyramid 12 cm high stands on a square base of sides 10 cm. Calculate its surface area.

Polyhedra: their dimensions and areas

is a measurement in one dimension (sides of a polyhedron).

is a measurement in two dimensions (faces and bases).

is a measurement in three dimensions (the full object, “its inside”).

area is the sum of all the areas of all the shapes that cover the surface of an object.

The surface area of any prism (cubes included)

of the areas of its faces (let’s say the floor, roof and walls).

Because the floor and the roof of a prism have the same shape,

the surface area can always be found as follows:

2 × Area of base + perimeter of base

As it is a 2D measurement, the area is expressed in square

metres, square kilometres, square centimetres, etc.

The surface area of a regular pyramid

1. The Base Area depends on the shape; there are

different formulas for triangles, squares, etc. Use the

apotheme to calculate the areas of polygons.

2. The Lateral Area is simpler. Just multiply the

perimeter by the side length and divide by 2 (t

are always triangles).

1/2 × Perimeter × [Side Length] + [Base Area]

1. Calculate the surface areas of these polyhedra:

2. What’s the surface area of this prism? (Round decimals

te the surface area of a cube whose edge measures 15 cm.

A right pyramid 12 cm high stands on a square base of sides 10 cm. Calculate its surface area.

is a measurement in three dimensions (the full object, “its inside”).

area is the sum of all the areas of all the shapes that cover the surface of an object.

included) equals the sum

of the areas of its faces (let’s say the floor, roof and walls).

Because the floor and the roof of a prism have the same shape,

the surface area can always be found as follows:

Area of base + perimeter of base × H (height)

D measurement, the area is expressed in square

metres, square kilometres, square centimetres, etc.

pyramid has two parts:

depends on the shape; there are

different formulas for triangles, squares, etc. Use the

to calculate the areas of polygons.

is simpler. Just multiply the

perimeter by the side length and divide by 2 (the sides

× Perimeter × [Side Length] + [Base Area]

A right pyramid 12 cm high stands on a square base of sides 10 cm. Calculate its surface area.

GEOMETRIC OBJECTS: SURFACES AND SOLIDS OF REVOLUTION

A solid of revolution is a figure obtained by

axis) that lies on the same plane. Its outer structure is called

name (solid of revolution) is mainly used when we are talking about volumes. The solids of

revolution can have only a round face (

(cone and cylinder). These are the most important ones, but there are many other solids, built

by mixing different shapes or by cutting a cone or a sphere:

A cone is a three-dimensional geometric shape

with a flat, round base and a

finishing in a point called the

The axis of a cone is the straight line passing

through the apex, about which the lateral surface

has its rotational symmetry.

A cylinder has two faces, zero vertices, and

zero right edges. Its axis is a straight line that

rotates 360º. The solid enclosed by this sur

and by two planes perpendicular to the axis is

consequently called a cylinder. If we cut a

cylinder and its cross section is an

parabola or hyperbola, we can find

parabolic or hyperbolic cylinders

A sphere is a symmetrical geometrical object. It

is built up by all the equid

another inner fixed point in three

space (R3). That distance r

of the sphere.

Activities

1. Describe these objects; say what solids of revolution are included in them:

2. Label the parts of the previous solids (base, lateral surfaces, edges, axis, radius, etc.

3. True or false? If false, correct the sentences:

- A cone has got an appex. A cylinder has got two lateral surfaces.

- A hemisphere is half of a sphere. A sphere hasn’t got any lateral surfaces.

- A conic section is a part of a full cone. A conic sec


is a figure obtained by rotating a 2D shape around a straight line (its

axis) that lies on the same plane. Its outer structure is called surface of revolution

) is mainly used when we are talking about volumes. The solids of

ve only a round face (sphere), or a combination of round and flat faces

). These are the most important ones, but there are many other solids, built

by mixing different shapes or by cutting a cone or a sphere: conic sections

dimensional geometric shape

and a lateral surface

finishing in a point called the apex or vertex.

of a cone is the straight line passing

through the apex, about which the lateral surface

tational symmetry.

has two faces, zero vertices, and

zero right edges. Its axis is a straight line that

rotates 360º. The solid enclosed by this surface

and by two planes perpendicular to the axis is

consequently called a cylinder. If we cut a

cylinder and its cross section is an ellipse,

, we can find elliptic

hyperbolic cylinders.

is a symmetrical geometrical object. It

is built up by all the equidistant points to

another inner fixed point in three-dimensional

r is called the radius

ibe these objects; say what solids of revolution are included in them:

2. Label the parts of the previous solids (base, lateral surfaces, edges, axis, radius, etc.


A cone has got an appex. A cylinder has got two lateral surfaces.

A hemisphere is half of a sphere. A sphere hasn’t got any lateral surfaces.

A conic section is a part of a full cone. A conic section has got two bases.


rotating a 2D shape around a straight line (its

surface of revolution; the general

) is mainly used when we are talking about volumes. The solids of

), or a combination of round and flat faces

). These are the most important ones, but there are many other solids, built

conic sections or spherical caps.

ibe these objects; say what solids of revolution are included in them:

2. Label the parts of the previous solids (base, lateral surfaces, edges, axis, radius, etc.)

A hemisphere is half of a sphere. A sphere hasn’t got any lateral surfaces.

tion has got two bases.

How to calculate surface area of a surface of revolution

Total Surface Area of a Cone:

If a cone of base radius r and slant height

then the radius of the sector formed is

surface area of a cone is equal to the area of sector ABC, then the area of the curved surface

is π · r · s. Then, the total surface area of the cone is:

Area of the base

Surface Area of a Sphere:

Its area is equal to the lateral area of the cylinder which

contains that sphere: 4 π r2

Surface Area of a Cylinder:

If h is the height of the cylinder and

then its surface area is:

Areas of top and bottom + Area of the side

π r2 + π r

2 π r

Activities

1. Calculate the surface areas of these solids of revolution:

2. A solid sphere has a radius of 8 m. Calculate its surface area.

How to calculate surface area of a surface of revolution

Total Surface Area of a Cone:

and slant height s is cut along the slant height and opened out flat,

then the radius of the sector formed is s and the arc length AB is 2 · π · r


. Then, the total surface area of the cone is:

+ Area of the curved surface: π r2 + π r s

Surface Area of a Sphere:

Its area is equal to the lateral area of the cylinder which

Surface Area of a Cylinder:

is the height of the cylinder and r is the radius of the base,

Areas of top and bottom + Area of the side

+ π r2 + 2 π r h

2 π r2 + 2 π r h

1. Calculate the surface areas of these solids of revolution:

r = 3 m r = 3 cm h = 4 cm

A solid sphere has a radius of 8 m. Calculate its surface area. (Take π

is cut along the slant height and opened out flat,

2 · π · r. If the curved


r = 3 cm h = 4 cm

π = 22/7)

Unit 10 Volumes

The volume of any solid or liquid object is

often quantified numerically. One

shapes (such as squares) are assigned z

commonly presented in units such as mL or cm

The standard unit of volume in the metric system is the

centimetres in volume. O

equivalents in litres are as follows:

1 millilitre = 0.001 litre

1 centilitre = 0.01 litre

From these units, we see that 1000 millilitres

centimetre in volume.

Activities

2. Calculate the equivalents to the

- How many litres does it contain?

- How many centilitres does it contain?

3. Explain the meaning of these three

cubes drawn on your right:

- Why is the green cube included in the

pink one, and both in the yellow one?

- What does m3 mean?

- How many pink cubes (dm

yellow cube (m3) contain?

4. About the English language:

- What’s the difference between litre

and liter, or centimetre and centimeter?

- Read aloud: 10 m3, 34 cm

Volumes

of any solid or liquid object is how much three-dimensional space it occupies

often quantified numerically. One-dimensional figures (such as lines) and two

shapes (such as squares) are assigned zero volume in the three-dimensional space. Volume is

commonly presented in units such as mL or cm3 (millilitres or cubic centimeters).

The standard unit of volume in the metric system is the litre. One litre is equal to 1000 cubic

centimetres in volume. Other units of volume (multiples and submultiples) and their

equivalents in litres are as follows:

1 centilitre = 0.01 litre

1 decilitre = 0.1 litre

1 kilolitre = 1000 litres

From these units, we see that 1000 millilitres equal 1 litre; so 1 millilitre equals 1 cubic

1. Look at this picture and

answer the questions about it:

- What is its measurement?

- Translate into Spanish:

• length:

• height:

• width:

- Is it a “cubic metre” or not?

- What type of polyhedra is it?

- How many faces has it got?

2. Calculate the equivalents to the cubic metre, as the one shown in the previous picture:

s does it contain? - How many millilitres does it contain?

How many centilitres does it contain? - How many kilolitres does it contain?

3. Explain the meaning of these three

cubes drawn on your right:

Why is the green cube included in the

, and both in the yellow one?

How many pink cubes (dm3) can a

4. About the English language:

What’s the difference between litre

and liter, or centimetre and centimeter?

, 34 cm3, 67 dm

3.

dimensional space it occupies,

dimensional figures (such as lines) and two-dimensional

dimensional space. Volume is

(millilitres or cubic centimeters).

. One litre is equal to 1000 cubic

ther units of volume (multiples and submultiples) and their

1 kilolitre = 1000 litres

equal 1 litre; so 1 millilitre equals 1 cubic

1. Look at this picture and

answer the questions about it:

What is its measurement?

Translate into Spanish:

length:

height:

width:

Is it a “cubic metre” or not?

What type of polyhedra is it?

How many faces has it got?

, as the one shown in the previous picture:

How many millilitres does it contain?

How many kilolitres does it contain?

How to calculate volumes of geometric objects

Volumes of some simple shapes, such as regular, straight-edged and circular shapes can be

easily calculated using arithmetic formulas. The units of volume depend on the units of

length. If the lengths are in metres, the volume will be in cubic metres. The most common

shapes and their formulas are the following:

Shape Equation Variables

A cube

a3 a = length of any side (or edge)

A rectangular prism:

l · w · h l = length, w = width, h = height; (l · w = area of base)

A non-right prism:

B · h B = area of the base, h = height (perpendicular distance)

A pyramid: Ah3

1 A = area of the base, h = height of pyramid

A cylinder:

hr ·2π r = radius of circular face, h = height

A cone: hr 2

3

1π r = radius of circle at base, h = distance from base to tip

A sphere: 3

3

4rπ r = radius of sphere

An ellipsoid: cba ···3

4π a, b, c = semi-axes of ellipsoid

Among the solids of revolution, the volume of

the cylinder is the largest. The volume of the

cone is one third of the volume of the

cylinder and it is the smallest. The volume of

a hemisphere is twice the volume of the cone

or two thirds the volume of the cylinder.

Activities

1. Calculate the volume of these solids:

(a) Cone: r = 6 cm, h = 8 cm

(b) Cylinder: r = 6 cm, h = 8 cm

(c) Hemisphere: r = 6 cm

2. Compare the volume of the above solids with the volume of a sphere whose radius is 6

cm:

3. Compare volumes of a prism and a pyramid: suppose they have the same measurements.

More activities about volumes


r = 4 cm

h = 7 cm

r = 200 m

2. Some other solids and their volume:

Regular hexagon

Side = 14 cm

Height = 25 cm

Cone frustum

A frustum is the portion of a solid (normally a cone or pyramid) which lies between two

parallel planes cutting the solid.

More activities about volumes


= 4 cm

= 7 cm

r = 5 cm

h = 9 cm

w = 4 m

h = 6 m

= 200 m

Diameter = 50 m

2. Some other solids and their volume:

Regular hexagon prism

Side = 14 cm

Height = 25 cm

Cone frustum

is the portion of a solid (normally a cone or pyramid) which lies between two

parallel planes cutting the solid.

= 5 cm

= 9 cm

w = 9 m

h = 4 m

l = 5 m

Length of d ?

Diameter = 50 m

Cylinder

r = 9 m h = 15 m

What is g?

Pyramid frustum

is the portion of a solid (normally a cone or pyramid) which lies between two

Unit 11 FUNCTIONS

Cartesian co-ordinate grid

system of axes where data are placed

both horizontal and vertically. The co

ordinates are written as pairs of

numbers inside brackets, like (4, 3):

- The first number, called x co

(abscissa), tells how many across from

zero to move.

- The second number is the

ordinate (ordinate) and it tells how

many up or down to move.

- The origin is the point (0, 0).

- With negative numbers inside the

brackets, we will need four

the axes are extended to create these

four sections: a negative x co

means that you move to the left from

the origin; a negative y co

means you move down from the origin.

FUNCTIONS

Things to remember:

ordinate grid is the

where data are placed

both horizontal and vertically. The co-

ordinates are written as pairs of

numbers inside brackets, like (4, 3):

x co-ordinate

), tells how many across from

ond number is the y co-

and it tells how

is the point (0, 0).

With negative numbers inside the

brackets, we will need four quadrants:

the axes are extended to create these

x co-ordinate

means that you move to the left from

y co-ordinate

means you move down from the origin.

A four-quadrants grid

Activities

1. Write the co-ordinates of the points shown on the left grid:

A: ( , ) B: ( , ) C: ( , )

2. Plot and label the following points on the left grid:

(1, 4), (4, 1), (0, -3), (-3, 0), (-4,

3. What’s the point of origin?

4. Describe the Cartesian grid; say the names of axes,

quadrants and how to place data.

5. A company called

Natural Soda, Inc. sells

Sodium Bicarbonate.

Have a look at its sales

graph and describe it.

………………………...

………………………...

………………………...

………………………...

………………………...

………………………...

quadrants grid

ordinates of the points shown on the left grid:

: ( , ) D: ( , )

2. Plot and label the following points on the left grid:

4, -2), (0, 0), (1, -1)

4. Describe the Cartesian grid; say the names of axes,

5. A company called

Natural Soda, Inc. sells

Sodium Bicarbonate.

Have a look at its sales

graph and describe it.

………………………...

………………………...

………………………...

………………………...

………………………...

………………………...

Variables and graph functions

How to read the information given in grids:

Real life situations can be represented in co

meaning of every axis: age / height, price / weight, speed / petrol, etc. The grid will show

both “variables” and their relationship (and this is called

• The independent variable

• The dependent variable

to be the function of x

x: for every value given to

When the values of the variables are real numbers, we can draw the

it shows more clearly how the function works.

• Increasing: if the value of

• Decreasing: if the value of

• Constant: if the value of

The largest value of y in a function is its

Activities

1. Say if the following graphs are functions or not; give your arguments:

It …… (is / isn’t) a function

because ……………………..

………………………………

2. Find data in the previous graphs:

Is it increasing or decreasing?

What is its maximum value?

What is its minimum value?

3. Draw a graph where there is more than one value for x, so it isn’t really a function:

h functions


Real life situations can be represented in co-ordinate grids; it is then necessary to know the


” and their relationship (and this is called “a function”):

independent variable is the one shown horizontally (on the

dependent variable is given vertically (on the y axis). And this

function of x: f(x). The dependent variable y depends on the independent one

: for every value given to x, there is only one value for y.

When the values of the variables are real numbers, we can draw the graph of the function

it shows more clearly how the function works. There are different types of functions:

: if the value of x increases, then the value of y increases, too.

: if the value of x increases, the value of y decreases.

: if the value of y is always the same, no matter what the val

in a function is its maximum; the smallest value is its



because ……………………..

………………………………


because …………………….

……………………………..


because ……...……………..

………………..……………

e previous graphs:








What is its maximum va




ordinate grids; it is then necessary to know the


is the one shown horizontally (on the x axis).

axis). And this variable y is said

depends on the independent one

graph of the function:

There are different types of functions:

increases, then the value of y increases, too.

is always the same, no matter what the value of x is.

; the smallest value is its minimum.



because ……...……………..

………………..……………





Building Function Graphs

Function graphs are made up of three important parts: the vertical axis (

axis (x), and the line showing the relationship between the figures on the vertical axis and

those on the horizontal. In order to build a graph:

- Determine the topic of the graph.

- Locate data on the axes.

- Plot the points where data from

- Join these points with a line to understand the relationship that is being illustrated.

Activity

Look at the following table and build its graph (give numbers from 1 to 12 for every month):

Months of the year

J

Temperature (Cº)

7

A functional equation expresses the relation between the value of a function (or functions)

at a point with its values at

determined by considering the types of functional equations they satisfy; there is an algebraic

relation between x and the values of

y = x + 2

Then, the graph made in the previous activity cannot be expressed as an equation, because

temperature and months of a year cannot have any algebraic relationship.

Activity

Represent the graph of the function whose equation is

Building Function Graphs

Function graphs are made up of three important parts: the vertical axis (

), and the line showing the relationship between the figures on the vertical axis and

those on the horizontal. In order to build a graph:

Determine the topic of the graph.

Locate data on the axes.

Plot the points where data from x and y meet.

e points with a line to understand the relationship that is being illustrated.


F M A M J J A

8 10 11 13 15 15 14 12

expresses the relation between the value of a function (or functions)

at a point with its values at other points. Properties of functions can for example be


and the values of y; consequently, we can find out values of

(if x = 1, then y =3; if x = 2, then y =4; and so on)



the function whose equation is y = ½ x. Is it increasing or decreasing?

Function graphs are made up of three important parts: the vertical axis (y), the horizontal

), and the line showing the relationship between the figures on the vertical axis and

e points with a line to understand the relationship that is being illustrated.


S O N D

12 11 10 6

expresses the relation between the value of a function (or functions)

other points. Properties of functions can for example be


; consequently, we can find out values of y. Example:

=4; and so on)



. Is it increasing or decreasing?

Linear functions

The term linear function refers to a first

degree polynomial function o

variable. It is called “linear” because it is

represented with a straight line in the

Cartesian coordinate graph. Such a

function can be written as:

f(x) = mx + n

The terms m and b are real constants and

x is a real variable. The constan

often called the slope or gradient

is the y-intercept, which gives the point

of intersection between the graph of the

function and the y-axis. Changing

makes the line steeper or shallower, while

changing n moves the line up or down.

Examples of functions whose graph is a

line include these:

f2(x) = x / 2 + 1 f3(x) = x / 2 − 1

As a summary, the gradient of the straight line

the slope of the line. A value close to 0 makes the line more horizontal whereas greater

values make the line more vertical.

Activities

1. Real life example:

Brad sells a vacuum cleaner and earns £4

as commission for each vacuum cleaner he

sells. The owner pays him dependin

how many vacuum cleaners he sells.

Identify the table that best suits the

situation, and also plot a graph for the

input - output table:

2. Answer the questions about the previous graph:

- Does it have a straight line? ……………….. What is its function called? ………...………..

- Can you write its function in algebraic language? ………….......……...…………………….

- What is its slope or gradient? ……...……. What is its

- Imagine Brad earns £8 for each vacuum cleaner. Will the graph line be steeper or not? …....

refers to a first-

degree polynomial function of one

variable. It is called “linear” because it is

represented with a straight line in the

Cartesian coordinate graph. Such a

n

are real constants and

is a real variable. The constant m is

gradient, while n

, which gives the point

of intersection between the graph of the

axis. Changing m

makes the line steeper or shallower, while

moves the line up or down.

les of functions whose graph is a

− 1 f1(x) = 2x + 1

the gradient of the straight line is m. The value of the

A value close to 0 makes the line more horizontal whereas greater

values make the line more vertical.

Brad sells a vacuum cleaner and earns £4

as commission for each vacuum cleaner he

sells. The owner pays him depending on

how many vacuum cleaners he sells.

Identify the table that best suits the

situation, and also plot a graph for the

2. Answer the questions about the previous graph:

Does it have a straight line? ……………….. What is its function called? ………...………..

n in algebraic language? ………….......……...…………………….

What is its slope or gradient? ……...……. What is its y-intercept? …………………………

8 for each vacuum cleaner. Will the graph line be steeper or not? …....

The value of the gradient determines

A value close to 0 makes the line more horizontal whereas greater

Does it have a straight line? ……………….. What is its function called? ………...………..

n in algebraic language? ………….......……...…………………….

intercept? …………………………

8 for each vacuum cleaner. Will the graph line be steeper or not? …....

Constant functions and identity f

A constant function is a type of linear function whose

value does not vary and thus is constant. For example,

if we have the function f(x) = 4, then

f maps any value to 4:

- A constant function is a linear function of th

y = b, where b is a constant.

- It is also written as f(x) =

- The graph of a constant function is a

An identity function is another type of linear

functions. Its value on the

same number on the y-

coordinates of each of the points are the same: (1,1),

(2,2), (5,5). Its graph is a straight line with a 45º slope.

Activities

1. Which of the following equations represent a constant function? Choose the right option:

ç

A. x = 3

2. Why have you chosen that option? (Give your arguments)

………………………………………………………….……

…………………………………………………………….…

3. Represent its graph on the right.

4. What’s the difference between a constant function and an

identity function? ……………………………………………

……………………………………………………………….

5. Plot the graph for the following function:

6. How many degrees does its angle measure?

7. Which straight line is steeper, the one

function or the one of an identity function?

Why? ………………………………………………………...

……………………………………………………………….

8. True or false? If false, corr

- All constant functions have straight lines. All constant functions have horizontal lines.

- All identity functions have straight lines. All identity functions have horizontal lines.

Constant functions and identity functions

is a type of linear function whose

value does not vary and thus is constant. For example,

) = 4, then f is constant since

A constant function is a linear function of the form

is a constant.

) = b.

The graph of a constant function is a horizontal line.

is another type of linear

functions. Its value on the x-axis corresponds to the

axis. In other words, the

coordinates of each of the points are the same: (1,1),

(2,2), (5,5). Its graph is a straight line with a 45º slope.

Example of a constant function: y

The value of y is always 12 for any


B. 2x + y = 4

C. y = 6

ou chosen that option? (Give your arguments)

………………………………………………………….……

…………………………………………………………….…

3. Represent its graph on the right. →

4. What’s the difference between a constant function and an

……………………………………………

………………………………………….

5. Plot the graph for the following function: f(x) = x →

6. How many degrees does its angle measure? ……………..

7. Which straight line is steeper, the one of a constant

function or the one of an identity function? ...........................

………………………………………………………...

……………………………………………………………….


All constant functions have straight lines. All constant functions have horizontal lines.

All identity functions have straight lines. All identity functions have horizontal lines.

mple of a constant function: y = 12

is always 12 for any x.


D. 2y + x = 12

All constant functions have straight lines. All constant functions have horizontal lines.

All identity functions have straight lines. All identity functions have horizontal lines.

The direct proportional function

The function y = mx implies that for every corresponding pair of values

other pairs: y1 / x1 = y2 / x2 = y

The function

If there is a direct proportionality

functional equation, because it can only be

Example: The salary for work done (

then, y = mx. You can determine the

a worker receives 20 euros (

equation is y = 10 · x. For

yields the value of y for unit

The functional equation has at the same time a

its graph). It will always be a straight line, crossing the point (0, 0).

Direct proportional functions are linear, but not all the linear functions are proportional.

Activities

1. Have a look at this graph and try

to explain its meaning, axes and

relation between them.

2. Questions about the graph:

a) Does it correspond to a direct

proportional function?

b) Why do you think so?

c) What is its functional equation?

d) What is its proportionality factor?

e) How many kilometres will be run

in 10 hours, for example?

3. Another real-life problem to work with:

Andrew’s computer downloads 3 songs a minute when it is

properly working:

a) What is its functional equation? ........................................

b) Is it a linear function? .............................................................

c) Is it a direct proportional function? .........................................

d) What’s its proportionality factor

e) Guess the value of y for x

f) Plot its functional graph.

The direct proportional function y = mx

implies that for every corresponding pair of values

= y3 / x3 = m, which is called the factor of proportionality

The function y = mx is the function of direct proportionality.

s a direct proportionality m between two quantities, you know immediately its

functional equation, because it can only be y = mx.

: The salary for work done (y euros) is proportional to the time worked (

. You can determine the constant m by means of a single pair of values (

a worker receives 20 euros (y) in 2 (x) hours, then 20 = m · 2 or m = 10, and the

. For x = 1, one has y = m; consequently, the proportionality factor

for unit x, which in the present problem is the salary per hour.

The functional equation has at the same time a graphical solution of the problem (by plotting

its graph). It will always be a straight line, crossing the point (0, 0).

ortional functions are linear, but not all the linear functions are proportional.

1. Have a look at this graph and try

to explain its meaning, axes and

2. Questions about the graph:

a) Does it correspond to a direct

c) What is its functional equation?

d) What is its proportionality factor?

e) How many kilometres will be run

life problem to work with:

Andrew’s computer downloads 3 songs a minute when it is

a) What is its functional equation? ..............................................

b) Is it a linear function? .............................................................

c) Is it a direct proportional function? .........................................

d) What’s its proportionality factor m? .......................................

x = 8. ..............................................

f) Plot its functional graph. → →

implies that for every corresponding pair of values y / x = m, we can get

factor of proportionality.

is the function of direct proportionality.

between two quantities, you know immediately its

euros) is proportional to the time worked (x hours);

by means of a single pair of values (x, y). If

= 10, and the functional

the proportionality factor m

, which in the present problem is the salary per hour.

of the problem (by plotting

ortional functions are linear, but not all the linear functions are proportional.

Unit 12 STATISTICS

Statistics is a mathematical science working with the collection, analysis, interpretation and

presentation of data.

Statistical analysis of data can reveal that

connected. If the variables conform only to nominal or ordinal measurements and cannot be

reasonably measured numerically, they are called

(about qualities, not numbers). Bu

as quantitative variables

Frequency is the number of times that an event occurs in an experiment or study. We can

also call it absolute frequency

Relative frequency refers to the proportion between the absolute frequency and the total

number of data (expressed in a fraction).

We can present data in different types of

Activities

1. Tell if these variables are

Favourite football team.

Places to go on holidays.

TV programmes spectators.

2. Have a look at these two frequency tables and answer the questions about them:

The rain in Glasgow during a winter week:

Day

Sunday

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Absolute frequency:

f (Sunday) = ______

f (Friday) = _______

Relative frequency:

fr (Monday) = _____

fr (Thursday) = ____

3. Label these types of charts:

STATISTICS

Things to remember about statistics:

is a mathematical science working with the collection, analysis, interpretation and

Statistical analysis of data can reveal that two variables vary together, as if they were


reasonably measured numerically, they are called categorical or

(about qualities, not numbers). But ratio and interval measurements can be grouped together

due to their numerical nature.

is the number of times that an event occurs in an experiment or study. We can

absolute frequency (because the number of times of the event is exactly shown).

refers to the proportion between the absolute frequency and the total

number of data (expressed in a fraction).

We can present data in different types of charts: pie charts, bar charts and line charts.

1. Tell if these variables are qualitative or quantitative:

TV programmes spectators.

Pupils in every classroom.

Number of subjects at school.

Height of mountains.

The children’s weight.

Births in a hospital.

Teenagers’ hobbies.


The rain in Glasgow during a winter week: Survey about number of TV sets at home:

Rain (l/m2)

8

12

0

5

23

15

12

Number of TV sets

Zero

One

Two

Three

Four

More than four

Relative frequency:

(Monday) = _____

(Thursday) = ____

Absolute frequency:

f (zero) = _______

f (three) = _______

f charts:

is a mathematical science working with the collection, analysis, interpretation and

vary together, as if they were


qualitative variables

t ratio and interval measurements can be grouped together

is the number of times that an event occurs in an experiment or study. We can

of times of the event is exactly shown).

refers to the proportion between the absolute frequency and the total

and line charts.

The children’s weight.

Births in a hospital.

Teenagers’ hobbies.


Survey about number of TV sets at home:

Number of people

5

45

32

14

5

2

Relative frequency:

fr (two) = _______

fr (four) = _______

Types of averages

One way of analysing data is to find averages; averages are used to represent a middle or

typical value in a set of numbers. There are three types of averages:

Mode Median Mean

It is the most popular or

frequent value in a list, the

most repeated one.

Example: with the numbers in

this list (5, 7, 3, 4, 7, 5, 7) the

mode is 7.

The numbers are placed in

order and then we get the

middle value; if there are two

middle values, the median is

the number halfway between

them; in our example: 3, 4, 5,

5, 7, 7, 7; and the median is 5.

We sum up all the numbers

and then divide the total by

the quantitity of numbers.

Ex: 3 + 4 + 5 + 5 +7 + 7 + 7 = 38

38 : 7 = 5.42

The mean is approximately 5.4

There is a fourth concept called range: it is the difference between the highest and the lowest

values in a set of numbers; it is useful because it gives us a sense of how the data differ.

Using the same example as above: the highest number was 7 and the lowest 3; then, 7 – 3 = 4

Activities

1. Find the mode, median, mean and range of this set of data:

Maria’s Maths test scores (out of 10): 6, 5, 2, 5, 6, 4, 8, 4, 5, 6, 6, 3, 7

Mode: Median: Mean: Range:

2. Do the same with this table about the number of pupils in every classroom in our school:

E.S.O. 1 – A: 30 E.S.O. 2 – A: 29 E.S.O. 3 – A: 25 E.S.O. 4 – A: 24

E.S.O. 1 – B: 28 E.S.O. 2 – B: 30 E.S.O. 3 – B: 27 E.S.O. 4 – B: 26

Mode: Median: Mean: Range:

3. Calculate averages and solve the following problems:

a) The arithmetic mean of 3 numbers is 60. If two of the numbers are 50 and 60, what is the

third number?

b) In the triangles below what is the mean of a , b, c, x, and y?

c) The mean of nine numbers is 9. When a tenth number is added the average of the ten

numbers is also 9. What is the tenth number?

d) A class of 25 students took a science test. 10 students had an average (mean) score of 80.

The other students had an average score of 60. What is the average score of the whole class?

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